---
abstract: |
    We consider neutralino-proton cross sections for halo dark matter neutralinos ($\tilde\chi_1^0$) within the framework of supergravity models with R-parity invariance for models with universal soft breaking (mSUGRA) and models with nonuniversal soft breaking. The analysis includes the necessary corrections to treat the large tan$\beta$ region (i.e. L-R mixing in the squark and slepton mass matrices, loop corrections to the $b$ and $\tau$ masses,etc) and includes all coannihilation phenomena. For mSUGRA, dark matter detectors with current sensitivity are seen to be probing the region where tan$\beta \stackrel{>}{\sim}$25, $\Omega_{\tilde\chi_1^0} h^2 <0.1$, $m_{\tilde\chi_1^0}\stackrel{<}{\sim}90$ GeV, and for the light Higgs, $m_h \stackrel{<}{\sim}120$ GeV. Nonuniversal models can have a much larger cross section, and current detectors can probe part of the parameter space where tan$\beta\stackrel{>}{\sim}4$. Minimum cross sections are generally greater than $10^{-9}$ pb to $10^{-10}$ pb for $m_{1/2} <600$ GeV (and hence accessible to planned future detectors), with the exception of a region when $\mu <0$ where for $m_{1/2}\stackrel{>}{\sim}450$ GeV, $4\stackrel{<}{\sim} \tan\beta
    \stackrel{<}{\sim}20$, the cross section drops to a minimum of about $1\times10^{-12}$ pb at $m_{1/2}=600$ GeV, $\tan \beta \simeq10$. In this region, the gluino and squarks lie above 1 TeV, but should still be accessible to the LHC.
author:
- 'R. Arnowitt'
- 'B. Dutta'
- 'Y. Santoso'
title: DARK MATTER IN SUPERGRAVITY
---

Introduction
============

If the dark matter that exists in the Milky Way is a supersymmetric weakly interacting particle (wimp), there are several ways in which it might be detected. Annihilation of two wimps in the halo might give rise to signals of gamma rays, anti-protons or positrons. Dark matter particles caught by the gravitational fields of the Sun or Earth would be expected to sink to the center, and there annihilate leading to neutrinos that might be detected on the surface of the earth. Finally direct detection of incident wimps from their scattering by nuclear targets on the Earth is possible. Of these, the last possibility appears most promising, and there are now a large number of detectors searching for supersymmetric wimps. We consider here what signals might be available within the framework of supergravity(SUGRA) models with grand unification of the gauge coupling constants at the GUT scale $M_G\cong2\times10^{16}$GeV. There are three different types of SUGRA models currently being investigated, which differ by the mechanisms used to achieve supersymmetry(SUSY) breaking in the physical sector. These are gravity mediated SUGRA models, gauge mediated models, and anomaly mediated models. Of these, the gravity mediated models with R-parity invariance have the most robust candidate for particle dark matter, and we will restrict our discussion here to such models. In gravity mediated models, the dark matter particle is the lightest supersymmetric particle, the LSP, (absolutely stable due to the R-parity invariance), and this is generally the lightest neutralino, $\tilde\chi_1^0$. The nucleus-$\tilde\chi_1^0$ scattering cross section contains a spin independent part and a spin dependent part. For heavy nuclear targets, the spin independent scattering dominates, and it is possible to extract from data the $\tilde\chi_1^0$-proton cross section, $\sigma_{\tilde\chi_1^0-p}$. There are a number of astronomical uncertainties, but making conventional assumptions, current detectors (DAMA, CDMS, UKCDM) are sensitive to halo $\tilde\chi_1^0$ if $$\sigma_{\tilde\chi_1^0-p} \stackrel{>}{\sim} 1\times10^{-6} {\rm pb}$$ and future detectors (GENIUS, Cryoarray) plan to achieve sensitivities of $$\sigma_{\tilde\chi_1^0-p} \stackrel{>}{\sim} (10^{-9} -10^{-10}){\rm pb}$$

