Dark matter direct detection of a fermionic singlet at one loop
Eur. Phys. J. C
Dark matter direct detection of a fermionic singlet at one loop
Juan HerreroGarcía 2
Emiliano Molinaro 0 1
Michael A. Schmidt 3
0 CP3Origins, University of Southern Denmark , Campusvej 55, 5230 Odense M , Denmark
1 Department of Physics and Astronomy, University of Aarhus , Ny Munkegade 120, 8000 Århus C , Denmark
2 ARC Center of Excellence for Particle Physics at the Terascale, University of Adelaide , Adelaide, South Australia 5005 , Australia
3 ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics, The University of Sydney , Physics Road, Sydney, New South Wales 2006 , Australia
The strong direct detection limits could be pointing to dark matter  nucleus scattering at loop level. We study in detail the prototype example of an electroweak singlet (Dirac or Majorana) dark matter fermion coupled to an extended dark sector, which is composed of a new fermion and a new scalar. Given the strong limits on colored particles from direct and indirect searches we assume that the fields of the new dark sector are color singlets. We outline the possible simplified models, including the wellmotivated cases in which the extra scalar or fermion is a Standard Model particle, as well as the possible connection to neutrino masses. We compute the contributions to direct detection from the photon, the Z and the Higgs penguins for arbitrary quantum numbers of the dark sector. Furthermore, we derive compact expressions in certain limits, i.e., when all new particles are heavier than the dark matter mass and when the fermion running in the loop is light, like a Standard Model lepton. We study in detail the predicted direct detection rate and how current and future direct detection limits constrain the model parameters. In case dark matter couples directly to Standard Model leptons we find an interesting interplay between lepton flavor violation, direct detection and the observed relic abundance. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2 Fermionic singlet dark matter . . . . . . . . . . . . . 2.1 Dirac dark matter . . . . . . . . . . . . . . . . . 2.2 Majorana dark matter . . . . . . . . . . . . . . . 2.3 Standard Model particles in the loop . . . . . . .

Contents
1 Introduction
Direct detection (DD) experiments search for dark
matter (DM) scatterings off nuclei in underground detectors.
The current limits impose very strong constraints on the
parameters of weakly interacting massive particles (WIMPs),
which are one of the prototype DM candidates. The current
most stringent DD limits for WIMPs in the mass range of
[
10, 1000
] GeV come from xenon experiments [
1–3
]. In this
work we hypothesize that the absence of DD signals may be
reconciled with the WIMP paradigm by generating the
scattering at oneloop order and thus with an extra 1/(16π 2)2
suppression of the cross section. As we will see, current
and nextgeneration experiments are able to test significant
regions in parameter space of this class of scenarios.
There have been several works in the literature on DD at
oneloop order. In Refs. [
4–7
] the authors studied DD limits
from photon interactions in the context of flavored DM and
in Ref. [8] in the context of a radiative neutrino mass model
(the scotogenic model [
9
]) with inelastic Majorana DM. In
Ref. [
10
] the authors performed a detailed study of oneloop
scenarios with a charged mediator directly coupled to
Standard Model (SM) fields, including the Z and Higgs boson
contributions. For couplings to the first and second
generation of quarks the dominant contribution may be due to
scattering at tree level, while box diagrams may be significant
for third generation quarks. Similarly, Ref. [
11
] studied direct
detection of Majorana DM directly coupled to both left and
righthanded SM leptons via two charged scalar mediators.
The Z and Higgs contributions were also computed for the
scotogenic model in Ref. [
12
] and also for DM connected to
the SM via a neutrinoportal in Ref. [
13
]. In Ref. [
14
] the
authors studied the oneloop contributions to DD in models
with pseudoscalar mediators or inelastic scattering. In the
context of supersymmetry detailed computations have been
performed for the bino [
15
] and wino [
16–18
] DM cases. In
the latter scenario loop contributions to DMnucleus
scattering due to gauge bosons may give significant corrections.
In this work we study the DD scattering rate for the case
of DM being a SM singlet Dirac or Majorana fermion ψ ,
which is coupled to a more complex dark sector. A
conserved global U(1) or Z2 symmetry is assumed in order to
S
F
γ/Z/h
ψ
q
ψ
q
F
S
γ/Z/h
ψ
q
stabilize the DM particle. In our scenario there are no
treelevel contributions to the DD cross section. The lowest order
scattering off nuclei occurs at oneloop order via the penguin
diagrams in Fig. 1, with a dark fermion F and a dark scalar
S running in the loop. We assume that the new particles are
color singlets, so that there are no flavor changing neutral
currents in the quark sector, and there are only weak limits
from direct production at the large hadron collider (LHC). In
this way boxdiagram contributions to the scattering
amplitude are absent. Our main goal is to study analytically the
different contributions to the DMnucleus scattering, as well
as to outline possible simplified models, including those with
SM fields. In addition we analyze the current limits from DD,
as well as constraints coming from the relic abundance,
lepton flavor violation (LFV) and anomalous magnetic dipole
moments (AMMs).1
The paper is structured as follows: In Sect. 2 we study
the UV completions of the fermionic DM scenario
including models with SM particles in the loop. In order to fix the
notation we review in Sect. 3 the relevant effective operators
for DD at the quark level and also their nonrelativistic (NR)
versions at the nucleon level. In Sect. 4 we derive analytical
expressions for the Wilson coefficients and provide compact
expressions in certain limits. In Sect. 5 we perform a
numerical analysis of the phenomenology relevant for DD. First we
show some numerical examples for the Wilson coefficients
at the quark and nucleon level (the latter in their NR
version). Afterwards we derive the current limits on the model
parameters and discuss future expected sensitivity. We also
discuss limits from LFV processes for models in which DM
is directly coupled to SM leptons. Sections 4 and 5 contain
the main results of this paper. We discuss other
phenomenological aspects of the proposed scenario, such as the DM
1 In our scenario leptonic electric dipole moments appear only at
twoloop order and are therefore suppressed.
relic abundance, invisible decays and searches at colliders in
Sect. 6. Finally we present our conclusions in Sect. 7.
The manuscript also includes several appendices with
technical details. The generalization to larger symmetry
groups in the dark sector is presented in Appendix A. In
Appendix B we show a compact expression for the
differential cross section in order to make contact with the
literature and we briefly review the differential event rate for
DD. In Appendix C we give generic expressions for Higgs
and Z boson invisible decays into DM. Relevant formulae
for LFV observables and for AMMs of leptons are provided
in Appendix D. Details about the calculation of the relic
abundance are collected in Appendix E and the numerical
expressions for the matching to NR operators are given in
Appendix F.
2 Fermionic singlet dark matter
In the following sections we first present simplified models of
Dirac and Majorana fermion DM with vectorlike fermions
in the loop and then discuss SM particles in the loop.
2.1 Dirac dark matter
The new particles can have different combinations of
quantum numbers as displayed in Table 1. We consider a global
U(1)dm symmetry in the dark sector to stabilize DM. It can
equally be replaced by a discrete Zn subgroup. Other
symmetry groups are discussed in Appendix A.
The interaction Lagrangian for the fields ψ , F and S reads
Lψ = i ψ ∂/ ψ − mψ ψ ψ + i F D/ F − m F F F
+
−
Dμ S † Dμ S − V (S, H )
y1 FR S ψL + y2 FL S ψR + H.c. ,
(1)
where H is the SM Higgs doublet2 and V (S, H ) denotes
the scalar potential. The DM ψ is a SM fermion singlet,
but is charged under the dark sector symmetry. The fields
F and S are charged under the electroweak gauge group.
Electroweak gauge invariance requires them to be in the same
2 We define the SM Higgs doublet H with hypercharge 1/2.
SU(2)L irreducible representation of dimension dF and to
have equal hypercharge YF . Notice that in some cases there
can be interactions with the SM fields which are subject to
strong constraints. We discuss such cases in Sect. 2.3.
In the case of a global symmetry, even if DM is stable at the
renormalizable level, higherorder Planckscale suppressed
operators may induce its decay [
19
]. In particular for a Dirac
fermion ψ the dimension5 operator ψ H˜ †( D/ L ) with the SM
lepton doublet L is one such example. One can construct UV
completions of such operators by softlybreaking the global
symmetry in the dark sector which induces decays, possibly
radiatively. The limits dramatically depend on the DM mass
and the Wilson coefficient of the operator. For the rest of the
paper we assume that DM is cosmologically stable and that
it satisfies all indirect detection constraints on decaying DM.
In our simplified scenario with the interactions given in
Eq. (1) it would seem that two of the three new states were
stable: ψ and one of S or F . For the following discussion let
us assume m S ≥ m F + mψ so that F is potentially stable,
while S can decay.3 Then, there are two possibilities: (i) If
the fermion F is a SM singlet (but charged under the dark
group), it also contributes to the DM relic density.4 Hence,
the DD rate of the ψ has to be rescaled by its smaller density
under the assumption that the global density scales as the
local one, and there is a similar DD rate for F via Higgs
penguins with ψ in the loop. (ii) If F is charged under the SM
group (SU(2)L charges and/or hypercharge) its electrically
charged components have to decay given the stringent limits
on charged stable particles [
21–24
]. If the components of F
mix with SM leptons, they decay like in the model discussed
in Refs. [
25, 26
]. Otherwise, as for the DM via the interaction
ψ H˜ †( D/ L ), the fermion F may also decay into SM particles
via nonrenormalizable operators, which are allowed on
general grounds, unless F carries fractional electric charge, or
other symmetries forbid them. In this case the fermion F has
to decay much faster than the longlived DM particle.5
If F is a SM lepton, a charged lepton or a neutrino, it may
be stable. Similarly, if F is a righthanded neutrino, it mixes
with SM neutrinos and decay. Also, in the case in which S
is the SM Higgs doublet and F a heavier fermion, the latter
may decay into the SM Higgs boson. We discuss all these
possibilities in more detail below.
In general the SM Higgs boson couples to the new scalar
multiplet S via a Higgs portal interaction in the scalar
potential V (S, H ). Depending on the quantum numbers of the
particles in the dark sector, it may also have an
interaction with the fermion F , for instance if the latter is a SM
3 Similar arguments apply to the other case where S might be stable.
4 In this case, if mψ m F , coannihilations play an important role [
20
].
5 Naively, the scalar S being lighter than the fermion F appears to be
more natural given that there are 13 dimension5 operators which induce
decay for a scalar compared to one for a fermion [
19
].
lepton. In the case of a charged lepton in the loop, the
largest Higgs interactions are proportional to the square of
its mass (m /v)2 1, with the electroweak vacuum
expectation value (VEV) v 246 GeV. Therefore this
contribution is suppressed and can be safely neglected. While
these interactions are even further suppressed for Dirac
neutrinos, in principle it is possible to have O(1) Yukawa
couplings for Majorana neutrinos (we discuss this case in
Sect. 2.3.3).
