SUSY Dark Matter in the Universe- Theoretical Direct Detection Rates
J. D. VERGADOS
Exotic dark matter together with the vacuum energy or cosmological constant seem to dominate in the Universe. An even higher density of such matter seems to be gravitationally trapped in our Galaxy. Thus its direct detection is central to particle physics and cosmology. Current fashionable supersymmetric models provide a natural dark matter candidate which is the lightest supersymmetric particle (LSP). Such models combined with fairly well understood physics like the quark substructure of the nucleon and the nuclear structure (form factor and/or spin response function), permit the evaluation of the event rate for LSP-nucleus elastic scattering. The thus obtained event rates are, however, very low or even undetectable. So it is imperative to exploit the modulation effect, i.e. the dependence of the event rate on the earth's annual motion. Also it is useful to consider the directional rate, i.e its dependence on the direction of the recoiling nucleus. In this paper we study such a modulation effect both in non directional and directional experiments. We calculate both the differential and the total rates using both isothermal, symmetric as well as only axially asymmetric, and non isothermal, due to caustic rings, velocity distributions. We find that in the symmetric case the modulation amplitude is small. The same is true for the case of caustic rings. The inclusion of asymmetry, with a realistic enhanced velocity dispersion in the galactocentric direction, yields an enhanced modulation effect, especially in directional experiments.
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\begin{document} 
\title{SUSY Dark Matter in the Universe- Theoretical  Direct Detection Rates}

\author{J. D. VERGADOS} 

\address{Theoretical Physics Section, University of Ioannina, GR-45110, 
Greece\\E-mail:Vergados@cc.uoi.gr} 

\maketitle\abstracts{
Exotic dark matter together with the vacuum energy or  cosmological constant 
seem to dominate in the Universe. An even higher
density of such matter seems to be gravitationally trapped in our Galaxy.
Thus its direct detection 
is central to particle physics and cosmology. Current fashionable supersymmetric
models provide a natural dark matter candidate which is the lightest
supersymmetric particle (LSP). Such models combined with fairly well 
understood physics like
the quark substructure of the nucleon and the nuclear structure (form factor 
and/or spin response function), permit the evaluation of
the event rate for LSP-nucleus elastic scattering. The thus obtained event rates
are, however, very low or even undetectable.
 So it is imperative to exploit the modulation effect, i.e. the dependence of
the event rate on  the earth's annual motion.  Also it is useful to consider
the directional rate, i.e its dependence on the direction of the recoiling
nucleus. In this paper we study
such a modulation effect both in non directional and directional experiments.
We calculate both the differential and the total rates using both isothermal, 
symmetric as well
as only axially asymmetric, and non isothermal, due to caustic rings, velocity
distributions. We find that in the symmetric case 
the modulation amplitude is small. The same is true for the case of caustic
 rings.  The inclusion of asymmetry, with a realistic enhanced velocity 
dispersion  in the galactocentric direction, yields an enhanced
modulation effect, especially in directional experiments.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
In recent years the consideration of exotic dark matter has become necessary
in order to close the Universe \cite{Jungm}. Furthermore in
in order to understand the large scale structure of the universe 
it has become necessary to consider matter
made up of particles which were 
non-relativistic at the time of freeze out. This is  the cold dark 
matter component (CDM). The COBE data ~\cite{COBE} suggest that CDM
is at least $60\%$ ~\cite {GAW}. On the other hand during the last few years
evidence has appeared from two different teams,
the High-z Supernova Search Team \cite {HSST} and the
Supernova Cosmology Project  ~\cite {SPF} $^,$~\cite {SCP} 
 which suggests that the Universe may be dominated by 
the  cosmological constant $\Lambda$.
As a matter of fact recent data the situation can be adequately
described by  a baryonic component $\Omega_B=0.1$ along with the exotic 
components $\Omega _{CDM}= 0.3$ and $\Omega _{\Lambda}= 0.6$
(see next section for the definitions).
In another analysis Turner \cite {Turner} gives 
$\Omega_{m}= \Omega _{CDM}+ \Omega _B=0.4$.
Since the non exotic component cannot exceed $40\%$ of the CDM 
~\cite{Jungm},~\cite {Benne}, there is room for the exotic WIMP's 
(Weakly  Interacting Massive Particles).
  In fact the DAMA experiment ~\cite {BERNA2} 
has claimed the observation of one signal in direct detection of a WIMP, which
with better statistics has subsequently been interpreted as a modulation signal
~\cite{BERNA1}.

The above developments are in line with particle physics considerations. Thus,
in the currently favored supersymmetric (SUSY)
extensions of the standard model, the most natural WIMP candidate is the LSP,
i.e. the lightest supersymmetric particle. In the most favored scenarios the
LSP can be simply described as a Majorana fermion, a linear 
combination of the neutral components of the gauginos and Higgsinos
\cite{Jungm,ref1,Gomez,ref2}. 

\section{Density Versus Cosmological Constant}

 The evolution of the Universe is governed by the General Theory of
Relativity. The most
  commonly used  model is that of Friedman, which utilizes the Robertson-
Walker metric
\begin{equation}
 (ds)^2=(dt)^2-R^2(t)[\frac{(dr)^2}{1-\kappa r^2}
          +r^2((d \theta)^2+ \sin ^2 \theta (d \phi)^2)]
\label{cosmo.1}
\end{equation}
The resulting Einstein equations are:
\begin{equation}
 R_{\mu \nu}-\frac{1}{2}g_{\mu \nu}R=-8\pi G_{N} 
T_{\mu \nu}+
                                      \Lambda g_{\mu \nu}
\label{kosmo.2}
\end{equation}
where $G_{N}$ is Newton's constant and $\Lambda$ is the
cosmological constant.
The equation for the scale factor $(t)$ becomes:
\begin{equation}
\frac{d^{2}R}{dt^{2}}=-\frac{4\pi}{3}G_{N}(\rho+3p)R=-\frac{4\pi
                   G_{N}\rho}{3}R+\frac{\Lambda}{3}
\label{cosmo.3}
\end{equation}
where $\rho$ is the mass density. Then the energy is
\beq
E=\frac{1}{2}m(\frac{dr}{dt})^{2}-G_{N}\frac{m}{R}(4\pi\rho
R^{3})+\frac{\Lambda}{6}mR^{2}=constant=-\frac{\kappa}{2}m
\label{kosmo.4}
\eeq
 This can be equivalently be written as
\beq
 H^{2}+\frac{\kappa}{R^{2}}=\frac{8\pi}{3}G_{N\rho}+\frac{\Lambda}{3}
\label{cosmo.5a}
\eeq
 where the quantity $H$ is Hubble's constant defined by
\beq
 H=\frac{1}{R}\frac{dR}{dt}
\label{cosmo.5b}
\eeq
Hubble's constant is perhaps the most important parameter of cosmology. In
fact it is not a constant but it changes with time. Its present day value
is given by
\beq
H_0 =( 65\pm 15)~km/s~M_{pc}^{-1}
\label{cosmo5.c}
\eeq
In other words $H_0^{-1}=(1.50\pm).35)\times10^{10}~y$, which is roughly equal
to the age of the Universe. Astrophysicists conventionally write it as
\beq
H_0 =100~h~km/s~M_{pc}^{-1}~~~~,~~~~~0.5<h<0.8
\label{cosmo5.d}
\eeq
 Equations \ref{cosmo.3}-\ref{cosmo.5a} coincide with those of the
Newtonian
  theory with the following two types of forces:
  An attractive force decreasing in absolute value with the scale
  factor (Newton)
%\beq
%  F=-G_{N} \frac{mM}{R^{2}}~~~~(Newton)
%\label{cosmo.6a}
%\eeq
  and a repulsive force increasing with the scale factor (Einstein)

\beq
  F=-G_{N} \frac{mM}{R^{2}}~~(Newton)~~,~~
  F=\frac{1}{3}\Lambda mR~~(Einstein)
\label{cosmo.6b}
\eeq
   Historically the cosmological constant was introduced by
   Einstein so that General Relativity yields a stationary
   Universe, i.e. one which satisfies the conditions:
\beq
    \frac{dR}{dt}=0~~~~\frac{d^{2}R}{dt^{2}}=0
\label{cosmo.7}
\eeq
    Indeed for $\kappa>0$, the above equations lead to
    $R=R_{c}=constant$ provided that
\beq
    \frac{1}{3}\Lambda R_{c}-\frac{4\pi}{3} G_{N}\rho R_{c}=0~~,~~
     \frac{1}{3}\Lambda R^{2}_{c}-\frac{4\pi}{3} G_{N}\rho R^{2}_{c}=\kappa
\label{cosmo.8}
\eeq
    These equations have a non trivial solution provided that the density
$\rho$ and the cosmological constant $\Lambda$ are related, i.e.
\beq
    \Lambda=4 \pi G_{N} \rho
\label{cosmo.9}
\eeq
    The radius of the Universe then is given by
\beq
    R_{c}=[\frac{\kappa}{4\pi G_{N}\rho}]^{1/2}
\label{cosmo.10}
\eeq
    Define now
\beq
\Omega_{m}=\frac{\rho}{\rho _{c}}~~,~~
          \Omega_{\Lambda}=\frac{\rho _{v}} {\rho _{c}}~~,~~
    \rho_{v}=\frac{\Lambda}{8 \pi G_{N}}~~("vacuum" density)
\label{cosmo.11}
\eeq
%    with
%\beq
%    \rho_{v}=\frac{\Lambda}{8 \pi G_{N}}~~~("vacuum" density)
%\label{cosmo.12}
%\eeq
    The critical density is
\beq
    \rho_{c}=1.8\times
    10^{-23}h^{2}\frac{g}{cm^{3}}=10h^{2}\frac{nucleons}{m^{3}}
\label{cosmo.13}
\eeq
    With these definitions Friedman's equation
    $E=-\kappa\frac{m}{2}$
    takes the form
\beq
    \frac{\kappa}{R^{2}}=(\Omega _{m} +\Omega _{\Lambda}-1)H^{2}
\label{cosmo.14}
\eeq
    Thus we distinguish the following special cases:
\beq
    \kappa >0~~~~\Leftrightarrow~~~~\Omega _{m} +\Omega
    _{\Lambda}>1~~~~\Leftrightarrow Closed~curved~Universe
\eeq
\beq
    \kappa =0~~~~\Leftrightarrow~~~~\Omega _{m} +\Omega
    _{\Lambda}=1~~~~\Leftrightarrow Open~ Flat~ Universe
\eeq
\beq
    \kappa <0~~~~\Leftrightarrow~~~~\Omega _{m} +\Omega
    _{\Lambda}<1~~~~\Leftrightarrow Open~ Curved~ Universe
\eeq
   In other words it is the combination of matter and "vacuum" 
   energy, which determines the fate of the our Universe.

    Before concluding this section we remark that the above
    equations do not suffice to yield a solution since the density
    is a function of the scale factor. An equation of state is in
    addition needed, but we are not going to elaborate further.

