4 C.-Y. Li et al. / Nuclear Physics B 945 (2019) 114678
G(q
2
,υ)=
(q
2
ref
−q
2
min
)|M
χN
|
2
q
2
ref
q
2
min
dq
2
|M
χN
(q
2
,υ
ref
)|
2
, (5)
and where q
2
ref
≡ 4μ
2
χN
υ
2
ref
.
Concretely, the corresponding formulae of G(q
2
, υ) can be explicitly obtained for different
operator. According to the momentum and velocity dependencies, the effective operators are
catalogued into six types. In this paper, we only consider spin-independent scattering, so we only
focus on the following three type operators [27],
Type-I operators
O
1(1)
=
1
q
2
+m
2
φ
¯χχ
¯
NN, O
1(2)
=
1
q
2
+m
2
φ
¯χγ
μ
χ
¯
Nγ
μ
N,
O
1(3)
=
2m
χ
q
2
+m
2
φ
χ
∗
χ
¯
NN,O
1(4)
=
1
q
2
+m
2
φ
(χ
∗
←→
∂
μ
χ)
¯
Nγ
μ
N. (6)
Type-II operators
O
2(1)
=
1
q
2
+m
2
φ
¯χγ
5
χ
¯
NN,O
2(2)
=
2m
χ
q
2
+m
2
φ
χ
∗
χ
¯
Nγ
5
N. (7)
Type-III operator
O
3
=
1
q
2
+m
2
φ
¯χγ
μ
γ
5
χ
¯
Nγ
μ
N. (8)
The factor G(q
2
, v) can be written as [27,29]
G
1
(q
2
) =
1
I
1
(m
2
φ
+q
2
)
2
,G
2
(q
2
) =
q
2
/m
2
φ
I
2
(m
2
φ
+q
2
)
2
,G
3
(q
3
) =
v
2
⊥
/v
2
ref
I
3
(m
2
φ
+q
2
)
2
, (9)
and
I
1
=
1
(1 +a)(1 + b)
,I
2
=
1
b − a
ln(
1 +b
1 +a
) −I
1
,I
3
= I
1
−I
2
/b, (10)
where a = q
2
min
, b = q
2
ref
, v
⊥
= v + q/(2μ
χN
) is the transverse velocity of the DM particle, and
v
2
⊥ = v
2
− q
2
/(4μ
2
χN
). For type-III, the reduced mass μ
χN
in the expression of v
⊥
will be
replaced by μ
χA
, because the nucleon velocity operator v acting on the nucleus wave function
will pick up the nucleus mass [27,29,50].
At nucleus le
vel, for the three type operators, the spin-independent WIMPs-nucleus differen-
tial cross section can be written as
d(σ
A
)
i
dq
2
=
¯σ
p
4μ
2
χp
v
2
[Z + ξ(A − Z)]
2
G
i
(q
2
,v)F
2
A
(q
2
), (11)
where i = I, II, III, Z is the number of protons and A is the number of atomic mass number of the
target nucleus, υ is the relative velocity of the WIMP in the nuclear rest frame, and ξ = f
n
/f
p
where f
p
(f
n
) are the DM couplings to protons (neutrons). In the simple model, the scattering is
isospin conserving (IC), ξ 1. However, in the general model with ξ = 1, the true value of σ
p
(the cross section for the DM particle scattering off a free nucleon) differs from σ
IC
p
(the cross
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