The Higgs portal contribution depends on the coupling of
the SM Higgs boson h to a pair of scalars S after electroweak
symmetry breaking. In the case of a complex scalar S, we
parameterize it in terms of
V(S, h) ⊃ λH S v h(S† S)
and similarly for a real scalar S with an additional factor 1/2
in order for the Feynman rule (and therefore the expression
of the Wilson coefficients) to be identical
V(S, h) ⊃ λH2Sv h S2. (3)
In the case of a complex scalar S, the Higgs couplings of
Eq. (2) are induced by SM gauge invariant Higgs portal
interactions such as
(4)
(5)
(6)
(7)
(H † H )(S† S) = hv S† S + · · ·
(H † S)(S† H ) = hvSd 2 + · · · ,
√
with Sd ≡ (Sd,r + i Sd,i )/ 2.
(H † S)2 + H.c. = hv Sd,r − Sd,i + · · ·
2 2
H †[S†, S]H = hv(S+2 − S−2) + · · ·
The term in the first line is always present, while those in
the second and third lines require S to be an SU(2)L doublet,
S ≡ (Su , Sd )T . Moreover the term in Eq. (6) assumes that S
has the same hypercharge as the SM Higgs doublet, which
we write after spontaneous electroweak symmetry breaking
as H ≡ (0, (h + v)/√2)T . Finally the term in the last line
exists for electroweak triplets S ≡ S · σ , where S± denotes
the coefficients of σ ±.6 In the following we parameterize all
the results in terms of λH S, which allows to easily generalize
the result of Higgs penguins for arbitrary combinations of
Higgs portals. If S± (Sd,r and Sd,i ) have the same mass,
their contribution from the interactions (5) and (6) to the
DD scattering amplitude exactly cancels due to the relative
minus sign in the interaction term.7 For equal masses the
6 σ ≡ (σ1, σ2, σ3) denote the Pauli matrices, with σ ± = (σ1 ± iσ2)/2.
7 This is not expected on general grounds, as the same terms in
the potential generate splittings after electroweak symmetry
breaking between the different components of the scalar multiplets. Also
a mass splitting, typically much smaller (O(100) MeV), is generated
2 λH S,1 + λH S,2 if dF = 2
dF λH S,1 otherwise
(9)
(10)
It is also interesting to study the case where one of the
particles in the loop is a SM state. As either the scalar S or the
fermion F need to be charged under the dark symmetry, only
Footnote 7 continued
radiatively by loops of gauge bosons between the neutral and the charged
components of the SU(2)L multiplets [
27
].
where λH S,1 (λH S,2) is the coupling of the quartic scalar
coupling in Eq. (4) (Eq. 5).
2.2 Majorana dark matter
If the DM particle is in a real representation of a
stabilizing dark sector group, it could be a Majorana particle
ψ ≡ ψL + (ψL )c (keeping the 4component notation). We
consider the simplest case of a Z2 symmetry in the dark
sector and comment on the general case in Appendix A. The
particle content for Majorana DM is listed in Table 2. The
Lagrangian is given by
1
Lψ = 2 ψ (i ∂/ − mψ ) ψ + i F D/ F − m F F F
+ Dμ S † Dμ S − V(S, H )
−
y1 FR S ψ + y2 FL S ψ + H.c. .
If additionally YF = 0 and consequently S and F both
transform according to a real representation, they can be chosen
to be a real scalar and a Majorana fermion F = FR + (FR )c,
respectively, and the fermionic part of the Lagrangian
simplifies to
1 1
Lψ = 2 ψ (i ∂/ − mψ ) ψ + 2 F (i ∂/ − m F ) F
− y F S ψ + H.c.
with y = y1 = y2.
2.3 Standard Model particles in the loop
one of them can be substituted by a SM field. We discuss in
the following the cases of S being the Higgs doublet H , and
F being the lepton doublet L , the righthanded (RH) charged
lepton eR or a righthanded neutrino νR . Interestingly, these
types of leptophilic models have some very nice features:
(i) the absence of charged stable particles; (ii) the
possibility to generate the correct relic abundance by annihilations
into leptons; (iii) an interplay with LFV and leptonic AMMs;
(iv) the possible relation to lepton number violation (LNV)
and neutrino masses; (v) other possible phenomenological
signals at future lepton colliders, like MET searches.
2.3.1 Lefthanded lepton doublet
The quantum numbers of the remaining states are fixed by
demanding that the fermion F in the loop is the SM lepton
doublet L , as can be seen in Table 3. Moreover y1 = 0
in Eq. (1) for Dirac DM (or Eq. (9) for Majorana DM),
because we are now considering only chiral lefthanded (LH)
fermions.
The coupling of the DM to the lepton doublets can lead
to new contributions to LFV processes as well as AMMs of
leptons, which are induced by loop diagrams with the dark
scalar and the DM in the loop. These pose strong constraints
on the flavor structure of the Yukawa couplings. However,
the flavor constraints can be easily circumvented if DM only
couples to the tau lepton.
In general, for direct couplings to leptons, it is possible
to assign lepton number either to the DM particle ψ or the
scalar S. An example with Majorana fermion DM ψ and a
discrete Z2 symmetry (S → −S, ψ → −ψ ) is the
wellknown scotogenic model, proposed in Ref. [
9
] and
extensively studied, e.g., in Refs. [
8, 28–37
]. See also the recent
review on radiative neutrino mass models [
38
]. In this case
lepton number is broken by the combination of the
Majorana mass term of ψ and the operator in Eq. (6). These
interactions generate neutrino masses and lepton mixing at
oneloop order, which significantly constrain the parameter
space of the model. However, in general DD and neutrino
masses decouple, because the LNV coupling in the potential
could be made arbitrarily small without affecting DD. For
fermionic DM, typically, either coannihilations [
20
] or the
freezein mechanism [
39, 40
] need to be invoked in order to
be compatible with low energy constraints, specially the limit
stemming from nonobservation of μ → eγ .
Dark matter
Dark scalar
RH charged lepton
ψ
S
eR
2.3.2 Righthanded charged lepton
If F is the SM righthanded (RH) charged lepton eR, the
quantum numbers are fixed as shown in Table 4. In this case
y2 = 0 in Eq. (1) for Dirac DM (or Eq. (9) for Majorana DM),
because the fermions have RH chirality. As in the previous
case one should expect new contributions to lepton AMMs
and LFV processes. By demanding that the scalar singlet S
has lepton number + 1, the total lepton number is conserved
at the renormalizable level (the term in Eq. (6) is absent) and
consequently no Majorana neutrino masses are induced.
2.3.3 Righthanded neutrino
Dark matter may also couple to righthanded neutrinos νR
with y2 = 0 in Eq. (1) for Dirac DM (or Eq. (9) for Majorana
DM). In this case the quantum numbers are fixed as shown
in Table 5. As all particles in the loop are neutral, the only
possible interactions are with the Z and Higgs bosons via
the mixing of left and righthanded neutrinos. This mixing
is induced after electroweak symmetry breaking by
LνR =
−L Yν νR H˜ − 21 νR MRνRc + H.c..
In this scenario there are two possibilities regarding the nature
of neutrinos: they are Dirac fermions for MR = 0, or
Majorana fermions for MR = 0. In the latter case, Majorana
masses for the active light neutrinos are generated via the
seesaw mechanism. In the seesaw scenario the activesterile
mixing angles are tiny, either due to small Yukawa couplings
or large righthanded Majorana neutrino masses, and thus
the Z penguin contributions and the additional Higgs
penguin contributions are extremely small, which agrees with
Eq. (19) of Ref. [
13
]. A possible wayout is to consider an
inverseseesaw scenario, where the suppression needed to
have small neutrino masses originates from a small LNV
(11)
3 Effective operators for dark matter direct detection
In the following sections we briefly review the effective
operators for DM DD. In Sect. 3.1 we show those involving DM
interactions with quarks, while in Sect. 3.2 we briefly discuss
their NR versions at the nucleon level.
3.1 Wilson coefficients at the quark level
Here we review the necessary notation for the effective
interactions of the DM with the quarks. The effective Lagrangian
at the quark level for a DM fermion ψ is8
ckq Okq +cg Og +c˜g O˜ g +μψ Omag +dψ Oedm,
(12)
8 We do not include twist2 operators involving quarks and gluons in the
effective Lagrangian. These are only generated by box diagrams, which
are absent in our simplified models. They are relevant for example for
wino DM in supersymmetric theories, see e.g. Refs. [
18,43
].
Majorana mass term, and not from small Yukawa couplings
and/or large righthanded Majorana masses.
As DM couples to the SM particles mainly via
neutrinos, this is known as the neutrino portal. It has been studied
in detail for general heavy SM singlet Dirac and Majorana
fermions νR in Ref. [
13
] and also in Refs. [
41, 42
].
2.3.4 Higgs doublet
Finally we consider the case of S being the SM Higgs. This
fixes the SM quantum numbers of the new particles, which
are shown in Table 6. This case is qualitatively different,
because the neutral component of the electroweak doublet
F and the fermion field ψ mix after electroweak symmetry
breaking. The lighter of the two neutral mass eigenstates is
the DM particle. The Yukawa interactions with the Higgs
necessarily induces treelevel contributions to DD via Higgs
and Z boson exchange. Although a treelevel contribution
exists, DD may still be dominated by the looplevel induced
electric or magnetic dipole moments, because they are
longrange interactions.
Leff =
k,q
where ckq are the dimensionful Wilson coefficients with the
quark q, cg and c˜g are the Wilson coefficients for gluon
operators and μψ and dψ magnetic and electric dipole moments.
We implicitly assume that the operators are generated at a
scale above the nuclear scale, ∼ 2 GeV. See Appendix F for
further details.
We focus on the contributions to spinindependent (SI) and
spindependent (SD) operators of photon, Z boson and Higgs
penguins which are not momentum or velocity suppressed.
The latter would yield very small rates, as there is already the
oneloop squared factor at cross section level, 1/(16π 2)2. We
start the discussion with the case of ψ being a Dirac fermion
and later on discuss the case of DM being a Majorana particle.
For SI scattering the relevant dimension6 effective
operators are
q
OSS = mq (ψ ψ )(qq),
q
OVV = (ψ γ μψ )(qγμq),
(13)
q
where q denotes the quark field. OSS is generated by the
gaugeinvariant dimension7 operators (ψ ψ )(Q L H˜ u R ) and
(ψ ψ )(Q L H dR ), where Q L , u R , dR represent the quark
flaq
vor eigenstates. OSS flips chirality and it is generated by
Higgs exchange and thus we factor out the quark mass mq .
q
OVV preserves chirality and is generated by photon or Z
exchange. The contribution from the photon penguin can
be related to the anapole moment ψ γ μψ ∂ ν Fμν and the
(nongauge invariant) millicharge operator ψ γ μψ Aμ via
the equations of motion for the photon.
There are also scatterings of the DM with gluons at
twoloop order which generate the dimension7 operators:
αs (ψ ψ )Gaμν Gaμν ,
Og = 12π
O˜ g = 8απs (ψ ψ )Gaμν G˜ aμν ,
(14)
where a = 1, . . . , 8 are the adjoint color indices, αs is the
strong coupling constant, Gμν the gluon field strength
ten1
sor and G˜ μν ≡ 2 μνρσ Gρσ its dual. Og is induced from
q
OSS after integrating out the heavy quarks. We explicitly
factorized out a loop factor, as these operators can never be
generated at tree level.
