Supersymmetric dark matter and neutralino-nucleon cross %section %%\footnote{Plenary review talk given %%by C. Mu\~noz %%at the CICHEP %%2001} %
D. G. Cerde\~no
In the light of recent experimental results for the direct detection of dark matter, we analyze in the framework of SUGRA the value of the neutralino-nucleon cross section. We study how this value is modified when the usual assumptions of universal soft terms and GUT scale are relaxed. In particular we consider scenarios with non-universal scalar and gaugino masses and scenarios with intermediate unification scale. We also study superstring constructions with D-branes, where a combination of the above two scenarios arises naturally. In the analysis we take into account the most recent experimental constraints, such as %These are the lower bound on the Higgs mass, the $b\to s\gamma$ branching ratio, and the %anomalous magnetic moment of the muon % $a_{\mu}$. muon $g-2$.
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\begin{document}

%\title{Supersymmetric dark matter and neutralino-nucleon cross
%section
%%\footnote{Plenary review talk given
%%by C. Mu\~noz 
%%at the CICHEP
%%2001}
%}

%\author{D. G. Cerde\~no }
%
%\address{Departamento de F\'{\i}sica
%Te\'orica C-XI, Universidad Aut\'onoma de Madrid,
%Cantoblanco, 28049 Madrid, Spain. }
%
%\author{S. Khalil}
%
%\address{Centre for Theoretical Physics, University of Sussex,
%Brighton BN1 9QJ, U.K.\\
%Ain Shams University, Faculty of Science, Cairo, 11566, Egypt.}
%
%\author{C. Mu\~noz}
%
%\address{Departamento de F\'{\i}sica
%Te\'orica C-XI and Instituto de F\'{\i}sica
%Te\'orica C-XVI, Universidad Aut\'onoma de Madrid,
%Cantoblanco, 28049 Madrid, Spain. }


\rightline{FTUAM 02/11}
\rightline{IFT-UAM/CSIC-02-11}
\rightline{HIP-2002-17/TH}
%\rightline{SUSX-TH/01-021}
\rightline{hep-ph/0204271}
\rightline{April 2002}   

\renewcommand{\thefootnote}{\fnsymbol{footnote}}
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\vspace{0.0cm}
\begin{center}
{\large{\bf Experimental Constraints on the Neutralino-Nucleon 
Cross Section\footnote{Talk given at Corfu Summer Institute 
on Elementary Particle Physics, August 31-September 20, 2001.}
\\[5mm]
%\\ and neutralino-nucleon (MODIFY TITLE)
%\\ cross section
}}
\vspace{0.5cm}
\mbox{
\sc{
%\small{
D.G. Cerde\~no$^{1}$,
%\footnote{dgarcia@delta.ft.uam.es}, 
E. Gabrielli$^{2}$,
%\footnote{emidio.gabrielli@helsinki.fi}, 
C. Mu\~noz$^{1}$
%\footnote{carlos.munnoz@uam.es},
% }
}
}
%\end{center}
\vspace{0.5cm}
%\begin{center}

{\small
{\it $^1$ Departamento de F\'{\i}sica
Te\'orica C-XI and Instituto de F\'{\i}sica
Te\'orica C-XVI,\\ 
Universidad Aut\'onoma de Madrid,
% [-0.1cm]
Cantoblanco, 28049 Madrid, Spain. \\
%\vspace*{2mm}
%\it $^2$ Instituto de F\'{\i}sica Te\'orica  C-XVI,
%Universidad Aut\'onoma de Madrid,\\[-0.1cm]
%Cantoblanco, 28049 Madrid, Spain.\\
\vspace*{2mm}
\it $^2$ Helsinki Institute of Physics,
%\\[-0.1cm]
P.O. Box 64, FIN-00014 Helsinki, Finland.
%\vspace*{2mm}
%\it $^4$ Centre for Theoretical Physics, University of Sussex, 
%Brighton BN1 9QJ, U.K.\\
%\vspace*{2mm}
%\it $^5$ Ain Shams University, Faculty of Science, Cairo
%11566, Egypt.
} 
}
\end{center}



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\vspace{0.2cm}

%\abstract
{\small 
In the light of recent experimental results for the direct detection
of dark matter,
we analyze in the framework of SUGRA the
value of the neutralino-nucleon cross section.
We study how this value is modified when the usual 
assumptions of universal soft terms and GUT scale are relaxed.
In particular we consider scenarios with non-universal
scalar and gaugino masses and scenarios 
with intermediate unification scale.
We also study superstring constructions with D-branes,
where a combination of the above two scenarios arises naturally.
In the analysis we take into account 
the most recent experimental constraints, such as
%These are 
the lower bound on the Higgs mass,
the $b\to s\gamma$ branching ratio, and the
%anomalous magnetic moment of the muon 
% $a_{\mu}$.
muon $g-2$.
}


\section{Introduction}

%One of the strongest motivations for physics beyond the standard model is the
%existence of dark matter. Substantial evidences exist suggesting that most
%of the mass in the Universe is some non-luminous matter, the so called
%`dark matter',
%of as yet unknown composition. Currently the most convincing observational 
%evidence for the existence of dark matter comes from the analysis
%of rotation curves of spiral Galaxies, i.e., measurements of the velocity 
%of isolated stars or gas clouds which orbit in the outer parts
%of spiral Galaxies. It
%has been noted that there is not enough luminous matter in those
%Galaxies to account for their observed rotation curves \cite{rotation}. 

%The detailed analysis of rotation curves offers convincing evidence that 
%90\% or more of the mass of the Galaxies is dark. 
%Other evidences arise from large scale measurements, as e.g. the
%motions
%of cluster members Galaxies.
%A key question is then, what 
%could this dark matter be? We don't know the answer
%yet, but we do know that not all of it can be ordinary (baryonic) matter,
%since measured abundances of helium, deuterium and lithium in the 
%scenario of big-bang nucleosynthesis 
%impose a strong upper bound on the baryon density in the Universe. This 
%density
%is too small to account for the whole dark matter in the Universe. 
%The conclusion is that baryonic objects, such as e.g. MACHOs, 
%can be components of the dark matter, but more candidates are 
%needed \cite{Sadoulet}.

%Fortunately, 
%particle physics offers various candidates for (non-baryonic) dark matter,
%all of which would indicate new physics beyond the well tested standard model
%of particle physics \cite{Drees}. For example, 

One of the most interesting candidates for dark matter is a 
long-lived or stable weakly-interacting massive 
particle (WIMP). WIMPs can remain from the earliest moments of the Universe in 
sufficient number to account for a significant fraction of relic density. 
These particles would form not only a background density in the Universe, but
also would cluster gravitationally with ordinary stars in the galactic halos.
This raises the hope of detecting relic 
WIMPs directly, by observing their elastic scattering on  
target nuclei through nuclear recoils \cite{contemporary}.
In fact, one of the current experiments, the DAMA collaboration,
reported recently \cite{experimento1}
data favoring the existence of a 
WIMP signal.
%When 
%uncertainties as e.g. the WIMP velocity or possible bulk halo rotation,
%are included, 
It was claimed that (at 4$\sigma$ C.L.) 
the preferred range of the WIMP-nucleon cross section
is 
$10^{-6}$-$10^{-5}$ pb 
for a WIMP mass between 30 and 200 GeV.
Unlike this spectacular result, other collaborations, 
CDMS \cite{experimento2}, EDELWEISS \cite{edelweiss}, 
and IGEX \cite{igex}, 
claim to have excluded regions of the DAMA 
parameter space.


%Since WIMPs interact with ordinary matter with very roughly weak
%strength, and assuming that their masses are of the order of  
%weak scale (i.e., between 10 GeV and a few TeV),
%it is natural to expect a WIMP-nucleus cross section of the same order
%as that of a weak process, which is around 1 pb.
%This would imply a WIMP-nucleon cross section around   
%$10^{-8}$ pb, too low to be detected by current dark matter
%experiments,
%DAMA, CDMS and UKCDM,
%which are sensitive to a cross section around $10^{-6}$ pb.
%Surprisingly, 
%the DAMA collaboration reported recently \cite{experimento1}
%data favouring the existence of a 
%WIMP signal in their search for annual modulation.
%When 
%uncertainties as e.g. the WIMP velocity or possible bulk halo rotation,
%are included, it was claimed that the preferred range of parameters
%is  (at 4$\sigma$ C.L.) $10^{-6}$ pb $\lsim \sigma\lsim 10^{-5}$ pb 
%for a WIMP mass 30 GeV $\lsim m \lsim 200$ GeV.
%Unlike this spectacular result, the CDMS collaboration 
%claims to have excluded \cite{experimento2} regions of the DAMA 
%parameter space.



In any case, due to these and other projected experiments, 
it seems very plausible that the dark matter 
will be found in the near future. 
In this situation, and assuming that the dark matter 
is a WIMP, it is natural to wonder how big 
the cross section for its direct detection can be.
The answer to this 
question depends on the particular WIMP considered.
The leading candidate in this class is the lightest 
neutralino
\cite{contemporary}, 
a particle 
predicted by the supersymmetric (SUSY) extension of the standard model (SM).


In the simplest SUSY extension, the so-called
minimal supersymmetric standard model (MSSM),
there are four neutralinos, 
$\tilde{\chi}^0_i~(i=1,2,3,4)$, 
since they
are the physical 
superpositions of the bino ($\tilde{B}^0$), of the neutral 
% ($\tilde{B}^0$) 
wino 
($\tilde{W}_3^0$), 
and of the
neutral Higgsinos ($\tilde{H}^0_u$, $\tilde{H}_d^0$). 
In most of the parameter space of
the MSSM,
the lightest neutralino, $\tilde{\chi}^0_1$,
is the lightest supersymmetric particle (LSP). When R-parity is
imposed this implies that $\tilde{\chi}^0_1$ is absolutely stable,
and therefore a dark matter candidate.





In this paper we will analyze this SUSY scenario
in the framework of supergravity (SUGRA), and in particular
several constructions proposed recently in the 
literature (for a review see ref.~\cite{darkcairo}), where the 
$\tilde{\chi}^0_1$-nucleon
cross section can be enhanced, and might be of the order of $10^{-6}$ pb,
i.e., where current dark matter detectors are 
sensitive.
First, in Section~2, we will briefly introduce the SUGRA framework.
In Section~3 we will describe the most recent experimental 
%and theoretical 
constraints which can affect the computation
of the cross section. Then, in the rest of the sections, we 
will re-evaluate previous computations, taking into account
these constraints. In particular, 
in Section~4
we will review the value of the cross section when
universal soft SUSY-breaking 
terms are assumed,
%\cite{Bottino}$^-$\cite{Mario}, 
and how
this value is modified when this assumption is 
relaxed 
\cite{Bottino,Arnowitt,Nath2,darkcairo,Arnowitt3}.
%-\cite{Arnowitt3}.
%\cite{Bottino,Arnowitt,Nath2,darkcairo,Arnowitt3}.
In Section~5 we will consider the case of an intermediate unification 
scale \cite{muas,bailin}.
Finally, in Section~6, we will study superstring scenarios with 
D-branes \cite{nosotros}.
 
%===================================

%This is the case of scenarios with large 
%$\tan \beta$ \cite{Bottino}$^-$\cite{Mario}, with
%non-universal soft SUSY-breaking terms \cite{Bottino,Arnowitt,Nath2}, 
%with multi-TeV masses
%for scalar superpartners known as `focus point' 
%supersymmetry \cite{focus}, and
%finally scenarios with intermediate unification 
%scale \cite{muas}$^-$\cite{nosotros}. 