\section{An Overview of Direct Detection - The Allowed SUSY Parameter Space.}

 Since this particle is expected to be very massive, $m_{\chi} \geq 30 GeV$, and
extremely non relativistic with average kinetic energy $T \leq 100 KeV$,
it can be directly detected ~\cite{JDV96,KVprd} mainly via the recoiling
of a nucleus (A,Z) in the elastic scattering process:
\begin{equation}
\chi \, +\, (A,Z) \, \to \, \chi \,  + \, (A,Z)^* 
\end{equation}
($\chi$ denotes the LSP). In order to compute the event rate one needs
the following ingredients:

1) An effective Lagrangian at the elementary particle 
(quark) level obtained in the framework of supersymmetry as described 
, e.g., in Refs.~\cite{Jungm,ref2}.

2) A procedure in going from the quark to the nucleon level, i.e. a quark 
model for the nucleon. The results depend crucially on the content of the
nucleon in quarks other than u and d. This is particularly true for the scalar
couplings as well as the isoscalar axial coupling ~\cite{Dree}$^-$\cite{Chen}.

3) Compute the relevant nuclear matrix elements \cite{KVdubna,DIVA00}
using as reliable as possible many body nuclear wave functions.
By putting as accurate nuclear physics input as possible, 
one will be able to constrain the SUSY parameters as much as possible.
The situation is a bit simpler in the case of the scalar coupling, in which
case one only needs the nuclear form factor.

Since the obtained rates are very low, one would like to be able to exploit the
modulation of the event rates due to the earth's
revolution around the sun \cite{Verg98,Verg99}$^-$\cite{Verg01}. To this end one
adopts a folding procedure
assuming some distribution~\cite{Jungm,Verg99,Verg01} of velocities for the LSP.
One also would like to know the directional rates, by observing the 
nucleus in a certain direction, which correlate with the motion of the
sun around the center of the galaxy and the motion of the Earth 
\cite {ref1,UKDMC}.

 The calculation of this cross section  has become pretty standard.
 One starts with   
representative input in the restricted SUSY parameter space as described in
the literature~\cite{Gomez,ref2}. 
We will adopt a phenomenogical procedure taking  universal soft 
SUSY breaking terms at $M_{GUT}$, i.e., a 
common mass for all scalar fields $m_0$, a common gaugino mass 
$M_{1/2}$ and a common trilinear scalar coupling $A_0$, which 
we put equal to zero (we will discuss later the influence of 
non-zero $A_0$'s). Our effective theory below $M_{GUT}$ then 
%depends on the parameters ($\mu_0=\mu(M_{GUT})$)
depends on the parameters \cite{Gomez}:
\[
m_0,\ M_{1/2},\ \mu_0,\ \alpha_G,\ M_{GUT},\ h_{t},\ ,\ h_{b},\ ,\ h_{\tau},\ 
\tan\beta~,  
\]
where $\alpha_G=g_G^2/4\pi$ ($g_G$ being the GUT gauge coupling 
constant) and $h_t, h_b, h_\tau $ are respectively the top, bottom and 
tau Yukawa coupling constants at $M_{GUT}$. The values of $\alpha_G$ and 
$M_{GUT}$ are obtained as described in Ref.\cite{Gomez}.
For a specified value of $\tan\beta$ at $M_S$, we determine $h_{t}$ at 
$M_{GUT}$ by fixing the top quark mass at the center of its 
experimental range, $m_t(m_t)= 166 \rm{GeV}$. The value
of  $h_{\tau}$ at $M_{GUT}$ is  fixed by using the running tau lepton
mass at $m_Z$, $m_\tau(m_Z)= 1.746 \rm{GeV}$. 
The value of $h_{b}$ at $M_{GUT}$ used is such that:
\[
m_b(m_Z)_{SM}^{\overline{DR}}=2.90\pm 0.14~{\rm GeV}.
\]
after including the SUSY threshold correction. 
%$m_b(m_Z)$ is at the center experimental upper bound \cite{mb}:
%\[
%m_b(m_Z)=2.67 \pm 0.50~{\rm GeV}.
%\]
The SUSY parameter space is subject to the
 following constraints:\\
1.) The LSP relic abundance will satisfy the cosmological constrain:
\begin{equation}
0.09 \le \Omega_{LSP} h^2 \le 0.22
\label{eq:in2}
\end{equation}
 2.) The Higgs bound obtained from recent CDF \cite{VALLS}
 and LEP2 \cite{DORMAN},
%\cite{VALDOR},
 i.e. $m_h~>~113~GeV$.\\
3.) We will limit ourselves to LSP-nucleon cross sections for the scalar
coupling, which gives detectable rates
\begin{equation}
4\times 10^{-7}~pb~ \le \sigma^{nucleon}_{scalar} 
\le 2 \times 10^{-5}~pb~
\label{eq:in3}
\end{equation}
 We should remember that the event rate does not depend only
 on the nucleon cross section, but on other parameters also, mainly
 on the LSP mass and the nucleus used in target. 
The condition on the nucleon cross section imposes severe constraints on the
acceptable parameter space. In particular in our model it restricts 
$tan \beta$ to values $tan \beta \simeq 50$. We will not elaborate further
on this point, since it has already appeared \cite{gtalk}. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%
\section{Expressions for the Differential Cross Section .} 
\bigskip

 The effective Lagrangian describing the LSP-nucleus cross section can
be cast in the form \cite {JDV96}
 \beq
{\it L}_{eff} = - \frac {G_F}{\sqrt 2} \{({\bar \chi}_1 \gamma^{\lambda}
\gamma_5 \chi_1) J_{\lambda} + ({\bar \chi}_1 
 \chi_1) J\}
 \label{eq:eg 41}
\eeq
where
\beq
  J_{\lambda} =  {\bar N} \gamma_{\lambda} (f^0_V +f^1_V \tau_3
+ f^0_A\gamma_5 + f^1_A\gamma_5 \tau_3)N~~,~~
J = {\bar N} (f^0_s +f^1_s \tau_3) N
 \label{eq:eg.42}
\eeq
%and
%\beq
%J = {\bar N} (f^0_s +f^1_s \tau_3) N
% \label{eq:eg.45}
%\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We have neglected the uninteresting pseudoscalar and tensor
currents. Note that, due to the Majorana nature of the LSP, 
${\bar \chi_1} \gamma^{\lambda} \chi_1 =0$ (identically).

 With the above ingredients the differential cross section can be cast in the 
form \cite{ref1,Verg98,Verg99}
\begin{equation}
d\sigma (u,\upsilon)= \frac{du}{2 (\mu _r b\upsilon )^2} [(\bar{\Sigma} _{S} 
                   +\bar{\Sigma} _{V}~ \frac{\upsilon^2}{c^2})~F^2(u)
%\frac{u}{\rangle~u~\rangle})\frac{\upsilon^2}{c^2}\frac{u}{\rangle~u~\rangle}) 
                       +\bar{\Sigma} _{spin} F_{11}(u)]
\label{2.9}
\end{equation}
\begin{equation}
\bar{\Sigma} _{S} = \sigma_0 (\frac{\mu_r(A)}{\mu _r(N)})^2  \,
 \{ A^2 \, [ (f^0_S - f^1_S \frac{A-2 Z}{A})^2 \, ] \simeq \sigma^S_{p,\chi^0}
        A^2 (\frac{\mu_r(A)}{\mu _r(N)})^2 
\label{2.10}
\end{equation}
\begin{equation}
\bar{\Sigma} _{spin}  =  \sigma^{spin}_{p,\chi^0}~\zeta_{spin}~~,~~
\zeta_{spin}= \frac{(\mu_r(A)/\mu _r(N))^2}{3(1+\frac{f^0_A}{f^1_A})^2}S(u)
\label{2.10a}
\end{equation}
%\begin{equation}
%\zeta_{spin}= \frac{(\mu_r(A)/\mu _r(N))^2}{3(1+\frac{f^0_A}{f^1_A})^2}S(u)
%\label{2.10b1}
%\end{equation}
\begin{equation}
S(u)=[(\frac{f^0_A}{f^1_A} \Omega_0(0))^2 \frac{F_{00}(u)}{F_{11}(u)}
  +  2\frac{f^0_A}{ f^1_A} \Omega_0(0) \Omega_1(0)
\frac{F_{01}(u)}{F_{11}(u)}+  \Omega_1(0))^2  \, ] 
\label{2.10b2}
\end{equation}
\begin{equation}
\bar{\Sigma} _{V}  =  \sigma^V_{p,\chi^0}~\zeta_V 
\label{2.10c}
\end{equation}
\begin{equation}
\zeta_V =  \frac{(\mu_r(A)/\mu _r(N))^2}{(1+\frac{f^1_V}{f^0_V})^2} A^2 \, 
(1-\frac{f^1_V}{f^0_V}~\frac{A-2 Z}{A})^2 [ (\frac{\upsilon_0} {c})^2  
[ 1  -\frac{1}{(2 \mu _r b)^2} \frac{2\eta +1}{(1+\eta)^2} 
\frac{\langle~2u~ \rangle}{\langle~\upsilon ^2~\rangle}] 
\label{2.10d}
\end{equation}
\\
$\sigma^i_{p,\chi^0}=$ proton cross-section,$i=S,spin,V$ given by:\\
$\sigma^S_{p,\chi^0}= \sigma_0 ~(f^0_S)^2~(\frac{\mu _r(N)}{m_N})^2$ 
 (scalar) , 
(the isovector scalar is negligible, i.e. $\sigma_p^S=\sigma_n^S)$\\
$\sigma^{spin}_{p,\chi^0}=\sigma_0~~3~(f^0_A+f^1_A)^2~(\frac{\mu _r(N)}{m_N})^2$ 
  (spin) ,
$\sigma^{V}_{p,\chi^0}= \sigma_0~(f^0_V+f^1_V)^2~(\frac{\mu _r(N)}{m_N})^2$ 
(vector)   \\
where $m_N$ is the nucleon mass,
 $\eta = m_x/m_N A$, and
 $\mu_r(A)$ is the LSP-nucleus reduced mass,  
 $\mu_r(N)$ is the LSP-nucleon reduced mass and  
\begin{equation}
\sigma_0 = \frac{1}{2\pi} (G_F m_N)^2 \simeq 0.77 \times 10^{-38}cm^2 
\label{2.7} 
\end{equation}
\begin{equation}
%u = q^2b^2/2~~,~~
Q=Q_{0}u~~, \qquad Q_{0} = \frac{1}{A m_{N} b^2}=4.1\times 10^4A^{-4/3}~KeV 
\label{2.15} 
\end{equation}
where
%b is (the harmonic oscillator) size parameter, 
%q is the momentum transfer to the nucleus, and
Q is the energy transfer to the nucleusr,
$F(u)$ is the nuclear form factor and
\begin{equation}
F_{\rho \rho^{\prime}}(u) =  \sum_{\lambda,\kappa}
\frac{\Omega^{(\lambda,\kappa)}_\rho( u)}{\Omega_\rho (0)} \,
\frac{\Omega^{(\lambda,\kappa)}_{\rho^{\prime}}( u)}
{\Omega_{\rho^{\prime}}(0)} 
, \qquad \rho, \rho^{\prime} = 0,1
\label{2.11} 
\end{equation}
are the spin form factors \cite{KVprd} ($\rho , \rho^{'}$ are isospin indices)
normalized to one at $u=0$.
$\Omega_0$ ($\Omega_1$) are the static isoscalar (isovector) spin 
matrix elements.
Note that the quantity S(u) is essentially independent of $u$. So the energy
transfer dependence is contained in the function $F_{11}(u)$. Note also that
$S(u)$ depends on the ratio of the isoscalar to isovector axial current 
couplings. These individual couplings can vary a lot within the SUSY parameter
space.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[t]  
\caption{
The static spin matrix elements for the light nuclei considered 
here. For comparison we also quote the results for the medium heavy nucleus 
$^{73}$Ge \protect\cite{Ress} and the heavy nucleus $^{207}$Pb
 \protect\cite{KVprd}.
}
\label{table.spin}
\begin{center}
\begin{tabular}{lrrrrr}
%\hline\hline
 &   &  &  &  &    \\
 & $^{19}$F & $^{29}$Si & $^{23}$Na  & $^{73}$Ge & $^{207}$Pb\\
\hline
    &   &  &  &  &    \\
$[\Omega_{0}(0)]^2$         & 2.610   & 0.207  & 0.477  & 1.157    & 0.305\\
$[\Omega_{1}(0)]^2$         & 2.807   & 0.219  & 0.346  & 1.005    & 0.231\\
$\Omega_{0}(0)\Omega_{1}(0)$& 2.707   &-0.213  & 0.406  &-1.078    &-0.266\\
$\mu_{th} $& 2.91   &-0.50  & 2.22  &    &\\
$\mu_{exp}$& 2.62   &-0.56  & 2.22  &    &\\
$\mu_{th}(spin)/ \mu_{exp}(spin)$& 0.91   &0.99  & 0.57  &    &\\
%\hline
%\hline
\end{tabular}
\end{center}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\epsfig{figure=fig5.ps,height=3.90in,angle=0}
\vspace*{-2.cm}
\caption{
The energy dependence of the coherent process, i.e. the square
 of the form factor, ($|F(u)|^2$),
for the isotopes $^{19}F$,$^{23}Na$ and $^{29}Si$.
The allowed range of  
$u$ for the above isotopes is $0.011\leq u \leq 0.17$, 
$0.015\leq u \leq 0.30$, and $0.021\leq u \leq 0.50$ respectively. 
This corresponds to energy transfers $8.9\leq Q \leq 140$, 
$9.5\leq Q \leq 190$, and $9.7\leq u \leq 230$ $KeV$ respectively. 
\label{cohff}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 Their ratio, however, is not changing very much. In fact actual
 calculations \cite{DIVA00} show that $3.0\leq S(0)\leq < 7.5$ for $^{19}F$,
$0.03\leq S(0)\leq 0.2$ for $^{29}Si$ and
$0.4\leq S(0)\leq 1.1$ for $^{23}Na$. The quantity $S(u)$ depends very
 sensitively on nuclear physics via the static spin ME. This is exhibited
in Table (\ref{table.spin}). As we can see from Table \ref{table.spin} the
spin matrix elements are very accurate. This is evident by comparing the
obtained magnetic dipole moments to experiment and noting that the magnetic
 moments, with the exception of $^{23}Na$  are dominated by the spin. From
 the same table we see that $^{19}F$ is
favored from the point of view of the spin matrix element. This advantage 
may be partially lost if the LSP is very heavy, due to the kinematic factor
$\mu_r (A)$, which tends to favor a heavy target.
The energy transfer dependence of the differential cross section for the
coherent mode is given
 by the square of the form factor,
 i.e. $|F(u)|^2$. These form factors
for the isotopes $^{19}F$,$^{23}Na$ and $^{29}Si$ were calculated by
Divari {\it et al} \cite{DIVA00} and are shown in Fig (\ref{cohff}).
The energy transfer dependence of the differential cross section due to 
spin is essentially given by $F_{11}(u)$.  These functions
for the isotopes $^{19}F,^{23}Na$ and $^{29}Si$ were calculated by
Divari {\it et al} \cite{DIVA00} and are shown in Fig (\ref{spinff}).
Note that the energy dependence of the coherent and the spin modes for 
light systems are not very different, especially if the PCAC corrections
on the spin response function are ignored.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\epsfig{figure=fig4.ps,height=3.90in,angle=0}
\vspace*{-2.cm}
\caption{
The energy dependence of the spin contribution (spin response function
 $F_{11}(u)$)
for the isotopes $^{19}F,^{23}Na$ and $^{29}Si$. The allowed range
of energy transfers is the same as in Table \ref{cohff}
%The behavior of the other isospin structure functions $F_{\rho \rho^{'}}$
%with $\rho, \rho^{'} = 0, 1$ is similar. 
In this figure we also plot  
$F_{11}(u)$ when the PCAC effect is  considered. 
\label{spinff}}
\end{figure}