For SD interactions the relevant dimension6 effective
operators are
whereq σμν = 2i [γμ, γν ]. Only the Z boson contributes
to OAA. In SM effective theory the tensor operator may
arise from one of the dimension7 operators (ψ σ μν ψ )(Q L H˜
σμν u R ) and (ψ σ μν ψ )(Q L H σμν dR ) which are however not
induced at leading order.
Photon penguins also generate longrange interactions
which are described by the magnetic (CPeven) and electric
(CPodd) dipole moments of the DM ψ , namely
e e
Omag = 8π 2 (ψσ μν ψ)Fμν , Oedm = 8π 2 (ψσ μν i γ5ψ)Fμν ,
(16)
(17)
with μψ and dψ the coefficients of the magnetic and electric
dipole moment operators introduced in Eq. (12), respectively.
The latter are generated radiatively and therefore it is
convenient to factorize a loop factor.
In the case of a Majorana DM particle there are only
operators with the bilinears ψ ψ , ψ γ5ψ and ψ γ μγ5ψ , so that the
vector OVqV, the tensor OTT and the dipole moment
operaq
tors, Omag and Oedm, vanish identically. Thus, for SI
scattering only the Higgs penguin which generates OSqS is present.
q
For SD scattering OAA generated by the Z boson can also
be nonvanishing. In this case we also compute the photonic
contribution to the anapole operator
q
OAV = (ψ γ μγ5ψ )(qγμq),
which gives rise to momentumsuppressed and
velocitysuppressed NR operators (both SI and SD). See also Ref. [
44
]
for a study of the phenomenology of Majorana DM in EFT.
In general the penguin contributions are isospinviolating,
i.e., with different couplings to protons and neutrons ( fn =
f p). This isospin violation is maximal for photon
contributions which only couple to protons. The latter dominate the
DMnucleus scattering via the dipole moments μψ and dψ .
Hence for SI DMnucleus scattering the enhancement due to
coherent scattering is Z 2 instead of A2 with Z ( A) being the
number of protons (nucleons) of the nucleus.
3.2 Nonrelativistic Wilson coefficients at the nucleon level
The previous Wilson coefficients at the quark level generate
nontrivial Wilson coefficients at the nucleon level [
45–47
].
The different contributions generally interfere. The matrix
elements of DMnucleon scattering can be written as a linear
combination of the following relevant NR operators
N
O1 = Iψ IN
N iq
O5 = Sψ · v⊥ × m N
N
O4 = Sψ · SN
IN
N
O6 =
q
Sψ · m N
N
O9 = Sψ ·
iq
m N × SN
N
O8 = Sψ · v⊥ IN
N iq
O11 = − Sψ · m N
IN
in the convention of Ref. [
48
]. Iψ (IN ) denotes the identity
operators for DM (nucleons), Sψ (SN ) denotes DM (nucleon)
spin, and q and v⊥ describe the momentum and velocity
exchange. We use DirectDM [
48
] to match the
simplified models onto the NR operators. The numerical
expressions for the matching to NR operators are collected in
Appendix F. The NR Wilson coefficients may depend on the
transferred momentum q. Note the different normalizations
of the spinors and the effective operators between Refs. [
45–
47,49
] and Refs. [
48,50–52
]. In addition to the different
definitions of the quark and nucleonlevel operators, in order
to translate between these conventions one needs to multiply
the NR Wilson coefficients of Refs. [
48,50–52
] by 4 mψ m N
(4 mψ q2) in the case of contact (longrange) interactions.
Further details can be found in the recent Refs. [
48–52
]. The
differential cross section for DM scattering off nuclei is given
in Appendix B.
4 Analytical results
The effective operators in Eq. (12) are generated at oneloop
order from penguin diagrams mediated by the photon and the
Z and Higgs bosons. We have computed the different
contributions using the Mathematica packages FeynRules [
53
],
FeynArts [
54
], FormCalc and LoopTools [
55–57
],
ANT [
58
] and Package X [
59,60
]. As we show below,
although the longrange interactions are expected to
dominate, the shortrange effective operators become relevant in
some cases. One obvious example is DMnucleus scattering
of Majorana DM, since the dipole moments vanish.
Therefore we show below all relevant contributions.
The interesting SI (SD) interactions in Eq. (12) are given
by the dipole moment operators Omag and Oedm as well as the
operators OSqS, Og and OVV ( q
q OAA). All the other operators
in Eq. (12) are suppressed in the limit of small momentum
transfer by a factor q2/m2N or q2/m2 , where m N is the
ψ
nucleon mass. In the following we express the SI and SD
Wilson coefficients in Eq. (12) in terms of the ratios
xψ ≡
mψ
m S
and the loop function
and
xF =
m F
m S
g xψ , xF
=
ln
1−xψ2 +xF2 + xψ4 +(1−xF2 )2−2xψ2 (1+xF2 )
2xF
x ψ4 + (1 − x F2 )2 − 2x ψ2 (1 + x F2 )
(18)
q
SN · m N
(19)
(20)
(21)
It is convenient to define the vector and axial Yukawa
couplings:
1
yV ≡ 2 (y1 + y2) ,
1
yA ≡ 2 (y2 − y1) .
(22)
Similarly, the interaction of the fermion F with the Z boson
in Eq. (1) may be written in terms of vector and axialvector
couplings, namely
4.1.1 Electromagnetic dipole moments
The magnetic and electric dipole moments are given by
(26)
and
(27)
(28)
LFZ = cwesw Z μ F γμ (zV − z A γ5) F,
where e > 0 is the proton electric charge, and sw (cw) denotes
the sine (cosine) of the weak mixing angle. If F is a
vectorlike fermion, then we have
zV = cw2 Q − YF ,
z A = 0,
where Q is the electric charge of the (component of the) field
F , in units of e, and YF is the corresponding hypercharge.
Conversely, for a SM lepton F we have
1
zV = 2
1
(1 − 2sw2) Q − YF , z A = 2 (Q − YF ) ,
and the Yukawa couplings are
yV = yA =
yV = − yA =
y2
2
y1
2
if S is a doublet of SU(2)L,
if S is a singlet of SU(2)L.
For simplicity of notation we report the full analytic results
for SU(2)L singlets F and S. In the case of no mass splittings
between the components of the SU(2)L multiplets of
dimension dF it is straightforward to generalize the results: The
expressions for photon penguins and electric and magnetic
dipole moments are generalized by replacing Q → dF YF .
Higgs penguins are generalized for different scalar multiplets
as in Eq. (8).
Most Z penguin contributions (apart from some with
chiral SM fermions) vanish at leading order. This is also the case
for other SU(2)L multiplets.
We summarize below the relevant contributions to the
(Dirac or Majorana) DM–quark scattering amplitude. We
have checked that our expressions agree with those reported
in the literature in the appropriate limits: dipole and anapole
moments in Refs. [
5,6,8,10
], and also for the Z boson
contributions in Refs. [
10,15
].
4.1 Dirac dark matter
The leading contributions for Dirac fermion DM are from
dipole moments, the operators OV V and OqA A, and the scalar
q
operator OqSS . Integrating out heavy quarks induces the gluon
operator Og.
(29)
(30)
(31)
(32)
Q
dψ = − 2 x ψ2 m S Im[yV y∗A] xF
× ln xF +
1 + xψ − x F2 g xψ , xF .
2
Both Omag and Oedm flip chirality and therefore the
dominant contributions to their coefficients are proportional to
the heaviest fermion mass, either mψ or m F . In the limit
mψ m F < m S these expressions reduce to
Q
μψ ≈ − 4 m S
Q
+ 8 m S
yV 2 − yA2 xF
yV 2 + yA2 xψ
2
1 − xF + 2 ln xF
2 2
(1 − xF )
1 − x F2 (x F2 − 4 ln xF ) ,
2 3
(1 − xF )
Q
dψ ≈ − 2 m S Im[yV∗ yA] xF
4.1.2 Photon penguin
2
1 − xF + 2 ln xF
2 2
(1 − xF )
.
Photon penguins induce the operator OVq V . The relevant
Wilson coefficient in the effective Lagrangian (12) is
q αem 1
cVV = − 24 π x ψ4 m2 Q Qq yV 2
S
×
+
− 3x ψ6 + 6x ψ5 xF + 12xψ xF (1 − xF )
2 2
+ 8(1 − xF ) + 2x ψ4 (5 + xF )
2 3 2
− 6x ψ3 xF (1 + 3x F2 ) − 3x ψ2 (5 − 2x F2 − 3x F4 )
g xψ , xF
× 1 − (xψ − xF )2
2x ψ2 (4 − 3x ψ2 + 6xψ xF − 4x F2 )
1 − (xψ − xF )2
+ (8 + xψ − 4xψ xF − 8x F2 ) ln xF
2
+ yV → yA, xψ → −xψ , xF → xF ,
(33)
where Qq is the electric charge of the quark q in units of
e > 0. In the limit mψ m F < m S the expression above
reduces to
of small DM mass, xψ
neutrinos is
1, the contribution of righthanded
×
.
For a vectorlike fermion the resulting SI and SD
scattering amplitudes are suppressed by q2/m2F and q2/m2S due
to a cancellation between the diagrams where the Z boson
couples to the scalar and to the fermion. Therefore, no strong
constraints on the model parameters can be obtained. For SM
leptons in the loop we distinguish two cases:
(i) If S is a singlet under SU(2)L, the axialvector
coupling in Eq. (27) is z A = 0 and both SI and SD scattering
amplitudes are suppressed as for a vectorlike fermion.
(ii) If S ≡ (S0, S−)T is a doublet under SU(2)L, there
are contributions from both diagrams where the Z boson is
attached to the SM lepton or the scalar in the loop
q
cVV =
f ={ ,ν}
×
(1 + 2 Q f )αem qV x 2f
16 π cw sw m2Z e x ψ2 y22
x 2f − 1 − x ψ2
g xψ , x f − ln x f ,
q q
cAA = cVV (qV → qA).
The couplings qV,A are qV /e = 3 − 8sw2/(12cwsw) and
qA/e = −1/(4cwsw) for uptype quarks and qV /e =
−3 + 4sw2/(12cwsw) and qA/e = 1/(4cwsw) for downtype
quarks. Q f denotes the electric charge of the lepton. We
define xψ ≡ mψ /m S− , x ≡ m /m S− and xν ≡ mν /m S0
with the charged lepton mass m and the neutrino mass mν .
This agrees with the expression in Ref. [
10
]. The contribution
with light active neutrinos in the loop is negligible because
it is proportional to xν2 and thus the contribution is entirely
determined by the charged lepton in the loop. However, for
models with a neutrino portal as outlined in Sect. 2.3.3 there
may be a sizable contribution from righthanded neutrinos
(mixed with lefthanded neutrinos) in the loop. In the limit
(34)
(35)
(36)
(37)
q xN2
cVV,N = 16απemcwssinw2mθ2Z qeV y22 (1 − xN2 )2
q q
cAA,N = cVV,N (qV → qA)
2
1 − xN + 2 ln xN
(38)
(39)
with the activesterile mixing angle θ . We define xN ≡
m N /m S with the heavy neutrino mass m N . These
interactions are also discussed in Ref. [
13
] (see also Ref. [
61
]). In
the case of mixing of vectorlike charged fermions with SM
charged leptons, there is an overall minus sign in the
expressions of Eqs. (38) and (39).