\section{SUGRA framework}


%The cross section for the elastic scattering of  $\tilde{\chi}^0_1$
%on protons and neutrons 
%has been examined exhaustively in the 
%literature (for a recent re-evaluation see ref.~\cite{Ellis}).
%%\cite{kami}.   
%This is for example the case in
Working in the framework of SUGRA one makes several 
assumptions. In particular, the soft parameters,
i.e., gaugino masses,
scalar masses, and 
trilinear couplings, are generated once SUSY is broken through
gravitational interactions.
%, are universal at the 
%grand unification 
%scale, $M_{GUT} \approx 2\times 10^{16}$ GeV.
They are denoted at the grand unification 
scale, $M_{GUT} \approx 2\times 10^{16}$ GeV,
by $M_{a}$,
$m_{\alpha}$, and $A_{\alpha\beta\gamma}$ 
respectively. 
Likewise, 
radiative electroweak symmetry breaking is imposed, i.e., 
the Higgsino mass parameter $\mu$ 
is determined by the minimization of the Higgs effective 
potential. This implies 
% 
\begin{equation}
\mu^2 = \frac{m_{H_d}^2 - m_{H_u}^2 \tan^2 \beta}{\tan^2 \beta -1 } - 
\frac{1}{2} M_Z^2\ ,
\label{electroweak}
\end{equation} 
%
where
$\tan\beta= \langle H_u^0\rangle/\langle H_d^0\rangle$ 
is the ratio of Higgs vacuum expectation values. 


The soft SUSY-breaking terms will have in general a
non-universal structure in the framework of SUGRA \cite{dilaton}. 
For the case of the observable scalar masses,
this is due to the non-universal couplings
in the K\"ahler potential
between the hidden sector fields breaking SUSY and the
observable sector fields.
For the case of the gaugino masses, this is due to the
non-universality of the gauge kinetic functions associated to the
different gauge groups.
Explicit string constructions, whose low-energy limit is SUGRA,
exhibit these properties \cite{dilaton}.
 

With these assumptions, the SUGRA framework still allows a 
large number of free 
parameters.
In particular, it contains more than 100 parameters.
In order to have predictive power one usually assumes in addition
that the above soft parameters are real and universal at $M_{GUT}$.
This is the so-called minimal supergravity (mSUGRA) scenario,
where there are only four free parameters: 
$m$, $M$, $A$, and $\tan \beta$. In addition, the
sign of $\mu$ remains also undetermined.  
It is worth noticing that explicit string constructions
with these characteristics 
can also be found \cite{dilaton}.

%In what follows we will review first 
%the mSUGRA scenario,
%%recently re-evaluated in ref.~\cite{Ellis}
%%\cite{Bottino}$^-$\cite{Mario}
%and finally we will relax the universality condition 
%allowing non-universal soft masses.
%Let us recall that
%the relevant effective Lagrangian describing
%the elastic scattering of  $\tilde{\chi}^0_1$ on protons and neutrons
%%in the MSSM 
%has a spin-independent (scalar) interaction and a spin-dependent
%interaction.
%However, the contribution of the scalar one is generically larger and
%therefore we will concentrate on it. 
%This scalar interaction includes contributions from squark exchange and
%neutral Higgs exchange.
%Let us finally remark that
%the scalar cross sections for both, protons and neutrons, 
%are basically equal.




\section{
%Phenomenological
Experimental constraints}

Here we list the most recent experimental 
%and theoretical
results which might be relevant when computing
the  $\tilde{\chi}^0_1$-nucleon
cross section in the context of SUGRA. In particular,
they may
produce important constraints in the
parameter space, forbidding for example points with a 
large cross section.




\begin{itemize}

\item Higgs mass

Whereas in the context of the SM,
the negative direct search for the Higgs at the LEP2 collider implies
a precise lower bound on its mass of 114.1 GeV,
the situation in SUSY scenarios is more involved.
In particular, in the framework of mSUGRA, 
one obtains \cite{Heinemeyer} for the lightest CP-even Higgs mass
$m_h \gsim 114.1$ GeV when $\tan\beta \lsim 50$,
and $m_h \gsim 91$ GeV when $\tan\beta$ is larger.
Let us remark that $\tan\beta$ is constrained to be 
smaller than 60 since otherwise the condition of
radiative electroweak symmetry breaking cannot be fulfilled.

On the other hand, when the mSUGRA framework is relaxed
the above SUSY bounds must be modified.
In particular, for the benchmark scenarios studied in 
ref.~\cite{experimentales} one obtains
$m_h \gsim 114.1$ GeV when $\tan\beta \lsim 8$,
and $m_h \gsim 91$ GeV when $\tan\beta$ is larger.

Let us finally remark that
in our computation below we evaluate $m_h$ using the 
program {\it FeynHiggsFast},
a simplified version of the updated program {\it FeynHiggs} \cite{FeynHiggs}
which contains the complete one-loop and dominant two-loop corrections.
The value of $m_h$ obtained with {\it FeynHiggsFast} is
approximately 1 GeV below the one obtained using {\it FeynHiggs}.
In addition, we should also keep in mind that 
the value of $m_h$ obtained with {\it FeynHiggs}
has an
uncertainty of about 3 GeV, due e.g. to higher-order corrections.
%Thus we will allow values of $m_h$ as low as 
%111 GeV (or 88 GeV depending on the value of $\tan\beta$ as explained above).

\item Top mass
%Top and bottom masses

Needless to say we use as input for the
top mass throughout this paper 
the central experimental value $m_t(pole)=175$ GeV.
However, let us remark that a modification in this mass by
$\pm 1$ GeV implies, basically, a modification also of $\pm 1$ GeV
in $m_h$.
%Concerning the bottom mass, we use 
%$m_b(m_b)^{\bar{MS}}_{SM}=4.25$ GeV.




\item SUSY spectrum

We impose
the present experimental lower
bounds on SUSY masses coming from LEP and Tevatron. In particular,
using the low-energy relation from mSUGRA,
$M_1=\frac{5}{3}\tan^2\theta_W M_2$,
one obtains for the lightest chargino mass the bound \cite{chargino}
$m_{\tilde\chi_1^{\pm}}>103$ GeV.
Likewise, one is also able to obtain the following 
bounds for sleptons masses \cite{sleptons}:
$m_{\tilde e}>99$ GeV,
$m_{\tilde\mu}>96$ GeV,
$m_{\tilde\tau}>87$ GeV.
Finally, we use the following 
bounds on the masses of sneutrino, the stop, the rest of squarks, and
gluinos: 
%are given by \cite{spectrum}
$m_{\tilde\nu}>50$ GeV,
$m_{\tilde t}>95$ GeV,
$m_{\tilde q}>150$ GeV,
$m_{\tilde g}>190$ GeV.


%\item Naturalness

%Since we do not want to re-introduce the hierarchy problem we will
%use an upper bound on SUSY masses of about 1 TeV.


\item $b\to s\gamma$ 

The measurements of $B\rightarrow X_s\gamma$ decays 
at 
CLEO \cite{cleo} 
%and LEP \cite{lep}
and BELLE \cite{belle},
lead to bounds on the branching ratio 
$b\to s\gamma$. In particular we impose in our computation
$2\times 10^{-4}\leq BR(b\to s\gamma)\leq 4.1\times
10^{-4}$, where the evaluation 
is carried out at the NLO accuracy in
the SM \cite{gambino}, while for the SUSY contributions only the 1-loop
diagrams have been taken into account \cite{GS}.


\item $g_{\mu}-2$

The new measurement of the anomalous
magnetic moment of the muon, $a_\mu=(g_{\mu}-2)/2$, 
in the E821 experiment at the 
BNL \cite{muon} implied apparently a 2.6$\sigma$ 
deviation from the SM predictions.
In particular, taking a 2$\sigma$ range around the E821 central value
one would have
$11\times 10^{-10}\leq a_{\mu}(E821)-a_{\mu}(SM)\leq 75\times 10^{-10}$.
However, recent theoretical computations \cite{rafael}
have shown that a significant part of this discrepancy was due to the
evaluation of the hadronic light-by-light 
contribution \cite{todos}.
%reduce this deviation to 1.6$\sigma$.
As a consequence, assuming that the possible 
new physics is due to SUGRA, we impose in our computation the current
constraint
$-7\times 10^{-10}\leq a_{\mu} (SUGRA)\leq 57\times 10^{-10}$ at the
2$\sigma$ level.

\item LSP

The LSP must be an electrically neutral
(also with no strong interactions) particle,
since otherwise it 
would bind to nuclei and would be excluded as a candidate
for dark matter from unsuccessful searches for exotic heavy 
isotopes \cite{isotopes}.
As mentioned in the Introduction, the $\tilde{\chi}_1^0$ is
the LSP in most of the parameter space of SUGRA.
However, in some regions the stau can be lighter 
than the $\tilde{\chi}_1^0$. Following the above argument,
we discard these regions.


Other cosmological constraints, as e.g. the observational bounds
on the relic $\tilde{\chi}_1^0$ density,
$0.1\lsim \Omega_{\tilde{\chi}_1^0}h^2\lsim 0.3$,
will not be applied. The computation of the relic  $\tilde{\chi}_1^0$
density depends on assumptions about the evolution of the early
Universe, and therefore different cosmological scenarios give
rise to different results \cite{relic}.




%\item CCB

%\item $\tilde{\chi}^0_1$ relic density 


\end{itemize}








\section{SUGRA predictions for the neutralino-nucleon cross section}


%In the simplest SUSY extension of the standard model, the so-called
%minimal supersymmetric standard model (MSSM)
%there are four neutralinos, 
%%$\tilde{\chi}^0_i~(i=1,2,3,4)$, 
%since they
%are the physical 
%superpositions of the fermionic partners of the neutral electroweak 
%gauge bosons, 
%called bino ($\tilde{B}^0$) and wino ($\tilde{W}_3^0$), and of the
%fermionic partners of the  
%neutral Higgs bosons, called Higgsinos ($\tilde{H}^0_u$, 
%$\tilde{H}_d^0$). 
%Therefore the lightest neutralino, $\tilde{\chi}^0_1$, will be the 
%dark matter candidate.
%The neutralino mass matrix 
%with the conventions for gaugino and Higgsino masses in the Lagrangian,
%${\cal L} =\frac{1}{2}\sum_{a} M_a {\lambda}_a \lambda_a + 
%\mu \tilde{H}^0_u \tilde{H}^0_d$ + h.c.,
%is given by
%%
%\begin{equation}
%{\small \bmat{cccc}M_1 & 0 &-M_Z\cos\beta \sin\theta_W &M_Z\sin\beta 
%\sin\theta_W \\ 0 & M_2 & M_Z \cos \beta \cos\theta_W & -M_Z \sin \beta 
%\cos\theta_W \\ -M_Z \cos \beta \sin\theta_W & M_Z \cos \beta\cos\theta_W&
%0& -\mu \\ M_Z \sin \beta \sin\theta_W & -M_Z \sin \beta\cos\theta_W&-\mu 
%& 0 \emat,}
%\label{mass}
%\end{equation}
%%
%in the above basis 
%($\tilde B^0=-i{\lambda}'$, $\tilde W_3^0=-i{\lambda}^3$, $\tilde H_u^0$, 
%$\tilde H_d^0$). 
%%and with the convention in the superpotential
%%$W=-\mu \epsilon_{ij} H_d^i H_u^j = -\mu (H_d^0 H_u^0-H_d^- H_u^+)$.
%Here $M_1$ and $M_2$ are the soft bino and wino masses respectively, 
%$\mu$ is the Higgsino mass parameter 
%and 
%$ \tan\beta= \langle H_u^0\rangle/\langle H_d^0\rangle$ 
%is the ratio of Higgs vacuum expectation values. 
%We parameterize the gaugino and Higgsino content
%of the lightest neutralino according to 
%%
%\begin{equation}
%\tilde{\chi}^0_1 = N_{11} \tilde{B}^0 +N_{12} \tilde{W}_3^0 +
%N_{13} \tilde{H}^0_d + N_{14} \tilde{H}^0_u\ .
%\label{lneu}
%\end{equation}
%%
%It is commonly defined that $\tilde{\chi}^0_1$ is mostly gaugino-like 
%if $P\equiv \vert N_{11}\vert^2 + \vert N_{12} \vert^2 > 0.9$, Higgsino-like
%if $P<0.1$, and mixed otherwise.