\section{Expressions for the Rates.} 
 The non-directional event rate is given by:
\begin{equation}
R=R_{non-dir} =\frac{dN}{dt} =\frac{\rho (0)}{m_{\chi}} \frac{m}{A m_N} 
\sigma (u,\upsilon) | {\boldmath \upsilon}|
\label{2.17} 
\end{equation}
 Where
 $\rho (0) = 0.3 GeV/cm^3$ is the LSP density in our vicinity and 
 m is the detector mass 
The differential non-directional  rate can be written as
\begin{equation}
dR=dR_{non-dir} = \frac{\rho (0)}{m_{\chi}} \frac{m}{A m_N} 
d\sigma (u,\upsilon) | {\boldmath \upsilon}|
\label{2.18}  
\end{equation}
where $d\sigma(u,\upsilon )$ was given above.

 The directional differential rate \cite{ref1},\cite{Verg01} in the
direction $\hat{e}$ is given by :
\begin{equation}
dR_{dir} = \frac{\rho (0)}{m_{\chi}} \frac{m}{A m_N} 
{\boldmath \upsilon}.\hat{e} H({\boldmath \upsilon}.\hat{e})
 ~\frac{1}{2 \pi}~  
d\sigma (u,\upsilon)
\label{2.20}  
\end{equation}
where H the Heaviside step function. The factor of $1/2 \pi$ is 
introduced, since  the differential cross section of the last equation
is the same with that entering the non-directional rate, i.e. after
an integration
over the azimuthal angle around the nuclear momentum has been performed.
In other words, crudely speaking, $1/(2 \pi)$ is the suppression factor we
 expect in the directional rate compared to the usual one. The precise 
suppression factor depends, of course, on the direction of observation.
In spite of their very interesting experimental signatures, we will
not be concerned here with directional rates.
The mean value of the non-directional event rate of Eq. (\ref {2.18}), 
is obtained by convoluting the above expressions with the LSP velocity
distribution $f({\bf \upsilon}, {\boldmath \upsilon}_E)$ 
with respect to the Earth, i.e. is given by:
\beq
\Big<\frac{dR}{du}\Big> =\frac{\rho (0)}{m_{\chi}} 
\frac{m}{A m_N}  
\int f({\bf \upsilon}, {\boldmath \upsilon}_E) 
          | {\boldmath \upsilon}|
                       \frac{d\sigma (u,\upsilon )}{du} d^3 {\boldmath \upsilon} 
\label{3.10} 
\eeq
 The above expression can be more conveniently written as
\beq
\Big<\frac{dR}{du}\Big> =\frac{\rho (0)}{m_{\chi}} \frac{m}{Am_N} \sqrt{\langle
\upsilon^2\rangle } {\langle \frac{d\Sigma}{du}\rangle } 
\label{3.11}  
\eeq
where
\beq
\langle \frac{d\Sigma}{du}\rangle =\int
           \frac{   |{\boldmath \upsilon}|}
{\sqrt{ \langle \upsilon^2 \rangle}} f({\boldmath \upsilon}, 
         {\boldmath \upsilon}_E)
                       \frac{d\sigma (u,\upsilon )}{du} d^3 {\boldmath \upsilon}
\label{3.12}  
\eeq


 After performing the needed integrations over the velocity distribution,
to first order in the Earth's velocity, and over the energy transfer u  the
 last expression takes the form
\beq
R =  \bar{R}~t~
          [1 + h(a,Q_{min})cos{\alpha})] 
\label{3.55a}  
\eeq
where $\alpha$ is the phase of the Earth ($\alpha=0$ around June 2nd)
and  $Q_{min}$ is the energy transfer cutoff imposed by the detector.
In the above expressions $\bar{R}$ is the rate obtained in the conventional 
approach \cite {JDV96} by neglecting the folding with the LSP velocity and the
momentum transfer dependence of the differential cross section, i.e. by
\beq
\bar{R} =\frac{\rho (0)}{m_{\chi}} \frac{m}{Am_N} \sqrt{\langle
v^2\rangle } [\bar{\Sigma}_{S}+ \bar{\Sigma} _{spin} + 
\frac{\langle \upsilon ^2 \rangle}{c^2} \bar{\Sigma} _{V}]
\label{3.39b}  
\eeq
where $\bar{\Sigma} _{i}, i=S,V,spin$ have been defined above, see Eqs
 (\ref {2.10}) - (\ref {2.10c}). It contains all the parameters of the
SUSY models. 
 The modulation is described by the parameter $h$ . Once
the rate is known and the parameters $t$ and $h$, which depend only
on the LSP mass, the nuclear form factor and the velocity distribution 
the nucleon cross section can be extracted and compared to experiment.

The total  directional event rates  can be obtained in a similar fashion by
by integrating Eq. (\ref {2.20}) 
with respect to the velocity as well as the energy transfer u.  We find

\barr
%R_{dir}& = &  \bar{R} [\rho^{'}(0)/(\rho(0)] \, (t^0/4 \pi) \, 
R_{dir}& = &  \bar{R} [(t^0/4 \pi) \, 
            |(1 + h_1(a,Q_{min})cos{\alpha}) {\bf e}~_z.{\bf e}
\nonumber\\  &-& h_2(a,Q_{min})\, 
cos{\alpha} {\bf e}~_y.{\bf e}
                      + h_3(a,Q_{min})\, 
sin{\alpha} {\bf e}~_x.{\bf e}|
\label{4.55}  
\earr
We remind that the z-axis is in the direction of the sun's motion, the y-axis
 is perpendicular to the plane of the galaxy and the x-axis is in the 
galactocentric direction.
 The effect of folding
with LSP velocity on the total rate is taken into account via the quantity
$t^0$, which depends on the LSP mass. All other SUSY parameters have been
 absorbed in $\bar{R}$.
 We see that the modulation of the directional total event rate can
be described in terms of three parameters $h_l$, l=1,2,3. 
 In the special case of $\lambda=0$ we essentially have  one 
parameter, namely $h_1$, since then we have $h_2=0.117$ and $h_3=0.135$.
Given the functions $h_l(a,Q_{min})$ one can plot the the expression in
Eq. (\ref {4.55}) as a function of the phase of the earth $\alpha$. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Scalar Contribution- The Role of the Heavy Quarks}
\bigskip

The coherent scattering can be mediated via the the neutral
intermediate Higgs particles (h and H), which survive as physical 
particles. It can also be mediated via s-quarks, via the mixing
of the isodoublet and isosinlet s-quarks of the same charge. In our
model we find that the Higgs contribution becomes dominant and, as a
matter of fact the heavy Higgs H is more important (the Higgs particle
$A$ couples in a pseudoscalar way, which does not lead to coherence).
It is well known that all quark flavors contribute
\cite{Dree}, since the relevant couplings are proportional to
the quark masses.
One encounters in the nucleon not only the usual 
sea quarks ($u {\bar u}, d {\bar d}$ and $s {\bar s}$) but the 
heavier quarks $c,b,t$ which couple to the nucleon via two gluon 
exchange, see e.g. Drees {\it et al}  ~\cite{Dree00} and references
therein.
 
