MODERN COSMOLOGY

Low-energy SUSY and DM 199

SUSY particle can never decay. This is the famous LSP (lightest SUSY particle)
candidate for CDM.
Note that proton decay does not call directly forRparity. Indeed this decay
entails the violation of bothBandL. Hence, to prevent a fast proton decay
one may impose a discrete symmetry which forbids all theBviolating terms in
the SUSY Lagrangian, while allowing for terms withLviolation (the reverse is
also viable). Models with such alternative discrete symmetries are called SUSY
models with brokenRparity. In such models the stability of the LSP is no longer
present and the LSP cannot be a candidate for stable CDM. We will comment
later on these alternative models in relation to the DM problem, but we turn now
to the more ‘orthodox’ situation withRparity. The favourite LSP is the lightest
neutralino.

5.5.2 Neutralinos in the minimal supersymmetric SM

If we extend the SM in the minimal way, adding for each SM particle a
supersymmetric partner with the same quantum numbers, we obtain the so
called Minimal Supersymmetric Standard Model (MSSM). In this context the
neutralinos are the eigenvectors of the mass matrix of the four neutral fermions
partners of the W 3 ,B,H^01 and H^02 called, respectively, winoW ̃ 3 ,binoB, higgsinos ̃

H ̃^01 andH ̃^02. There are four parameters entering the mass matrix,M 1 ,M 2 ,μand
tanβ:

M=





M 2 0 mZcosθwcosβ −mZcosθwsinβ

0 M 1 −mZsinθwcosβ mZsinθwsinβ

mZcosθwcosβ −mZsinθwcosβ 0 −μ

−mZcosθwsinβ mZsinθwsinβ −μ 0





(5.37)

wheremZ= 91. 19 ± 0 .002 GeV is the mass of the Z boson,θwis the weak mixing
angle, tanβ≡v 2 /v 1 withv 1 ,v 2 vevs of the scalar fieldsH 10 andH 20 respectively.
In generalM 1 andM 2 are two independent parameters, but if one assumes
that a grand unification scale takes place, then at grand unificationM 1 =M 2 =
M 3 ,whereM 3 is the gluino mass at that scale. Then at theMWscale one obtains:

M 1 =^53 tan^2 θwM 2 ^12 M 2 , (5.38)

M 2 =

g 22

g 32

mg ̃mg ̃/ 3 , (5.39)

whereg 2 andg 3 are theSU( 2 )andSU( 3 )gauge coupling constants, respectively,
andm ̃gis the gluino mass.
The relation (5.38) betweenM 1 andM 2 reduces to three the number of
independent parameters which determine the lightest neutralino composition and
mass: tanβ,μandM 2. The neutralino eigenstates are usually denoted byχ ̃i^0 ,χ ̃ 10
being the lightest one.