We discuss here how these sensitivities might relate to supergravity models. In particular, we consider the minimal supergravity GUT model (mSUGRA)\[1\] which has universal soft breaking masses at $M_G$, and nonuniversal soft breaking models\[2\] which allow nonuniversal Higgs masses and nonuniversal third generation squark and slepton masses at $M_G$ (but keep the gaugino masses universal at $M_G$). While the models are physically different, they lead to qualitatively similar results: Current detectors are sensitive to a significant part of the SUSY parameter space, and future detectors should be able to cover all of the parameter space, except for special regions where there is an accidental cancelation of terms making $\sigma_{\tilde\chi^0_1-p}$ anomalously small. Each of the above models contains a number of arbitrary new parameters. In spite of this they can still make relevant predictions for two main reasons: (i) Using the renormalization group equations (RGE) starting from $M_G$, they allow for radiative breaking of $SU(2)\times U(1)$ at the electroweak scale (and thus furnish a natural explanation for the Higgs mechanism); (ii) Along with being able to calculate $\sigma_{\tilde\chi^0_1-p}$, the models can also calculate the relic density of neutralinos, i.e. $\Omega_{\tilde\chi_1^0} h^2$, where $\Omega_{\tilde\chi_1^0}
= \rho_{\tilde\chi_1^0}/\rho_c$, $\rho_{\tilde\chi_1^0}$ is the relic mass density of the ${\chi_1^0}$, and $\rho_c=3H_0/8\pi G_N$. Here $H_0=h(100\mbox{km/s Mpc})$ is the Hubble constant, and $G_N$ is the Newton constant. Both of the above lead to important constraints on the SUSY parameter space. Thus one has that $$\Omega_{\tilde\chi^0_1}h^2\sim(\int^{x_f}_0 dx\langle\sigma_{\rm
ann}v\rangle)^{-1}$$ where $\sigma_{\rm ann}$ is the annihilation cross section in the early universe, v is the relative neutralino velocity at annihilation, and $<...>$ means thermal average. The dominant Feynman diagrams for $\sigma_{\rm ann}$ and spin independent $\sigma_{\tilde\chi^0_1-p}$ are shown in Fig.\[fig1\], and roughly speaking $\sigma_{\rm ann}$ depends on the crossed diagrams relative to $\sigma_{\tilde\chi^0_1-p}$. Thus usually, then, when $\sigma_{\tilde\chi^0_1-p}$ is large, $\sigma_{\rm ann}$ will also be large, and hence by Eq(3), $\Omega_{\tilde\chi_1^0}h^2$ will be small. Thus lower bounds on $\Omega_{\tilde\chi_1^0}h^2$ will produce upper bounds on $\sigma_{\tilde\chi^0_1-p}$. In the following, we will assume $h=0.70\pm0.07$ and for matter(m) and baryonic matter(b) the values $\Omega_m=0.3\pm0.1$, and $\Omega_b=0.04$. (This corresponds to a dark energy amount of $\Omega_\Lambda \sim0.65$). For the dark matter then one has $\Omega_{\tilde\chi_1^0}=0.26\pm 0.10$, and if one combines errors in quadrature, one finds $\Omega_{\tilde\chi_1^0} h^2 =0.13\pm0.05$. Since there is undoubtedly a large amount of systematic error in the above estimates, in the following we will assume the approximately 2$\sigma$ spread of $$0.02 < \Omega_{\tilde\chi_1^0} h^2 < 0.25$$ The lower bound of Eq.(4) lies somewhat below other estimates. However, it also allows for the possibility that not all the dark matter in the Galaxy are neutralinos (e.g. some may be machos). In addition to the above, there are accelerator bounds that constrain the SUSY parameter space. In the following we use the LEP bounds for the light Higgs ($h$) of $m_h > 104$ GeV for tan$\beta$ = 3, $m_h > 102$ GeV for tan$\beta=$5 and for the light chargino $ m_{\tilde\chi_1^\pm} > 102$ GeV. (For tan$\beta > $5, the Higgs mass bounds do not restrict the parameter space significantly.) The Tevatron gives the gluino ($\tilde g$) mass bound of $m_{\tilde g} > 270$ GeV (for gluino and squarks nearly degenerate). In addition there is the CLEO measurment of the $b\rightarrow s +
\gamma$ decay. We take here a 2$\sigma$ range around the experimental central value of the $b\rightarrow  X_s + \gamma$ branching ratio\[3\]: $$1.8 \times10^{-4} < B(B\rightarrow X_s \gamma) < 4.5 \times10^{-4}$$ Of the above, the most significant constraints come from the Higgs mass bounds and the $b \rightarrow s\gamma$ branching ratio.