As a result  one obtains an effective scalar Higgs-nucleon
%coupling $f^0_s(H)$ by using  effective quark masses as follows
coupling by using  effective quark masses as follows
\begin{center}
$m_u \ra f_u m_N, \ \ m_d \ra f_d m_N. \ \ \  m_s \ra f_s m_N$   
\end{center}
\begin{center}
$m_Q \ra f_Q m_N, \ \ (heavy\ \  quarks \ \ c,b,t)$   
\end{center}
where $m_N$ is the nucleon mass. The isovector contribution is now
negligible. The parameters $f_q,~q=u,d,s$ can be obtained by chiral
symmetry breaking 
terms in relation to phase shift and dispersion analysis.
Following Cheng and Cheng ~\cite{Chen} we obtain:
\begin{center}
$ f_u = 0.021, \quad f_d = 0.037, \quad  f_s = 0.140$ 
\quad  \quad  (model B)   
\end{center}
\begin{center}
$ f_u = 0.023, \quad f_d = 0.034, \quad  f_s = 0.400$ 
\quad  \quad  (model C)   
\end{center}
 We see that in both models the s-quark is dominant.
Then to leading order via quark loops and gluon exchange with the
nucleon one finds:
\begin{center}
\quad $f_Q= 2/27(1-\quad \sum_q f_q)$   
\end{center}
This yields:
\begin{center}
\quad $ f_Q = 0.060$    (model B),   
\quad $ f_Q = 0.040$    (model C)   
\end{center}
 There is a correction to the above parameters coming from loops
involving s-quarks \cite {Dree00} and due to QCD effects. 
 Thus for large $tan \beta$ we find \cite {ref1}:
\begin{center}
\quad $f_{c}=0.060 \times 1.068=0.064,
       f_{t}=0.060 \times 2.048=0.123,
       f_{b}=0.060 \times 1.174=0.070$  \quad (model B)
\end{center}
\begin{center}
\quad $f_{c}=0.040 \times 1.068=0.043,
       f_{t}=0.040 \times 2.048=0.082,
       f_{b}=0.040 \times 1.174=0.047$  \quad (model B)
\end{center}
For a more detailed discussion we refer the reader to 
Refs~\cite{Dree,Dree00}.

\section{Results and Discussion}
The three basic ingredients of our calculation were 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
the input SUSY parameters (see sect. 1), a quark model for the nucleon
(see sect. 3) and the velocity distribution combined with the structure of
 the nuclei involved (see sect. 2). we will focus our attention on the
coherent scattering and present results for the popular target $^{127}I$.
We have utilized two nucleon models indicated by B and C which take into
account the presence of heavy quarks in the nucleon. We also considered
energy cut offs imposed by the detector,  
 by considering two  typical cases $Q_{min}=10,~20$ KeV. The thus obtained
results for the unmodulated total non directional event rates $\bar{R}t$
 in the case
 of the symmetric isothermal model for a typical SUSY parameter choice
\cite{Gomez} are shown in Fig. \ref{rate}. 
\begin{figure}
\epsfig{figure=rates.eps,height=3.90in,angle=0}
\caption{The Total detection rate per $(kg-target)yr$ vs the LSP mass
in GeV for a typical solution in our parameter space in the case of 
$^{127}I$ corresponding to  model B (thick line) and Model C (fine 
line). For the definitions see text.
\label{rate}}
\end{figure}
Special attention was paid to the the directional rate and its  
modulation due to the annual motion of the earth in the case of isothermal
models. The case of non isothermal models, e.g. caustic rings, is more
complicated \cite{Verg01} and it will not be further discussed here.
 As expected, the parameter $t_0$, which contains the 
effect of the nuclear form factor and the LSP velocity dependence,
decreases as the reduced mass increases.
 
We will focus to the discussion of the directional rates
 described in terms of $t_0$
and  $h_i,~i=1,2,3$ (see Eq.  (\ref{4.55})) 
 and limit ourselves to directions of observation 
close to the coordinate axes. 
 As expected, the parameter $t_0$,
decreases as the reduced mass increases.
%These are shown in tables 1-2 for various values of $Q_{min}$ and $\lambda$. 
% $h_2$ and $h_3$ 
The quantity $t^0$ is shown in Fig. (\ref{t0}),  
for three values of the detector energy cutoff
, $Q_{min}=0,10$ and $20~KeV$. Similarly we show the quantity
$h_1$ in Fig. (\ref{h1}). The quanities 
$h_2$ and $h_3$  are shown in Fig (\ref{h23}) for $\lambda=1$.
For $\lambda=0$ they are not shown, since they are 
essentially constant and equal to $0.117$ and $0.135$ respectively. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setlength{\unitlength}{1mm}
\begin{figure}
\begin{picture}(150,50)
%origin is lower left corner
\put(0,04){\epsfxsize=5cm \epsfbox{t0.ps}}
\put(8,04) {$\lambda=0$}
\put(60,04){\epsfxsize=5cm \epsfbox{t1.ps}}
\put(85,04){$\lambda=1$}
\end{picture}
\caption[]{
The dependence of the quantity $t^0$ on the LSP mass
 for the symmetric case ($\lambda=0$) as well as for the maximum axial
 asymmetry ($\lambda=1$) in the case of the target $^{127}I$.
For orientation purposes  three  detection cutoff energies are exhibited,
$Q_{min}=0$ (thick solid line),$ Q_{min}=5~keV$ (thin solid line) and
$ Q_{min}=10~keV$ (dahed line).
 As expected $t^0$ decreases as the cutoff energy increases. 
}
\label{t0}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setlength{\unitlength}{1mm}
\begin{figure}
\begin{picture}(150,50)
%origin is lower left corner
\put(0,04){\epsfxsize=5cm \epsfbox{hz0.ps}}
\put(8,04) {$\lambda=0$}
\put(60,04){\epsfxsize=5cm \epsfbox{hz1.ps}}
\put(85,04){$\lambda=1$}
\end{picture}
\caption[]{
The same as in the previous figure for the modulation amplitude $h_1$.
}
\label{h1}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setlength{\unitlength}{1mm}
\begin{figure}
\begin{picture}(150,50)
%origin is lower left corner
\put(0,04){\epsfxsize=5cm \epsfbox{hy1.ps}}
\put(6,04) {$h_2$ (for $\lambda=1$)}
\put(60,04){\epsfxsize=5cm \epsfbox{hx1.ps}}
\put(75,04){$h_3$ (for $\lambda=1$)}
\end{picture}
\caption[]{
The same as in the previous figure for  the modulation amplitudes $h_2$
and  $h_3$ for $\lambda=1$.
}
\label{h23}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 As expected, the parameter $t_0$, decreases as the reduced mass increases.
It also decreasres as the cutoff energy $Q_{min}$ increases.
 We notice that $t^0$ is affected little by the presence of asymmetry.
 On the other
hand $h_1,h_2$ and $h_3$ substantially increase in the presence of asymmetry.
Sometimes they increase as the cutoff energy increases (at the expense, of
 course, of the total number of counts. 
For the differential rate the reader is 
referred to our previous work \cite {Verg99,Verg00}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
In the present paper we have discussed the parameters, which describe
the event rates for direct detection of SUSY dark matter.
Only in a small segmant of the allowed parameter space the rates are above
 the present experimental goals.
 We thus looked for
characteristic experimental signatures for background reduction, i.e.
a) Correlation of the event rates
with the motion of the Earth (modulation effect) and b)
 the directional rates (their correlation  both with the velocity of the sun
 and that of the Earth.)

A typical graph for the total unmodulated rate is shown Fig. \ref{rate}. 
We will concentrate here on the directional rates, described in
terms of the parameters $t_0,h_1,h_2$ and $h_3$.
For simplicity these parameters are given in Figs (\ref{t0})-(\ref{h23})
for directions
of observation close to the three axes $x,y,z$.  We see that the unmodulated 
rate scales by the $cos\theta_s$, with $\theta_s$ (the angle between
the direction of observation and the velocity of the sun).
 The reduction factor, $f_{red}=t_0/(4 \pi~t_0)=\kappa/(2 \pi)$, 
 of the total directional
rate, along the sun's direction of motion, compared to the total non
directional rate depends on the
nuclear parameters, the reduced mass and the asymmetry parameter $\lambda$
\cite{Verg00}.
 We find
that $\kappa$ is around 0.6 (no asymmetry) and around 0.7 (maximum
asymmetry, $\lambda=1.0$), i.e. not very different from the 
naively expected $f_{red}=1/( 2 \pi)$, i.e.  $\kappa=1$. 
The modulation of the directional rate 
increases with the asymmetry parameter $\lambda$ and it also depends
of the direction of observation. For $Q_{min}=0$ it can reach values up 
to $23\%$. Values up to $35\%$ are possible for large values of
 $Q_{min}$, but they occur at the expense of the total
number of counts.

\par
This work was supported by the European Union under the contracts 
RTN No HPRN-CT-2000-00148 and TMR 
No. ERBFMRX--CT96--0090 and $\Pi E N E \Delta~95$ of the Greek 
Secretariat for Research.