%The relevant effective Lagrangian describing
%the elastic $\tilde{\chi}^0_1$-nucleon scattering in the MSSM is given by
%%
%\begin{equation}
%\mathcal{L}_{eff} = \sum_{q} \left(\alpha_{q} \bar{\chi}\chi \bar{q} q
%+
%\beta_q \bar{\chi} \gamma^{\mu}  \gamma^{5} 
%\chi \bar{q} \gamma_{\mu}  \gamma^{5} q\right)\ ,
%\end{equation}
%%
%where  $\alpha_q$, $\beta_q$ are given e.g. in ref.\cite{Ellis} 
%with the sign conventions for Yukawa couplings in the Lagrangian,
%${\cal L}=-h_u H_u^0 \bar u_L u_R -h_d H_d^0 \bar d_L d_R 
%- h_e H_d^0 \bar e_L e_R $ + h.c.,
%and the sum 
%runs over the six quarks. 
%The contribution of the scalar (spin-independent) interaction,
%the one proportional to $\alpha_q$, to the $\tilde{\chi}^0_1$-nucleon 
%cross section is generically larger than the spin-dependent
%interaction,
%the one proportional to $\beta_q$.
%One can concentrate then on the scalar 
%%In particular, in the context of DAMA and CDMS experiments,
%% $\sigma_{spin}/\sigma_{scalar} \sim 1-5 \%$. 
%$\tilde{\chi}^0_1$-nucleon
%cross section. This is given by
%%
%\begin{equation}
%\sigma
%%_{\tilde{\chi}_1^0-p} 
%= \frac{4 m_{r}^2}{\pi} f^2\ ,
%\label{crosssection}
%\end{equation}
%%
%where $m_{r}$ is the reduced $\tilde{\chi}^0_1$ mass and
%%
%\begin{equation}
%\frac{f}{m} = \sum_{q} f_q \frac{\alpha_q}{m_q}
%\end{equation}
%%
%with $m$ the mass of the nucleon and $m_q$ the mass of the quarks. 
%The scalar 
%coefficients $\alpha_{q}$ include contributions from 
%squark ($\tilde q$) exchange and neutral Higgs 
%$(h, H)$ exchange, as illustrated in Fig.~\ref{Feynman}.  
%The numerical values of the hadronic matrix elements $f_q$ 
%for the proton are as follows \cite{Ellis}:
%%
%\begin{equation}
%f_u= 0.020 \pm 0.004\ , \hspace{0.5cm} f_d= 0.026 \pm
%0.005\ ,
%\hspace{0.5cm} f_s= 0.118 \pm 0.062\ ,
%\label{central}
%\end{equation}
%%
%and \footnote{Larger values,
%as for example $f_s=0.455$, have also been used in the 
%literature, see e.g. ref.~\cite{Arnowitt}.} 
%$f_{c,b,t}= \frac{2}{27} (1-\sum_{q=u,d,s} f_q)$.
%For the neutron the value of $f_s$ is the same and therefore
%the scalar cross sections for both, protons and neutrons, are
%basically equal. 

%\begin{figure}[t]
%\begin{center}
%\epsfig{file= feynman.eps,
%%width=7cm,
%height=3.5cm,angle=0}
%%\hspace{0.5cm}\epsfig{file= p24f008.eps, width=5.75cm, height=4.5cm,angle=0}
%\end{center}
%\caption{Feynman diagrams contributing to neutralino-nucleon cross section.}
%\label{Feynman}
%\end{figure}


%==========================================


The cross section for the elastic scattering of  $\tilde{\chi}^0_1$
on nucleons  
has been examined exhaustively in the 
literature (for a recent re-evaluation see ref.~\cite{Ellis}).
Let us recall that
the relevant effective Lagrangian describing
the elastic scattering of  $\tilde{\chi}^0_1$ on protons and neutrons
%in the MSSM 
has a spin-independent (scalar) interaction and a spin-dependent
interaction.
However, the contribution of the scalar one is generically larger and
therefore we will concentrate on it. 
This scalar interaction includes contributions from squark exchange and
neutral Higgs exchange.
Let us also remark that
the scalar cross sections for both, protons and neutrons, 
are basically equal.


%\cite{kami}.   
%This is for example the case in
%the framework of SUGRA.
%Let us recall that in this framework one makes several 
%assumptions. In particular, the soft parameters,
%i.e., gaugino masses,
%scalar masses, and 
%trilinear couplings, are generated once SUSY is broken through
%gravitational interactions.
%%, are universal at the 
%%grand unification 
%%scale, $M_{GUT} \approx 2\times 10^{16}$ GeV.
%They are denoted at the grand unification 
%scale, $M_{GUT} \approx 2\times 10^{16}$ GeV,
%by $M_{i}$,
%$m_{\alpha}$, and $A_{\alpha\beta\gamma}$ 
%respectively. 
%Likewise, 
%radiative electroweak symmetry breaking is imposed, i.e., 
%the Higgsino mass parameter $\mu$ 
%is determined by the minimization of the Higgs effective 
%potential. This implies 
%% 
%\begin{equation}
%\mu^2 = \frac{m_{H_d}^2 - m_{H_u}^2 \tan^2 \beta}{\tan^2 \beta -1 } - 
%\frac{1}{2} M_Z^2\ ,
%\label{electroweak}
%\end{equation} 
%%
%where
%$\tan\beta= \langle H_u^0\rangle/\langle H_d^0\rangle$ 
%is the ratio of Higgs vacuum expectation values. 
%With these assumptions, the SUGRA framework still allows a 
%large number of free 
%parameters.
%In particular, it contains 100??? free parameters.
%In order to have predictive power one usually assumes in addition
%that the above soft parameters are real and universal at $M_{GUT}$.
%This is the mSUGRA scenario,
%where there are only four free parameters: 
%$m$, $M$, $A$, and $\tan \beta$. In addition, the
%sign of $\mu$ remains also undetermined.  
%It is worth noticing that explicit string constructions
%can exhibit these properties \cite{dilaton}.
In what follows we will review the possible value of the cross section 
in the framework of SUGRA. First
we will consider 
the mSUGRA scenario, where the soft terms are assumed to be universal,
and we 
%recently re-evaluated in ref.~\cite{Ellis}
%\cite{Bottino}$^-$\cite{Mario}
will later relax this condition 
allowing non-universal soft masses.

 
\subsection{mSUGRA scenario}



In the mSUGRA scenario $\tilde{\chi}^0_1$ is mainly bino.
This result can be qualitatively understood from the 
well known evolution of
$m_{H_u}^2$ with the scale.
It becomes large and negative at low scales. Then 
$|\mu|$ given by eq.~(\ref{electroweak}) 
becomes also large, and in particular much larger than
$M_1$ and $M_2$. Thus,  
the lightest neutralino will be mainly gaugino, and in particular
bino, since, as mentioned in the previous section, 
at low energy $M_1=\frac{5}{3}\tan^2\theta_W M_2\approx 0.5 M_2$.

When  $\tilde{\chi}^0_1$ is basically bino
scattering channels through Higgs ($h$, $H$) exchange are highly 
suppressed, and
as a consequence, the predicted scalar $\tilde{\chi}^0_1$-proton cross section
%\begin{figure}[t]
%\begin{center}
%\epsfig{file= running2.eps,width=7cm,height=3.5cm,angle=0}
%%\hspace{0.5cm}\epsfig{file= p24f008.eps, width=5.75cm, height=4.5cm,angle=0}
%\end{center}
%\caption{
%Running of the soft Higgs masses-squared with energy.}
%\label{run}
%\end{figure}
%It was observed (for a recent re-evaluation see ref.\cite{Ellis})
%that for low and moderate values of
%$\tan \beta$ the predicted scalar neutralino-proton cross sections
is well below the accessible experimental regions for low and moderate
values of $\tan\beta$
\cite{Ellis}.
Although
the cross section increases entering in the DAMA 
region \cite{experimento1,halo} when the value
of $\tan\beta$ increases \cite{Bottino,Arnowitt}, 
the experimental constraints discussed in Section~3
exclude this possibility \cite{Ellis2,Arnowitt3}.
%\cite{Ellis,Ellis2,Arnowitt3}.
%We show this fact in Fig.~\ref{ellisfig}a, 
%where for $\tan\beta=3$ the gaugino-Higgsino 
%components-squared $N_{1i}^2$ of the lightest neutralino as a function
%of its mass $m_{\tilde{\chi}_1^0}$ 
%are exhibited. Here we are using as an example $m_0=150$ GeV, and
%$M_{1/2}$ is essentially fixed for a given
%$m_{\tilde{\chi}_1^0}$.
%Note that 
%%$m_{\tilde{\chi}_1^0}>70$, 
%$N_{11}$ is 
%extremely large and therefore $P\gsim 0.9$.
%Then, the scattering channels through Higgs exchange shown in 
%Fig.~\ref{Feynman} are suppressed (recall that the Higgs-neutralino-neutralino
%couplings are proportional to $N_{13}$ and $N_{14}$) and therefore
%the cross section is small.
%As a matter of fact, the scattering channels through squark 
%exchange, shown also in Fig.~\ref{Feynman}, 
%are also suppressed by the mass of the squarks. Indeed
%in this limit the cross section (\ref{crosssection}) can be approximated as
%%
%\bea
%\sigma_{\tilde\chi_1^0-p}
%\approx {m_r^2}\ \frac{\alpha^2}{\left(m_{\tilde s}^2-
%m_{\tilde{\chi}_1^0}^2\right)^2}\  f_s^2\   |N_{11}|^4
%\ .
%\label{cross}
%\eea
%%
%Thus it can be roughly estimated to be 
%$\sigma_{\tilde{\chi}_1^0-p}\approx 10^{-8}$ pb,
%for $\alpha\approx 10^{-2}$, 
%$f_s\approx 10^{-1}$,
%$m_r\approx 1$ GeV and 
%$m_{\tilde s}\approx 300$ GeV. This is precisely the value of
%the cross section 
%expected for a weak process as discussed in the Introduction.
We show this fact in
Fig.~\ref{ellisfig}. There the  $\tilde{\chi}^0_1$-proton cross section
$\sigma_{\tilde{\chi}_1^0-p}$ as a function of the neutralino mass
$m_{\tilde{\chi}_1^0}$ for
$\tan \beta=20$ and 35 is plotted.

Concerning the parameter space of the figure,
we use the following values. For the soft scalar mass
0$\leq m \leq 600$ GeV, where 
the curve associated to $m=600$ GeV 
corresponds to the minimum values of 
$\sigma_{\tilde{\chi}_{1}^{0}-p}$ in the figure.
Note that 
in order to avoid the stau being the LSP not very small values of
$m$ are in fact allowed.
%for all values of 
%$m_{\tilde{\chi}_1^0}$.
%For the trilinear coupling we use $A = M$ 
%because this value corresponds to the maximum value of
%$\sigma_{\tilde{\chi}_{1}^{0}-p}$
%in the wide range $\mid A/M\mid \lsim 4$.
%, although in fact
%the variations are small.
The cross section is not very sensitive to 
the specific values of $A$ in a wide 
range. In particular,
we have checked that 
this is so for $\mid A/M\mid \lsim 4$. In the figure we fix
$A = M$,
corresponding to the maximum value of $\sigma_{\tilde{\chi}_{1}^{0}-p}$.
%and $m_0$ 
%As mentioned above 
On the other hand,
the gaugino mass
$M$ is essentially fixed for a given $\tan\beta$ and 
$m_{\tilde{\chi}_1^0}$. 
We consider only positive values of $M$, since negative ones
correspond to smaller cross sections.
Finally, 
we choose to plot only the case $\mu > 0$ since for negative values of
$\mu$ the cross section is 
smaller. In addition, constraints
coming from
the $b\rightarrow s\gamma$ and $g_{\mu}-2$ processes highly reduce
the $\mu<0$ parameter space. 
%In addition,
%recent measurements of the anomalous magnetic moment of the
%muon \cite{marci} seems to exclude all cases with $\mu<0$.
%In what follows we will always use
%$\mu > 0$.
Let us remark that we are using the conventions
for the SUGRA parameters entering in the Lagrangian
explained in ref.~\cite{darkcairo}.