(diag.eps width 10 cm)

 Theoretical Analysis
=====================

In order to get accurate results, it is necessary to include a number of corrections in the calculations. We list some of these here: (1) One needs to run the two loop gauge RGE and one loop Yukawa RGE from $M_G=2\times10^{16}$ GeV down to the electroweak scale, iterating to get a consistent SUSY mass spectrum. (2) Below the SUSY scale $M_S$, one runs the QCD RGE for contributions dominated by light quarks. (3) It turns out that results are somewhat sensitive to bounds on the Higgs mass $m_h$, and so one needs to use the one loop, two loop and pole mass corrections to accurately calculate the value of $m_h$. (4) L-R mixing in the sfermion mass matrices must be included. These are important for large tan$\beta$ and in the third generation. (5) One loop corrections to $m_b$ and $m_\tau$ are included. This is needed to get the correct value of the $b$ and $\tau$ Yukawa coupling constants, and again are important for large $\tan \beta$. (6) Leading order (LO)\[4\] and some next to leading order (NLO)\[5\] corrections to the $b \rightarrow s\gamma$ decay are included. All of the above are under good theoretical controll except perhaps for the $b \rightarrow s\gamma$ constraint[^1]. We note that we do not make any assumptions on the nature of the GUT group at grand unification. Hence we do not impose $b-\tau$ (or $ b-\tau-t$) Yukawa unification at $M_G$, and do not impose proton decay constraints. Such phenomena depend sensitively on unknown post-GUT physics, and so the validity of these constraints are unclear. For example, string models in which there is Wilson line breaking of the GUT group to the Standard Model group at $M_G$, require gauge coupling constant unification but neither Yukawa unifications implied by the GUT group nor the SUGRA proton decay constraints need hold\[8\].

SUSY theory allows one to calculate neutralino-quark scattering (Fig.\[fig1\]), and one must convert this to neutralino-proton scattering to compare with experiment. To do this we follow the proceedures of \[9\], which requires three parameters: (i) the pion-nucleon sigma term, $\sigma_{\pi N}= 1/2(m_u +
m_d)<p|u{\bar u} +d{\bar d}|p>$, (ii) $\sigma_0 = \sigma_{\pi N} - (m_u + m_d)<p|s
{\bar s}|p>$, and (iii) the quark mass ratio $r=2m_s/(m_u + m_d)$. We use here the values $\sigma_{\pi N} = 65$ MeV (based on analyses\[10\] making use of recent $\pi-N$ scattering data), $\sigma_0 = 30$ MeV \[11\], and $r=24.4\pm1.5$\[12\]. If one were to use instead the value $\sigma_{\pi N}=45$ MeV (based on older $\pi-N$ data) then the value of $\sigma_{\tilde\chi^0_1}-p$ would be reduced by a factor of about 3.

 mSUGRA Model
=============

Introduction
------------

The mSUGRA model has universal soft breaking and so depends on a minimum number of new parameters i.e. four parameters and one sign. These are (1) $m_0$, the universal scalar particle mass at $M_G$. (2) $m_{1/2}$, the universal gaugino mass at $M_G$. (Alternately, one may use $m_{\tilde\chi^0_1} $ or $m_{\tilde g}$ since these scale approximately with $m_{1/2}$, i.e. $m_{\tilde\chi^0_1} \simeq 0.4m_{1/2}$, and m$_{\tilde g} \simeq 2.8 m_{1/2})$. (3) $A_0$, the universal cubic soft breaking mass at $M_G$. (4) $\tan \beta =
<H_2>/<H_1>$, where $<H_2>$ gives mass to the $u$-quarks and $<H_1>$ gives mass to the $d$-quarks and charged leptons. In addition, the sign of the Higgs mixing parmeter $\mu$ is undetermined. ($\mu$ appears in the superpotential $W$ as $\mu H_1H_2$.) We take for this parameter space the following ranges: $$m_0  \leq \,1 \mbox{ TeV}$$ $$m_{1/2} \leq 600 \mbox{ GeV}\,\, 
        \mbox{(which corresponds to } m_{\tilde g}\leq1.5 \mbox{ TeV},\,\,
         m_{\tilde\chi^0_1} \leq
240\, \mbox{ GeV)}$$ $$2 < \tan\beta < 50$$ $$|A_0/m_0| \leq5$$

If one increases the $m_{1/2}$ bound to $m_{1/2}=$ 1 TeV (corresponding to $m_{\tilde g}$ =2.5 TeV, which is the upper detection limit for gluinos at the LHC), the neutralino-proton cross section will drop by a factor of about 2-3 at the high end of the parameter space.

