Although the existence of dark matter is well motivated by several lines of evidence, the exact nature of dark matter remains elusive. Dark matter candidates are generically referred to as WIMPs (Weakly Interacting Massive Particles); in other words, they are massive particles that are electrically neutral which do not interact very strongly with other matter. In this section we will explore some possible particle candidates for dark matter and the theories that lie behind them. But to begin with, we give a brief review of the Standard Model of particle physics.
A. The Standard Model and the Neutrino
The Standard Model (SM) is the quantum field theory that describes three
of the four fundamental forces in nature: electricity and magnetism, the
weak nuclear force, and the strong nuclear force. Gravitational
interactions are not part of the SM; at energies below the Planck scale
gravity is unimportant at the atomic level. There are sixteen confirmed
particles in the SM, seven of which were predicted by the model before
they were found experimentally, and one particle yet to be seen: the
Higgs boson, which is believed to be the mediator of the Higgs field
responsible for giving all other SM particles mass. In the SM, there are
six quarks (up, down, top, bottom, charm, and strange), six leptons
(electron, mu, tau, and their respective neutrinos), and five force
carriers (photons, gluons, W±, Z, and the
Higgs boson). Quarks and leptons are classified as fermions with half
integer spins and are split into three generations, where force carriers
are classified as gauge bosons with integer spins. Each of these
particles also has a corresponding antiparticle, denoted with a bar (for
example, the up antiquark's symbol is
), with opposite
charge. Table I arranges the SM fundamental
particles and some of their basic qualities.
| Generation 1 | Generation 2 | Generation 3 | ||||||
| Particle | Mass (MeV) | Charge | Particle | Mass (MeV) | Charge | Particle | Mass (MeV) | Charge |
| up quark (u) | 2.55 | +2/3 | charm quark (c) | 1270 | +2/3 | top quark (t) | 171200 | +2/3 |
| down quark (d) | 5.04 | -1/3 | strange quark (s) | 104 | -1/3 | bottom quark (b) | 4200 | -1/3 |
| electron (e-) | 0.511 | -1 | muon (µ-) | 105.7 | -1 | tau
( -) |
1776.8 | -1 |
| e neutrino (ve) | < 2.0 × 10-6 | 0 | µ neutrino (vµ) | <0.19 | 0 |
neutrino
(v) |
< 18.2 | 0 |
| Particle | Force | Acts through | Acts on | Mass (MeV) | Charge |
Photon
( ) |
Electromagnetic | Electric charge | Electrically charged particles | < 1× 10-24
 0 |
0 |
| Z boson (Z) | Weak nuclear | Weak interaction | Quarks and leptons | 91188 | 0 |
| W± bosons (W±) | Weak nuclear | Weak interaction | Quarks and leptons | 80398 | ±1 |
| Gluon (g) | Strong nuclear | Color charge | Quarks and gluons | 0 | 0 |
| Higgs boson (H0) | Higgs force | Higgs field | Massive particles | > 114400 | 0 |
Table I: The particles predicted by the Standard Model. Approximate masses of particles as last reported by the Particle Data Group. [28]
SM particle interactions obey typical conservation of momentum and
energy laws as well as conservation laws for internal gauge symmetries
like conservation of charge, lepton number, etc. The model has been
thoroughly probed up to energies of
1 TeV, and has led to
spectacular results such as the
precision measurement of the anomalous magnetic moment of the electron
(analogous to measuring the distance between New York and Los Angeles to
the width of a human hair).
The final undiscovered particle, the Higgs boson, is thought to be extremely massive; the latest bounds from the CDF and DØ collaborations at the Tevatron have restricted the mass of the Higgs to two regions: 114-160 GeV and 170-185 Gev. [29] Since the Higgs boson couples very weakly to ordinary matter it is difficult to create in particle accelerators. Hopefully, the powerful Linear Hadron Collider (LHC) in Geneva, Switzerland, will confirm the existence of the Higgs boson, the final particle of the SM.