Concerning the experimental constraints, the lower
bound in the figure $m_{\tilde{\chi}_1^0}\gsim 50$ GeV is obtained
imposing the experimental bound on the 
lightest chargino mass 
$m_{\tilde\chi_1^{\pm}}>90$ GeV. However, as mentioned in Section~3,
the most recent bound in the context of mSUGRA is now \cite{chargino}  
$m_{\tilde\chi_1^{\pm}}>103$ GeV, and we show it with
a dotted line.
We also show in Fig.~1 several values of the Higgs mass with 
dashed lines.
In addition the region to the left of the dot-dashed curve
is excluded by $b\to s\gamma$,
and the region above the solid line is excluded by $g_{\mu}-2$.



%
\begin{figure}[t]
\begin{center}
\epsfig{file= universal.eps,
%width=11cm,
height=7cm
}
%\hspace{0.5cm}\epsfig{file=ellis_m30-550.eps, width=5.75cm, height=4.5cm
%,angle=0
% }
\end{center}

\vspace{-0.5cm}

\caption{
% (a) Gaugino-Higgsino components-squared
%$N_{1i}^2$ 
%of the lightest neutralino as a function of its mass.
%for the case
%$\tan\beta=10$, $M_I=10^{16}$ GeV and
%$M_I=10^{11}$ GeV
% (b) 
Scatter plot of the neutralino-proton cross section
$\sigma_{\tilde{\chi}_1^0-p}$ 
as a function of 
the neutralino mass $m_{\tilde{\chi}_1^0}$ in the mSUGRA scenario, for
$\tan\beta=20$ and 35.
%discussed in the text.
%$m_{\tilde{\chi}_1^0}$
The dotted line corresponds to the LEP lower bound
$m_{\tilde\chi_1^{\pm}}=103$ GeV. Dashed lines correspond to different
values of the Higgs mass, 110 and 114 GeV.
The region to the left of the dot-dashed curve is excluded by
$b\to s\gamma$,
and the region above the solid line is excluded by $g_{\mu}-2$.
DAMA and CDMS current 
experimental limits and projected GENIUS limits are shown.
}
\label{ellisfig}
\end{figure}
%


As we can see, 
the present experimental lower limit for the Higgs mass 
in mSUGRA when $\tan\beta\lsim 50$, $m_h\gsim 114.1$ GeV,
is the most important constraint, implying that only the
solid region where  $\sigma_{\tilde{\chi}_1^0-p} \lsim 10^{-8}$ pb
survives. 
%although the cross section increases when 
%the value of $\tan \beta$ increases, still for $\tan \beta=20$  
%$\sigma_{\tilde{\chi}_1^0-p} < 10^{-7}$ pb.
Obviously, in this mSUGRA case, more sensitive detectors
producing further data are needed.
Fortunately, many dark matter detectors are being projected.
%This is the case e.g. of DAMA 250 kg. and CDMS Soudan, but
Particularly interesting is the GENIUS detector \cite{GENIUS},
where values of the cross section as low as 
$10^{-9}$ pb will be accessible, as shown in the figure. 



%%Fortunate
% and
%therefore
%for these values of the parameters
%%Given this situation, 
%we would have to wait in principle for the 
%projected GENIUS detector \cite{GENIUS} to be able to
%test the neutralino as a dark-matter candidate.



%Clearly, more sensitive detectors
%producing further data are needed to solve this contradiction.
%Fortunately, many dark matter detectors are being projected.
%This is the case e.g. of DAMA 250 kg. and CDMS Soudan, but
%particularly interesting is the projected GENIUS \cite{GENIUS}
%detector where values of the cross section as low as 
%$10^{-9}$ pb will be accesible. 





%It was in fact
%pointed out in refs.\cite{Bottino,Arnowitt} that the large
%$\tan\beta$ regime allows regions where 
%$\sigma_{\tilde{\chi}_1^0-p}\approx 10^{-6}$ pb is reached. 
%We can see in Fig.~\ref{ellisfig2} that this is so for 
%$\tan\beta \gsim 20$.

Very large values of 
$\tan\beta$, 
like $\tan \beta \simeq 50$, 
%which 
%correspond approximately to the unification of the tau and top Yukawa 
%couplings at $M_{GUT}$,
have also been considered \cite{Mario,Ellis2,Arnowitt3}.
Although it was found, as expected, that the 
cross section is enhanced,
the 
well known
experimental limits coming from 
$b \to s \gamma$ for large $\tan\beta$, 
lead to severe constraints on the parameter space. In particular,
these constraints imply $\sigma_{\tilde{\chi}_1^0-p} \lsim 
10^{-7}$ pb. 




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\epsfig{figure=nuniversal.eps, 
%width=7cm, 
height=7cm}
\end{center}

\vspace{-0.5cm}

\caption{
Scatter plot of the neutralino-proton cross section
$\sigma_{\tilde{\chi}_1^0-p}$ 
as a function of 
the neutralino mass $m_{\tilde{\chi}_1^0}$ in the SUGRA scenario
with non-universal soft scalar masses, for
$\tan\beta=20$ and 35.
Only points of the parameter space fulfilling
$b \to s \gamma$ and $g_{\mu}-2$ are plotted. 
Big (red) dots correspond to points 
with $m_h\geq 114$ GeV. Small (blue) dots correspond to points
with $91 <m_h < 114$ GeV. 
%The same as in Fig.~\ref{ellisfig}
%$\sigma_{\tilde{\chi}_{1}^{0}-p}$ vs.
%$m_{\tilde{\chi}_{1}^{0}}$ 
%but 
%for $\tan{\beta}=10$ and 
%using non-universal soft 
%scalar masses.
The universal case (green region) 
is shown for comparison.
DAMA and CDMS current 
experimental limits and projected GENIUS limits are also shown.
}
\label{arnowit}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\subsection{Scenario with non-universal soft terms}

%As mentioned above, the soft SUSY-breaking terms will have in general a
%non-universal structure in the framework of SUGRA \cite{dilaton}. 
%For the case of the observable scalar masses,
%this is due to the non-universal couplings
%in the K\"ahler potential
%between the hidden sector fields breaking SUSY and the
%observable sector fields.
%For the case of the gaugino masses, this is due to the
%non-universality of the gauge kinetic functions associated to the
%different gauge groups.
%Explicit string constructions, whose low-energy limit is SUGRA,
%exhibit these properties \cite{dilaton}.
 
It was shown recently
that non-universality allows to 
increase 
%\cite{Bottino}-\cite{Arnowitt3}
%\cite{Bottino,Arnowitt,Nath2}
the neutralino-proton cross section. 
%for moderate values of $\tan\beta$.
The key point
consists of reducing the value of $|\mu|$.
Following the discussion in the previous subsection, 
the smaller $|\mu|$ is,
the larger the Higgsino components of the lightest neutralino become.
Eventually, $|\mu|$ will be of the order of $M_1$, $M_2$
and $\tilde{\chi}_1^0$ will be a mixed Higgsino-gaugino state.
Indeed scattering channels through Higgs exchange 
%in 
%Fig.~\ref{Feynman}
are now important and their contributions to the cross section
will increase it.
This can be carried out with non-universal 
scalar masses \cite{Bottino,Arnowitt,Arnowitt3}
and/or gaugino masses \cite{Nath2,darkcairo}.
%, as we will discuss below.



%====================================

%Recently there has been some theoretical 
%activity \cite{Bottino}$^-$\cite{nosotros} trying to obtain regions
%in the parameter space of supergravity (SUGRA) scenarios 
%compatible with the sensitivity of
%current dark matter detectors, DAMA and CDMS. 

%==================================

%\section{Large neutralino-nucleon cross section in supergravity scenarios}
 
%The key point in most of these scenarios 
%to carry it out consists of reducing the value of $|\mu|$.
%Following the discussion below eq.(\ref{electroweak}) in the previous section, 
%the smaller $|\mu|$ is,
%the larger the Higgsino components of the lightest neutralino become.
%Eventually, $|\mu|$ will be of the order of $M_1$, $M_2$
%and $\tilde{\chi}_1^0$ will be a mixed Higgsino-gaugino state.
%Indeed scattering channels through Higgs exchange in 
%Fig.~\ref{Feynman}
%are now important and their contributions to the cross section
%(\ref{crosssection}) can
%be schematically approximated as 
%%
%\bea
%\sigma_{\tilde\chi_1^0-p}
%\approx {m_r^2}\
%\frac{\lambda_s^2 \alpha}{m_h^4}\
%f_s^2\   |N_{13} N_{11}|^2 
%\ ,
%\label{cross2}
%\eea
%%
%where $\lambda_s$ is the strange quark Yukawa coupling and
%$m_h$ represent the Higgs masses.
%With the same rough estimate as in eq.(\ref{cross}),
%one obtains $\sigma_{\tilde{\chi}_1^0-p}\approx 10^{-6}$ pb for 
%$m_h\approx 100$ GeV.











%Non-universality in the Higgs sector, concerning dark matter, was studied in 
%refs.\cite{Ole,Fornengo,Bottino}. Subsequently, non-universality in
%the sfermion sector was added in the analysis \cite{arna,Arnowitt}.
%Thus the soft masses can be parameterized as follows:
%\begin{eqnarray}
%m_{H_{d}}^2&=&m_{0}^{2}(1+\delta_{1}), \quad m_{H_{u}}^{\ 2}=m_{0}^{2}
%(1+ \delta_{2}),\nonumber \\
%m_{Q_{L}}^2&=&m_{0}^{2}(1+\delta_{3}), \quad m_{u_{R}}^{\ 2}=m_{0}^{2}
%(1+\delta_{4}), \nonumber\\
%m_{e_{R}}^2&=&m_{0}^{2}(1+\delta_{5}),  \quad m_{d_{R}}^{\ 2}=m_{0}^{2}
%(1+\delta_{6}), \nonumber\\
%m_{L_{L}}^2&=&m_{0}^{2}(1+\delta_{7}),     
%\end{eqnarray}
%where 
%$Q_{L}=(\tilde{t}_{L}, \tilde{b}_{L})$, $L_{L}=(\tilde{\nu}_{L},
%\tilde{\tau}_{L})$, $u_{R}=\tilde{t}_{R}$, $e_{R}=\tilde{\tau}_{R}$, and 
%it is assumed that $ -1 \leq \delta_{i} \leq 1 $. 




%As explained above 
%%at the beginning of this section~3,
%$|\mu|$ is one of the important parameters when
%computing
%the  $\tilde{\chi}^0_1$-nucleon cross section.
%Its value is determined by condition (\ref{electroweak}) and can
%be significantly reduced for some choices of $\delta$'s. 
We can have
a qualitative
understanding 
%of the effects of the $\delta$'s on $\mu$ 
from the following.
When $m_{H_u}^2$($m_{H_d}^2$) at $M_{GUT}$ increases(decreases),
its negative(positive) contribution at low energy in eq.~(\ref{electroweak})
is less important. Likewise, when $m_{Q_{L}}^2$ and $m_{u_{R}}^2$
at $M_{GUT}$ decrease, due to their contribution proportional
to the top Yukawa coupling in the RGE of $m_{H_u}^2$ the negative
contribution of the latter to $\mu^2$ is again less important.
%The importance of including both effects was pointed out in
%ref.\cite{Arnowitt}.
%can be read from
%the following analytic expression \cite{Arnowitt} for low and
%intermediate $\tan\beta$:
%
%\begin{eqnarray}
%\mu^{2} & = & \frac{t^{2}}{t^{2}-1} \left[ \left\{ \frac{1-3 D_{0}}{2} +
%\frac{1}{t^{2}} \right\} + \left\{
%\frac{1-D_{0}}{2}(\delta_{3}+\delta_{4})-\frac{1+D_{0}}{2} \delta_{2} +
%\frac{\delta_{1}}{t^{2}} \right\} \right] m_{0}^{2} \nonumber \\ & & + 
%\mbox{universal parts + loop corrections}       
%\end{eqnarray}
%
%where $t \equiv \tan{\beta}$ and $D_{0} \cong 1 - 
%(\frac{m_{t}}{200 \sin{\beta}})^{2}$. In general $D_{0}$ is small, $D_{0}
%\leq 0.2$, and hence
%the squark non-universality, $\delta_{3}$ and $\delta_{4}$, produce 
%comparable size effects as the Higgs non-universalities $\delta_{1}$ and 
%$\delta_{2}$, so that both must be included. 
%Thus one can deduce that 
%$\mu^2$ will be reduced 
%and hence $\sigma_{\tilde{\chi}_1^0-p}$ increased.
%by choosing $\delta_3, \delta_4, \delta_1 < 0$ and $\delta_2 >0$.