4.1.4 Higgs penguin
At leading order in q2 there is only the contribution to the
SI scattering amplitude. The relevant Wilson coefficient
generated by the Higgsportal interaction is
cSqS = − 16 πλ2HxψS3 m2h m1S yV 2
× xψ2 + 1 − xψ2 − xψ xF − xF2 ln xF
2 2
+ 1 − xψ − 2xψ xF − xF
2 2
1 − xψ + xψ xF − xF
×g xψ , xF
− yV → yA, xψ → −xψ , xF → xF .
(40)
As previously mentioned we neglect the contribution from
the Higgs penguin where the Higgs boson couples to a SM
lepton in the loop, because it is suppressed by (m /v)2 1.9
q
As in the case of Omag and Oedm, the operator OSS flips
chirality, and therefore the dominant contribution to its Wilson
coefficients is proportional to either mψ or m F . If both F
and ψ are charged under U(1)dm, then mψ < m F and thus
the largest contribution comes with the chirality flip on the
fermion line of F . On the contrary if F is a SM lepton the
largest Higgs contribution is proportional to mψ . In the limit
mψ m F < m S, Eq. (40) simplifies to
9 For the case of SM leptons in the loop this other contribution of the
Higgs coupling to the leptons is given in Ref. [
10
]. For the neutrino
portal these interactions are given in Ref. [
13
].
.
For Majorana DM ψ the electromagnetic dipole moments
identically vanish and the only allowed electromagnetic form
factor is the anapole moment. This gives rise to the effective
q
operator OAV in Eq. (17). We obtain
q
cSS ≈ − 16 π 2 m2h m S (x F2 − 1)2
λH S
cAqV = − 2QπQxqψ2αmem2 Re[yV∗ yA] ln xF
S
x ψ2 2
+ 1 + 3 − xF
.
In the limit mψ
m F < m S this simplifies to
q
cAV ≈
In the case the mass of the fermion in the loop is much smaller
than the momentum transfer, m F −q2, we have
q Q Qq αem Re[yV∗ yA]
cAV = − 18 π m2
S
2x ψ2 (5 − 6 ln xq ) + 3(3 + x ψ2 ) ln 1 − xψ ,
2
x ψ2 (1 − x ψ2 )
where xq ≡
For a vectorlike fermion F in the loop the Z penguin
diagram does not contribute to the SI scattering amplitude,
because the DM vector current identically vanishes for
Majorana fermions. The SD scattering amplitude is suppressed by
q2/m2F and q2/m2S due to a cancellation similar to that
occurring in the case of Dirac fermion DM, see Sect. 4.1.3.
If F is a lefthanded lepton doublet, and consequently S ≡
(S0, S−)T is an SU(2)L doublet, we find at leading order in
q2: cVqV = 0 and cAqA is a factor of two larger than result for
(41)
(42)
(43)
(44)
the Dirac case provided in Eq. (37). If F is a righthanded
charged lepton or a righthanded neutrino, the scalar S is
necessarily an SU(2)Lsinglet and thus the axialvector
coupling z A in Eq. (27) is zero and both SI and SD scattering
Z mediated amplitudes are suppressed as for a vectorlike
fermion. In some models, like with righthanded neutrinos
or with vectorlike fermions, there can be mixing with SM
leptons. These generate couplings to the Z and the Higgs
bosons, see discussion around Eqs. (38) and (39), and
footnote 9.
For the Higgs penguin there is only a contribution to the SI
amplitude cSqS at leading order in q2, which again is a factor
of two larger than in the Dirac DM case, given in Eq. (40).
The fact that the h and the Z penguin contributions to the
nonzero Wilson coefficients, cSqS and cAqA, are a factor of 2 larger
for Majorana than for Dirac DM, can be understood from the
presence of extra crossed diagrams for Majorana particles,
where the initial and final DM particles are interchanged.
5 Numerical analysis
We use LikeDM [
62,63
] to compute the differential rates and
the experimental upper bounds on our scenarios. We have
also performed cross checks with the program of Ref. [49].
First we show results for the event rates and upper limits for
Dirac and Majorana DM, having either vectorlike fermions
or SM leptons in the loop. For the latter case we also show
upper limits from LFV signals. In the following we
parameterize the vector and axial Yukawa couplings of Eq. (24) in
terms of their absolute value and phase as yV = yV eiφV
and yA = yAeiφA .
5.1 Wilson coefficients at the quark level
In order to illustrate the relative weight of the different
contributions, we plot in Fig. 2 the long and shortrange
contributions with uptype quarks for vectorlike fermions (upper
panel) and for a SM lefthanded lepton doublet (lower panel)
in the loop. The plots on the left correspond to Dirac DM,
while the plots on the right are for Majorana DM. Unless
otherwise stated we always set the dark charge Qψ to one and fix
the Higgs portal coupling, λH S = 3. The Wilson coefficients
of the shortrange interactions (dimension6 operators) have
been rescaled by the nuclear magneton μN = e/(2m p) to
compare them to the (dimension5) dipole moments.
For Dirac DM with vectorlike fermions in the loop (top
left) we show the magnetic moment μψ (in solid green), the
dipole moment dψ (dashed orange), as well as the shortrange
contributions mediated by the photon (dotdashed blue) and
the Higgs (dotted purple). We have fixed m F = 600 GeV,
m S = 500 GeV, yV = 1 and yA = 1.3 e1.4 i . The DM
electric dipole moment dψ is around 10−4 fm and it
dominates, followed closely by the magnetic moment. The Higgs
and the shortrange photon interactions are always very
suppressed, below 10−9 fm. All Wilson coefficients increase for
mψ ∼ m F + m S (not shown as we demand ψ to be the
lightest particle charged under U(1)dm), when the particles in the
loop are almost onshell. For this example the Wilson
coefficients μψ and cSuS change sign at particular values of the DM
mass and thus there is a dip in their absolute magnitude. The
case of Majorana DM with vectorlike fermions (top right)
only shows the shortrange Higgs and photon contributions,
the latter being the anapole moment (dotdashed dark blue).
These Wilson coefficients are of similar size as in the Dirac
case, although the photon anapole (Higgs) contribution is
smaller (a factor of two larger) than the photon shortrange
(Higgs) Wilson coefficient of the Dirac case.
For Dirac DM with SM lepton doublets in the loop
(bottom left) we show the magnetic moment μψ (in solid green),
the shortrange contributions mediated by the photon
(dotdashed blue), the Z penguin SI (dashed brown) and SD
(dashed red) scattering and the Higgs penguin (dotted
purple). The electric dipole moment dψ vanishes at oneloop
order. We have fixed m S = 1000 GeV and yV = yA =
y2/2 = 1/2. In the case of the (light) SM leptons in the loop,
the photon penguin contribution cVuV,γ depends on the
transferred momentum √2m A E R ,10 for which we use E R = 8.59
keV (which is a reasonable value for xenon nuclei, with mass
mXe 132 GeV). The magnetic dipole moment dominates,
followed by the photon shortrange contribution which is
roughly ∼ 10−8 fm. The increase of μψ and the Higgs
contribution with mψ is easily understood from chirality
arguu
ments. This also implies that μψ and cSS are suppressed
with respect to the case of vectorlike leptons (cf. upperleft
panel of Fig. 2) by the DM mass, except in the region of mψ
close to m S. The Higgs and the Z penguin interactions are
always very suppressed (for the Z penguin the SD amplitude
is smaller than the SI contribution, due to the factors qV,A/e
in Eqs. (36) and (37)), below 10−11 fm, and therefore they
can be safely neglected. All Wilson coefficients increase for
mψ ∼ m F + m S.
For Majorana DM with SM lepton doublets in the loop
(bottom right) the Higgs and the Z SD amplitudes are a
factor of two larger than in the Dirac case and with the same
dependence on mψ , while the anapole Wilson coefficient
(dotdashed purple) is slightly larger than the photon
shortrange contribution present in the Dirac case. Notice that this
is the opposite behavior of the case with vectorlike fermions.
5.2 Wilson coefficients at the nucleon level
The previous Wilson coefficients at the quark level can
interfere and generate nontrivial effective operators at the
10 m A is the nucleus mass and E R the recoil energy.
nucleon level, see Sect. 3.2. We plot in Fig. 3 the NR Wilson
coefficients with protons (neutrons) in dotted (dashed) lines
(N = n, p for neutrons and protons). All Wilson coefficients
are displayed in dimensionless units, by rescaling them with
the square of the electroweak VEV, v = 246.2 GeV. As for
Fig. 2, the upper panel is for vectorlike fermions and the
lower panel for SM lefthanded lepton doublets. The plots
on the left correspond to Dirac DM, while the plots on the
right are for Majorana DM.
For a vectorlike fermion F (upper panels of Fig. 3), we fix
m F = 600 GeV, m S = 500 GeV, yV = 1 and yA = 1.3 e1.4 i .
For Dirac DM (left plot), we show the coefficients
shortrange SI c1N (black) and the SD scattering c4N (blue), and
the longrange contributions c5N (red), c6N (orange) and c1N1
(green). Notice that both c5N and c1N1 are generated by the
electric and magnetic DM dipole moments proportionally to the
nucleon charge and they are therefore absent for neutrons.
The longrange Wilson coefficients cN , c6N and c1N1
domi5
nate. The SD coefficients c4N are more than two orders of
magnitude smaller and very similar for protons and neutrons,
although slightly larger for the former. The SI coefficients c1N
p
are the smallest ones, and c1 decreases with the DM mass up
to mψ 500 GeV. The difference in behavior of c1p and c1n
stems mainly from the nonzero contribution of μψ to the
former (c1p ∝ μψ /mψ ). In this example the Wilson coefficients
c1N change sign at about mψ = 500 GeV. For Majorana DM
with vectorlike fermions (top right) the c1N contributions
generated by the Higgs penguin diagram (black) are very
similar for protons and neutrons (they are superimposed in
the plot). The anapole moment generates c8p (solid purple),
c9p (solid magenta) and c9n (dashed magenta) which are very
similar, specially c8p, to the c1N contributions (black). All the
Wilson coefficients are in the range 10−4 − 10−3, except in
the region of DM mass when a Wilson coefficient changes
sign.
