Etude de bruit de fond induit par les muons dans l'expérience ...

tel-00724955, version 1 - 23 Aug 2012
2.3 Direct detection 37
we must generally consider a three-parameter space: ap, an, and Mχ. Furthermore,
it means that the finite-momentum-transfer effects of the form factor cannot be
factored out of the cross section in a model-independent fashion. The distributions
of proton and neutron spin may be very different in a given nucleus, and so finite
momentum effects may be very model-dependent. The preferred way to deal with
this is to follow [106] by writing the WIMP-nucleus differential cross section in
the form
dσSD
=
dq2 where v is the incident velocity and
8G2 F
S(q) (2.7)
(2J + 1)v2 S(q) ≡ a 2 0S00(q) + a0a1S01(q) + a 2 1S11(q) (2.8)
with a0 ≡ ap+an and a1 ≡ ap−an. S(q) encompasses the effects of finite momentum
transfer, as well as values for the neutron and proton spin expectations ∗ . There is
no universal form of S(q). It must be computed separately for each nuclide using
nuclear structure models [104, 105].
Although neutralinos often have intrinsically larger spin-dependent than spin
independent couplings to nucleons, due to the great power of coherent enhancement
– Equation 2.4 shows that scalar interactions are enhanced by the square of the
target nuclear mass, while the spin-dependent cross-section in Equation 2.7 does
not increase with A –, spin independent WIMP-nucleus cross sections are generally
much greater. Because of this, and because of the relative rarity of heavy spinsensitive
isotopes, spin-independent interactions are targeted by most leading direct
searches.
2.3.4 Event rate
The differential rate for scalar interactions can be written in terms of σ0 SI from
Equation 2.3:
dR
dE = ρ0σ0 SI |F (q)|2
2Mχµ 2
f(v, t)
d
v
3 v (2.9)
v>q/2µ
The lower limit of integration is the minimum WIMP velocity required in order
to be kinematically possible for an energy E to be transferred to the nucleus. For
the velocity profile, we assume a Maxwellian distribution truncated at the galactic
escape velocity vesc. However, vesc is large enough, so that it has little effect on the
calculation, and so we have omitted it here. The Maxwellian distribution is
f(v)d 3 v = 1
v 3 0π 3/2 e−v2 /v 2 0d 3 v (2.10)
with a characteristic velocity v0 = 270 km/s in the solar neighborhood and truncated
at a galactic escape velocity of vesc ≈ 650 km/s. Substituting this form into
Equation 2.9 gives an energy spectrum that is a falling exponential modified by
F (q). Then, the differential rate is
dR
dE = ρ0σ0
SI |F (E)|2
√ exp −
πv0Mχµ 2 EMN
2µ 2v2
(2.11)
0
∗ There is a variation on this method advocated by some authors [58, 101] in which the scattering
expressions are made to look more similar to those in the SI case by multiplying the above
expression for σSD by a model-dependent form factor F 2 (q) = S(q)/S(0). This is equivalent in
principle, but the division by S(0) may increase numerical errors in practice.
2