\def\ijmp#1#2#3{{ Int. Jour. Mod. Phys. }{\bf #1~}(#2)~#3}
\def\pl#1#2#3{{ Phys. Lett. }{\bf B#1~}(#2)~#3}
\def\zp#1#2#3{{ Z. Phys. }{\bf C#1~}(#2)~#3}
\def\prl#1#2#3{{ Phys. Rev. Lett. }{\bf #1~}(#2)~#3}
\def\rmp#1#2#3{{ Rev. Mod. Phys. }{\bf #1~}(#2)~#3}
\def\prep#1#2#3{{ Phys. Rep. }{\bf #1~}(#2)~#3}
\def\pr#1#2#3{{ Phys. Rev. }{\bf D#1~}(#2)~#3}
\def\np#1#2#3{{ Nucl. Phys. }{\bf B#1~}(#2)~#3}
\def\npps#1#2#3{{ Nucl. Phys. (Proc. Sup.) }{\bf B#1~}(#2)~#3}
\def\mpl#1#2#3{{ Mod. Phys. Lett. }{\bf #1~}(#2)~#3}
\def\arnps#1#2#3{{ Annu. Rev. Nucl. Part. Sci. }{\bf
#1~}(#2)~#3}
\def\sjnp#1#2#3{{ Sov. J. Nucl. Phys. }{\bf #1~}(#2)~#3}
\def\jetp#1#2#3{{ JETP Lett. }{\bf #1~}(#2)~#3}
\def\app#1#2#3{{ Acta Phys. Polon. }{\bf #1~}(#2)~#3}
\def\rnc#1#2#3{{ Riv. Nuovo Cim. }{\bf #1~}(#2)~#3}
\def\ap#1#2#3{{ Ann. Phys. }{\bf #1~}(#2)~#3}
\def\ptp#1#2#3{{ Prog. Theor. Phys. }{\bf #1~}(#2)~#3}
\def\plb#1#2#3{{ Phys. Lett. }{\bf#1B~}(#2)~#3}
\def\apjl#1#2#3{{ Astrophys. J. Lett. }{\bf #1~}(#2)~#3}
\def\n#1#2#3{{ Nature }{\bf #1~}(#2)~#3}
\def\apj#1#2#3{{ Astrophys. Journal }{\bf #1~}(#2)~#3}
\def\anj#1#2#3{{ Astron. J. }{\bf #1~}(#2)~#3}
\def\mnras#1#2#3{{ MNRAS }{\bf #1~}(#2)~#3}
\def\grg#1#2#3{{ Gen. Rel. Grav. }{\bf #1~}(#2)~#3}
\def\s#1#2#3{{ Science }{\bf #1~}(19#2)~#3}
\def\baas#1#2#3{{ Bull. Am. Astron. Soc. }{\bf #1~}(#2)~#3}
\def\ibid#1#2#3{{ ibid. }{\bf #1~}(19#2)~#3}
\def\cpc#1#2#3{{ Comput. Phys. Commun. }{\bf #1~}(#2)~#3}
\def\astp#1#2#3{{ Astropart. Phys. }{\bf #1~}(#2)~#3}
\def\epj#1#2#3{{ Eur. Phys. J. }{\bf C#1~}(#2)~#3}
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%\np{353}{1991}{591}; 
%R. Barbieri and G. F. Giudice, \pl{309}{1993}{86}; 
%R. Garisto and J. N. Ng, \pl{315}{1993}{372}; 
%A. J. Buras, M. Misiak, M. M\"{u}nz and S. Pokorski, 
%\np{424}{1994}{374}; 
%B. de Carlos and J. A. Casas, \pl{349}{1995}{300}; 
%S. Bertolini and F. Vissani, \zp{67}{1995}{513}; 
%V. Barger, M. S. Berger, P. Ohmann and R. J. N. Phillips, 
%\pr{51}{1995}{2438}.
%\bibitem{borzumati} F. M. Borzumati, M. Olechowski
%and S. Pokorski, \pl{349}{1995}{311}.
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%\pr{58}{1998}{015007}.
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%V. C. Spanos, \pl{464}{1999}{213}; hep-ph/9909497.  
%\bibitem{coan} K. Griest and D. Seckel, \pr{43}{1991}{3191}.
\bibitem{drees} M. Drees and M. M. Nojiri, 
\pr{47}{1993}{376}; 
%S. Mizuta and M. Yamaguchi, \pl{298}{1993}{120}; 
%P. Gondolo and J. Edsj\"o, \pr{56}{1997}{1879};
%M. Drees, hep-ph/9703260;  
%\bibitem{Hab-Ka}H.E. Haber and G.L. Kane, {\it Phys. Rep.} {\bf  117} 75 (1985).
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M. Spira {it et al}, {\it Nucl. Phys.} {\bf B453},17 (1995).
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H -Y. Cheng, Phys. Lett. {\bf B 219} 347 (1989).
\bibitem{Ress}M.T. Ressell {\it et al.}, {\it Phys. Rev.} {\bf D 48}, 5519 (1993);
%M.T. Ressell and D.J. Dean, hep-ph$/9702290$.
%\bibitem{Dimit}V.I. Dimitrov, J. Engel and S. Pittel, 
%{\it Phys. Rev.} {\bf D 51}, R291 (1995).
%\bibitem{Engel}J. Engel, {\it Phys. Lett.} {\bf B 264}, 114 (1991).
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%{\bf A 345}, 373 (1993); {\it Phys. Lett} {\bf 329 B}, 5 (1993);
%{\it  Phys. Rev.} {\bf D 50}, 7128 (1995).
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{\it  Phys. Rev.} {\bf C 61} (2000), 044612-1.
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%\bibitem{SIKIVIE}P. Sikivie, I. Tkachev and Y. Wang, {\it Phys. Rev. Let.}
% {\bf 75}, 2911 (1995; {\it Phys. Rev.} {\bf D 56}, 1863 (1997)\\
%P. Sikivie, {\it Phys. Let.} {\bf b 432}, 139 (1998); astro-ph/9810286 
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%J.I. Collar et al., {\it Phys. Lett} {\bf B 275}, 181 (1992).
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%               {\it Phys.Rep.} {\bf 307}, 253 (1999);
%J.J. Quenby et al Astropart. Phys. {\bf 5}, 249 (1996); Phys. Lett. 
%{\bf B 351}, 70 (1995).
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%Part. Sci. {\bf 38}, 751 (1988).
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%1st Int. Workshop on IDM}, p. 475, Ed. N.J.C. Spooner,
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%S. Cabrera, \npps{64}{1998}{376}.
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%\bibitem{VALDOR} CDF Collaboration, FERMILAB-Conf-99/263-E CDF;
%http://fnalpubs.fnal.gov/archive/1999/conf/Conf-99-263-E.html.
\bibitem{DORMAN}
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%\bibitem{DORMAN}, P.J. Dorman, {\it ALEF Collaboration}, March 2000,\\
%http:\/\/aalephwww.cern.ch\/ALPUB\/sseminar\/lepc_mar200\/lepc2000.pdf.
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%J. Ellis and M. Carliuer, The Strange Spin of the nucleon, CERN-Th/95334,
%Hep-ph 960 1280.
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%A next-to-Next-to Leading Order in QCD Analysis of the Ellis-Jaffe Sum Rule,
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%B. C., Canada, July 23-29 1998;
%ALEPH Collaboration (R. Barate {\it{et al.}}),\pl{429}{1998}{169}.
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%K. A. Olive and M. Schmitt, \pr{58}{1998}{095002}; 
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%\pl{474}{2000}{314}.
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%\np{491}{1997}{3}.
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%\pr{50}{1994}{5980}.
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%\pl{380}{1996}{385}; \pr{54}{1996}{3350}; K. Chetyrkin, 
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%\np{567}{2000}{153}.
\end{thebibliography}
\end{document}

psfig.tex

% Psfig/TeX 
\def\PsfigVersion{1.9}
% dvips version
%
% All psfig/tex software, documentation, and related files
% in this distribution of psfig/tex are 
% Copyright 1987, 1988, 1991 Trevor J. Darrell
%
% Permission is granted for use and non-profit distribution of psfig/tex 
% providing that this notice is clearly maintained. The right to
% distribute any portion of psfig/tex for profit or as part of any commercial
% product is specifically reserved for the author(s) of that portion.
%
% *** Feel free to make local modifications of psfig as you wish,
% *** but DO NOT post any changed or modified versions of ``psfig''
% *** directly to the net. Send them to me and I'll try to incorporate
% *** them into future versions. If you want to take the psfig code 
% *** and make a new program (subject to the copyright above), distribute it, 
% *** (and maintain it) that's fine, just don't call it psfig.
%
% Bugs and improvements to trevor@media.mit.edu.
%
% Thanks to Greg Hager (GDH) and Ned Batchelder for their contributions
% to the original version of this project.
%
% Modified by J. Daniel Smith on 9 October 1990 to accept the
% %%BoundingBox: comment with or without a space after the colon.  Stole
% file reading code from Tom Rokicki's EPSF.TEX file (see below).
%
% More modifications by J. Daniel Smith on 29 March 1991 to allow the
% the included PostScript figure to be rotated.  The amount of
% rotation is specified by the "angle=" parameter of the \psfig command.
%
% Modified by Robert Russell on June 25, 1991 to allow users to specify
% .ps filenames which don't yet exist, provided they explicitly provide
% boundingbox information via the \psfig command. Note: This will only work
% if the "file=" parameter follows all four "bb???=" parameters in the
% command. This is due to the order in which psfig interprets these params.
%
%  3 Jul 1991	JDS	check if file already read in once
%  4 Sep 1991	JDS	fixed incorrect computation of rotated
%			bounding box
% 25 Sep 1991	GVR	expanded synopsis of \psfig
% 14 Oct 1991	JDS	\fbox code from LaTeX so \psdraft works with TeX
%			changed \typeout to \ps@typeout
% 17 Oct 1991	JDS	added \psscalefirst and \psrotatefirst
%

% From: gvr@cs.brown.edu (George V. Reilly)
%
% \psdraft	draws an outline box, but doesn't include the figure
%		in the DVI file.  Useful for previewing.
%
% \psfull	includes the figure in the DVI file (default).
%
% \psscalefirst width= or height= specifies the size of the figure
% 		before rotation.
% \psrotatefirst (default) width= or height= specifies the size of the
% 		 figure after rotation.  Asymetric figures will
% 		 appear to shrink.
%
% \psfigurepath#1	sets the path to search for the figure
%
% \psfig
% usage: \psfig{file=, figure=, height=, width=,
%			bbllx=, bblly=, bburx=, bbury=,
%			rheight=, rwidth=, clip=, angle=, silent=}
%
%	"file" is the filename.  If no path name is specified and the
%		file is not found in the current directory,
%		it will be looked for in directory \psfigurepath.
%	"figure" is a synonym for "file".
%	By default, the width and height of the figure are taken from
%		the BoundingBox of the figure.
%	If "width" is specified, the figure is scaled so that it has
%		the specified width.  Its height changes proportionately.
%	If "height" is specified, the figure is scaled so that it has
%		the specified height.  Its width changes proportionately.
%	If both "width" and "height" are specified, the figure is scaled
%		anamorphically.
%	"bbllx", "bblly", "bburx", and "bbury" control the PostScript
%		BoundingBox.  If these four values are specified
%               *before* the "file" option, the PSFIG will not try to
%               open the PostScript file.
%	"rheight" and "rwidth" are the reserved height and width
%		of the figure, i.e., how big TeX actually thinks
%		the figure is.  They default to "width" and "height".
%	The "clip" option ensures that no portion of the figure will
%		appear outside its BoundingBox.  "clip=" is a switch and
%		takes no value, but the `=' must be present.
%	The "angle" option specifies the angle of rotation (degrees, ccw).
%	The "silent" option makes \psfig work silently.
%

% check to see if macros already loaded in (maybe some other file says
% "\input psfig") ...
\ifx\undefined\psfig\else\endinput\fi

%
% from a suggestion by eijkhout@csrd.uiuc.edu to allow
% loading as a style file. Changed to avoid problems
% with amstex per suggestion by jbence@math.ucla.edu

\let\LaTeXAtSign=\@
\let\@=\relax
\edef\psfigRestoreAt{\catcode`\@=\number\catcode`@\relax}
%\edef\psfigRestoreAt{\catcode`@=\number\catcode`@\relax}
\catcode`\@=11\relax
\newwrite\@unused
\def\ps@typeout#1{{\let\protect\string\immediate\write\@unused{#1}}}
\ps@typeout{psfig/tex \PsfigVersion}

%% Here's how you define your figure path.  Should be set up with null
%% default and a user useable definition.