For Dirac DM with SM leptons (bottom left) the
phenomenology is very rich. The longrange Wilson coefficients
c5p (dotted red), c6p (dotted orange) and c6n (dashed orange)
dominate (c5n = 0, as it is proportional to the electric charge
of the nucleon). They increase with the DM mass as they
are generated dominantly by μψ , i.e., chirality needs to be
violated. Similarly c4N (blue) with p (dotted) and n (dashed)
have a dominant contribution from μψ and therefore increase
with mψ . Regarding c1n (dashed black), the increase of its
slope reflects the fact that the shortrange Higgs
contribution (which grows with mψ ) increasingly becomes more and
more comparable to the photon shortrange coefficient, but
in any case c1n remains very suppressed. c1p is dominated
by the photon penguin, and both the shortrange
contribuq
tion parameterized by cV V,γ and the longrange
contribution from the magnetic moment μψ /mψ are important. Due
the dependence of the quarklevel Wilson coefficients on the
DM mass, the NR Wilson coefficient c1p is basically
connuclear magneton μN = e/(2m p). The photon penguin contribution
u
cVV,γ depends on the transferred momentum q for light SM leptons:
We choose a recoil energy E R = 8.59 keV for 15342Xe which results in
q2 = 2.11 × 10−3 GeV2
stant with respect to it. For Majorana DM with SM leptons
(bottom right) the Wilson coefficients c8p (solid purple), c9
p
(solid magenta) and c9n (dashed magenta), which are
generated by the anapole operator, dominate. c6N (c6p and c6n are
superimposed in the plot) do not increase with the DM mass,
unlike in the Dirac case, because here they come from cANA
and not from μψ ; c1N , generated by the Higgs penguin
diagram, increases with mψ and is very similar for n and p (c1n
p
and c1 are superimposed in the plot). Finally, c4N , generated
by the Z penguin, are similar for both n and p (superimposed
in the plot) and very suppressed, as expected.
The different Wilson coefficients are expected to generate
different features in the DD differential spectrum. In the
upperleft panel of Fig. 4 we plot the DD differential event rates
in xenon versus the recoil energy E R for Dirac DM with a
vectorlike fermion F (solid blue) and with a righthanded
tau (dotted green), and for Majorana DM with a vectorlike
fermion (dashed red) and with a righthanded tau lepton
(dotdashed purple). For details on the astrophysical assumptions
used in the numerical analysis see Ref. [
63
]. The rate for
Dirac DM with a vectorlike fermion is roughly 9 orders of
magnitude larger than that with a SM lepton (a tau lepton
in this case), because in the latter case the magnetic dipole
471
moment μψ is suppressed by the DM mass mψ . The
smallest rate occurs for Majorana DM with a righthanded tau in
the loop. The relative size of the spectra is obvious from the
relative size of the NR Wilson coefficients discussed in the
previous section.
In the upperright panel we show the spectrum normalized
to the maximum value (5.7 × 105 [9.7 × 10−4] t−1 day−1
keV−1 for Dirac [Majorana] DM) for the case with
vectorlike fermions in the loop, for Dirac DM (solid blue) and
Majorana DM (dashed red). The spectral shapes are quite
different, which is mainly due to the fact that there is no
magnetic moment for Majorana DM.
In the bottom panel of Fig. 4 we plot the DD differential
rates for Dirac DM with coupling to a righthanded electron
(solid blue), muon (dashed red) and tau (dotted green). The
coefficients are displayed in dimensionless units by rescaling with the
square of the electroweak VEV v = 246.2 GeV
spectra are the largest for the electron (the lightest lepton),
with maxima at roughly the same recoil energy. The maxima
go approximately in the ratios ∼ (4 : 2 : 1) for e, μ, τ . This
is due to the dependence on the shortrange contribution of
q
the photon penguin, cV V,γ , via the NR Wilson coefficient
c1p. The spectqra are dominated by the photon shortrange
contribution cV V,γ for this choice of parameters.
5.4 Direct detection limits
Next we study the upper limits that current DD experiments
can impose on the scenarios discussed so far. In order to
illustrate current direct detection limits, we consider different
scenarios of TeVscale dark sectors. We also discuss how the
limits vary with the masses of the particles in the loop. We
F and a scalar S in the loop we fix m F = 600 GeV, mS = 500 GeV,
yV = 1 and yA = 1.3 ei 1.4. In the case of a righthanded τ lepton we
fix mS = 1000 GeV and y1 = 1
show the 90% C.L. upper limits from current DD experiments
that have xenon as a target, which provide the most stringent
limits for SI interactions for our range of DM masses. We
show mψ 5 GeV, as very light DM does not produce recoils
at energies above the threshold of the DD experiments. The
limits are subject to large uncertainties from nuclear physics
and astrophysics as well as from experimental uncertainties.
In the following we do not show limits from Higgs and Z
boson invisible decay widths into DM, as those are weaker
than the ones coming from DD in our scenarios. In Sect. 6.2
we discuss some examples where these limits can be relevant,
and complementary to DD, specially for light DM masses,
and in Appendix C we provide the relevant expressions for
the Higgs and Z boson invisible decay rates.
In the left panel of Fig. 5 we plot the upper limits for
Dirac DM in the plane yV  versus mψ for XENON1T (solid
brown), PandaX (dashed green) and LUX (dotted purple),
together with their combined limit (thicker solid red line). We
have fixed m S = 500 GeV, m F = 600 GeV, λH S = 0.1, the
ratio of Yukawa couplings yA/yV  = 1.3 and the phases
of the Yukawa couplings φV = 0 and φA = 1.4. As expected
the bounds are weakened at very large and very small DM
masses. At large DM masses the limits appear to approach
a constant value, instead of decreasing as 1/mψ as expected
from the DM number density. This is due to the nontrivial
dependence of the Wilson coefficients on mψ . In particular
the Wilson coefficients generally increase for mψ → m F +
m S. The yV  limits are of the order of ∼ 10−2 for a large
φV = 0 and φA = 1.4. Unless specified the Higgs portal coupling is
λH S = 0.1. We highlight in gray the region where the Yukawa coupling
is nonperturbative, yV  > √4π
range of DM masses between 10 GeV and 500 GeV. This
is a clear example of the superb sensitivity achieved by DD
experiments, which are able to probe such small Yukawa
couplings for loopinduced scenarios of Dirac DM.
In the right panel of Fig. 5 we show the limits for Majorana
DM with λH S = 0.1 (dashed red) and λH S = 3 (dotted
green), together with those for Dirac DM (solid blue). The
current limits for Majorana DM are very weak, close to the
naive perturbativity limit. Notice that the Higgs interactions
are nonnegligible: changing λH S = 0.1 to λH S = 3 the
upper bound on the Yukawa couplings improves by a factor
of ∼ 6 (at the level of the rate, the scalar quartic coupling
enters quadratically, while the Yukawa couplings enter to
the fourth power). The difference with respect to the strong
limits for Dirac DM stems, of course, from the absence of
dipole moments for Majorana DM. In the gray shaded region,
the Yukawa coupling is nonperturbative, yV  > √4π , and
therefore the oneloop computation cannot be trusted.
5.5 Interplay with lepton flavor violation and relic
abundance
When there are SM charged leptons running in the loop,
there may also be limits from LFV processes. We provide the
relevant expressions for α → β γ , μ − e conversion and
α → β γ δ in Appendix D.11 It is therefore interesting to
study the interplay between both types of signals. Although
one may naively expect that LFV limits are stronger (because
an accidental symmetry of the SM is violated), we see in the
following that this is not the case in all scenarios.
11 In the following, we do not show results for LFV Higgs and Z boson
decays, as the experimental limits on these are weaker than limits from
leptonic LFV decays.
In Fig. 6, topleft panel, we plot the DD upper limits12 in
the plane y1 versus mψ , assuming equal couplings to all
leptons, i.e., y1e = y1μ = y1τ = y1 (we denote this the
“symmetric” case). In Fig. 6 topright, middleleft and middleright
panels we show the cases of no couplings to taus, electrons
and muons, respectively. Lefthanded and righthanded
leptons in the loop lead to the same result. We show the cases
of Dirac DM (solid red), and Majorana DM with λH S = 0.1
(dashed light blue) and λH S = 3 (dashed green). We have
fixed m S = 1000 GeV for the four upper plots. The most
relevant 90% C.L. LFV limits are shown using dotted lines:
μ → eγ (green), μ−e conversion (orange), μ → 3e (black)
and aμ (brown).13 Notice that LFV limits do not depend
on whether the DM is a Dirac or Majorana fermion. Also,
we emphasize once more that DD limits are subject to large
astrophysical and nuclear uncertainties, which are absent in
the case of LFV experiments.
In addition we plot the contour of the DM relic abundance,
set by tchannel DM annihilations ψ ψ → α β mediated by
the scalar S, with a dotdashed navy blue (purple) line for
Dirac (Majorana) DM, whose leading contribution is from
swave (pwave) scattering. We use the instantaneous
freezeout approximation which is sufficient for our purposes (see
Sect. 6.3.1 and Appendix E for more details and the
relevant expressions). Above the h2 contour the DM would be
under abundant and requires an additional component of DM
to account for the observed relic abundance. Below the h2
12 In the following we only show the combined limit from all xenon
experiments, like the thicker solid red line shown in the left panel in
Fig. 5, but for the case of SM leptons in the loop.
13 This corresponds to the 4σ limit coming from the AMM of the muon
aμ. This discrepancy with respect to the SM cannot be explained in
our model, because the additional contribution is negative and thus leads
to a larger departure from the experimental value.
annihilations mediated by the scalar S are shown as dotdashed navy
blue (purple) line for Dirac (Majorana) dark matter. The dotted lines
indicate constraints from the relevant LFV processes. In the gray shaded
region the Yukawa coupling is nonperturbative, y1 > √4π
contour DM is over abundant if its abundance is solely set by
freezeout, and thus there has to be a mechanism to further
deplete its density. It could be reduced via coannihilation
and resonant effects [
20
], multibody scatterings [
64–68
], or
a nontrivial thermal evolution in the early universe [
69
]. In
case ψ does not account for all of the DM abundance the DD
limits have to be rescaled appropriately. Assuming thermal
freezeout reproducing the correct relic abundance imposes a
lower bound on the DM mass. In the case of equal couplings
to all leptons with m S = 1000 GeV, mψ 10 (25) GeV
for Dirac (Majorana) DM. When one final channel is closed,
the lower limits increase by roughly 5 (15) GeV for Dirac
(Majorana) DM. For light scalar mass (see bottomleft panel
of Fig. 6), all Yukawa couplings are perturbative. However,
for heavy m S, bottomright panel, the Yukawa couplings are
perturbative only for very heavy masses, above 0.4 (1) TeV,
as in this case the tchannel interaction is significantly
suppressed by the mass of the mediator.
The main changes in the case of no couplings to taus,
electrons and muons (topright, middleleft and middleright
panels in Fig. 6) are in the LFV limits, as depending on the
flavor structure, different processes are possible. In these panels
the relic abundance contours are almost identical, as the SM
leptons are always much lighter than the DM (and therefore
phase space plays no significant role). Of course, the contours
are at somewhat larger Yukawa couplings than for the
“symmetric” scenario, as in the latter there were more available
annihilation channels. The DD limits are also slightly
modified due to the different masses of SM leptons in the loop (see
also the lower panel of Fig. 4). When there are no couplings
to taus (topright panel), the LFV limits are almost identical
to the “symmetric” scenario, because they are driven by the
first family. However, for no couplings to electrons or muons
(middle panels), DD limits are more stringent than LFV
limits for Dirac DM with a mass above a certain value. This is
quite remarkable: DD experiments are able to better constrain
scenarios where an accidental symmetry of the SM is
violated than experiments directly searching for it. Interestingly,
limits on y1 from trilepton τ decays (τ → 3 ) dominate
over radiative τ decays (τ → γ ) in contrast to the limits
from muon decays. As the limits from τ decays are generally
weaker and thus the corresponding Yukawa couplings larger,
boxdiagram contributions to trilepton decays may give a
sizable contribution and thus break the dipole dominance.