This effect can be seen in Fig.~\ref{arnowit}, where
a scatter plot of
$\sigma_{\tilde{\chi}_{1}^{0}-p}$ as a function of $m_{\tilde{\chi}_{1}^{0}}$
for a scanning of the soft scalar masses is shown.
This scanning is explained in detail e.g. in ref.~\cite{darkcairo}.
%In order to avoid potential problems
%with flavour changing neutral currents, it was 
%assumed that
%the first two generations of squarks
%and sleptons have a common scalar mass
%$m_0$ at $M_{GUT}$, and 
%non-universalities were allowed in the third generation and
%Higgs sector.
%Following this analysis \cite{Bottino,Arnowitt} we show in Fig.~\ref{arnowit},
%FOR $\tan\beta=10$ ????
%a scatter plot of
%$\sigma_{\tilde{\chi}_{1}^{0}-p}$ as a function of $m_{\tilde{\chi}_{1}^{0}}$
%for a scanning of the parameters as follows:
%$0 \leq \delta_2 \leq 1$, $-1 \leq \delta_{1,3,4} \leq 0$.
%The other $\delta'$s are not so important in the computation and we take
%them vanishing.
%As in Fig.~1 we are taking $A=M_{1/2}$
%, 30 GeV $\leq m_0\leq 550 $ GeV as in
As in Fig.~\ref{ellisfig}, we are taking $A=M$.
%and we have to take now 140 GeV $\lsim M_{1/2} \lsim
%450$ GeV.
For comparison we superimpose also the region (green area) 
obtained in Fig.~\ref{ellisfig} 
%for $\tan\beta=10$
with universality.
%, $\delta_i=0$.
We see that non-universal scalar masses help in increasing the value of
$\sigma_{\tilde{\chi}_{1}^{0}-p}$.
In the figure we only plot the points of the parameter space fulfilling
$b \to s \gamma$ and $g_{\mu}-2$. In particular,
the big (red) dots correspond to points 
with $m_h\geq 114$ GeV. As a conservative approach,
we also plot with small (blue) dots those points
with $91 <m_h < 114$ GeV (as discussed in Section~3,
this possibility may arise when the mSUGRA scenario is relaxed).
For $\tan\beta=20$, both kind of points enter in the DAMA region.
In particular, those with $m_h < 114$ GeV have an upper bound for
the cross section 
%$\sigma_{\tilde{\chi}_{1}^{0}-p}> } \lsim 10^{-5}$ pb, whereas
$\approx 10^{-5}$ pb, whereas
for those with $m_h > 114$ GeV the upper bound is
$\approx 10^{-6}$ pb.
For $\tan\beta=35$, the area of points entering in the DAMA region
increases a lot, and now, even points with $m_h > 114$ GeV
can have $\sigma_{\tilde{\chi}_{1}^{0}-p} \approx 10^{-5}$ pb.
%Note that the line associated to the 
%universal case $\delta_i=0$ corresponds to the minimum value of 
%$\sigma_{\tilde{\chi}_{1}^{0}-p}$ (see also  Fig.~\ref{ellisfig}).
Let us finally remark that the 
non-universality in the Higgs sector gives the most important
effect, and the non-universality in the sfermion sector only increases
slightly the cross section.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\epsfig{figure=nuniversalg.eps,
%width=9cm, 
height=7cm}
\end{center}

\vspace{-0.5cm}

\caption{
The same as in Fig.~\ref{arnowit}
but using non-universal soft 
gaugino masses. 
%The universal case (green region) 
%is shown for comparison.
%corresponds to the
%universal case.  
\label{nunivg.eps}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




On the other hand, concerning gaugino masses,
%Let us discuss now which values of the parameters are interesting
%in order to increase the cross section, with respect to the
%universal case $\delta'_i=0$. 
%In this sense, 
it is worth noticing that
$M_3$ appears in the RGEs of squark masses, so e.g.
their contribution proportional
to the top Yukawa coupling in the RGE of $m_{H_u}^2$ will
do this less negative if $M_3$ is small, and therefore $\mu^2$ will become 
smaller in this case.
Taking into account this effect, 
we show in Fig.~\ref{nunivg.eps},
for $\tan\beta=20$ and 35,
a scatter plot of
$\sigma_{\tilde{\chi}_{1}^{0}-p}$ as a function of $m_{\tilde{\chi}_{1}^{0}}$
for the scanning of the parameters $M_{3}$, $M_{2}$, $M_{1}$  explained
in ref.~\cite{darkcairo}.
%:
%$ -1 \leq \delta'_{1,2} \leq 1 $
%%$ -1 \leq \delta'_{2} \leq 1 $
%and $ -2/3 \leq \delta'_{3} \leq 0 $ (where the lower bound
%is due to the experimental bound on gluino masses), imposing 
%$M_3<M_1, M_2$.
%We
%fix 
%$M_{1/2}= 225$ GeV and $A=M_{1}$.
%As in the previous figures
%we are 
%taking 30 GeV $\leq m_0\leq 550$ GeV.
%We also superimpose for comparison the region (green area) 
%obtained in Fig.~\ref{ellisfig2} for $\tan\beta=10$
%with universality, $\delta'_i=0$.
Although points with a large cross section entering in the DAMA region
do exist \cite{darkcairo}, they are, at the end of the day, forbidden 
when 
$b \to s \gamma$ and $g_{\mu}-2$ constraints are imposed.
%, unlike
%the case with non-universal scalars. 

Clearly, in this sense, SUGRA scenarios with non-universal scalars
are favored with respect to scenarios with non-universal gauginos.

%Clearly non-universal gaugino masses produce a large 
%increment in the cross section.

%Let us finally remark that the upper and right hand side regions with
%large cross sections in the
%figure correspond to points of the parameter space fulfilling 
%$M_2<M_1$ (and $M_3$ close to its lower bound).
%As discussed in Section~2,
%for
%low or moderate $\tan\beta$ the lightest neutralino is mainly gaugino $P>0.9$,
%and in particular bino since $N_{11}\gg N_{12}$. However, relaxing the
%universality condition for gaugino masses, $M_2<M_1$
%will produce $N_{11}<N_{12}$, and therefore
%the lightest neutralino will have an important wino 
%component\footnote{It is worth noticing that this situation
%also arises in anomaly-mediated SUSY-breaking scenarios, where
%the cross section can also be increased \cite{anomaly,anomaly2}.}.
%Since the latter couples through $SU(2)$
%interactions, unlike the bino component which couples through $U(1)$
%ones, the cross section increases.\\
%% (see eq. ????????????).



%\hspace{-0.65cm}{\it $(iii)$ Non-universal gaugino and scalar masses}\\

%Given the above situation concerning the enhancement of
%neutralino-proton cross section through non-universality,
%%the most simple way
%%to obtain $\sigma_{\tilde{\chi}_{1}^{0}-p}>10^{-6}$ pb
%%for moderate $\tan\beta$, 
%it is worth analyzing the combination of both possibilities,
%non-universal gaugino and scalar masses.
%However,
%%Thus we show in Fig.~\ref{nunivgs} 
%the final result 
%for the cross section 
%%of combining the parameters studied in
%% {\it (i)} and {\it (ii)} 
%is not going to be very different from the one obtained
%in Fig.~\ref{nunivg.eps}.
%We have explicitly checked that the qualitative pattern is basically
%the same. 
%%We find that in this class of scenarios, it is possible to enhance the 
%%neutralino-proton cross section 
%%and to obtain $\sigma_{\tilde{\chi}_{1}^{0}-p}$ 
%%within the experimental DAMA region with moderate $\tan \beta$. 




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[t]
%\begin{center}
%\epsfig{figure=nonunivgs_m30-550.eps,
%%width=9cm, 
%height=7cm}
%\end{center}
%\caption{
%The same as in Fig.~\ref{ellisfig}b
%but only for $\tan{\beta}=10$ and using non-universal soft 
%gaugino and scalar masses 
%as explained in the text. The line ????????corresponds to the
%universal case.  
%\label{nunivgs}}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%






%=================================================
%WRITTEN BY SHAABAN IN THE PREVIOUS VERSION

%It was realized that the reason for the failure of low $\tan\beta$ mSUGRA
%to solve the dark matter problem is the low annhiliation cross section 
%for the neutralino. The only channel capable of providing a suitable 
%cross section
%is the t--channel sfermion exchange which is only effeicent in a small
%region of parameter space. If we relax the GUT relationalship between the 
%gaugino masses but still remain in the large $\vert \mu \vert$ limit 
%(low $\tan \beta$) then we will continue to have a predominantly gaugino--like
%LSP ($P \sim 1$) with the relative values of $N_{11}$ and $N_{12}$ governed
%by the relative values of $M_1$ and $m_2$. Decreasing $M_2$ relative to $M_1$
%at the electroweak scale increases the wino content of the LSP.
%This also could help in enhancing the LSP--nucleon cross section.

%Here, We consider as an example the following  non--universal soft breaking 
%terms at $M_{GUT}$ where the gaugino masses are non--universal, namelly
%we assumed that $M_2 < M_1 < M_3$ and the soft scalar masses for the first
%and second generations are given by $m_{1,2}$ and for the third generation 
%is given by $m_{3}$ where  
%\begin{eqnarray}
%&& m_{H_2}^2 = m_3^2 =  a~ m^2~,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~a > 1
%\;, 
%\nonumber \\
%&& m_{H_1}^2 = m_{l,2}^2  = b~ m^2 ~,~~~~~~~~~~~~~~~~~~~~~~~~~~~0 \leq b < 1
%\;.
%\label{mimi}
%\end{eqnarray}
%Since the smaller(bigger) $m_{H_1}^2$($m_{H_2}^2$) at $M_{GUT}$ is,
%the less important the positive(negative) contribution
%to $\mu$ at the electroweak scale becomes,
%the above non-universality for Higgs masses will decrease the value of
%$\mu$. Reducing the soft scalar masses for the first generation also 
%help in enhancing the cross section. We have fixed $M_2$ such that the 
%lightest chargino mass at the weak scale is of the order of 
%the current experimental limit, i.e. $\mathcal{O}(100)$ GeV and assume that 
%$M_1 = 2 M_2$ while $M_3 = \sqrt{3} m_0$. 
%Finally we assume that the A-terms to be of order $-3 m_0$. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[ht]
%\begin{center}
%\epsfig{figure=nuniv.eps,width=13cm, height=5.5cm}
%\end{center}
%\caption{Values for $\sigma_{\tilde{\chi}_{1}^{0}-p}$ vs. 
%$m_{\tilde{\chi}_1^0}$.  
%\label{nuniv.eps}}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%We find that in this class of models, it is possible to enhance the 
%LSP--nucleon cross section and to obtain $\sigma_{\tilde{\chi}_{1}^{0}-p}$ 
%within the experimental DAMA region with low $\tan \beta$. 