 Maximum Cross Section
----------------------

We examine first the maximum cross section the model can achieve. $\sigma_{\tilde\chi^0_1-p}$ is an increasing function of tan$\beta$, and a decreasing function of m$_{1/2}$ and $m_0$. Thus the maximum $\sigma_{\tilde\chi^0_1-p}$ should occur at large tan$\beta$ and small $m_{\tilde\chi^0_1}$. This is illustrated in Fig.\[fig2\] where the maximum cross section is plotted at a function of $m_{\tilde\chi^0_1}$ for tan$\beta$ = 20, 30, 40, 50, in the range of cross sections acessible to current detectors. One sees that current detectors with the sensitivity of Eq.(1), have begun to sample part of the parameter space for tan$\beta\stackrel{>}{\sim}$ 25. Further, from the

(aads20304050.epsf width 10 cm)

maximum tan$\beta=$ 50 curve, one sees that only neutralinos with mass $m_{\tilde\chi^0_1}\leq$ 90 GeV are accessible to such detectors. Fig.\[fig3\] shows $\Omega_{\tilde\chi^0_1-p}h^2$ as a function of $m_{\tilde\chi^0_1}$ for $\tan\beta =30$, when $\sigma_{\tilde\chi^0_1-p}$ takes on its maximum value of Fig.\[fig2\]. One sees that $\Omega_{\tilde\chi^0_1}h^2$ is an increasing function

(aads30om.epsf width 10 cm)

of $m_{\tilde\chi^0_1}$ as expected from the discussion in Sec.1, i.e. since $\sigma_{\tilde\chi^0_1-p}$ decreases with $m_{\tilde\chi^0_1}$, one expects that the early universe annihilation cross will similarly decrease, and hence $\Omega_{\tilde\chi^0_1}h^2$ will increase by Eq.(3). Since $m_{\tilde\chi^0_1}\leq$ 90 GeV for current detector sensitivities, we see that current detectors are accessing only the region where $\Omega_{\tilde\chi^0_1}h^2\leq 0.1$. It is clear that the future very accurate determinations of $\Omega_{\rm CDM}$ by the MAP and Planck satellites will greatly sharpen the predictions of the SUGRA models. Fig.\[fig4\] shows the light Higgs mass for tan$\beta=$30 when $\sigma_{\tilde\chi^0_1-p}$ takes on its maximum value. For $m_{\tilde\chi^0_1}< $90GeV (the range accesible by current detectors) one has

(aads30higgs.epsf width 10 cm)

$m_h\leq$120 GeV. Such a range of Higgs mass would be accessible to RUN2 at the Tevatron, if the run achieves maximum luminosity.

 Minimum Cross Sections
-----------------------

We turn next to consider how small the $\sigma_{\tilde\chi^0_1-p}$ cross sections can get to see how sensitive future detectors must be to cover the full parameter space. It is convenient to divide the discussion into the region below coannihilation effects and the region where coannihilation can take place.

### Below coannihilation ($m_{\tilde\chi^0_1}\leq150$ GeV)

In this region there is no coannihilation, and the smallest cross sections occur at the smallest values of tan$\beta$. Fig.\[fig5\] shows the minimium value of the $\sigma_{\tilde\chi^0_1-p}$ cross section as a function of $m_{\tilde\chi^0_1}$ for tan$\beta$=3. One sees that the cross section

(aadslowlimit.epsf width 10 cm)

decreases with increasing $m_{\tilde\chi^0_1}$ as expected, and in this domain we have $$\sigma_{\tilde\chi^0_1-p}\stackrel{>}{\sim} 4\times 10^{-9} {\rm
     pb}\,\, \mbox{ for }
\, m_{\tilde\chi^0_1} \leq 140 \mbox{ GeV}.$$which would be accessible to planned future experiments (such as GENIUS).