A few interesting remarks can be drawn from these plots.
First, note that the DD limits with SM leptons in the loop,
even for Dirac DM, are much weaker than in the scenario with
vectorlike fermions in the loop, as also demonstrated in the
topleft panel of Fig. 4. Second, clearly the LFV limits are
the strongest ones, with μ → eγ the most stringent among
them. Its limit on the Yukawa coupling y1 is a factor of a
few stronger than the one of DD for Dirac DM. Again, the
DD limits become very strong close to mψ → m S +m F as in
the case with vectorlike fermions. Third, for scalar masses at
the TeV scale, the DD limit already excludes the production
via thermal freezeout for Majorana DM, and also for Dirac
DM in the mass range 5 GeV mψ 200 GeV. Finally, the
muon AMM constraint is always very weak, being the limit
above the perturbativity bound.
In Fig. 6, bottom panels, we show two examples of a scalar
S in the loop with a different mass: m S = 300 GeV (left plot)
and m S = 5000 GeV (right plot). All limits are generically
stronger for m S = 300 GeV and weaker for m S = 5000
GeV compared to m S = 1000 GeV. In particular, the relative
contribution of the box diagrams and the dipole moment for
the trilepton τ decay changes: for m S = 300 GeV τ → eγ
sets a stronger limit than τ → 3e. Similarly the Yukawa
coupling required to explain the observed relic abundance
also has to be larger for heavier scalar masses, as already
discussed. Indeed, for m S = 5000 GeV almost all the limits
on the Yukawa couplings are in the nonperturbative region.
In summary, strong limits can be set for Dirac DM with
vectorlike fermions in the loop. For Dirac DM with SM
leptons in the loop LFV limits or DD limits may set the
strongest bounds depending on the flavor structure and the
DM mass. Therefore, the two limits are complementary: LFV
limits are more important for DM coupling to both muons
and electrons, whereas DD limits dominate if there are no
LFV processes of type μ → e X , X being anything, and
the DM mass is not too small (mψ 5 GeV). For Majorana
DM, LFV limits, if present, are generally more stringent than
constraints from DD. Future DD experiments and LFV
limits on τ decays are expected to improve by 1–2 orders of
magnitude and hence the situation is not expected to change
dramatically. If μ − e conversion in nuclei and/or μ → 3e
expected sensitivities (by several orders of magnitude) are
achieved, LFV limits will continue to dominate and even
increase their difference with respect to DD.
6 Other phenomenological aspects
6.1 LHC searches
Generally colliders may only set competitive limits via
missing energy searches for light DM and SD interactions. In the
scenarios discussed here, naively the production of DM
particles at the LHC occurs at oneloop level via the penguin
diagrams in Fig. 1 and is therefore suppressed. For example,
Ref. [
25
] showed that there are only very weak collider limits
on a model with a magnetic moment interaction. Thus it is
more promising to search for the mediators S and F at
colliders via qq¯ → F F¯ , S S∗ mediated by the photon, the Z boson
and/or the Higgs. If the new fermion and scalar have electric
charge, the production is dominated by the DrellYan
process. Higgsmediated production of exotic particles has been
Fig. 7 Branching ratios of the Z and the Higgs bosons decaying
invisibly into DM (Dirac in solid, Majorana in dashed). We show in black the
case of vectorlike fermions in the loop, in red the case of a taulepton
doublet and in blue the case of taulepton singlet. The experimental
upper limits on nonSM invisible decays are displayed as horizontal
gray lines. See the text for details
discussed in e.g. Ref. [
70
]. As we are assuming that the new
particles are not colored, only modest lower limits (below 1
TeV) are expected, unless very large SM quantum numbers
(for instance electric charges) are invoked. The dark sector
particles may decay invisibly into DM and a lighter dark
sector state. The phenomenology of these decays are however
modeldependent, see discussion in Sect. 2. Another
interesting option would be to search for DM in models with
electrons/muons running in the loop at future lepton
colliders. The main production process is via t channel exchange
of the scalar, + − → ψ ψ¯ with = e, μ.
6.2 Z and Higgs boson invisible decays
If the DM ψ is sufficiently light [mψ < m H /2 (m Z /2)]
there is an additional contribution to the invisible width of
the Higgs (Z ) boson. In Appendix C we present the
relevant expressions for these processes. We find that there
are no limits from Z or Higgs boson decays into DM for
the parameter values used in Figs. 5 and 6. However, there
may be limits for small scalar/fermion/DM masses and large
Yukawa couplings. To illustrate this point we plot in Fig. 7 the
branching ratios Br(Z → ψ ψ ) (left plot) and Br(h → ψ ψ )
(right plot), for Dirac (Majorana) DM with solid (dashed)
lines. For the SM widths we use h,SM = 4.1 MeV and
Z,SM = 2.495 GeV, such that the Higgs branching ratio
reads Br(h → ψ ψ ) = h→ψψ /( h,SM + h→ψψ ) and
similarly for the Z boson. We show the cases of different
particles running in the loop with solid lines: in black the
case of vectorlike fermions with Q F = YF = −1 and
in red (blue) the case of a taulepton doublet (singlet). For
Higgs decays the taulepton doublet and the singlet
generate the same branching ratio, shown in blue. The
experimental upper limits on invisible nonSM decays are shown
as horizontal gray lines: solid for the Z boson from LEP
(the total invisible width of the Z including neutrinos is
Z→inv = 499.1 ± 1.5 MeV [
71
]), and dotdashed (dashed)
for the Higgs from CMS [
72
] (ATLAS [
73
]), which reads
Br(h → inv) < 0.24 (0.28) at 95% CL. We used m S = 120
GeV and m F = 150 GeV and a Higgs portal coupling
λH S = 0.2. For vectorlike fermions in the loop we used
yV = 4 eiπ/3 and yA = 3 eiπ/4, while for SM taulepton
doublets [singlets] we fixed yV = [−] yA = 4 eiπ/3.
In Fig. 7 one can observe that the limits for Dirac DM are
stronger than those for Majorana DM in the case of Z boson
decays, independently of the particles in the loop, while the
situation is the opposite in the case of Higgs decays. Also,
invisible Z boson decays constrain light DM which couples
to SM leptons (the tau in this case). For Dirac DM the limits
exclude DM masses below 14 (36) GeV in the case of
couplings to tau singlets (doublets). The width is dominated by
cV and cA, while dA 0 and dV is suppressed by mψ . For
vectorlike fermions the width is dominated by dV and dA
with dV > dA, and there are no relevant limits. For Majorana
DM the limits are weaker than for Dirac DM, demanding
mψ 6 (21) GeV in the case of couplings to tau singlets
(doublets), with no limits in the case of couplings to
vectorlike leptons.
As in the case of the Z boson, the decays of the Higgs
boson do not pose limits on the scenario with vectorlike
fermions in the loop. For the taulepton bA = 0 and the
dominant contribution to bV is proportional to mψ , as m F mψ .
The branching ratio increases with the DM mass for low DM
masses, while at some DM mass value ( 40 GeV in the
plot) the phase space suppression dominates and the
branching ratio decreases again. Therefore there is a constraint on
an intermediate DM mass range of [
25, 53
] GeV ([
22, 55
]
GeV) by ATLAS (CMS) for Dirac DM and [
16, 57
] GeV
([
14, 58
] GeV) by ATLAS (CMS) for Majorana DM.
To summarize, while for vectorlike fermions there are no
limits, for SM particles in the loop there may be interesting
constraints in the absence of LFV. Indeed, there is a
wellknown complementarity between invisible decays and DD.
The experimental energy threshold of DD experiments limits
their ability to impose limits for arbitrarily low DM masses
and thus invisible decays may set competitive limits for low
DM masses.
6.3 Relic abundance
The production of the correct relic DM density in the early
universe is generally modeldependent. Although it is not the
main focus of this work, we briefly outline different avenues
to obtain the correct relic density. See e.g. Ref. [
74
] for a
connection of DD with thermal freezeout.
6.3.1 Thermal freezeout
If mψ > m S , m F (but of course mψ < m S + m F ), the
relic abundance can be set via the tchannel interactions
ψ ψ¯ → S S∗ or ψ ψ¯ → F F¯ . Subsequently, S and F can
decay to SM particles, in some cases at loop level or via
nonrenormalizable operators. In particular if F is a SM lepton
α , DM annihilations to SM leptons ψ ψ¯ → α ¯β may set
the relic abundance. For Dirac DM the cross section is not
velocity suppressed and thus the leading (swave) part of the
thermally averaged annihilation cross section14 is given by
m2
ψ
σ v D = 32π(m2ψ + m2 )2
S
α,β
yi,β yi∗,α 2,
where we have summed over all possible final state leptons
(neutrinos and charged leptons) in the limit of vanishing
lepton masses. Here i = 1 (2) for couplings to LH (RH) leptons,
see Eq. (1). For Majorana DM the annihilation cross section
is velocity suppressed and the leading contribution is due to
pwave scattering15
where x = mψ / T .
As discussed in Appendix E, for DM masses in the range
10 GeV mψ 104 GeV we obtain the correct relic
abundance for cross sections σ v D [
2, 3
] · 10−26 cm3 s−1 for
Dirac DM and σ v M [0.5, 1] · 10−23 cm3 s−1 for
Majorana DM. Equating these values to Eq. (45) and Eq. (46),
respectively, we plot in Fig. 6 the relic abundance contours
in the y1 − mψ plane.
If ψ is the lightest particle in the dark sector (i.e., mψ <
m S , m F ), DM may annihilate at oneloop order into quarks
via the penguin diagrams in Fig. 1. However this is very
suppressed and results in an over abundance of DM and requires
another mechanism: (i) In a larger dark sector DM may
annihilate into other lighter dark particles, ψ ψ → X X which
subsequently decay to SM particles. These new light
particles may lead to large DM selfinteractions, see for instance
Ref. [
78
]. (ii) Coannihilation and resonant effects [
20
]
may increase the effective thermal annihilation cross
section. For example processes like ψ F¯ → S∗ → H H with
(m F − mψ )/mψ 1/20 could be induced by a coupling
of S to the SM Higgs.16 Similarly there may be
coannihilations with S. If S has gauge interactions the dominant
channel may be S S → SM SM (see for instance Ref. [
79
])
if (m F − mψ )/mψ 1/20 and ψ and S are in thermal
equilibrium. (iii) Multibody scatterings may also increase
the effective thermal annihilation cross section [
64–68
]. (iv)
A nontrivial thermal evolution in the early universe may
depopulate an initially over abundant DM relic density [
69
].