%======================================================


\begin{figure}[t]
\begin{center}
\epsfig{file= scales2.eps, 
height=7cm}
\end{center}
\vspace{-0.5cm}
\caption{
The same as in Fig.~\ref{ellisfig}
but including the case 
with intermediate scale,
%$M_I=10^{16}$ GeV and 
$M_I=10^{11}$ GeV.
%, and for $\tan\beta=3$ and 10.
Now the Higgs masses 105 and 110 GeV are shown. 
%The universal case (green region) 
%is shown for comparison.
%corresponds to the
%universal case.  
}
\label{sigmaM_I}
\end{figure}  


\section{Scenario with intermediate unification scale}


The analyses of 
$\sigma_{\tilde{\chi}_1^0-p}$ 
%the neutralino-nucleon cross section 
in SUGRA, described in the previous section,
were performed assuming the
unification scale
$M_{GUT} \approx 10^{16}$ GeV.
However, there are several interesting arguments in favor of SUGRA scenarios
with scales 
$M_I\approx 10^{10-14}$ GeV.
These can be found e.g. in ref.~\cite{darkcairo} and references therein.
Thus to use the value of the initial scale,
say $M_I$, as a free parameter for the running of the soft terms
is particularly interesting.
In this sense, it was recently 
pointed out \cite{muas} 
that 
$\sigma_{\tilde{\chi}_1^0-p}$
is very sensitive to the variation of the initial scale 
for the running of the soft terms.
For instance, by taking $M_I=10^{10-12}$ GeV rather than 
$M_{GUT}$, regions in the parameter space of mSUGRA can be  
found 
where  $\sigma_{\tilde{\chi}_1^0-p}$ is two orders of magnitude
larger than for $M_{GUT}$.

%is in the expected range of sensitivity of present detectors,
%and this
%even for moderate values of $\tan\beta$ ($\tan\beta\gsim 3$). 
%In contrast, in the usual case with initial scale at $M_{GUT}$,
%this large 
%$\sigma_{\tilde{\chi}_1^0-p}$
%is achieved only for $\tan\beta \gsim 20$.





The fact that smaller initial scales imply a larger 
$\sigma_{\tilde{\chi}_1^0-p}$ 
%neutralino-proton cross section 
can be understood from the variation in the value of 
$\mu$ with $M_I$.
One observes that, for $\tan\beta$ fixed, the smaller the initial
scale for the running is, the smaller the numerator in the
first piece of eq.(\ref{electroweak}) becomes. 
This can be understood qualitatively from 
%Fig.~\ref{run}, where
the well known evolution of $m_{H_d}^2$ and $m_{H_u}^2$ with the
scale.
%is schematically shown. 
Clearly, the smaller the initial scale is, the shorter the
running becomes. As a consequence, 
also the less important the positive(negative) contribution 
$m_{H_d}^2$($m_{H_u}^2$) to $\mu^2$ in eq.(\ref{electroweak}) becomes.
Thus 
%The above conclusion has important consequences
%for the cross section.
%As is well known, and can easily be understood from the neutralino mass
%matrix in
%eq.(\ref{mass}), when 
$|\mu|$ decreases, and therefore, as discussed in the previous section,
$\sigma_{\tilde{\chi}_1^0-p}$ increases.
This is shown in 
Fig.~\ref{sigmaM_I}, where the result
%, where
%the cross section as a function
%of the lightest neutralino mass $m_{\tilde{\chi}_1^0}$ is plotted 
for the scale $M_I=M_{GUT}$ is compared 
%studied in Figs.~\ref{ellisfig} and \ref{ellisfig2} 
with the result for 
the intermediate scale $M_I=10^{11}$ GeV.






Clearly, in the latter scenario the Higgs mass decreases
and all points would be excluded if we impose the mSUGRA bound
$m_h>114$ GeV.
On the other hand, taking into account the conservative approach
discussed in Section~3, since all points in the figure have $m_h>100$ GeV
we will not discard them.
Then, the most important constraint for $\tan\beta=20$
is due to $g_{\mu}-2$, excluding a large area of points in the
DAMA region. 
We can understand this from the results of ref.~\cite{muonmio}.
There, $a_{\mu}$ versus $m_{\tilde{\chi}_1^0}$ is plotted, and one
can observe that for $m_{\tilde{\chi}_1^0}\leq 150$ GeV and $m$ small 
too large values of $a_{\mu}$, beyond
the experimental upper bound, are obtained. 
Note that the area left in DAMA by this constraint
and the mSUGRA one
$m_{\tilde\chi_1^{\pm}}>103$ GeV
may be increased if we relaxed the latter bound to e.g. 90 GeV.
We see in Fig.~\ref{sigmaM_I}
that the range 75 GeV $\lsim m_{\tilde{\chi}_1^0}\lsim$ 125 GeV is now
consistent 
with DAMA limits. 
However, for $\tan\beta=35$ all points in the DAMA region are
excluded because of $b \to s \gamma$.




%This corresponds to $M_I\lsim 10^{14}$ GeV.
%For $\tan\beta=20$, although not shown in the figure,
%it is obtained that the above range 
%Finally, we show in Fig.~3 the case increases to 
%50 GeV $\lsim m_{\tilde{\chi}_1^0}\lsim$ 170 GeV, corresponding
%now to  $M_I\lsim 10^{16}$ GeV.




%Let us remark that these figures have been obtained \cite{muas}
%taking $30\lsim m_0\lsim 550$ GeV and $A_0=M_{1/2}$.
%In any case, the cross section is 
%not very sensitive to the specific values
%of $A_0$. In particular it was checked that 
%this is so for 
%$\mid A_0/M_{1/2}\mid \lsim 1$.
%%For example, relation $A_0=-M_{1/2}$ 
%is particularly interesting since it arises naturally
%in several string models \cite{dilaton,Rigolin}. 

%and\footnote{It is worth noticing however,
%that for $m_0$ in the range $50-100$ GeV the neutralino is not the
%LSP in the whole parameter space. In some regions the stau is the LSP.}
%50 GeV $\lsim m_0\lsim 250$ GeV.
%It was also checked that the results 
%are consistent with present 
%bounds coming from accelerators.
%These are
%LEP and Tevatron bounds on SUSY masses and 
%CLEO $b\to s\gamma$ branching ratio measurements.




%Let us finally recall that
%the above computations have been carried out for the case of
%universal soft terms. This is not only the most simple
%possibility in the framework of SUGRA, but  
%is also allowed in the context of superstring models. This is e.g.
%the case of the dilaton-dominated SUSY-breaking 
%scenario \cite{dilaton}
%or weakly and strongly coupled heterotic models with one K\"ahler 
%modulus \cite{softM}.
%In this sense the analysis of neutralino-nucleon
%cross sections of those models 
%is included in the above analyses. This is exactly true for
%$M_I=M_{GUT}$. For intermediate unification scale,
%note that we are assuming gaugino mass universality
%at the high energy scale, 
%although in this scenario gauge couplings do not unify. 
%This situation is in principle
%possible in generic supersymmetric models, however it is not so natural in
%supersymmetric models from supergravity 
%where gaugino masses and gauge couplings
%are related through the gauge kinetic function. 
%Since an explicit string construction with nonuniversal
%gauge couplings and gaugino masses will be analyzed in detail below,
%we have chosen to simplify the discussion here
%assuming gaugino mass universality.

Given the above situation concerning the enhancement of the 
neutralino-proton cross section through non-universal scalars
and intermediate scales, it is worth analyzing the
combination of both possibilities.
This is shown in Fig.~\ref{sigmaM_I2}.
Although no points with $m_h > 114$ GeV enter in the DAMA region,
many points with 
$91 <m_h < 114$ GeV do enter and 
are allowed by all the other constraints. For
$\tan\beta=35$ the allowed area is extremely large and
we can even have points entering in the region
excluded by CDMS.



%
\begin{figure}[t]
\begin{center}
\epsfig{file= nuniversalinterm.eps, 
height=7cm}
\end{center}
\vspace{-0.5cm}
\caption{
Scatter plot of the neutralino-proton cross section
$\sigma_{\tilde{\chi}_1^0-p}$ 
as a function of 
the neutralino mass $m_{\tilde{\chi}_1^0}$ in the SUGRA scenario
with the intermediate scale
$M_I=10^{11}$ GeV and 
non-universal soft scalar masses, for
$\tan\beta=20$ and 35.
Only points of the parameter space fulfilling
$b \to s \gamma$ and $g_{\mu}-2$ are plotted. 
(Blue) dots correspond to points
with $91 <m_h < 114$ GeV. 
%The same as in Fig.~\ref{ellisfig}
%$\sigma_{\tilde{\chi}_{1}^{0}-p}$ vs.
%$m_{\tilde{\chi}_{1}^{0}}$ 
%but 
%for $\tan{\beta}=10$ and 
%using non-universal soft 
%scalar masses.
The universal case (green region) 
is shown for comparison.
DAMA and CDMS current 
experimental limits and projected GENIUS limits are also shown.
%The same as in Fig.~\ref{ellisfig}
%but for two values of the initial scale,
%$M_I=10^{16}$ GeV and $10^{11}$ GeV, and for $\tan\beta=3$ and 10. 
}
\label{sigmaM_I2}
\end{figure}  
%


\section{Superstring scenario with D-branes}




In the previous section the analyses were performed assuming intermediate
unification scales.
In fact, this situation can be inspired by superstring theories, since
it was recently 
realized that 
the string scale may be anywhere between the weak and the Plank 
scale.
% \cite{Lykken}-\cite{weakandstronghete}. 
For example, embedding the SM inside D3-branes in type I
strings, the string scale is given by
$M_I^4= \alpha M_{Planck} M_c^3/{\sqrt 2}$,
%%
%\bea
%M_I^4= \frac{\alpha M_{Planck}}{\sqrt 2} M_c^3\ ,
%\label{gravitino2}
%\eea
%%
where $\alpha$ is the gauge coupling and $M_c$ is the compactification scale. 
Thus one gets $M_I\approx 10^{10-12}$ GeV with $M_c\approx 10^{8-10}$ GeV.
%Then, to use the value of the initial scale,
%say $M_I$, as a free parameter for the running of the soft terms
%is particularly interesting.
%In addition, 
%there are several interesting arguments in favour of SUSY scenarios
%with scales 
%$M_I\approx 10^{10-14}$ GeV.
%These can be found e.g. in ref.~\cite{darkcairo} and references therein.
%These scales were suggested \cite{stronghete,typeIinter}
%to explain many experimental observations as
%neutrino masses or the scale for axion physics. 
%With the string scale of the order of
%$10^{10-12}$ GeV
%one is also able
%to attack the hierarchy problem of unified theories 
%without invoking any hierarchically suppressed non-perturbative
%effect \cite{stronghete,typeIinter}. 
%In supergravity models supersymmetry can be spontaneously broken in a 
%hidden sector of the theory and the gravitino mass,
%which sets the overall scale of the soft terms, is given by
%%%
%%\bea
%%m_{3/2}\approx \frac{F}{M_{Planck}}\ ,
%%\label{gravitino}
%%\eea
%%%
%$m_{3/2}\approx \frac{F}{M_{Planck}}$,
%where $F$ is the auxiliary field whose vacuum expectation value
%breaks supersymmetry. 
%Since in supergravity one would expect $F\approx M_{Planck}^2$, one  
%obtains
%$m_{3/2}\approx M_{Planck}$ and therefore 
%the hierarchy problem solved in principle by supersymmetry
%would be re-introduced.
%However, if the scale
%of the fundamental theory is $M_I\approx 10^{10-12}$ GeV instead of
%$M_{Planck}$,
%then $F\approx M_I^2$
%and one gets $m_{3/2}\approx M_W$ in a natural way.
%There are other arguments in favour 
%of scenarios with initial scales $M_I$ smaller than $M_{GUT}$.
%For example,
%charge and color breaking constraints,
%which are very strong with the usual scale $M_{GUT}$ \cite{servidor},
%become less important \cite{Allanach}.
%These scales might also
%explain the observed ultra-high energy ($\approx 10^{20}$ eV) cosmic rays
%as products of long-lived massive string mode decays.
%\cite{stronghete,typeIinter} (see ref.\cite{cosmic} for more
%details about this possibility). Besides,
%several models of chaotic inflation favour also these scales \cite{caos}.
%%
%\begin{figure}[ht]
%\begin{center}
%\begin{tabular}{c}
%\epsfig{file= branes.eps, width=9cm}\\
%%\epsfig{file= p24f006.eps,width=10cm,height=7.0cm}
%\end{tabular}
%\end{center} 
%\caption{A generic D-brane scenario giving rise to the gauge bosons
%and matter of the stanard model. It contains three D$p_3$-branes,
%two D$p_2$-branes and one D$p_1$-brane, where $p_N$ may be
%either 9 and $5_i$ or 3 and $7_i$. The presence of extra D-branes,
%say D$q$-branes, is also
%necessary. For each set the D$p_N$-branes
%are in fact on the top of each other.} 
%\label{branas}
%\end{figure}
%%
In addition,
D-brane constructions are explicit scenarios where not only
intermediate scales arise naturally but also 
non-universality.