### Coannihilation region ($m_{\tilde\chi^0_1}\stackrel{>}{\sim}150$ GeV)

Coannihilation in the early universe occurs when a second SUSY particle becomes nearly degenerate with the neutralino LSP, and hence increase the annihilation cross section. This effect is significant in mSUGRA due to two “accidents": (1) The ${\tilde\chi^0_1}$ is a Majorana spinor and so its early universe annihilation cross section $\sigma_{\tilde\chi^0_1-\tilde\chi^0_1}$ is supressed relative to e.g. R-slepton ($\tilde l_R$) annihilations: $$\sigma_{\tilde\chi^0_1-\tilde\chi^0_1}\simeq (1/10) 
      (\sigma_{\tilde\chi^0_1-\tilde l_R},\, \sigma_{\tilde l_R\tilde l_R})$$

\(2) There is an accidental near degeneracy between $l_R$ and ${\tilde\chi^0_1}$ in a small region of parameter space. To see this, one may look at low tan$\beta$, where one has the analytic formulae for the selectron and neutralino masses of $$m_{e_R}^2 = m_0^2  + (6/5)(\alpha_G/4\pi)f_1m_{1/2}^2 -\sin^2\theta_W M_W^2
\cos 2\beta$$ $$m_{\tilde\chi^0_1}\simeq (\alpha_1/\alpha_G)m_{1/2}$$ where $f_1=(1/\beta_1)[1-(1/(1+\beta_1 t)]$ and $\beta_1$ is the $U(1)$ beta function and $t=2\mbox{ln}[M_G/M_Z]$. Numerically, Eqs.(13) and (14) give $$m_{eR}^2 \simeq m_0^2 \,+\, 0.15 m_{1/2}^2\, +\, (40\, \rm GeV)^2$$

$$m_{\tilde\chi^0_1}^2\simeq 0.16 m_{1/2}^2$$

One sees that for small $m_0$, the $\tilde e_R$ can become degenerate with the $\tilde\chi^0_1$, and as $m_{1/2}$ increases, $m_0$ correspondingly increases to maintain the region of near degeneracy. Thus one has a corridor in the $m_0-m_{1/2}$ plane of increasing $m_0$ and $m_{1/2}$ where coannihilation effects can occur, extending the region where Eq(4) can be satisfied. The importance of this effect has been stressed in \[14\], where the analysis has been carried out for low and intermediate tan$\beta$, an example of

(toby.eps width 10 cm)

which is shown in Fig.\[fig6\]. One sees that the coannihilation effect begins at $m_{1/2}\stackrel{>}{\sim}400$ GeV (i.e. $m_{\tilde\chi^0_1}\stackrel{>}{\sim}150$GeV). For large tan$\beta$, the sitiuation is more complicated as L-R mixing in the slepton mass matrices becomes important, particularly for the third generation, and generally the light $\tilde\tau_1$ is the lightest slepton, considerably lighter than the other sleptons. We consider first the case where $\mu > 0$. (We use the Isajet sign convention for $\mu$). Fig.\[fig7\] shows the allowed region in the $m_0-m_{1/2}$ plane where Eq.(4) is satisfied for tan$\beta=$ 40. One sees that there is a significant $A_0$

(aadcoan40.epsf width 10 cm)

dependence, with larger $A_0$ allowing for larger $m_0$. In general the thickness of the corridor is $\delta m_0\simeq25$ GeV. There is no longer any non coannihilation region left, as the corridors terminate at $m_{1/2}$ above the non coannihilation domain (for large tan$\beta$). (The termination is due to the $m_h$ and $b\rightarrow s\gamma$ constraints.) The minimum value of $m_{1/2}$ decreases with increasing $A_0$, as does the thickness of the allowed corridor. Thus for very large $A_0$, the existence of these corridors eventually becomes a fine tuning. Since larger values of $A_0$ allow for larger values of $m_0$, one expects the $\sigma_{\tilde\chi^0_1-p}$ cross section to decrease with $A_0$. This is illustrated in Fig.\[fig8\] where $\sigma_{\tilde\chi^0_1-p}$ is given as a function of $m_{1/2}$ for $\tan
\beta = 40$ for two values of $A_0$. Thus one expects

(aadcoan40A.epsf width 10 cm)

the minimum detection cross section to occur at largest $A_0$ and smallest tan$\beta$. This is illustrated in Fig.\[fig9\]. where the cross section is plotted for $A_0=4m_{1/2}$, $\mu>$0, tan$\beta=40$ and tan$\beta=3$. Because the higher tan$\beta$ allows $m_0$ to become

(aadcoan403.epsf width 10 cm)

larger (compare with Fig.\[fig6\]) which also reduces the cross section, the tan$\beta$ dependence is mostly neutralized for large $m_{1/2}$. One has, however, the lower bound on the cross section of $$\sigma_{\tilde\chi^0_1-p}\stackrel{>}{\sim} 1\times10^{-9}\,{\rm pb},\,\,
       {\rm for}\,\,
        m_{1/2} <600\,\mbox{GeV }\, (m_{\tilde\chi^0_1}<240\,{\rm GeV}),\, \mu>0$$ which should still be accesible to the proposed future detectors.