Despite its success, the SM does not contain any particle that could act
as the dark matter. The only stable, electrically neutral, and weakly
interacting particles in the SM are the neutrinos. Can the neutrinos be
the missing dark matter? Despite having the "undisputed virtue of being
known to exist" (as put so well by Lars Bergstrom), there are two major
reasons why neutrinos cannot account for all of the universe's dark
matter. First, because neutrinos are relativistic, a neutrino-dominated
universe would have inhibited structure formation and caused a
"top-down" formation (larger structures forming first, eventually
condensing and fragmenting to those we see today).
[30]
However, galaxies have been observed
to exist less than a billion years after the big bang and, together with
structure formation simulations, a "bottom-up" formation (stars then
galaxies then clusters etc.) seems to be the most likely.
[31]
Second, Spergel et al. ruled
out neutrinos as the entire solution to missing mass using cosmological
observations: WMAP combined with large-scale structure data constrains
the neutrino mass to mv < 0.23 eV, which in
turn makes the cosmological density
v
h2 < 0.0072.
[16]
While neutrinos do account for a small fraction of dark matter, they
clearly cannot be the only source.
The lack of a dark matter candidate does not invalidate the SM, but rather suggests that it must be extended. Perhaps the SM is only a valid theory at low energies, and that there is new physics "beyond the Standard Model;" that is, new theories may supplement, rather than replace, the SM. Such new theories have already been proposed, the most promising being supersymmetry, which also yields a viable dark matter candidate called the neutralino or LSP.
B. Problems of the Standard Model
Although very successful, the SM has two flaws which hint at the need
for new solutions: the hierarchy problem and the fine-tuning
problem. The hierarchy problem arises from the SM's prediction of the
Higgs vacuum expectation value (vev), which is about 246 GeV. Theorists
have predicted that at high enough energies
( 1 TeV) the
electromagnetic
and weak forces act as a single unified force called the electroweak
force (this has also been experimentally verified). However at smaller
energies, the single unified force breaks down into two separate forces:
the electromagnetic force and the weak force. It turns out that after
this breaking, the Higgs field's lowest energy state is not zero, but
the 246 GeV vacuum expectation value. It is precisely this non-zero
value that gives other particles mass through their interactions with
the Higgs field. The 246 GeV vev is at the weak scale (the typical
energy of electroweak processes); however, the Planck scale (the energy
at which quantum effects of gravity become strong) is around
1019 GeV. The basic question, then, is why is the Planck
scale 1016 times larger than the weak scale? Is there simply
a "desert" between 103 and 1019 GeV in which no
new physics enters?
There is an additional difficulty with the Standard Model. Most
calculations in a quantum field theory are done perturbatively. For
example, the scattering cross section of two electrons at a given energy
can be calculated up to a certain power of
, the fine structure
constant, which is the coupling constant for electromagnetism. The
calculation is represented pictorially with Feynman diagrams; the number
of particle interaction vertices is related to the power of
. However, virtual
particles and more complicated diagrams can
also contribute to the process with higher powers of
.
 |
Figure 4. The tree level diagram on the
left represents the electron-electron scattering process to the lower
order in perturbation theory. The two graphs on the right represent
higher order processes (which can be thought of as loop corrections)
and enter with higher powers of
|
As an example, the best anomalous magnetic moment of the electron
calculation involves 891 diagrams.
[32]
Thus most quantities have so-called "quantum-loop" corrections (although
some quantities, like the photon's mass, are protected by
symmetries). The fine-tuning problem arises when trying to calculate the
mass of the Higgs particle; quantum loop corrections to the Higgs mass
are quadratically divergent. If one uses 1016 GeV as the
scale at which the electroweak and strong forces combine to become a
single unified force (which has been theorized but not seen), one
requires an almost perfect cancellation on the order of 1 part in
1014 for the Higgs mass to come out at the electroweak scale
( 150 GeV). This
unnatural cancellation is a source of alarm for theorists and signifies
that we lack an understanding of physics beyond the SM.