\begin{figure}[t]
\begin{center}
%\begin{tabular}{c}
\epsfig{file= dbranes.eps, 
height=7cm
%width=13cm
}
\end{center}
\vspace{-0.5cm}
\caption{
Scatter plot of the neutralino-proton cross section
$\sigma_{\tilde{\chi}_1^0-p}$ 
as a function of 
the neutralino mass $m_{\tilde{\chi}_1^0}$ in the D-brane scenario
with 
the string scale $M_I=10^{12}$ GeV
discussed
in the text, and for $\tan\beta= 10$ and 15.
Only the big (red and blue) dots 
%and the small (blue) dots
fulfil $b \to s \gamma$ and $g_{\mu}-2$ constraints.
The red ones 
correspond to points 
with $m_h\geq 114$ GeV whereas the blue ones correspond
to points
with $91 <m_h < 114$ GeV. 
DAMA and CDMS current 
experimental limits and projected GENIUS limits are also shown.
}
\label{dbrane}
\end{figure}
%




The crucial point in these analyses 
%The first attempts to study dark matter within these constructions
%were carried out in scenarios with the unification scale  
% $M_{GUT} \approx 10^{16}$ GeV
%as the initial scale.
%\cite{khalil,Nath2,Arnowitt2}
%and dilaton-dominated SUSY-breaking scenarios 
%with an intermediate scale as the initial scale \cite{bailin}.
is the D-brane origin of the $U(1)_Y$ gauge group
as a combination of other $U(1)$'s, 
and its influence on the matter distribution in these scenarios.
%was not included in the above analyses.
%When this 
%is taken into account, interesting results are 
%obtained \cite{nosotros}. 
In particular, 
scenarios with the gauge group and particle content of the
SUSY SM lead naturally to intermediate values for the
string
scale, in order to reproduce the value of gauge couplings
deduced from experiments. In addition, the soft terms 
turn out to be generically non-universal.
Due to these results, 
large cross sections
%in the small $\tan\beta$ regime
can be obtained \cite{nosotros}.


Let us consider for example a type I string scenario 
%\cite{nosotros}
where the gauge group 
$U(3)\times U(2)\times U(1)$, giving rise to
$SU(3)\times SU(2)\times U(1)^3$, arises
from three different types of D-branes, 
and therefore the gauge couplings are non-universal. 
%as shown schematically in Fig.~\ref{branas}.
%For the sake of visualization each set is
%depicted at parallel locations, but in fact they are intersecting each
%other.
%and therefore
%the gauge couplings are
%non-universal
%as in Fig.~\ref{uni}a\footnote{To be precise, the running of the 
%$U(1)_Y$ gauge coupling is not exactly like in the figure since, due to the
%D-brane origin of the $U(1)$ gauge groups, relation (\ref{couplings})
%must be fulfilled. See in this respect Fig.~2 in ref.\cite{nosotros}.}.
%Other examples 
%with the standard model gauge group embedded in D-branes in 
%a different way 
%can be found in ref.\cite{nosotros}.
Here
$U(1)_Y$ is a linear combination
of the three $U(1)$ gauge groups arising from $U(3)$, $U(2)$ and
$U(1)$ within the three different D-branes. 
%where $\alpha_k$ correspond to the gauge couplings of the $U(k)$ branes.
As shown in ref.~\cite{nosotros},  
%$\alpha_1(M_I) = 0.1$ 
this leads
to the string scale $M_I = 10^{12}$ GeV. 
Likewise,
assuming 
%The analysis of the soft terms has been done under
%the assumption 
that only the
dilaton ($S$) and moduli ($T_i$) fields contribute to SUSY
breaking, it was found  that the soft terms  are generically
non-universal.



Fig.~\ref{dbrane} displays a scatter plot of 
$\sigma_{\tilde\chi_1^0-p}$ as a function of the
neutralino mass $m_{\tilde\chi_1^0}$ for a scanning of the parameter
space
described in ref.~\cite{nosotros}. In particular,
the gravitino mass $m_{3/2}\leq 300$ GeV is taken, since
larger values will always produce a cross section below DAMA limits.
%discussed above. 
The cases $\tan\beta=10$ and 15 are shown in Fig.~\ref{dbrane}.
Although regions of the parameter
space consistent with DAMA limits exist, the 
$b \to s \gamma$ and $g_{\mu}-2$ constraints forbid 
most of them. The latter are shown with small (green) points,
and they have $91 <m_h < 114$ GeV. For example, 
to understand this result for $g_{\mu}-2$ is simple.
In ref.~\cite{muonmio} $a_{\mu}$ versus $m_{3/2}$ is plotted, and one
can observe that for $m_{3/2}\lsim 300$ GeV 
too large values of $a_{\mu}$, beyond
the experimental upper bound, are obtained. 
In Fig.~\ref{dbrane} only regions with big (red and blue) dots 
%and the small (blue) dots
fulfil the above mentioned constraints.
The red ones
correspond to points 
with $m_h\geq 114$ GeV whereas the blue ones correspond
to points
with $91 <m_h < 114$ GeV. 
It is worth noticing that the larger $\tan\beta$ is, 
the smaller the regions allowed by the experimental constraints become.
For example, increasing
$\tan\beta$ the value of 
$a_{\mu}$ turns out to be larger and may exceed the experimental bounds.




\section{Conclusions}

%There is overwhelming evidence that most of the matter in the Universe
%is dark matter.
%In the present paper w
We have reviewed the direct detection of 
supersymmetric dark matter in the light of recent experimental results.
%In particular, DAMA collaboration using a NaI detector
%has reported recently \cite{experimento1}
%data favouring the existence of a WIMP signal in their search for
%annual modulation. 
They require a large cross section of the order of
$10^{-6}$ pb. 
Although in the context of the mSUGRA scenario one cannot 
obtain this value once the experimental constraints are imposed,
this is possible for other scenarios.
This is the case of SUGRA models with non-universal soft scalar masses.
Likewise, scenarios with intermediate unification scale
may also produce this value if some experimental bounds are relaxed.
%Superstring constructions with D-branes are specially interesting
%in this sense, since both possibilities can be present naturally.
%We have observed that there are regions in the parameter space
%of SUGRA scenarios 
%\cite{Bottino}$^-$\cite{nosotros} 
%where such a value can be obtained
%, although it is fair to say that
%smaller values
%can also be obtained and even more easily.
%The latter result may be important since 
%CDMS collaboration using a germanium detector
%has reported a null result for part of the region explored by DAMA. 
%Clearly, more sensitive detectors
%producing further data are needed to solve this contradiction.
%Fortunately, many dark matter detectors are being projected.
%This is the case e.g. of DAMA 250 kg. and CDMS Soudan, but
%particularly interesting is the projected GENIUS \cite{GENIUS}
%detector where values of the cross section as low as 
%$10^{-9}$ pb will be accesible. 
%In summary, underground physics as the one discussed here in order
%to detect dark matter is crucial. Even if neutralinos are discovered
%at future particle accelerators such as LHC, only their direct detection
%due to their presence in our galactic halo will confirm that they
%are the sought-after dark matter of the Universe.
\\



\noindent {\bf Acknowledgements}

\noindent
We thank S. Khalil and E. Torrente-Lujan
as co-authors of some of the works reported in
this paper. D.G. Cerde\~no acknowledges the financial support
of the Comunidad de Madrid through a FPI grant.
E. Gabrielli is grateful to the CERN Theory Division and
the Department of Theoretical Physics of Madrid 
Aut\'onoma University, where 
part of this work has been done, for their warm hospitality.
%The work of E. Gabrielli was supported by PPARC.
The work of C. Mu\~noz was supported in part by the Ministerio de 
Ciencia y Tecnolog\'{\i}a under contract FPA2000-0980, 
and the European Union under contract 
HPRN-CT-2000-00148. 

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Phys. Rev. D59 (1999) 095004.

\bibitem{Arnowitt} E. Accomando, R. Arnowitt, B. Dutta and Y. Santoso, 
Nucl. Phys. B585 (2000) 124.

\bibitem {Nath2}
A. Corsetti and P. Nath, .
Phys. Rev. D64 (2001) 125010.


\bibitem{Arnowitt3} R. Arnowitt and B. Dutta, hep-ph/0112157.

\bibitem{muas} E. Gabrielli, S. Khalil, C. Mu\~noz
and E. Torrente-Lujan, Phys. Rev. D63 (2001) 025008. 

\bibitem{bailin} D. Bailin, G.V. Kraniotis and A. Love,
Phys. Lett. B491 (2000) 161.

\bibitem{nosotros} D.G. Cerde\~no, E. Gabrielli, S. Khalil, C. Mu\~noz
and E. Torrente-Lujan, Nucl. Phys. B603 (2001) 231.

\bibitem{dilaton} 
For a review, see, A. Brignole, L.E. Ibanez and C. Mu\~noz,
in the 
book `Perspectives on Supersymmetry', World Scientific Publ. Co. (1998)
125,
hep-ph/9707209.

\bibitem{Heinemeyer} S. Ambrosiano, A. Dedes, S. Heinemeyer, S. Su and
G. Weiglein, Nucl. Phys. B624 (2002) 3.

\bibitem{experimentales} The ALEPH, DELPHI, L3 and OPAL Collaborations,
and the LEP Higgs working group, hep-ex/0107030.

\bibitem{FeynHiggs} S. Heinemeyer, W. Hollik and G. Weiglein,
Comp. Phys. Comm. 124 (2000) 76;
hep-ph/0002213;  
M. Frank, S. Heinemeyer, W. Hollik and G. Weiglein, hep-ph/0202166. 


%\bibitem{kami} For a review, see, e.g., 
%G. Jungman, M. Kamionkowski and K. Griest, Phys. Rept. 267 (1996) 195,
%and references therein.

\bibitem{chargino} Joint LEP 2 supersymmetry working group,
http://lepsusy.web.cern.ch/lepsusy /www/inos$_{-}$moriond01/
charginos$_{-}$pub.html

\bibitem{sleptons} Joint LEP 2 supersymmetry working group,
http://alephwww.cern.ch/$\sim$ganis /SUSYWG/SLEP/sleptons$_{-}$2k01.html

%\bibitem{spectrum}


\bibitem{cleo} 
CLEO Collaboration, S. Chen et al., 
hep-ex/0108032.
%Phys. Rev. Lett. 74 (1995) 2885.
%S. Chen et al., CLEO CONF 01-16.


\bibitem{belle}
BELLE Collaboration, H. Tajima, 
hep-ex/0111037.

%\bibitem{lep}
%ALEPH Collaboration, R. Barate et al.,
%Phys. Lett. B429 (1998) 169.


\bibitem{gambino}
K. Chetyrkin, M. Misiak and M. Munz, Phys. Lett. B400 (1997) 206,
Erratum-ibid. B425 (1997) 414;
P. Gambino and M. Misiak, Nucl. Phys. B611 (2001) 338.



\bibitem{GS}
S. Bertolini, F. Borzumati, A. Masiero and G. Ridolfi,
Nucl. Phys. B353 (1991) 591;
E. Gabrielli and U. Sarid, Phys. Rev. Lett. 79 (1997) 4752.


%\bibitem{belle}





\bibitem{muon} Muon $g-2$ Collaboration, 
H.N. Brown et al., Phys. Rev. Lett. 86 (2001) 2227.

%\bibitem{revi} For a recent review, see, 
%J. Prades, hep-ph/0108192, and references therein.

\bibitem{rafael}
M. Knecht and A. Nyffeler, hep-ph/0111058;
M. Knecht, A. Nyffeler, M. Perrottet and E. de Rafael, Phys. Rev. Lett.
88 (2002) 071802;
Hayakawa and Kinoshita, hep-ph/0112102;
I. Blokland, A. Czarnecki and K. Melnikov, Phys. Rev. Lett. 88 (2002) 071803;
J. Bijnens, E. Pallante and J. Prades, Nucl. Phys. B626 (2002) 410.



