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Fig. 6. Regions of the (m₀, M_1/2) parameter space in minimal supergravity with A₀ = 0, tanβ=10, and μ > 0. The lower shaded region is excluded by the LEP chargino mass limit. The stau is the LSP in the narrow upper shaded region. In the rest of parameter space, the LSP is the lightest neutralino, and contours of its gaugino-ness R_χ (in percent) are shown. From [12].

There are, of course, many other models besides minimal supergravity. Phenomena that do not occur in minimal supergravity may very well occur or even be generic in other supersymmetric frameworks. On the other hand, if one looks hard enough, minimal supergravity contains a wide variety of dark matter possibilities, and it will serve as a useful framework for illustrating many points below.

2.7. Summary

• Supersymmetry is a new spacetime symmetry that predicts the existence of a new boson for every known fermion, and a new fermion for every known boson.

• The gauge hierarchy problem may be solved by supersymmetry, but requires that all superpartners have masses at the weak scale.

• The introduction of superpartners at the weak scale mediates proton decay at unacceptably large rates unless some symmetry is imposed. An elegant solution, R-parity conservation, implies that the LSP is stable. Electrically neutral superpartners, such as the neutralino and gravitino, are therefore promising dark matter candidates.

• The superpartner masses depend on how supersymmetry is broken. In models with high-scale supersymmetry breaking, such as supergravity, the gravitino may or may not be the LSP; in models with low-scale supersymmetry breaking, the gravitino is the LSP.

• Among standard model superpartners, the lightest neutralino naturally emerges as the dark matter candidate from the simple high energy framework of minimal supergravity.

• Supersymmetry reduces fine tuning in the cosmological constant from 1 part in 10¹²⁰ to 1 part in 10⁶⁰ to 10⁹⁰, and so does not provide much insight into the problem of dark energy.

3. Neutralino cosmology

Given the motivations described in Section 2 for stable neutralino LSPs, it is natural to consider the possibility that neutralinos are the dark matter [13], [14] and [15]. In this section, we review the general formalism for calculating thermal relic densities and its implications for neutralinos and supersymmetry. We then describe a few of the more promising methods for detecting neutralino dark matter.

3.1. Freeze out and WIMPs

Dark matter may be produced in a simple and predictive manner as a thermal relic of the Big Bang. The very early universe is a very simple place—all particles are in thermal equilibrium. As the universe cools and expands, however, interaction rates become too low to maintain this equilibrium, and so particles “freeze out.” Unstable particles that freeze out disappear from the universe. However, the number of stable particles asymptotically approaches a non-vanishing constant, and this, their thermal relic density, survives to the present day.

This process is described quantitatively by the Boltzmann equation

where n is the number density of the dark matter particle χ, H is the Hubble parameter, σ_Av is the thermally averaged annihilation cross-section, and n_eq is the χ number density in thermal equilibrium. On the right-hand side of Eq. (18), the first term accounts for dilution from expansion. The n² term arises from processes that destroy χ particles, and the term arises from the reverse process , which creates χ particles.

It is convenient to change variables from time to temperature,

where m is the χ mass, and to replace the number density by the co-moving number density

where s is the entropy density. The expansion of the universe has no effect on Y, because s scales inversely with the volume of the universe when entropy is conserved. In terms of these new variables, the Boltzmann equation is

In this form, it is clear that before freeze out, when the annihilation rate is large compared with the expansion rate, Y tracks its equilibrium value Y_eq. After freeze out, Y approaches a constant. This constant is determined by the annihilation cross-section σ_Av . The larger this cross-section, the longer Y follows its exponentially decreasing equilibrium value, and the lower the thermal relic density. This behavior is shown in Fig. 7.

Fig. 7. The co-moving number density Y of a dark matter particle as a function of temperature and time. From [16].

Let us now consider WIMPs—weakly interacting massive particles with mass and annihilation cross-section set by the weak scale: . Freeze out takes place when

Neglecting numerical factors, n_eq   (mT)^3/2e^−m/T for a non-relativistic particle, and H   T²/M . From these relations, we find that WIMPs freeze out when

Since , WIMPs freeze out with velocity v   0.3.

One might think that, since the number density of a particle falls exponentially once the temperature drops below its mass, freeze out should occur at T   m. This is not the case. Because gravity is weak and M is large, the expansion rate is extremely slow, and freeze out occurs much later than one might naively expect. For a m   300 GeV particle, freeze out occurs not at T   300 GeV and time t   10⁻¹² s, but rather at temperature T   10 GeV and time t   10⁻⁸ s.

With a little more work [17], one can find not just the freeze out time, but also the freeze out density

A typical weak cross-section is

corresponding to a thermal relic density of Ωh²   0.1. WIMPs therefore naturally have thermal relic densities of the observed magnitude. The analysis above has ignored many numerical factors, and the thermal relic density may vary by as much as a few orders of magnitude. Nevertheless, in conjunction with the other strong motivations for new physics at the weak scale, this coincidence is an important hint that the problems of electroweak symmetry breaking and dark matter may be intimately related.

3.2. Thermal relic density

We now want to apply the general formalism above to the specific case of neutralinos. This is complicated by the fact that neutralinos may annihilate to many final states: , W⁺W⁻, ZZ, Zh, hh, and states including the heavy Higgs bosons H, A, and H^±. Many processes contribute to each of these final states, and nearly every supersymmetry parameter makes an appearance in at least one process. The full set of annihilation diagrams is discussed in [18]. Codes to calculate the relic density are publicly available [19].

Given this complicated picture, it is not surprising that there are a variety of ways to achieve the desired relic density for neutralino dark matter. What is surprising, however, is that many of these different ways may be found in minimal supergravity, provided one looks hard enough. We will therefore consider various regions of minimal supergravity parameter space where qualitatively distinct mechanisms lead to neutralino dark matter with the desired thermal relic density.

3.2.1. Bulk region

As evident from Fig. 6, the LSP is a Bino-like neutralino in much of minimal supergravity parameter space. It is useful, therefore, to begin by considering the pure Bino limit. In this case, all processes with final state gauge bosons vanish. This follows from supersymmetry and the absence of 3-gauge boson vertices involving the hypercharge gauge boson.

The process through a t-channel sfermion does not vanish in the Bino limit. This process is the first shown in Fig. 8. This reaction has an interesting structure. Recall that neutralinos are Majorana fermions. If the initial state neutralinos are in an S-wave state, the Pauli exclusion principle implies that the

New ways to solve the Schroedinger equation

R. Friedberg and T.D. Lee

^aDepartment of Physics, Columbia University, New York, NY 10027, USA
^bChina Center of Advanced Science and Technology (CCAST) (World Laboratory), P.O. Box 8730, Beijing 100080, People’s Republic of China
^cRIKEN BNL Research Center (RBRC), Brookhaven National Laboratory, Upton, NY 11973, USA

Received 27 July 2004;  accepted 5 August 2004.  Available online 15 December 2004.

Abstract

We discuss a new approach to solve the low lying states of the Schroedinger equation. For a fairly large class of problems, this new approach leads to convergent iterative solutions, in contrast to perturbative series expansions. These convergent solutions include the long standing difficult problem of a quartic potential with either symmetric or asymmetric minima.

Article Outline

1. Introduction

2. Construction of trial functions

2.1. A new formulation of perturbative expansion

2.2. Trial function for the quantum double-well potential

3. Hierarchy theorem and its generalization

4. Asymmetric quartic double-well problem

4.1. Construction of the first trial function