We next turn to the case of $\mu <0$. As pointed out in \[15\], at low and intermediate tan$\beta$, an accidental cancellation can occur in part of the parameter space in the coannihilation region which can greatly reduce $\sigma_{\tilde\chi^0_1-p}$. We investigate here whether this cancellation continues to occur in the high tan$\beta$ region. What occurs is seen in Fig.\[fig10\] where $\sigma_{\tilde\chi^0_1-p}$ is shown for tan$\beta=$20, 5 and 10 (in descending order). One sees that

( aadcoan51020.epsf width 10 cm)

the cross section decreases between tan$\beta$ = 5 and tan$\beta$ = 10, but then rises again at higher tan$\beta$. Thus one has that $$\sigma_{\tilde\chi^0_1-p}< 10^{-10}\mbox{pb } \,{\rm for}
 \,\, 4\stackrel{<}{\sim}\tan\beta\stackrel{<}{\sim}20;\,\, m_{1/2}>450\,{\rm GeV}
  (m_{\tilde g}\stackrel{>}{\sim}\,1.1\mbox{ TeV});\, \mu>0$$ and the minimum cross section occurs at tan$\beta\cong10$: $$(\sigma_{\tilde\chi^0_1-p})_{\rm min}
      \simeq1\times10^{-12}\,{\rm pb}\,\, {\rm at}\,\,
       \tan\beta=10,\, m_{1/2}\simeq 600\, {\rm GeV}$$

Thus in this region of parameter space the proposed future detectors would not be able to detect mSUGRA ${\tilde\chi^0_1}$ wimps. However, the absence of detection of halo wimps would then imply that squarks and gluinos should lie above 1 TeV, but at masses still acessible to the LHC. Also then, mSUGRA would require that tan$\beta$ would be in the restricted range given in Eq(18), and $\mu$ be negative. This would allow a number of cross checks on the validity of the mSUGRA model.

Nonuniversal SUGRA Models
=========================

In most discussions of SUGRA models with nonuniversal soft breaking terms, the universality of the the soft breaking masses at $M_G$ of the first two generations of squarks and sleptons is maintained to suppress flavor changing neutral currents. However, one may allow both the Higgs masses and the third generation squark and slepton masses to become nonuniversal at $M_G$. One can parameterize this situation at $M_G$ as follows: $$\begin{aligned}
 m_{H_{1}}^{\
2}&=&m_{0}^{2}(1+\delta_{1}); 
\quad m_{H_{2}}^{\ 2}=m_{0}^{2}(1+ \delta_{2});\\\nonumber m_{q_{L}}^{\
2}&=&m_{0}^{2}(1+\delta_{3}); \quad m_{t_{R}}^{\ 2}=m_{0}^{2}(1+\delta_{4});
\quad m_{\tau_{R}}^{\ 2}=m_{0}^{2}(1+\delta_{5});   \\\nonumber m_{b_{R}}^{\
2}&=&m_{0}^{2}(1+\delta_{6}); \quad m_{l_{L}}^{\ 2}=m_{0}^{2}(1+\delta_{7}).
\label{eq18}\end{aligned}$$ where $q_{L}\equiv (\tilde t_L, \tilde b_L)$ squarks, $l_{L}\equiv (\tilde \nu_\tau, \tilde \tau_L)$ sleptons, etc. and $m_0$ is the universal mass for the first two generations of squarks and sleptons. The $\delta_i$ are the deviations from universality (and if one were to impose SU(5) or SO(10) symmetry one would have $\delta_3$=$\delta_4$= $\delta_5$, and $\delta_6$=$\delta_7$.) In the following we limit the $\delta_i$ to obey: $$-1 \leq \delta_i \leq +1$$ and maintain gauge coupling constant unification and gaugino mass unification at $M_G$.