\def\figurepath{./}
\def\psfigurepath#1{\edef\figurepath{#1}}

%
% @psdo control structure -- similar to Latex @for.
% I redefined these with different names so that psfig can
% be used with TeX as well as LaTeX, and so that it will not 
% be vunerable to future changes in LaTeX's internal
% control structure,
%
\def\@nnil{\@nil}
\def\@empty{}
\def\@psdonoop#1\@@#2#3{}
\def\@psdo#1:=#2\do#3{\edef\@psdotmp{#2}\ifx\@psdotmp\@empty \else
    \expandafter\@psdoloop#2,\@nil,\@nil\@@#1{#3}\fi}
\def\@psdoloop#1,#2,#3\@@#4#5{\def#4{#1}\ifx #4\@nnil \else
       #5\def#4{#2}\ifx #4\@nnil \else#5\@ipsdoloop #3\@@#4{#5}\fi\fi}
\def\@ipsdoloop#1,#2\@@#3#4{\def#3{#1}\ifx #3\@nnil 
       \let\@nextwhile=\@psdonoop \else
      #4\relax\let\@nextwhile=\@ipsdoloop\fi\@nextwhile#2\@@#3{#4}}
\def\@tpsdo#1:=#2\do#3{\xdef\@psdotmp{#2}\ifx\@psdotmp\@empty \else
    \@tpsdoloop#2\@nil\@nil\@@#1{#3}\fi}
\def\@tpsdoloop#1#2\@@#3#4{\def#3{#1}\ifx #3\@nnil 
       \let\@nextwhile=\@psdonoop \else
      #4\relax\let\@nextwhile=\@tpsdoloop\fi\@nextwhile#2\@@#3{#4}}
% 
% \fbox is defined in latex.tex; so if \fbox is undefined, assume that
% we are not in LaTeX.
% Perhaps this could be done better???
\ifx\undefined\fbox
% \fbox code from modified slightly from LaTeX
\newdimen\fboxrule
\newdimen\fboxsep
\newdimen\ps@tempdima
\newbox\ps@tempboxa
\fboxsep = 3pt
\fboxrule = .4pt
\long\def\fbox#1{\leavevmode\setbox\ps@tempboxa\hbox{#1}\ps@tempdima\fboxrule
    \advance\ps@tempdima \fboxsep \advance\ps@tempdima \dp\ps@tempboxa
   \hbox{\lower \ps@tempdima\hbox
  {\vbox{\hrule height \fboxrule
          \hbox{\vrule width \fboxrule \hskip\fboxsep
          \vbox{\vskip\fboxsep \box\ps@tempboxa\vskip\fboxsep}\hskip 
                 \fboxsep\vrule width \fboxrule}
                 \hrule height \fboxrule}}}}
\fi
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% file reading stuff from epsf.tex
%   EPSF.TEX macro file:
%   Written by Tomas Rokicki of Radical Eye Software, 29 Mar 1989.
%   Revised by Don Knuth, 3 Jan 1990.
%   Revised by Tomas Rokicki to accept bounding boxes with no
%      space after the colon, 18 Jul 1990.
%   Portions modified/removed for use in PSFIG package by
%      J. Daniel Smith, 9 October 1990.
%
\newread\ps@stream
\newif\ifnot@eof       % continue looking for the bounding box?
\newif\if@noisy        % report what you're making?
\newif\if@atend        % %%BoundingBox: has (at end) specification
\newif\if@psfile       % does this look like a PostScript file?
%
% PostScript files should start with `%!'
%
{\catcode`\%=12\global\gdef\epsf@start{%!}}
\def\epsf@PS{PS}
%
\def\epsf@getbb#1{%
%
%   The first thing we need to do is to open the
%   PostScript file, if possible.
%
\openin\ps@stream=#1
\ifeof\ps@stream\ps@typeout{Error, File #1 not found}\else
%
%   Okay, we got it. Now we'll scan lines until we find one that doesn't
%   start with %. We're looking for the bounding box comment.
%
   {\not@eoftrue \chardef\other=12
    \def\do##1{\catcode`##1=\other}\dospecials \catcode`\ =10
    \loop
       \if@psfile
	  \read\ps@stream to \epsf@fileline
       \else{
	  \obeyspaces
          \read\ps@stream to \epsf@tmp\global\let\epsf@fileline\epsf@tmp}
       \fi
       \ifeof\ps@stream\not@eoffalse\else
%
%   Check the first line for `%!'.  Issue a warning message if its not
%   there, since the file might not be a PostScript file.
%
       \if@psfile\else
       \expandafter\epsf@test\epsf@fileline:. \\%
       \fi
%
%   We check to see if the first character is a % sign;
%   if so, we look further and stop only if the line begins with
%   `%%BoundingBox:' and the `(atend)' specification was not found.
%   That is, the only way to stop is when the end of file is reached,
%   or a `%%BoundingBox: llx lly urx ury' line is found.
%
          \expandafter\epsf@aux\epsf@fileline:. \\%
       \fi
   \ifnot@eof\repeat
   }\closein\ps@stream\fi}%
%
% This tests if the file we are reading looks like a PostScript file.
%
\long\def\epsf@test#1#2#3:#4\\{\def\epsf@testit{#1#2}
			\ifx\epsf@testit\epsf@start\else
\ps@typeout{Warning! File does not start with `\epsf@start'.  It may not be a PostScript file.}
			\fi
			\@psfiletrue} % don't test after 1st line
%
%   We still need to define the tricky \epsf@aux macro. This requires
%   a couple of magic constants for comparison purposes.
%
{\catcode`\%=12\global\let\epsf@percent=%\global\def\epsf@bblit{%BoundingBox}}
%
%
%   So we're ready to check for `%BoundingBox:' and to grab the
%   values if they are found.  We continue searching if `(at end)'
%   was found after the `%BoundingBox:'.
%
\long\def\epsf@aux#1#2:#3\\{\ifx#1\epsf@percent
   \def\epsf@testit{#2}\ifx\epsf@testit\epsf@bblit
	\@atendfalse
        \epsf@atend #3 . \\%
	\if@atend	
	   \if@verbose{
		\ps@typeout{psfig: found `(atend)'; continuing search}
	   }\fi
        \else
        \epsf@grab #3 . . . \\%
        \not@eoffalse
        \global\no@bbfalse
        \fi
   \fi\fi}%
%
%   Here we grab the values and stuff them in the appropriate definitions.
%
\def\epsf@grab #1 #2 #3 #4 #5\\{%
   \global\def\epsf@llx{#1}\ifx\epsf@llx\empty
      \epsf@grab #2 #3 #4 #5 .\\\else
   \global\def\epsf@lly{#2}%
   \global\def\epsf@urx{#3}\global\def\epsf@ury{#4}\fi}%
%
% Determine if the stuff following the %%BoundingBox is `(atend)'
% J. Daniel Smith.  Copied from \epsf@grab above.
%
\def\epsf@atendlit{(atend)} 
\def\epsf@atend #1 #2 #3\\{%
   \def\epsf@tmp{#1}\ifx\epsf@tmp\empty
      \epsf@atend #2 #3 .\\\else
   \ifx\epsf@tmp\epsf@atendlit\@atendtrue\fi\fi}


% End of file reading stuff from epsf.tex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% trigonometry stuff from "trig.tex"
\chardef\psletter = 11 % won't conflict with \begin{letter} now...
\chardef\other = 12

\newif \ifdebug %%% turn me on to see TeX hard at work ...
\newif\ifc@mpute %%% don't need to compute some values
\c@mputetrue % but assume that we do

\let\then = \relax
\def\r@dian{pt }
\let\r@dians = \r@dian
\let\dimensionless@nit = \r@dian
\let\dimensionless@nits = \dimensionless@nit
\def\internal@nit{sp }
\let\internal@nits = \internal@nit
\newif\ifstillc@nverging
\def \Mess@ge #1{\ifdebug \then \message {#1} \fi}

{ %%% Things that need abnormal catcodes %%%
	\catcode `\@ = \psletter
	\gdef \nodimen {\expandafter \n@dimen \the \dimen}
	\gdef \term #1 #2 #3%
	       {\edef \t@ {\the #1}%%% freeze parameter 1 (count, by value)
		\edef \t@@ {\expandafter \n@dimen \the #2\r@dian}%
				   %%% freeze parameter 2 (dimen, by value)
		\t@rm {\t@} {\t@@} {#3}%
	       }
	\gdef \t@rm #1 #2 #3%
	       {{%
		\count 0 = 0
		\dimen 0 = 1 \dimensionless@nit
		\dimen 2 = #2\relax
		\Mess@ge {Calculating term #1 of \nodimen 2}%
		\loop
		\ifnum	\count 0 < #1
		\then	\advance \count 0 by 1
			\Mess@ge {Iteration \the \count 0 \space}%
			\Multiply \dimen 0 by {\dimen 2}%
			\Mess@ge {After multiplication, term = \nodimen 0}%
			\Divide \dimen 0 by {\count 0}%
			\Mess@ge {After division, term = \nodimen 0}%
		\repeat
		\Mess@ge {Final value for term #1 of 
				\nodimen 2 \space is \nodimen 0}%
		\xdef \Term {#3 = \nodimen 0 \r@dians}%
		\aftergroup \Term
	       }}
	\catcode `\p = \other
	\catcode `\t = \other
	\gdef \n@dimen #1pt{#1} %%% throw away the ``pt''
}

\def \Divide #1by #2{\divide #1 by #2} %%% just a synonym

\def \Multiply #1by #2%%% allows division of a dimen by a dimen
       {{%%% should really freeze parameter 2 (dimen, passed by value)
	\count 0 = #1\relax
	\count 2 = #2\relax
	\count 4 = 65536
	\Mess@ge {Before scaling, count 0 = \the \count 0 \space and
			count 2 = \the \count 2}%
	\ifnum	\count 0 > 32767 %%% do our best to avoid overflow
	\then	\divide \count 0 by 4
		\divide \count 4 by 4
	\else	\ifnum	\count 0 < -32767
		\then	\divide \count 0 by 4
			\divide \count 4 by 4
		\else
		\fi
	\fi
	\ifnum	\count 2 > 32767 %%% while retaining reasonable accuracy
	\then	\divide \count 2 by 4
		\divide \count 4 by 4
	\else	\ifnum	\count 2 < -32767
		\then	\divide \count 2 by 4
			\divide \count 4 by 4
		\else
		\fi
	\fi
	\multiply \count 0 by \count 2
	\divide \count 0 by \count 4
	\xdef \product {#1 = \the \count 0 \internal@nits}%
	\aftergroup \product
       }}

\def\r@duce{\ifdim\dimen0 > 90\r@dian \then   % sin(x+90) = sin(180-x)
		\multiply\dimen0 by -1
		\advance\dimen0 by 180\r@dian
		\r@duce
	    \else \ifdim\dimen0 < -90\r@dian \then  % sin(-x) = sin(360+x)
		\advance\dimen0 by 360\r@dian
		\r@duce
		\fi
	    \fi}