The DM abundance may also be produced nonthermally. If
DM is only very weakly coupled to the SM thermal bath and
it has not been produced during reheating, DM may be slowly
produced via the freezein mechanism [
39, 40
]. Ref. [30]
discussed the phenomenology of the freezein mechanism in the
scotogenic model [
9
] with fermionic DM, one of the
examples where DMnucleus scattering occurs at oneloop level.
(45)
6.3.2 Nonthermal production
σ v M =
m2
ψ
8π x
m4S + m4ψ
m2S + m2ψ
4
α,β
yi,β yi∗,α 2,
(46)
7 Conclusions
14 The thermally averaged cross section σ v = a + 6b/x with x =
mψ / T is obtained by integrating over the annihilation cross section
σ v = a + bv2, after it has been expanded up to second order in the
relative of velocity of the two DM particles in the center of mass frame
v = v. Note that, although the DM is nonrelativistic at freezeout,
the relative velocity is not small, v f = 12/x f 0.7 c in terms of the
speed of light c.
15 Annihilation channels with 3body final states which lift the
velocity suppression are generally not important during freezeout due to the
additional phase space suppression, but they are very important for
indirect detection. Their importance for indirect detection has been pointed
out in several papers [
75,76
], see also Refs. [
6,77
].
Direct detection of DM may not have been observed yet
because it is absent at tree level, occurring only at the loop
level. In this work we have studied the case of a fermionic
singlet DM ψ , which is a simple scenario where DD is naturally
induced at oneloop order. The type of scenario considered
appears in supersymmetric extensions where the neutralino
is pure bino [
15
] (notice that in this case its mass is
typically very heavy, larger than 2 TeV), and also in connection
to neutrino masses, in particular in the seesaw model [
13
]
16 If S carries a dark charge it may be a softbreaking term.
Hopefully a positive DD signal in the next years will serve as a
motivation and guidance to continue exploring the WIMP DD
theory space and its interplay with other beyond the Standard
Model probes.
Acknowledgements We thank YueLing Sming Tsai to provide a
preliminary version of LikeDM [
63
] and for answering many questions.
EM is grateful to Viviana Niro, Paolo Panci and Francesco Sannino for
useful discussions. JHG acknowledges Fady Bishara for providing an
earlier version of DirectDM [
48
]. MS thanks Yi Cai for numerous
useful discussions. This work has been supported in part by the Australian
Research Council. JHG acknowledges the support from the Australian
Research Council through the ARC Centre of Excellence for Particle
Physics at the Terascale (CoEPP) (CE110001104). All Feynman
diagrams were generated using the TikZFeynman package for LATEX [
82
].
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: Larger dark matter groups
In the main part of the text we restricted ourselves to a global
U(1) symmetry for a Dirac DM and to a discrete Z2
symmetry for a Majorana DM. Our results can be easily generalized
if the DM forms a larger nontrivial representation of the dark
symmetry group and there are multiple degenerate
components of the DM multiplet. As the dark symmetry commutes
with the SM gauge group it simply leads to an overall factor
of
(C γ †C γ )α α ≡
Cβγα∗ Cβα
γ
γ
β,γ
and in some radiative neutrino mass models [
8, 9, 12, 80
]. We
have considered a simplified scenario with a dark sector made
of a vectorlike (or a SM) fermion and a (complex) scalar.
We presented general analytical expressions for the
different contributions as well as current limits on the dark sector
parameters. We have outlined the possible UV completions
of the corresponding penguin diagrams, also those involving
SM fields, and we summarize the different possibilities in the
following:
(i) If the fermion is a SM lepton and thus leptophilic, the
DM interactions are generically flavored [
5
] and there is an
interesting phenomenology. There may be new contributions
to the anomalous magnetic moment, but the limit is very
weak. If there are couplings to at least two different flavors,
there are strong limits from LFV, especially for couplings
to both electrons and muons. In this case the limits from
LFV processes such as μ → eγ and μ → 3e are much
stronger than DD. In the absence of one of these couplings
DD limits are stronger above a certain DM mass given by
the experimental energy thresholds of the DD experiments.
In some cases the same particles entering in the DD loop
may naturally violate lepton number (specially if the DM
couples to the lefthanded lepton doublets) and give rise to
radiative neutrino mass models such as the scotogenic model
with Majorana DM [
9
] or the generalized scotogenic model
with Dirac DM [
80
].
(ii) If the dark fermion is a righthanded neutrino, it may be
a Majorana fermion and an active Majorana neutrino mass
term is generated via the seesaw mechanism [
81
]. As the
particles in the loop are neutral, DD is generated via Z and
Higgs penguin diagrams [
13
], which are very suppressed.
Although the DM may be assigned lepton flavor and lepton
number, there are no strong limits from LFV or lepton number
violation beyond those already present in seesaw scenarios.
This scenario is normally referred to as the neutrinoportal
to DM [
13, 41, 42
].
(iii) If the scalar is the SM Higgs, there is mixing between
the DM and the neutral component of the fermion in the
loop, which generates treelevel contributions mediated by
the Z boson and the Higgs. The Z mediated treelevel DD
is expected to dominate with respect to the dipole moment
contributions arising at loop level. In fact, elastic Z mediated
contributions are already ruledout by DD experiments.
While the correct relic abundance is easily achieved in
models with DM couplings to SM leptons (or not too heavy
righthanded neutrinos), it requires further modelbuilding in
the case of DM couplings to vectorlike fermions. We have
also found that the invisible loopinduced Z and Higgs boson
decays may sometimes impose restrictions in the case of light
DM.
In this work we studied the prototypical case of fermion
singlet DM with the simplest dark sector, where the loop
suppression still allows reasonably large DM interactions.
(47)
(48)
(49)
to the Wilson coefficients of a DM particlenucleus
scattering, ψα N → ψα N , where the ClebschGordan coefficients
γ
Cβα are defined such that the scalar and the two fermions are
invariant under the dark sector symmetry:
Cβγα F¯β Sγ (y1 PL + y2 PR ) ψα .
Thus for a general DM candidate with N components ψα
the DD cross section is obtained by summing over the final
states and averaging over the initial state and thus
σ
σ → N
γ ,δ
Tr C γ †C γ C δ†C δ
.
Note that a larger dark sector symmetry may lead to multiple
DM candidates, which requires to go beyond the discussed
scenario, see for instance Ref. [
83
].
N,V is induced by both interactions
The vector operator OSI
with a photon and a Z boson.
Once the differential cross section is computed via
Eq. (50), the differential event rate per unit detector mass
(for a detector with just one type of nucleus A) is given by:
d R
d E R
ρψ
= mψ m A vmin(ER) d E R
dσ
v fdet(v) d3v,
where ρψ is the local WIMP density, fdet(v) is the WIMP
velocity distribution in the detector rest frame and vmin is the
minimum WIMP velocity required to produce a recoil with
energy E R
vmin(ER) =
E R m A
2μψA
.
The velocity distribution in the detector rest frame is related
to the velocity distribution in the galaxy frame fgal(v, t ) by
a simple Galilean transformation, fdet(v) = fgal(v + vE (t )),
where vE (t ) is the velocity of the Earth in the galactic frame.
In our analysis we use LikeDM and refer to [
62, 63
] for the
technical details of the different detectors and astrophysical
assumptions.
Appendix B: Direct detection differential cross section and event rate
The differential cross section for fermionic DM may be
written in terms of NR operators at the nucleon level [
52
]
dσ
d E R
RτMτ W Mττ (q)+ Rτ τ W τ τ (q)+ Rτ τ W τ τ (q)
Rτ τ W τ τ (q) + Rτ τ
W τ τ (q)
(50)
with the nucleus mass m A and spin JA. The coefficients RX
are given in terms of the NR Wilson coefficients ci0,1 = (cip ±
cin )/2 and WX denote the nuclear response functions. The
explicit forms of RX and WX are given in Ref. [
50
]. For
q → 0, the long wavelength limit, WM (0) ∝ A2 counts the
number of nucleons in the nucleus, W and W measure
the nucleon spin content of the nucleus, W measures the
nucleon angular momentum and W the interference.
In the literature it is also common to show the differential
cross section as the sum of different dipole and charge
contributions. Neglecting the Z contributions to SD interactions,
which are suppressed with respect to the longrange
interactions, and taking dψ = 0, the differential cross section can
be written as [
84
]:
(54)
(55)
(56)
(57)
(58)
Appendix C: Expressions for Z and Higgs boson decays into dark matter
The relevant interactions of the DM ψ with the Higgs and
the Z boson can be parameterized as17
LH ψ
and
LZ ψ
= ψ (bV + bA γ5) ψ h + H.c.,
= ψ (cV γ μ + cA γ μγ5 + dV p2μ
μ
+dA p2 γ5)ψ Zμ + H.c.,
where p2μ is the 4momentum of the outgoing DM ψ . We
define xh ≡ mψ /mh and x Z ≡ mψ /m Z . The partial Higgs
decay width into the DM ψ is nonzero for mψ < mh /2 and
reads:
h→ψψ =
S Nψ mh
2π
+[Im(bA)]2
[Re(bV )]2 1 − 4 xh2
1 − 4 xh2 1/2 .
(51)
(52)
17 In the case of radiative neutrino mass models such as the scotogenic
model [
9
] and its variants [
38,80
], there are extra (lepton number
conserving) invisible Higgs boson decays into neutrinos at one loop, which
are not suppressed by phase space and could therefore be larger than
those into DM.
FS2I (E R )
2
+ αem
μ2Aμ2ψ m A JA + 1
4π 2v2 3 JA
m A
+ 2π v2 Ae2ff FS2I (E R ) ,
FS2D (E R )
where μψ A = mψ m A/(mψ + m A) is the DMnucleus
reduced mass and Aeff encodes the DMnucleus couplings
(see e.g. Ref. [
10
]):
Aeff = Z
cSp,IZ + cSp,Iγ + cSp,IH
αemμψ
− 2π mψ
+( A − Z ) cSn,IZ + cSn,IH
.
The first line in Eq. (51) corresponds to the dipole–charge
(D–C), the second line to the dipole–dipole (D–D) and the
third line to the chargecharge (C–C) interaction. FSI (E R )
and FSD (E R ) are the nuclear form factors. cSNI with N = n, p
are the relativistic Wilson coefficients at the nucleon level for
the operators
OSNI,V = ψ¯ γμψ N¯ γ μ N ,
OSNI,H = ψ¯ ψ N¯ N .
(53)
Similarly the partial width of the Z is given by
Z→ψψ =
for mψ < m Z /2. S is the symmetry factor, equal to 1/2
for identical final states (Majorana DM), and equal to 1 for
Dirac DM. The coefficients relevant for the decays of the
Higgs boson to Dirac DM can be expressed in terms of the
PassarinoVeltman functions
bV = λ32HπS2v m F yA2 − yV 2 C0 m2ψ , m2h, m2ψ , m F , m S, m S
+2mψ yA2 + yV 2 C1 m2ψ , m2h, m2ψ , m F , mS, mS , (60)
bA = i 1λ6HπS2v m F Im yV y∗A C0(m2ψ , m2h, m2ψ , m F , mS, mS) .
(61)
The mass insertions, mψ and/or m F are needed in order to
flip chirality. We do not report the expressions for the decays
of the Z boson, as they are very long and not illustrative.