\bibitem{todos} 
J. Bijnens, E. Pallante and J. Prades, Phys. Rev. Lett. 75 (1995) 1447,
Erratum-ibid. 75 (1995) 3781, Nucl. Phys. B474 (1996) 379;
M. Hayakawa, T. Kinoshita and A.I. Sanda, 
Phys. Rev. Lett. 75 (1995) 790, Phys. Rev D54 (1996) 3137;
M. Hayakawa and T. Kinoshita,
Phys. Rev D57 (1998) 465.







\bibitem{isotopes} See, e.g., A. Kudo and M. Yamaguchi,
Phys. Lett. B516 (2001) 151.
hep-ph/0103272, and references therein.



\bibitem{relic} See, e.g., S. Khalil, C. Mu\~noz and E. Torrente-Lujan, 
hep-ph/0202139, and references therein.
 

\bibitem{Ellis} 
J. Ellis, A. Ferstl and K. Olive, Phys. Lett. B481 (2000) 304.



\bibitem{Ellis2} 
J. Ellis, A. Ferstl and K. Olive, hep-ph/0111064.

\bibitem{halo} P. Belli, R. Cerulli, N. Fornengo and S. Scopel,
hep-ph/0203242. 


%\bibitem{focus} J.L. Feng, K.T. Matchev and F. Wilczek, 
%Phys. Lett. B482 (2000) 388.


%\bibitem{rges} L.E. Iba\~nez and C. L\'opez, 
%Nucl. Phys. B233 (1984) 511;
%L.E. Iba\~nez, C. L\'opez and C. Mu\~noz, 
%Nucl. Phys. B256 (1985) 218.



%\bibitem{Bottino} A. Bottino, F. Donato, N. Fornengo and S. Scopel, 
%\PRD{59}{99}{095004}.

%\bibitem{arna} R. Arnowitt and P. Nath, \PRD{60}{99}{044004}.

%%\bibitem{Ellis} 
%%J. Ellis, A. Ferstl and K. Olive, hep-ph/0001005.

%\bibitem{arnowitt}
%E. Accomando, R. Arnowitt, B. Dutta and Y. Santoso, 
%Nucl. Phys. B585 (2000) 124.

%%\bibitem{nath}
%%A. Corsetti and P. Nath, hep-ph/0003186.

%\bibitem{mario} 
%M.E. G\'omez and J.D. Vergados, hep-ph/0012020.



%\bibitem{marci} For recent reviews, see, e.g., 
%A. Czarnecki and W.J. Marciano, hep-ph/0102122;
%S. Narison, hep-ph/0103199.

%\bibitem{lep} DELPHI collaboration, hep-ex/0104020, 0104021.

\bibitem{GENIUS} H.V. Klapdor-Kleingrothaus, L. Baudis, G. Heusser,
B. Majorovits and H. Paes, hep-ph/9910205.


\bibitem{Mario}
M.E. G\'omez and J.D. Vergados,  Phys. Lett. B512 (2001) 252.


\bibitem{muonmio} D.G. Cerde\~no, E. Gabrielli, S. Khalil, C. Mu\~noz
and E. Torrente-Lujan, Phys. Rev. D64 (2001) 093012.


\end{thebibliography}




\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% End of  Rinton-P9x6.tex  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


hep-ph0204271.extracted_bib
\begin{thebibliography}{99}

\bibitem{contemporary}
For a recent review, see, S. Khalil and C. Mu\~noz, 
Contemp. Phys. 43 (2002) 51, hep-ph/0110122.

%\bibitem{rotation} For a recent review, see, e.g., 
%Y. Sofue and V. Rubin, astro-ph/0010594, and references therein.

%\bibitem{Sadoulet} For a review, see, e.g.,
%B. Sadoulet, Rev. Mod. Phys. 71 (1999) 197, and references therein.

%\bibitem{Drees} For a review, see, e.g., M. Drees, hep-ph/9804231, and
%references therein.

\bibitem{experimento1}
DAMA Collaboration, R. Bernabei et al., Phys. Lett. B480 (2000) 23.

\bibitem{experimento2} CDMS Collaboration, 
R. Abusaidi et al., Phys. Rev. Lett. 84 (2000) 5699.

\bibitem{edelweiss} EDELWEISS Collaboration,  
A. Benoit et al., Phys. Lett. B513 (2001) 15.

\bibitem{igex} IGEX Collaboration, 
A. Morales et al., hep-ex/0110061.



\bibitem{darkcairo}  D.G. Cerde\~no, S. Khalil and C. Mu\~noz,
Proceedings of CICHEP Conference in Cairo, Rinton Press (2001) 214, 
hep-ph/0105180.

\bibitem{Bottino} A. Bottino, F. Donato, N. Fornengo and S. Scopel, 
Phys. Rev. D59 (1999) 095004.

\bibitem{Arnowitt} E. Accomando, R. Arnowitt, B. Dutta and Y. Santoso, 
Nucl. Phys. B585 (2000) 124.

\bibitem {Nath2}
A. Corsetti and P. Nath, .
Phys. Rev. D64 (2001) 125010.


\bibitem{Arnowitt3} R. Arnowitt and B. Dutta, hep-ph/0112157.

\bibitem{muas} E. Gabrielli, S. Khalil, C. Mu\~noz
and E. Torrente-Lujan, Phys. Rev. D63 (2001) 025008. 

\bibitem{bailin} D. Bailin, G.V. Kraniotis and A. Love,
Phys. Lett. B491 (2000) 161.

\bibitem{nosotros} D.G. Cerde\~no, E. Gabrielli, S. Khalil, C. Mu\~noz
and E. Torrente-Lujan, Nucl. Phys. B603 (2001) 231.

\bibitem{dilaton} 
For a review, see, A. Brignole, L.E. Ibanez and C. Mu\~noz,
in the 
book `Perspectives on Supersymmetry', World Scientific Publ. Co. (1998)
125,
hep-ph/9707209.

\bibitem{Heinemeyer} S. Ambrosiano, A. Dedes, S. Heinemeyer, S. Su and
G. Weiglein, Nucl. Phys. B624 (2002) 3.

\bibitem{experimentales} The ALEPH, DELPHI, L3 and OPAL Collaborations,
and the LEP Higgs working group, hep-ex/0107030.

\bibitem{FeynHiggs} S. Heinemeyer, W. Hollik and G. Weiglein,
Comp. Phys. Comm. 124 (2000) 76;
hep-ph/0002213;  
M. Frank, S. Heinemeyer, W. Hollik and G. Weiglein, hep-ph/0202166. 


%\bibitem{kami} For a review, see, e.g., 
%G. Jungman, M. Kamionkowski and K. Griest, Phys. Rept. 267 (1996) 195,
%and references therein.

\bibitem{chargino} Joint LEP 2 supersymmetry working group,
http://lepsusy.web.cern.ch/lepsusy /www/inos$_{-}$moriond01/
charginos$_{-}$pub.html

\bibitem{sleptons} Joint LEP 2 supersymmetry working group,
http://alephwww.cern.ch/$\sim$ganis /SUSYWG/SLEP/sleptons$_{-}$2k01.html

%\bibitem{spectrum}


\bibitem{cleo} 
CLEO Collaboration, S. Chen et al., 
hep-ex/0108032.
%Phys. Rev. Lett. 74 (1995) 2885.
%S. Chen et al., CLEO CONF 01-16.


\bibitem{belle}
BELLE Collaboration, H. Tajima, 
hep-ex/0111037.

%\bibitem{lep}
%ALEPH Collaboration, R. Barate et al.,
%Phys. Lett. B429 (1998) 169.


\bibitem{gambino}
K. Chetyrkin, M. Misiak and M. Munz, Phys. Lett. B400 (1997) 206,
Erratum-ibid. B425 (1997) 414;
P. Gambino and M. Misiak, Nucl. Phys. B611 (2001) 338.



\bibitem{GS}
S. Bertolini, F. Borzumati, A. Masiero and G. Ridolfi,
Nucl. Phys. B353 (1991) 591;
E. Gabrielli and U. Sarid, Phys. Rev. Lett. 79 (1997) 4752.


%\bibitem{belle}





\bibitem{muon} Muon $g-2$ Collaboration, 
H.N. Brown et al., Phys. Rev. Lett. 86 (2001) 2227.

%\bibitem{revi} For a recent review, see, 
%J. Prades, hep-ph/0108192, and references therein.

\bibitem{rafael}
M. Knecht and A. Nyffeler, hep-ph/0111058;
M. Knecht, A. Nyffeler, M. Perrottet and E. de Rafael, Phys. Rev. Lett.
88 (2002) 071802;
Hayakawa and Kinoshita, hep-ph/0112102;
I. Blokland, A. Czarnecki and K. Melnikov, Phys. Rev. Lett. 88 (2002) 071803;
J. Bijnens, E. Pallante and J. Prades, Nucl. Phys. B626 (2002) 410.



\bibitem{todos} 
J. Bijnens, E. Pallante and J. Prades, Phys. Rev. Lett. 75 (1995) 1447,
Erratum-ibid. 75 (1995) 3781, Nucl. Phys. B474 (1996) 379;
M. Hayakawa, T. Kinoshita and A.I. Sanda, 
Phys. Rev. Lett. 75 (1995) 790, Phys. Rev D54 (1996) 3137;
M. Hayakawa and T. Kinoshita,
Phys. Rev D57 (1998) 465.







\bibitem{isotopes} See, e.g., A. Kudo and M. Yamaguchi,
Phys. Lett. B516 (2001) 151.
hep-ph/0103272, and references therein.



\bibitem{relic} See, e.g., S. Khalil, C. Mu\~noz and E. Torrente-Lujan, 
hep-ph/0202139, and references therein.
 

\bibitem{Ellis} 
J. Ellis, A. Ferstl and K. Olive, Phys. Lett. B481 (2000) 304.



\bibitem{Ellis2} 
J. Ellis, A. Ferstl and K. Olive, hep-ph/0111064.

\bibitem{halo} P. Belli, R. Cerulli, N. Fornengo and S. Scopel,
hep-ph/0203242. 


%\bibitem{focus} J.L. Feng, K.T. Matchev and F. Wilczek, 
%Phys. Lett. B482 (2000) 388.


%\bibitem{rges} L.E. Iba\~nez and C. L\'opez, 
%Nucl. Phys. B233 (1984) 511;
%L.E. Iba\~nez, C. L\'opez and C. Mu\~noz, 
%Nucl. Phys. B256 (1985) 218.



%\bibitem{Bottino} A. Bottino, F. Donato, N. Fornengo and S. Scopel, 
%\PRD{59}{99}{095004}.

%\bibitem{arna} R. Arnowitt and P. Nath, \PRD{60}{99}{044004}.

%%\bibitem{Ellis} 
%%J. Ellis, A. Ferstl and K. Olive, hep-ph/0001005.

%\bibitem{arnowitt}
%E. Accomando, R. Arnowitt, B. Dutta and Y. Santoso, 
%Nucl. Phys. B585 (2000) 124.

%%\bibitem{nath}
%%A. Corsetti and P. Nath, hep-ph/0003186.

%\bibitem{mario} 
%M.E. G\'omez and J.D. Vergados, hep-ph/0012020.



%\bibitem{marci} For recent reviews, see, e.g., 
%A. Czarnecki and W.J. Marciano, hep-ph/0102122;
%S. Narison, hep-ph/0103199.

%\bibitem{lep} DELPHI collaboration, hep-ex/0104020, 0104021.

\bibitem{GENIUS} H.V. Klapdor-Kleingrothaus, L. Baudis, G. Heusser,
B. Majorovits and H. Paes, hep-ph/9910205.


\bibitem{Mario}
M.E. G\'omez and J.D. Vergados,  Phys. Lett. B512 (2001) 252.


\bibitem{muonmio} D.G. Cerde\~no, E. Gabrielli, S. Khalil, C. Mu\~noz
and E. Torrente-Lujan, Phys. Rev. D64 (2001) 093012.


\end{thebibliography}