While there are a large numbers of new parameters, one can get an understanding of what effect they produce from the following. The neutralino ${\tilde\chi^0_1}$ is a mixture of gaugino (mostly bino) and higgsino parts: $${\tilde\chi^0_1}= \alpha \tilde W_3 + \beta \tilde B + \gamma \tilde H_1 +
       \delta \tilde H_2$$ The dominant spin independent $\sigma_{\tilde\chi^0_1-p}$ cross section is proportional to the interference between the gaugino and higgino amplitudes, and this interference is largely governed by the size of $\mu^2$. As $\mu^2$ decreases, the interference increases, and hence $\sigma_{\tilde\chi^0_1-p}$ increases. Radiative breaking of $SU(2)\times U(1)$ determines the value of $\mu^2$ at the electroweak scale. To see the general nature of the effects of nonuniverality, we consider low and intermediate tan$\beta$ where an analytic form exists for $\mu^2$ (see e.g. Arnowitt and Nath, Ref\[2\]): $$\begin{aligned}
\mu^2&=&{t^2\over{t^2-1}}[({{1-3 D_0}\over 2}-{1\over
t^2})+({{1-D_0}\over2}(\delta_3+\delta_4)-
{{1+D_0}\over2}\delta_2+{\delta_1\over
t^2})]m_0^2\\\nonumber&+& {\rm {universal \, parts +loop \, corrections}}.\end{aligned}$$ Here $t=\tan\beta$ and $D_0\cong1-(m_t/200\,{\rm GeV}\, \sin\beta)^2 \leq 0.2$ (Note that the Higgs and squark nonuniversalties enter coherrently, roughly in the combination $\delta_3+\delta_4-\delta_2$.) We see from Eq.(22) that $\mu^2$ is reduced, and hence $\sigma_{\tilde\chi^0_1-p}$ incresed for $\delta_3$, $\delta_4$, $\delta_1<0$, $\delta_2>0$, and $\mu^2$ is increased for $\delta_3$, $\delta_4$, $\delta_1>0$, $\delta_2<0$. Thus one can get significantly larger cross sections in the nonuniversal models with the first choice of signs for the $\delta_i$, and one can reduce the cross sections (though not by such a large amount) with the second choice. The above analytic results are illustrated in Fig.\[fig11\], where the maximum $\sigma_{\tilde\chi^0_1-p}$ for the universal and nonuniversal models are plotted for tan$\beta=7$. One sees that

(aads7uni7nonuni.epsf width 10 cm)

one can increase the cross section by a factor of 10 to 100 by an appropriate choice of signs. Thus current detectors can probe regions of lower tan$\beta$ for nonuniversal models than for the universal one. The allowed range can be seen from Fig.\[fig12\], where the maximum cross sections are plotted for tan$\beta =$ 5,7,and 15.

(aads5715nonuni.epsf width 10 cm)

Current detectors with sensitivity of Eq.(1) thus can probe parts of the parameter space with tan$\beta \stackrel{>}{\sim} 4$, and from the tan$\beta=$ 15 curve, we see that parts of the high tan$\beta$ part of the parameter space has already been eliminated. However, the very low tan$\beta$ values are on the edge of being eliminated by the LEP constraint on the light Higgs mass. Thus Fig.\[fig13\] shows that $m_h$ is quite small if

(aads7higgs1.epsf width 10 cm)

$m_{\tilde\chi^0_1}$ is light, and one would have to raise the lower bound on tan$\beta$ as LEP raises the lower bound on $m_h$.

As in mSUGRA, the minimum cross sections occur for the largest $m_{1/2}$ and smallest tan$\beta$, and so they occur in the coannihilation region. We consider here only the case where the Higgs masses are nonuniversal i.e. $\delta_{1,2} \neq0$ (the other $\delta_i$ set to zero). Results then are similar to the mSUGRA case. For $\mu>0$ we find $$\sigma_{\tilde\chi^0_1-p}\stackrel{>}{\sim} 1\times 10^{-9} {\rm pb};\,\,{\rm
for}\,\,
 m_{1/2} \leq 600\, {\rm GeV},\,\mu>0$$ For $\mu<$0 one again can get a cancelation reducing the cross section to a minimum near tan$\beta=$ 10: $$\sigma_{\tilde\chi^0_1-p}\stackrel{>}{\sim} 1\times 10^{-12} {\rm pb} \,
    \,{\rm at}\,\,
 m_{1/2} =600\,{\rm GeV},\, \tan\beta\cong10,\, \mu<0$$

