2.3. The neutral supersymmetric spectrum
Supersymmetric particles that are electrically neutral, and so promising dark matter candidates, are shown with their standard model partners in Fig. 2. In supersymmetric models, two Higgs doublets are required to give mass to all fermions. The two neutral Higgs bosons are Hd and Hu, which give mass to the down-type and up-type fermions, respectively, and each of these has a superpartner. Aside from this subtlety, the superpartner spectrum is exactly as one would expect. It consists of spin 0 sneutrinos, one for each neutrino, the spin 3/2 gravitino, and the spin 1/2 Bino, neutral Wino, and down- and up-type Higgsinos. These states have masses determined (in part) by the corresponding mass parameters listed in the top row of Fig. 2. These parameters are unknown, but are presumably of the order of the weak scale, given the motivations described above.
Fig.
2. Neutral particles in the supersymmetric spectrum. M1,
M2,
μ,
,
and m3/2
are unknown weak-scale mass parameters. The Bino, Wino, and down- and
up-type Higgsinos mix to form neutralinos.
The
gravitino is a mass eigenstate with mass m3/2.
The sneutrinos are also mass eigenstates, assuming flavor and
R-parity
conservation (see Section 2.4).
The spin 1/2 states are differentiated only by their electroweak
quantum numbers. After electroweak symmetry breaking, these gauge
eigenstates therefore mix to form mass eigenstates. In
the basis
the
mixing matrix is
|
|
(7) |
where
cW≡cosθW,
sW≡sinθW,
and β
is another unknown parameter defined by tanβ≡
Hu
/
Hd
,
the ratio of the up-type to down-type Higgs scalar vacuum expectation
values (vevs). The mass eigenstates are called neutralinos and
denoted χ ≡ χ1, χ2, χ3, χ4
, in order of increasing mass. If M1 
M2, |μ|,
the lightest neutralino χ
has a mass of approximately M1
and is nearly a pure Bino. However, for M1 
M2 
|μ|,
χ
is a mixture with significant components of each gauge eigenstate.
Finally, note that neutralinos are Majorana fermions; they are their own anti-particles. This fact has important consequences for neutralino dark matter, as will be discussed below.
2.4. R-Parity
Weak-scale superpartners solve the gauge hierarchy problem through their virtual effects. However, without additional structure, they also mediate baryon and lepton number violation at unacceptable levels. For example, proton decay p → π0e+ may be mediated by a squark as shown in Fig. 3.
Fig. 3. Proton decay mediated by squark.
An elegant way to forbid this decay is to impose the conservation of R-parity Rp ≡ (−1)3(B−L) + 2S, where B, L, and S are baryon number, lepton number, and spin, respectively. All standard model particles have Rp = 1, and all superpartners have Rp = −1. R-parity conservation implies ΠRp = 1 at each vertex, and so both vertices in Fig. 3 are forbidden. Proton decay may be eliminated without R-parity conservation, for example, by forbidding B or L violation, but not both. However, in these cases, the non-vanishing R-parity violating couplings are typically subject to stringent constraints from other processes, requiring some alternative explanation.
An immediate consequence of R-parity conservation is that the lightest supersymmetric particle (LSP) cannot decay to standard model particles and is therefore stable. Particle physics constraints therefore naturally suggest a symmetry that provides a new stable particle that may contribute significantly to the present energy density of the universe.