\def\Sine#1%
       {{%
	\dimen 0 = #1 \r@dian
	\r@duce
	\ifdim\dimen0 = -90\r@dian \then
	   \dimen4 = -1\r@dian
	   \c@mputefalse
	\fi
	\ifdim\dimen0 = 90\r@dian \then
	   \dimen4 = 1\r@dian
	   \c@mputefalse
	\fi
	\ifdim\dimen0 = 0\r@dian \then
	   \dimen4 = 0\r@dian
	   \c@mputefalse
	\fi
%
	\ifc@mpute \then
        	% convert degrees to radians
		\divide\dimen0 by 180
		\dimen0=3.141592654\dimen0
%
		\dimen 2 = 3.1415926535897963\r@dian %%% a well-known constant
		\divide\dimen 2 by 2 %%% we only deal with -pi/2 : pi/2
		\Mess@ge {Sin: calculating Sin of \nodimen 0}%
		\count 0 = 1 %%% see power-series expansion for sine
		\dimen 2 = 1 \r@dian %%% ditto
		\dimen 4 = 0 \r@dian %%% ditto
		\loop
			\ifnum	\dimen 2 = 0 %%% then we've done
			\then	\stillc@nvergingfalse 
			\else	\stillc@nvergingtrue
			\fi
			\ifstillc@nverging %%% then calculate next term
			\then	\term {\count 0} {\dimen 0} {\dimen 2}%
				\advance \count 0 by 2
				\count 2 = \count 0
				\divide \count 2 by 2
				\ifodd	\count 2 %%% signs alternate
				\then	\advance \dimen 4 by \dimen 2
				\else	\advance \dimen 4 by -\dimen 2
				\fi
		\repeat
	\fi		
			\xdef \sine {\nodimen 4}%
       }}

% Now the Cosine can be calculated easily by calling \Sine
\def\Cosine#1{\ifx\sine\UnDefined\edef\Savesine{\relax}\else
		             \edef\Savesine{\sine}\fi
	{\dimen0=#1\r@dian\advance\dimen0 by 90\r@dian
	 \Sine{\nodimen 0}
	 \xdef\cosine{\sine}
	 \xdef\sine{\Savesine}}}	      
% end of trig stuff
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\psdraft{
	\def\@psdraft{0}
	%\ps@typeout{draft level now is \@psdraft \space . }
}
\def\psfull{
	\def\@psdraft{100}
	%\ps@typeout{draft level now is \@psdraft \space . }
}

\psfull

\newif\if@scalefirst
\def\psscalefirst{\@scalefirsttrue}
\def\psrotatefirst{\@scalefirstfalse}
\psrotatefirst

\newif\if@draftbox
\def\psnodraftbox{
	\@draftboxfalse
}
\def\psdraftbox{
	\@draftboxtrue
}
\@draftboxtrue

\newif\if@prologfile
\newif\if@postlogfile
\def\pssilent{
	\@noisyfalse
}
\def\psnoisy{
	\@noisytrue
}
\psnoisy
%%% These are for the option list.
%%% A specification of the form a = b maps to calling \@p@@sa{b}
\newif\if@bbllx
\newif\if@bblly
\newif\if@bburx
\newif\if@bbury
\newif\if@height
\newif\if@width
\newif\if@rheight
\newif\if@rwidth
\newif\if@angle
\newif\if@clip
\newif\if@verbose
\def\@p@@sclip#1{\@cliptrue}


\newif\if@decmpr

%%% GDH 7/26/87 -- changed so that it first looks in the local directory,
%%% then in a specified global directory for the ps file.
%%% RPR 6/25/91 -- changed so that it defaults to user-supplied name if
%%% boundingbox info is specified, assuming graphic will be created by
%%% print time.
%%% TJD 10/19/91 -- added bbfile vs. file distinction, and @decmpr flag

\def\@p@@sfigure#1{\def\@p@sfile{null}\def\@p@sbbfile{null}
	        \openin1=#1.bb
		\ifeof1\closein1
	        	\openin1=\figurepath#1.bb
			\ifeof1\closein1
			        \openin1=#1
				\ifeof1\closein1%
				       \openin1=\figurepath#1
					\ifeof1
					   \ps@typeout{Error, File #1 not found}
						\if@bbllx\if@bblly
				   		\if@bburx\if@bbury
			      				\def\@p@sfile{#1}%
			      				\def\@p@sbbfile{#1}%
							\@decmprfalse
				  	   	\fi\fi\fi\fi
					\else\closein1
				    		\def\@p@sfile{\figurepath#1}%
				    		\def\@p@sbbfile{\figurepath#1}%
						\@decmprfalse
	                       		\fi%
			 	\else\closein1%
					\def\@p@sfile{#1}
					\def\@p@sbbfile{#1}
					\@decmprfalse
			 	\fi
			\else
				\def\@p@sfile{\figurepath#1}
				\def\@p@sbbfile{\figurepath#1.bb}
				\@decmprtrue
			\fi
		\else
			\def\@p@sfile{#1}
			\def\@p@sbbfile{#1.bb}
			\@decmprtrue
		\fi}

\def\@p@@sfile#1{\@p@@sfigure{#1}}

\def\@p@@sbbllx#1{
		%\ps@typeout{bbllx is #1}
		\@bbllxtrue
		\dimen100=#1
		\edef\@p@sbbllx{\number\dimen100}
}
\def\@p@@sbblly#1{
		%\ps@typeout{bblly is #1}
		\@bbllytrue
		\dimen100=#1
		\edef\@p@sbblly{\number\dimen100}
}
\def\@p@@sbburx#1{
		%\ps@typeout{bburx is #1}
		\@bburxtrue
		\dimen100=#1
		\edef\@p@sbburx{\number\dimen100}
}
\def\@p@@sbbury#1{
		%\ps@typeout{bbury is #1}
		\@bburytrue
		\dimen100=#1
		\edef\@p@sbbury{\number\dimen100}
}
\def\@p@@sheight#1{
		\@heighttrue
		\dimen100=#1
   		\edef\@p@sheight{\number\dimen100}
		%\ps@typeout{Height is \@p@sheight}
}
\def\@p@@swidth#1{
		%\ps@typeout{Width is #1}
		\@widthtrue
		\dimen100=#1
		\edef\@p@swidth{\number\dimen100}
}
\def\@p@@srheight#1{
		%\ps@typeout{Reserved height is #1}
		\@rheighttrue
		\dimen100=#1
		\edef\@p@srheight{\number\dimen100}
}
\def\@p@@srwidth#1{
		%\ps@typeout{Reserved width is #1}
		\@rwidthtrue
		\dimen100=#1
		\edef\@p@srwidth{\number\dimen100}
}
\def\@p@@sangle#1{
		%\ps@typeout{Rotation is #1}
		\@angletrue
%		\dimen100=#1
		\edef\@p@sangle{#1} %\number\dimen100}
}
\def\@p@@ssilent#1{ 
		\@verbosefalse
}
\def\@p@@sprolog#1{\@prologfiletrue\def\@prologfileval{#1}}
\def\@p@@spostlog#1{\@postlogfiletrue\def\@postlogfileval{#1}}
\def\@cs@name#1{\csname #1\endcsname}
\def\@setparms#1=#2,{\@cs@name{@p@@s#1}{#2}}
%
% initialize the defaults (size the size of the figure)
%
\def\ps@init@parms{
		\@bbllxfalse \@bbllyfalse
		\@bburxfalse \@bburyfalse
		\@heightfalse \@widthfalse
		\@rheightfalse \@rwidthfalse
		\def\@p@sbbllx{}\def\@p@sbblly{}
		\def\@p@sbburx{}\def\@p@sbbury{}
		\def\@p@sheight{}\def\@p@swidth{}
		\def\@p@srheight{}\def\@p@srwidth{}
		\def\@p@sangle{0}
		\def\@p@sfile{} \def\@p@sbbfile{}
		\def\@p@scost{10}
		\def\@sc{}
		\@prologfilefalse
		\@postlogfilefalse
		\@clipfalse
		\if@noisy
			\@verbosetrue
		\else
			\@verbosefalse
		\fi
}
%
% Go through the options setting things up.
%
\def\parse@ps@parms#1{
	 	\@psdo\@psfiga:=#1\do
		   {\expandafter\@setparms\@psfiga,}}
%
% Compute bb height and width
%
\newif\ifno@bb
\def\bb@missing{
	\if@verbose{
		\ps@typeout{psfig: searching \@p@sbbfile \space  for bounding box}
	}\fi
	\no@bbtrue
	\epsf@getbb{\@p@sbbfile}
        \ifno@bb \else \bb@cull\epsf@llx\epsf@lly\epsf@urx\epsf@ury\fi
}	
\def\bb@cull#1#2#3#4{
	\dimen100=#1 bp\edef\@p@sbbllx{\number\dimen100}
	\dimen100=#2 bp\edef\@p@sbblly{\number\dimen100}
	\dimen100=#3 bp\edef\@p@sbburx{\number\dimen100}
	\dimen100=#4 bp\edef\@p@sbbury{\number\dimen100}
	\no@bbfalse
}
% rotate point (#1,#2) about (0,0).
% The sine and cosine of the angle are already stored in \sine and
% \cosine.  The result is placed in (\p@intvaluex, \p@intvaluey).
\newdimen\p@intvaluex
\newdimen\p@intvaluey
\def\rotate@#1#2{{\dimen0=#1 sp\dimen1=#2 sp
%            	calculate x' = x \cos\theta - y \sin\theta
		  \global\p@intvaluex=\cosine\dimen0
		  \dimen3=\sine\dimen1
		  \global\advance\p@intvaluex by -\dimen3
% 		calculate y' = x \sin\theta + y \cos\theta
		  \global\p@intvaluey=\sine\dimen0
		  \dimen3=\cosine\dimen1
		  \global\advance\p@intvaluey by \dimen3
		  }}
\def\compute@bb{
		\no@bbfalse
		\if@bbllx \else \no@bbtrue \fi
		\if@bblly \else \no@bbtrue \fi
		\if@bburx \else \no@bbtrue \fi
		\if@bbury \else \no@bbtrue \fi
		\ifno@bb \bb@missing \fi
		\ifno@bb \ps@typeout{FATAL ERROR: no bb supplied or found}
			\no-bb-error
		\fi
		%
%\ps@typeout{BB: \@p@sbbllx, \@p@sbblly, \@p@sbburx, \@p@sbbury} 
%
% store height/width of original (unrotated) bounding box
		\count203=\@p@sbburx
		\count204=\@p@sbbury
		\advance\count203 by -\@p@sbbllx
		\advance\count204 by -\@p@sbblly
		\edef\ps@bbw{\number\count203}
		\edef\ps@bbh{\number\count204}
		%\ps@typeout{ psbbh = \ps@bbh, psbbw = \ps@bbw }
		\if@angle 
			\Sine{\@p@sangle}\Cosine{\@p@sangle}
	        	{\dimen100=\maxdimen\xdef\r@p@sbbllx{\number\dimen100}
					    \xdef\r@p@sbblly{\number\dimen100}
			                    \xdef\r@p@sbburx{-\number\dimen100}
					    \xdef\r@p@sbbury{-\number\dimen100}}
%
% Need to rotate all four points and take the X-Y extremes of the new
% points as the new bounding box.
                        \def\minmaxtest{
			   \ifnum\number\p@intvaluex<\r@p@sbbllx
			      \xdef\r@p@sbbllx{\number\p@intvaluex}\fi
			   \ifnum\number\p@intvaluex>\r@p@sbburx
			      \xdef\r@p@sbburx{\number\p@intvaluex}\fi
			   \ifnum\number\p@intvaluey<\r@p@sbblly
			      \xdef\r@p@sbblly{\number\p@intvaluey}\fi
			   \ifnum\number\p@intvaluey>\r@p@sbbury
			      \xdef\r@p@sbbury{\number\p@intvaluey}\fi
			   }
%			lower left
			\rotate@{\@p@sbbllx}{\@p@sbblly}
			\minmaxtest
%			upper left
			\rotate@{\@p@sbbllx}{\@p@sbbury}
			\minmaxtest
%			lower right
			\rotate@{\@p@sbburx}{\@p@sbblly}
			\minmaxtest
%			upper right
			\rotate@{\@p@sbburx}{\@p@sbbury}
			\minmaxtest
			\edef\@p@sbbllx{\r@p@sbbllx}\edef\@p@sbblly{\r@p@sbblly}
			\edef\@p@sbburx{\r@p@sbburx}\edef\@p@sbbury{\r@p@sbbury}
%\ps@typeout{rotated BB: \r@p@sbbllx, \r@p@sbblly, \r@p@sbburx, \r@p@sbbury}
		\fi
		\count203=\@p@sbburx
		\count204=\@p@sbbury
		\advance\count203 by -\@p@sbbllx
		\advance\count204 by -\@p@sbblly
		\edef\@bbw{\number\count203}
		\edef\@bbh{\number\count204}
		%\ps@typeout{ bbh = \@bbh, bbw = \@bbw }
}
%
% \in@hundreds performs #1 * (#2 / #3) correct to the hundreds,
%	then leaves the result in @result
%
\def\in@hundreds#1#2#3{\count240=#2 \count241=#3
		     \count100=\count240	% 100 is first digit #2/#3
		     \divide\count100 by \count241
		     \count101=\count100
		     \multiply\count101 by \count241
		     \advance\count240 by -\count101
		     \multiply\count240 by 10
		     \count101=\count240	%101 is second digit of #2/#3
		     \divide\count101 by \count241
		     \count102=\count101
		     \multiply\count102 by \count241
		     \advance\count240 by -\count102
		     \multiply\count240 by 10
		     \count102=\count240	% 102 is the third digit
		     \divide\count102 by \count241
		     \count200=#1\count205=0
		     \count201=\count200
			\multiply\count201 by \count100
		 	\advance\count205 by \count201
		     \count201=\count200
			\divide\count201 by 10
			\multiply\count201 by \count101
			\advance\count205 by \count201
			%
		     \count201=\count200
			\divide\count201 by 100
			\multiply\count201 by \count102
			\advance\count205 by \count201
			%
		     \edef\@result{\number\count205}
}
\def\compute@wfromh{
		% computing : width = height * (bbw / bbh)
		\in@hundreds{\@p@sheight}{\@bbw}{\@bbh}
		%\ps@typeout{ \@p@sheight * \@bbw / \@bbh, = \@result }
		\edef\@p@swidth{\@result}
		%\ps@typeout{w from h: width is \@p@swidth}
}
\def\compute@hfromw{
		% computing : height = width * (bbh / bbw)
	        \in@hundreds{\@p@swidth}{\@bbh}{\@bbw}
		%\ps@typeout{ \@p@swidth * \@bbh / \@bbw = \@result }
		\edef\@p@sheight{\@result}
		%\ps@typeout{h from w : height is \@p@sheight}
}
\def\compute@handw{
		\if@height 
			\if@width
			\else
				\compute@wfromh
			\fi
		\else 
			\if@width
				\compute@hfromw
			\else
				\edef\@p@sheight{\@bbh}
				\edef\@p@swidth{\@bbw}
			\fi
		\fi
}
\def\compute@resv{
		\if@rheight \else \edef\@p@srheight{\@p@sheight} \fi
		\if@rwidth \else \edef\@p@srwidth{\@p@swidth} \fi
		%\ps@typeout{rheight = \@p@srheight, rwidth = \@p@srwidth}
}
%		
% Compute any missing values
\def\compute@sizes{
	\compute@bb
	\if@scalefirst\if@angle
% at this point the bounding box has been adjsuted correctly for
% rotation.  PSFIG does all of its scaling using \@bbh and \@bbw.  If
% a width= or height= was specified along with \psscalefirst, then the
% width=/height= value needs to be adjusted to match the new (rotated)
% bounding box size (specifed in \@bbw and \@bbh).
%    \ps@bbw       width=
%    -------  =  ---------- 
%    \@bbw       new width=
% so `new width=' = (width= * \@bbw) / \ps@bbw; where \ps@bbw is the
% width of the original (unrotated) bounding box.
	\if@width
	   \in@hundreds{\@p@swidth}{\@bbw}{\ps@bbw}
	   \edef\@p@swidth{\@result}
	\fi
	\if@height
	   \in@hundreds{\@p@sheight}{\@bbh}{\ps@bbh}
	   \edef\@p@sheight{\@result}
	\fi
	\fi\fi
	\compute@handw
	\compute@resv}