For Majorana DM cV = dV = dA = 0 and the
remaining nonzero Wilson coefficients are a factor of two larger,
cAMajorana = 2 cADirac, bV Majorana = 2 bV Dirac, and
bAMajorana = 2 bADirac due to the presence of crossed
diagrams. This is analogous to direct detection: cSqS and cAqA for
Majorana DM are a factor 2 larger than for Dirac DM (see
Sect. 4.2.2).
Appendix D: Lepton flavor violation and anomalous dipole moments
If the DM couples to SM leptons there may be LFV processes
and anomalous electric and magnetic dipole moments. We
provide the relevant expressions for DM coupling to either
the lefthanded SM doublets or the righthanded SM singlets.
The results are identical for Dirac or Majorana DM.
Appendix D.1: Lefthanded lepton doublet
The relevant interaction term for LFV processes is with the
charged scalars:
LL L = − y2 LL S ψR + H.c. = y2eL S− ψR + H.c. + · · · .
(62)
The most general amplitude for the electromagnetic charged
lepton flavor transition α( p) → β (k) γ ∗(q) can then be
parameterized as [
85
]
where
A2L = 0,
with
f (x ) =
Aγ = e ρ∗(q) u(k) q2 γ ρ
A1L PL + A1R PR
+ mβ i σ ρσ
A2L PL + A2R PR qσ u( p) ,
(63)
where e > 0 is the proton electric charge, p (k) is the
momentum of the initial (final) charged lepton α ( β ), and q = p−k
is the momentum of the photon. As is well known, the charged
lepton radiative decays are mediated by the electromagnetic
dipole transitions in Eq. (63) and the corresponding
branching ratio (Br) for α → β γ is given by
Br( α →
β γ ) =
48 π 3 αem
G2F
×Br α →
A2L 2
+
β να νβ .
AR 2
2
AR
,
For trilepton decays we consider only the contributions from
the photon penguin and from boxtype diagrams, as the Z
penguin is suppressed by charged lepton masses. Box
diagrams may be the dominant contribution in absence of the
contributions from photon and Z penguins. The amplitude
from the box diagrams is given by
ABOX = e2 B u(k1) γ α PL u( p) u(k3) γα PL v(k2).
(67)
For sameflavor leptons in the final state the branching ratio
of α → β β β reads:
Br( α → β β β )
=
6π 2αe2m
G2F
A1L 2 + A2R 2 136 ln mmβα − 232
1 B 2 1
+ 6   − 4 Re A1L∗ A2R − 6
×Br α → β να νβ .
A1L − 2 A2R B∗
For α− →
β− γ− γ+ with β = γ the branching ratio reads:
Br( α → β γ γ )
=
6π 2αe2m 2
3
G2F
A1L 2 + A2R 2 136 ln mmγα − 8
+ 112 B2 − 83 Re A1L A2R∗ − 18
×Br α → β να νβ .
A1L − 2 A2R B∗
(64)
(65)
(66)
(68)
(69)
because there are only contributions from box diagrams. The
coefficients AL, R are given in Eq. (65) and
2
A1L = − 481π 2
y2β y2α∗ g
m2S±
m2
ψ
m2S±
,
AR
1 = 0 ,
.
The contribution from box diagrams B for α− →
reads
β− γ− γ+
y2α∗ y2β y2γ y2γ ∗ h
m2S±
y2α∗ y2β∗(y2γ )2
m2S±
h
m2
ψ
m2S±
m2
ψ
m2S±
,
and for α− →
γ− γ− β+ it is given by
1 − x 2 + 2x ln x
.
2(x − 1)3
All the external momenta and masses have been neglected.
Of course for α → β β β both Eqs. (73) and (74) agree
with γ = β.
For μ − e conversion in nuclei we only consider coherent
scattering via photon contributions, but include both
shortand longrange contributions [
86
]:18
e
Lint = − 2
L
mμ A2 e σ μν PL μ Fμν
+mμ A2R e σ μν PR μ Fμν + h.c.
γ
gLV (q) eγ α PL μ
+gRV (q) eγ α PR μ qγαq + h.c. .
The μ − e conversion rate is
ωconv = 4
8e A2R D + g˜(LpV) V ( p) + g˜(LnV) V (n) 2 ,
18 We neglect the Z boson contribution which is proportional to the
square of the charged lepton masses and thus negligible compared to
the photon penguin diagram.
(70)
(71)
(72)
(73)
(74)
(75)
(76)
(77)
Table 7 The overlap integrals in the units of m5μ/2 and the total capture
rates for different nuclei [
86
]. The total capture rates are taken from
Table 8 in [
86
]. The overlap integrals of 17997Au as well as 2173Al are taken
from Table 2 and 4282Ti are taken from Table 4 of Ref. [
86
]
17997Au
4282Ti
1237Al
V (p)
with
with
h(x ) =
(81)
g˜(LpV) ≈ 2 gγLV (u) + gLV (d) = e2 A1L , g˜(LnV)
γ
≈ gLV (u) + 2 gγLV (d) = 0,
γ
with
γ L
gLV (q) = e2 Qq A1 .
The coefficients AL, R are given in Eq. (71), and Qq is the
1
quark electric charge of the quark q in units of e > 0. The
numerical values of the overlap integrals D and V ( p,n) and
the total capture rate for each nucleus are reported in Table 7
for three different nuclei. As we only consider the photon
contribution and thus only couplings to the electric charge of
the quarks, there is no effective coupling to neutrons.
Even if lepton flavor is conserved there are processes that
can bound the DM interactions with the leptons. Electric
dipole moments for the leptons occur in these simplified
models only at the twoloop level. However leptonic magnetic
dipole moments occur at oneloop order via photon penguin
diagrams, similarly to μ → eγ transitions. They receive two
independent contributions from the charged scalars running
in the loop, which are given by [
87
]:
a ≡
A2R is the diagonal part (α = β ≡ ) of the coefficient given
in Eq. (65) and the loop function is defined in Eq. (66). Our
expression agrees with Ref. [
4
]. In the case of the muon
magnetic dipole moment, the discrepancy with the SM has the
opposite sign and hence the model cannot explain it. However
this can be used to (very weakly) bound the model. Electron
and tau AMMs do not lead to any relevant constraints.
Appendix D.2: Righthanded charged lepton
The relevant interaction term for LFV processes is with the
charged scalars:
LL L = − y1 eR S− ψL + H.c..
(78)
(79)
(80)
All the expressions are the same as for the lefthanded lepton
doublets after substituting the righthanded superscript by
the lefthanded one, i.e., A1R, 2 ↔ A1L, 2, gγLV ↔ gRV and the
γ
Yukawa couplings y1 ↔ y2.
Appendix E: Computation of the relic abundance
In this appendix we review the computation of the relic
abundance, see for instance Refs. [
20,88
]. We use the
instantaneous freezeout approximation which is sufficient for our
purposes. The final DM abundance is determined by
ψ =
n +λ 1 x nf +1 mρψcrs0 ,
with λ = [x s σ v /H ]x=1 and n = 0 (1) for swave (
pwave) DM annihilation. The entropy density is denoted by
s, with today’s value given in terms of the CMB temperature
Tγ,0 = 2.73 K as s0 = 2π 2/45 (43/11) Tγ3,0, where we have
used Neff = 3.
Equating the interaction rate ann for the process ψ ψ¯ ↔
α ¯β with the Hubble rate, H (T f ) = ann(T f ) we obtain a
condition for the freezeout temperature
where mP is the Planck mass, g∗ is the number of relativistic
degrees of freedom at freezeout (g∗ = 106.75 in the SM),
and gψ is the DM number of degrees of freedom, which is
equal to 2 (4) for Majorana (Dirac) DM.
The annihilation cross section may implicitly depend on
the freezeout temperature, and it is useful to factorize out
this dependence. Then Eq. (83) can be written in terms of λ
as
4 π 2 gs (2π )3/2
∗
3 30 gψ λ
= x f21 −ne−x f .
Solving for λ in Eq. (84), plugging it in Eq. (82), and imposing
that relic abundance matches the observed value ψ h2 =
0.12 [
89
], one can numerically obtain the value of x f . We
get values of 23 x f 30 for 10 GeV mψ 104 GeV,
which turn out to be identical for Dirac and for Majorana
DM. We also note that x f increases roughly logarithmically
with the DM mass mψ .
For a given (mψ , x f ) pair Eq. (84) allows one to compute
the annihilation cross section averaged over velocity, σ v ,
which depends exponentially on x f . For the range of DM
masses given above the dependence on the DM mass is very
mild. We obtain that the required thermally averaged
annihilation cross sections to reproduce the observed DM
abundance are in the range 1.8 1026 σ v D (cm3 s−1) 2.4
and 4 1024 σ v M (cm3 s−1) 9 for Dirac and Majorana
DM, respectively.
(82)
(84)
Appendix F: Matching onto nonrelativistic operators
We use DirectDM [
48
] which follows the normalization
of the NR operators in Ref. [
50
] to match our oneloop
calculation of DM scattering off quarks onto the NR
operators using 3 flavor QCD without running, i.e. the matching
occurs at μ = 2 GeV. This is justified as the relevant
relativistic operators are renormalization group invariant under
oneloop QCD corrections. There are no additional
significant contributions, because the particles in the loop are color
singlets. There can be sizable renormalization group
corrections, if there are colored particles in the loop, see e.g. the
discussion of bino DM in the minimal supersymmetric SM
in Ref. [
15
].
Note that the coefficients ciq depend on the 3momentum
transfer q = √2 m A E R with the target nucleus mass m A
and the recoil energy E R . In the numerical examples in the
figures we use E R = 8.59 keV for 15342Xe which results in
q2 = 2.11 × 10−3 GeV2. The exact numerical expressions
used in the code are given below. All quantities are defined
in units of GeV. All NR Wilson coefficients have
dimension GeV−2. Higgs penguins with heavy SM quarks Q are
described by the Wilson coefficient of the gluon operator
(83)
cg = −
Q=t,b,c
cSQS .
Appendix F.1: Dirac dark matter
NR Wilson coefficients for protons
+ 0.017cSuS + 2cVuV −
mψ
c1 = 0.036cSdS + 2cVdV − 0.0628148cg + 0.0413cSsS
n
(85)
(86)
(88)
(89)
(90)
+ 0.015cSuS + cVV
u
c1p = 0.064cSdS − 0.12563cg + 0.0826cSsS + 0.034cSuS (94)
c4p = 3.008cAdA + 0.248cAsA − 7.176cAuA
c9p = −4.12cAdV + 0.876cAsV + 14.72cAuV
and neutrons
c1n = 0.072cSdS − 0.12563cg + 0.0826cSsS + 0.03cSuS
c4n = −7.176cAdA + 0.248cAsA + 3.008cAuA
(99)
(100)
4.48648cAdA
c6n = q2 + 0.0182187 + q2 + 0.300153
1.37054cAsA 4.48648cAuA
c9n = 14.72cAdV + 0.876cAsV − 4.12cAuV
(93)
(95)
(96)
(97)
(98)
(101)
(102)
(103)
Page 26 of 27
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