Conclusions
===========

We have examined here the predictions of several SUGRA models which possess gauge coupling constant unification at $M_G\cong2\times10^{16}$ GeV, to see what parts of the SUSY parameter space are accessible to current detectors obeying Eq.(1), and what will be accessible to future detectors with the sensitivity of Eq.(2). For the minimal SUGRA model with universal soft breaking parameters, mSUGRA, current detectors are scanning parts of the parameter space where tan$\beta\stackrel{>}{\sim} 25$, $m_{\tilde\chi^0_1}\stackrel{<}{\sim} 90$ GeV and $\Omega h^2\stackrel{<}{\sim}0.1$. In addition, the light Higgs obeys $m_h\stackrel{<}{\sim} 120$ GeV, and hence possibly would be accessible to RUN2 at the Tevatron. For nonuniversal models, where one allows the Higgs and third generation squark and slepton softbreaking masses to be nonuniversal, the neutralino-proton cross section can be significantly increased, by a factor of 10-100, with an appropriate choice of sign in the soft breaking deviations from universality. Thus current detectors here could scan regions of parameter space as low as tan$\beta\simeq4$, though in these regions $m_h$ is very light, and the the minimum allowed tan$\beta$ may have to be raised as LEP raises the the bound on the Higgs mass. However, the possibility of large cross section here has already allowed current detectors to exclude parts of the high tan$\beta$ region, e.g. when tan$\beta \stackrel{>}{\sim}$15. How low SUGRA cross sections can lie is complicated by the existance of coannihilation effects where the R-sleptons (particularly the $\tilde\tau_1$) can become nearly degenerate with the neutralino. This allows the relic density constraint Eq.(4) to be satisfied in a narrow rising corridor (about 25 GeV wide in $m_0$) in the $m_0-m_{1/2}$ plane rising to relatively large $m_0$, and thus reducing the size of the ${\tilde\chi^0_1}-p$ cross section. For large tan$\beta$, the effect is sensitive to the value of $A_0$, the range of $m_0$ increasing with $A_0$. Thus for $\mu >$0, one finds for $m_{1/2}=$600 GeV, the minimum cross section at e.g. tan$\beta=40$, $A_0=4 m_{1/2}$ is almost the same as that at tan$\beta=3$, the increase of the cross section due to the increase in tan$\beta$ being offset by the decrease due to the allowed large value of $m_0$. One finds however, that for $\mu>0$, at m$_{1/2}$ = 600 GeV ($m_{\tilde g}\stackrel{\sim}{=}1.5$ TeV), the minimum cross sections would still be accessible to detectors with the sensitivity of Eq.(2). The minimum cross sections for $\mu < 0$ is complicated by the possibility of accidental cancelations in the scattering amplitudes, allowing the cross section to sink below 10$^{-10}$ pb in certain regions of the parameter space. Thus one finds for mSUGRA that $\sigma_{\tilde\chi^0_1-p }< 1\times 10^{-10}$ pb for $m_{1/2} \stackrel{>}{\sim}450$ GeV ($m_{\tilde g}\stackrel{>}{\sim}1.1$ TeV) when $4 \stackrel{<}{\sim}\tan\beta \stackrel{<}{\sim}$ 20, with a minmum cross section of $1\times
10^{-12}$ pb reached at $\tan\beta\cong$10. Similar results hold for the nonuniversal models. In this domain SUGRA models imply that halo dark matter would not be accessible to detectors with sensitivities of Eq.(2), and that the gluino and squarks would be quite heavy. They would still however be observable at the LHC which can detect gluinos with mass $m_{\tilde g}\stackrel{<}{\sim}$ 2.5 TeV \[16\].

Acknowledgement
===============

This work was supported in part by National Science Foundation Grant No. PHY-9722090.

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[^1]: Recent analyses\[6\] appear to have calculated the most important NLO corrections to the branching ratio for $b \rightarrow s\gamma$ for large tan$\beta$. These corrections have not been treated here, but will be included in \[7\] (where the bounds on $m_h$ will also be updated). We do not believe this will effect the predictions of the maximum and minimum cross sections given below, but may modify which regions of parameter space get excluded.