%
% \psfig
% usage : \psfig{file=, height=, width=, bbllx=, bblly=, bburx=, bbury=,
%			rheight=, rwidth=, clip=}
%
% "clip=" is a switch and takes no value, but the `=' must be present.
\def\psfig#1{\vbox {
	% do a zero width hard space so that a single
	% \psfig in a centering enviornment will behave nicely
	%{\setbox0=\hbox{\ }\ \hskip-\wd0}
	%
	\ps@init@parms
	\parse@ps@parms{#1}
	\compute@sizes
	%
	\ifnum\@p@scost<\@psdraft{
		%
		\special{ps::[begin] 	\@p@swidth \space \@p@sheight \space
				\@p@sbbllx \space \@p@sbblly \space
				\@p@sbburx \space \@p@sbbury \space
				startTexFig \space }
		\if@angle
			\special {ps:: \@p@sangle \space rotate \space} 
		\fi
		\if@clip{
			\if@verbose{
				\ps@typeout{(clip)}
			}\fi
			\special{ps:: doclip \space }
		}\fi
		\if@prologfile
		    \special{ps: plotfile \@prologfileval \space } \fi
		\if@decmpr{
			\if@verbose{
				\ps@typeout{psfig: including \@p@sfile.Z \space }
			}\fi
			\special{ps: plotfile "`zcat \@p@sfile.Z" \space }
		}\else{
			\if@verbose{
				\ps@typeout{psfig: including \@p@sfile \space }
			}\fi
			\special{ps: plotfile \@p@sfile \space }
		}\fi
		\if@postlogfile
		    \special{ps: plotfile \@postlogfileval \space } \fi
		\special{ps::[end] endTexFig \space }
		% Create the vbox to reserve the space for the figure.
		\vbox to \@p@srheight sp{
		% 1/92 TJD Changed from "true sp" to "sp" for magnification.
			\hbox to \@p@srwidth sp{
				\hss
			}
		\vss
		}
	}\else{
		% draft figure, just reserve the space and print the
		% path name.
		\if@draftbox{		
			% Verbose draft: print file name in box
			\hbox{\frame{\vbox to \@p@srheight sp{
			\vss
			\hbox to \@p@srwidth sp{ \hss \@p@sfile \hss }
			\vss
			}}}
		}\else{
			% Non-verbose draft
			\vbox to \@p@srheight sp{
			\vss
			\hbox to \@p@srwidth sp{\hss}
			\vss
			}
		}\fi	



	}\fi
}}
\psfigRestoreAt
\let\@=\LaTeXAtSign





hep-ph0201014.extracted_bib
\begin{thebibliography}{99}
%\bibitem{KTP} E.W. Kolb and M.S. Turner, {\it The Early Universe}, Addison, 
%Wesley, Redwood City, CA ,1990.\\
%P.J.E. Peebles, {\it Principles of Physical Cosmology}, Princeton 
%University Press, 1993.
%M.S. Turner, {\it Cosmology:going beyond the big bang}, {\it Physics World},
% September 1996, p. 31.
\bibitem{Jungm}For a review see e.g.
G. Jungman {\it et al.},{\it  Phys. Rep.} {\bf 267}, 195  (1996).
\bibitem{COBE}G.F. Smoot et al., (COBE data), {\it Astrophys. J.} {\bf 396} 
(1992) L1.
\bibitem{GAW}E. Gawiser and J. Silk,{\it {Science}} {\bf 280}, 1405 (1988);
M.A.K. Gross, R.S. Somerville, J.R. Primack, J. Holtzman and A.A. Klypin,
{\it Mon. Not. R. Astron. Soc.} {\bf 301}, 81 (1998).
\bibitem{HSST}A.G. Riess {\it et al}, {\it Astron. J.}
{\bf 116} (1998), 1009.
\bibitem{SPF} R.S. Somerville, J.R. Primack and S.M. Faber, astro-ph/9806228;
{\it Mon. Not. R. Astron. Soc.} (in press).
\bibitem{SCP}Perlmutter, S. {\it et al} (1999) {\it Astrophys. J.}
{\bf 517},565; (1997) {\bf 483},565 ({\it astro-ph}/9812133).\\
S. Perlmutter, M.S. Turner and M. White, {\it Phys. Rev. Let.} {\bf 83},
670 (1999).
%C. Krauss, The cosmological constant and dark matter,2nd Int.
%Workshop on the Identification of Dark Matter, Buxton, UK, 7-11 September
%1998 (to appear in the proceedings).
\bibitem{Turner} M.S. Turner, astro-ph/9904051;  Phys. Rep. {\bf 333-334}
 (1990), 619.
\bibitem{Benne} D.P. Bennett {\it et al.}, (MACHO collaboration), A binary
lensing event toward the LMC: Observations and Dark Matter Implications, 
Proc. 5th Annual Maryland Conference, edited by S. Holt (1995);\\
C. Alcock {\it et al.}, (MACHO collaboration), {\it Phys. Rev. Lett.} {\bf 74}
, 2967 (1995). 
\bibitem{BERNA2} R. Bernabei et al., INFN/AE-98/34, (1998);
R. Bernabei et al., {it Phys. Lett.} {\bf B 389}, 757 (1996).
\bibitem{BERNA1} R. Bernabei et al., {\it Phys. Lett.} {\bf B 424}, 195 (1998);
{\bf B 450}, 448 (1999).
\bibitem{ref1}For more references see e.g. our previous report:\\
J.D. Vergados, Supersymmetric Dark Matter Detection-
The Directional Rate and the Modulation Effect, hep-ph/0010151;
\bibitem{Gomez} M.E. G\'{o}mez, J.D. Vergados, hep-ph/0012020.\\
M.E. G\'{o}mez, G. Lazarides and C. Pallis, \pr{61}{2000}{123512} and
 {\it Phys. Lett.} {\bf B 487}, 313 (2000).
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