188
Supersymmetric dark matter
However, it is possible that the much larger low-energy background might be
subject to other modulating effects. The direction of the WIMP 'wind' felt by
the detectors is strongly peaked in the direction opposite to the solar motion, so
the recoil directions will be strongly peaked in the same direction. If observed,
this feature, combined with the annual modulation, would be the most convincing
demonstration of the existence of WIMPs.
6.6.1 Neutralin~nucleon elastic scattering
If WIMPs solve the dark matter problem, they must have some small but finite
coupling to ordinary matter-otherwise, they would not have annihilated in the
early universe and would be overabundant today. By crossing symmetry, the
amplitude for WIMP annihilation into a quark-antiquark pair is related to the
elastic scattering of the WIMPs by quarks. Thus, we expect that WIMPs will
have a small coupling to nuclei and may, therefore, be detectable by nuclear
scattering. The calculations are considerably simplified because the WIMP
velocity (vx/c ,.., 10- 3 ) is extremely non-relativistic. This feature also simplifies
the conversion of the scattering cross sections from quarks into the scattering
cross sections from the nuclei making the detector. In the non-relativistic limit,
the axial vector current XfY",)'5Xf is just the WIMP spin and it is coupled to the
nucleon spin. Since the neutralino is a Majorana fermion, it has no vector current
Xfy",Xf = O. So the only other possible term in the effective interaction is the
scalar xfxf which couples to the mass of the nucleus-in the non-relativistic
limit the 'tensor' current reduces to the scalar.
At tree level, the axial vector (spin) interaction receives contributions from tchannel Z boson and s-channel squark exchange, while the scalar interaction gets
contributions from t-channel Higgses H, h as well as s-channel quark exchange.
See figure 6.3. Since the lightest Higgs might be considerably lighter than the
lightest squarks, its contribution to the scalar interaction could be significant if
the neutralino state (6.50) has a substantial Higgsino component. Because it
is proportional to the mass number A of the nucleus, the scalar amplitude will
dominate for heavy nuclei. For a neutralino that is a pure Bino B, this occurs for
A ~ 20 in the large squark mass limit and this is confirmed by numerical surveys
of the supersymmetric parameter space where scalar dominance for A ~ 30 is
almost always found [27].
It is of interest to examine the theoretical implications for the direct detection
experiments of restricting the parameters of the CMSSM to the region allowed
by the cosmological and other constraints. This programme has so far been
undertaken [28] only in the case that A = O. The LEP lower limit on mh and
the b -. sy data provide upper limits on the cross sections, while the g - 2 data
provide lower limits, at least if J.I. > O. In that case, the overall conclusion is that
the spin-independent cross section (lSI satisfies
2 x 10- 10 pb ~ (lSI ~ 6 x 10- 8 pb
(6.62)
Supersymmetric dark matter
However, it is possible that the much larger low-energy background might be
subject to other modulating effects. The direction of the WIMP 'wind' felt by
the detectors is strongly peaked in the direction opposite to the solar motion, so
the recoil directions will be strongly peaked in the same direction. If observed,
this feature, combined with the annual modulation, would be the most convincing
demonstration of the existence of WIMPs.
6.6.1 Neutralin~nucleon elastic scattering
If WIMPs solve the dark matter problem, they must have some small but finite
coupling to ordinary matter-otherwise, they would not have annihilated in the
early universe and would be overabundant today. By crossing symmetry, the
amplitude for WIMP annihilation into a quark-antiquark pair is related to the
elastic scattering of the WIMPs by quarks. Thus, we expect that WIMPs will
have a small coupling to nuclei and may, therefore, be detectable by nuclear
scattering. The calculations are considerably simplified because the WIMP
velocity (vx/c ,.., 10- 3 ) is extremely non-relativistic. This feature also simplifies
the conversion of the scattering cross sections from quarks into the scattering
cross sections from the nuclei making the detector. In the non-relativistic limit,
the axial vector current XfY",)'5Xf is just the WIMP spin and it is coupled to the
nucleon spin. Since the neutralino is a Majorana fermion, it has no vector current
Xfy",Xf = O. So the only other possible term in the effective interaction is the
scalar xfxf which couples to the mass of the nucleus-in the non-relativistic
limit the 'tensor' current reduces to the scalar.
At tree level, the axial vector (spin) interaction receives contributions from tchannel Z boson and s-channel squark exchange, while the scalar interaction gets
contributions from t-channel Higgses H, h as well as s-channel quark exchange.
See figure 6.3. Since the lightest Higgs might be considerably lighter than the
lightest squarks, its contribution to the scalar interaction could be significant if
the neutralino state (6.50) has a substantial Higgsino component. Because it
is proportional to the mass number A of the nucleus, the scalar amplitude will
dominate for heavy nuclei. For a neutralino that is a pure Bino B, this occurs for
A ~ 20 in the large squark mass limit and this is confirmed by numerical surveys
of the supersymmetric parameter space where scalar dominance for A ~ 30 is
almost always found [27].
It is of interest to examine the theoretical implications for the direct detection
experiments of restricting the parameters of the CMSSM to the region allowed
by the cosmological and other constraints. This programme has so far been
undertaken [28] only in the case that A = O. The LEP lower limit on mh and
the b -. sy data provide upper limits on the cross sections, while the g - 2 data
provide lower limits, at least if J.I. > O. In that case, the overall conclusion is that
the spin-independent cross section (lSI satisfies
2 x 10- 10 pb ~ (lSI ~ 6 x 10- 8 pb
(6.62)