where we show only the most relevant contribution, coming
from top loops. In the previous expression, α
t
≡ y
2
t
=4π,
with y
t
the top Yukawa coupling. Notice that the top
contribution is present only down to the top scale, where
the top quark is integrated out. As already stressed, in
addition to the SD operators of the first two lines, the
running has generated the SI operators of the last two lines.
Even though the Wilson coefficients of the SI operators are
smaller than those of the SD ones, it has been shown in
Ref. [35] that they are sufficient to put bounds on g
2
=Λ
2
which are up to a factor of 100 stronger with respect to the
typical bounds that can be obtained from SD experiments
(i.e., considering only the operators in the first two lines).
This shows clearly the importance of the running in setting
consistent bounds on the parameter space of the DM EFT.
Let us point out however that for the effect to be numerically
relevant, it is instrumental for the coupling between the DM
and the top currents to be switched on in Eq. (13). If this is not
the case, the SI operators are still generated in the running,
but with much smaller Wilson coefficients and weaker
bounds (see Ref. [35] for more details). In the analysis of
Sec. V we will numerically implement the running and the
mixing of the currents using the runDM code [33–35,50],
which takes into account all the contributions.
We also perform analysis of DM direct detection signals
from operators whose contribution is momentum sup-
pressed. These operators vanish in the limit of zero
momentum transfer limit, but they are not exactly negli-
gible at actual DM DD experimental transferred energies in
10–100 KeV energy range. We have used XENON100
constraints [51] on the relevant non-relativistic nucleon
operators O
6
, O
7
, O
10
, O
11
, O
12
and Refs. [46,52] to
translate them into limits on C4, C6, D2, and D10 operators
for scalar and fermions DM. Since XENON100 are very
sensitive only to O
10
, O
11
, and O
12
nucleon operators, the
only non-negligible constraints can be set on D2 and D10
operators as we discuss below in Sec. V. Since, to the best
of our knowledge, there is no studies on connection of
vector DM operators to nonrelativistic operators, we did not
produce the respective limits of vector DM.
IV. ANALYSIS SETUP AND CONSTRAINTS
In this section we describe the analysis setup and
constraints used in this study. In particular we delineate
the limits originating from CMB, direct detection experi-
ments, and collider searches.
Direct and indirect detection constraints are affected by
uncertainties of astrophysical nature. On one hand, the
scattering of DM off nuclei on the Earth depends on the
DM local density and velocity distributions around Earth.
On the other hand, the DM self-annihilation rate in our
galaxy depends on its particle density distribution therein.
For what is concern of this paper, whenever possible we
make the conservative choice to select targets that
can reduce as much as possible the uncertainties, and
thoroughly account for the remaining ones. In practice, this
means that: (i) for indirect searches we adopt CMB limits,
as the energy injection of DM annihilation is unaffected by
the usual unknowns related to DM density profile within
structures; (ii) for direct searches we explicitly take into
account the systematic effects generated by the astrophysi-
cal uncertainties in the determination of the local DM
density ρ
0
.
A. CMB constraints
The observation of byproducts of DM annihilation
(or decay) in astrophysical targets can be used to determine
(or constrain, in case of missing observations) relevant
DM properties such as its mass and annihilation cross
section. Such bounds depend on the unknown DM dis-
tribution within the astrophysical objects chosen as targets.
A detailed analysis has shown that, choosing the CMB as
target, the leading signal of DM annihilation is produced
around redshift z ∼ 600 [53]. This makes the CMB a
quantitatively competitive channel for indirect searches
[1,19,20], since at z ∼ 600 the DM has not fallen into
structures yet, and the observation is free from the usual
astrophysical sources of uncertainties (density profile
within a halo, distribution and density of subhalos, mass
of the smallest bound halo). Moreover, additional sources
of systematics affecting the CMB constraints have also
been thoroughly examined [54,55], and shown to affect the
results below the sensitivity needed for this paper. In the
following, we will neglect them leaving our conclusions
unaltered.
In order to set the CMB bounds on the quantity of our
interest Λ, we first obtain the observational bound on the
thermally averaged annihilation cross-section at redshift
600:
p
ann
¼
X
j
f
j
ð600;m
DM
Þ
hσvi
j
ð600Þ
m
DM
; ð14Þ
where hσvi
j
ð600Þ is the thermally averaged partial anni-
hilation cross section for the jth channel at redshift 600 and
f
j
ðz; m
DM
Þ is the fraction of annihilation energy that is
absorbed by the plasma at redshift z. The quantity p
ann
is
constrained by Planck TT, TE, EE, and lowP data [1]:
p
ann
< 4.1 × 10
−28
cm
3
sGeV
at 95% C:L:; ð15Þ
and the values of the variable f
j
ðz; m
DM
Þ are taken
form Ref. [56].
In order to bound the new physics scale Λ, we numeri-
cally compute the velocity dependent annihilation cross
section with micrOMEGAs [45], obtaining the relationship
between hσvi
j
ð600Þ and Λ for each effective operator and
each final state. It may be noted that for those operators for
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