180
Supersymmetric dark matter
and 9. Since there are two Higgs doublet chiral superfields HII and Bd in the
MSSM, there are two VEVs, Vu and Vd. The combination v~ + v~ is fixed by
the measured value of mz and is, therefore, not a free parameter but the ratio
tan fJ == VII/Vd is a free parameter. The MSSM has five physical Higgs particles
of which two (H,z) are charged, two (hO, HO) are neutral scalars and one (Ao)
is a neutral pseudoscalar. The masses are all fixed in terms of the (known) gauge
coupling strengths and three parameters, two of which (lL, tan P) have already
been defined. Without loss of generality, the third may be taken to be mA. Apart
from 9 and the CKM mixing angles and phase, the unknown parameters of the
MSSM therefore consist of 63 masses and mixing angles and 43 phases. Even
if all of the phases are set to zero, it is still not feasible to explore the remaining
parameter space (MI.2.3, 11-, tan p, mA, 21 scalar squark and slepton masses and
36 mixing angles).
The number of parameters is, therefore, drastically reduced by making
further assumptions. In the low-energy approach [16], special phenomenolgically
viable points in the parameter space are selected. For example, the five scalar
(squark and slepton) symmetric 3 x 3 mass matrices and the three trilinear
coupling A-matrices might be assumed to be generation independent or that they
are flavour-diagonal in a basis where the quark and lepton mass matrices are
diagonal. Neither of these has any strong theoretical motivation. Alternatively,
in the high-energy approach that we shall follow, the parameters of the MSSM
are treated as running parameters. In other words, the parameters 'run' or evolve
with the renormalization scale in a way determined by the renormalization group
equations. Then a structure is imposed on the parameters at some high energy
scale. This would be the case if there is an underlying GUT symmetry, for
example. In such a model, it is assumed that the gauginos all have a common
mass m 1/2 at some (a priori unknown) unification scale m x:
MJ (mx) = M2(mx) = M3(mx) = m1/2.
(6.45)
The gaugino masses at the electroweak scale are determined using renormalization group equations and at the electroweak scale their ratios are determined by
the gauge coupling strengths:
M3
a3
MI
Sal
- = -
- = -
(6.46)
M2
a2
M2
3a2
where a3 == gi/41r etc. Similarly, it is also assumed that all scalars, except
possibly the Higgs soft masses squared mt2' have a common squared-mass m~
and that the trilinear cofficients have a common value A, at the unification scale.
m 2
Q _ (mx) = m~e (mx) = mt(mx) = mal3
(6.47)
L
L
Gi
mi(mx) = m~ (mx) = m~13
(6.48)
All = Ad = A. = A13.
(6.49)
Supersymmetric dark matter
and 9. Since there are two Higgs doublet chiral superfields HII and Bd in the
MSSM, there are two VEVs, Vu and Vd. The combination v~ + v~ is fixed by
the measured value of mz and is, therefore, not a free parameter but the ratio
tan fJ == VII/Vd is a free parameter. The MSSM has five physical Higgs particles
of which two (H,z) are charged, two (hO, HO) are neutral scalars and one (Ao)
is a neutral pseudoscalar. The masses are all fixed in terms of the (known) gauge
coupling strengths and three parameters, two of which (lL, tan P) have already
been defined. Without loss of generality, the third may be taken to be mA. Apart
from 9 and the CKM mixing angles and phase, the unknown parameters of the
MSSM therefore consist of 63 masses and mixing angles and 43 phases. Even
if all of the phases are set to zero, it is still not feasible to explore the remaining
parameter space (MI.2.3, 11-, tan p, mA, 21 scalar squark and slepton masses and
36 mixing angles).
The number of parameters is, therefore, drastically reduced by making
further assumptions. In the low-energy approach [16], special phenomenolgically
viable points in the parameter space are selected. For example, the five scalar
(squark and slepton) symmetric 3 x 3 mass matrices and the three trilinear
coupling A-matrices might be assumed to be generation independent or that they
are flavour-diagonal in a basis where the quark and lepton mass matrices are
diagonal. Neither of these has any strong theoretical motivation. Alternatively,
in the high-energy approach that we shall follow, the parameters of the MSSM
are treated as running parameters. In other words, the parameters 'run' or evolve
with the renormalization scale in a way determined by the renormalization group
equations. Then a structure is imposed on the parameters at some high energy
scale. This would be the case if there is an underlying GUT symmetry, for
example. In such a model, it is assumed that the gauginos all have a common
mass m 1/2 at some (a priori unknown) unification scale m x:
MJ (mx) = M2(mx) = M3(mx) = m1/2.
(6.45)
The gaugino masses at the electroweak scale are determined using renormalization group equations and at the electroweak scale their ratios are determined by
the gauge coupling strengths:
M3
a3
MI
Sal
- = -
- = -
(6.46)
M2
a2
M2
3a2
where a3 == gi/41r etc. Similarly, it is also assumed that all scalars, except
possibly the Higgs soft masses squared mt2' have a common squared-mass m~
and that the trilinear cofficients have a common value A, at the unification scale.
m 2
Q _ (mx) = m~e (mx) = mt(mx) = mal3
(6.47)
L
L
Gi
mi(mx) = m~ (mx) = m~13
(6.48)
All = Ad = A. = A13.
(6.49)