The scaleinvariant scotogenic model
Received: May
The scaleinvariant scotogenic model
Amine Ahriche 0 1 3 4 5 6
Kristian L. McDonald 0 1 3 6
Salah Nasri 0 1 2 3 6
0 School of Physics, The University of Sydney
1 Strada Costiera 11 , I34014, Trieste , Italy
2 Physics Department, UAE University
3 PB 98 Ouled Aissa , DZ18000 Jijel , Algeria
4 The Abdus Salam International Centre for Theoretical Physics
5 Department of Physics, University of Jijel
6 POB 17551 , Al Ain , United Arab Emirates
We investigate a minimal scaleinvariant implementation of the scotogenic model and show that viable electroweak symmetry breaking can occur while simultaneously generating oneloop neutrino masses and the dark matter relic abundance. The model predicts the existence of a singlet scalar (dilaton) that plays the dual roles of triggering electroweak symmetry breaking and sourcing lepton number violation. Important constraints are studied, including those from lepton avor violating e ects and dark matter directdetection experiments. The latter turn out to be somewhat severe, already excluding large regions of parameter space. None the less, viable regions of parameter space are found, corresponding to dark matter masses below (roughly) 10 GeV and above 200 GeV.
Beyond Standard Model; Neutrino Physics

The scaleinvariant scotogenic model
2
3
5
6
7
8
1
1 Introduction
2.1
2.2
Symmetry breaking
The scalar spectrum
Neutrino mass
4 Invisible Higgs decays Lepton avor violating decays Dark matter
The discovery of the Higgs boson provides an explanation for the origin of mass in the
origin of the O(100) GeV massparameter that determines the weak scale in the SM also
remains a mystery. Thus, with regard to the mechanisms of mass in the universe, there
remains much to be discovered.
The scotogenic model is a simple framework that aims to address some of these
shortcomings [1]. It o ers an explanation for the origin of neutrino mass and the nature of DM
by proposing a common or uni ed solution to these puzzles. In this approach, neutrinos
acquire mass as a radiative e ect, at the oneloop level, due to interactions with a Z2odd
sector that includes DM candidates. The resulting theory gives a simple model for neutrino
mass and DM, and has been wellstudied in the literature [2{8].
Motivated by the simplicity of the scotogenic model, and our inadequate understanding
O(TeV), enhancing the prospects for testing the model. The resulting theory provides a
common framework for the aforementioned problems relating to mass  namely the origin
of neutrino mass, the origin of the weak scale, and the nature of DM.
We investigate the SI scotogenic model in detail, demonstrating that viable electroweak
symmetry breaking can be achieved, while simultaneously generating neutrino masses and
the DM relic abundance. The model predicts a singlet scalar (dilaton) that plays two
important roles  it triggers electroweak symmetry breaking and sources the lepton number
violation that allows radiative neutrino mass. Important constraints are studied, including
those from lepton avor violating e ects, DM directdetection experiments, and the Higgs
sector, such as the invisible Higgs decay width and Higgsdilaton mixing. Directdetection
constraints turn out to be rather severe and we nd that large regions of parameter space
are already excluded. None the less, viable parameter space is found with a DM mass
below (roughly) 10 GeV or above 200 GeV. The model can be experimentally probed in
! e +
a number of ways, including:
searches, future directdetection experiments,
precision studies of the Higgs decays h !
and h !
Z, and collider searches for an
inert doublet.
Before proceeding we note that a number of earlier papers have studied relationships
between neutrino mass and DM; see e.g. refs. [10{64], and also ref. [65], in which DM
stability follows from an accidental symmetry. Earlier works investigating SI extensions of
the SM appear in refs. [66{81] and, in particular, studies of SI models for neutrino mass
can be found in refs. [82{92].
The structure of this paper is as follows. In section 2 we introduce the model and
detail the symmetry breaking sector. We turn our attention to the origin of neutrino mass
in section 3 and discuss various constraints in sections 4 and 5. Dark matter is discussed
in section 6 and our main analysis and results appear in section 7. Conclusions are drawn
in section 8.
2
The scaleinvariant scotogenic model
The minimal SI implementation of the scotogenic model is obtained by extending the SM to
include three generations of gaugesinglet fermions, NiR
(1; 1; 0), where i = 1; 2; 3; labels
generations, a second SMlike scalar doublet, S
(1; 2; 1), and a singlet scalar
(1; 1; 0).
A Z2 symmetry with action fNR; Sg !
fNR; Sg is imposed on the model.1 The scalar
, as well as the SM
elds, transform trivially under this symmetry. The lightest Z2odd
particle is stable and may be a DM candidate; this should be taken as either the lightest
singlet fermion N1 or a neutral component of the the doublet S, as discussed below. The
scalar
plays the dual roles of sourcing lepton number violation, to allow neutrino mass,
and triggering electroweak symmetry breaking.
1This model was also mentioned in refs. [93, 94].
{ 2 {
With this eld content, the mostgeneral Lagrangian consistent with both the SI and
Z2 symmetries contains the terms
L
iNR
yi NicR NiR
gi NiRL S
V ( ; S; H);
(2.1)
where L
letters, ;
(1; 2; 1) denotes the SM lepton doublets, with generations labeled by Greek
= e; ; . We denote the SM scalar doublet as H
(1; 2; 1) and V ( ; S; H)
is the mostgeneral scalar potential consistent with the symmetries. The SI symmetry
precludes any dimensionful parameters in the model, including bare Majorana mass terms
In the absence of dimensionful parameters, the scalar potential contains only quartic
in(2.2)
(2.4)
HJEP06(21)8
for the fermions Ni.
2.1
Symmetry breaking
teractions:
V ( ; S; H) =
G (X) =
X2
5 can be taken real without loss of generality. The desired VEV pattern has
hSi = 0, to preserve the Z2 symmetry, with hHi 6= 0 and h i 6= 0, to break both the SI
and electroweak symmetries. In addition to the doublet scalar S, we shall see that the
spectrum contains an SMlike scalar h1 and a dilaton h2.
Radiative corrections play an important role in triggering the desired symmetry
breaking pattern. A full analysis of the potential requires the inclusion of leadingorder loop
corrections; however, in general, the full oneloop corrected potential is not analytically
tractable. None the less, as discussed in ref. [92] (and guided by ref. [95]), simple analytic
expressions can be obtained by noting the following. Loop corrections involving SM
elds
are dominated by topquark loops, due to the large Yukawa coupling. To allow viable
electroweak symmetry breaking and give a positivelyvalued dilaton mass, these corrections
must be dominated by loop corrections from a beyondSM scalar, namely S. Thus, loop
corrections from S and t are expected to dominate and, to reasonable approximation, one
can neglect loop corrections involving the light scalars (namely the SMlike Higgs and the
H = (0; h=p2), the oneloop corrected potential for h and
is
dilaton). More precisely, this gives an approximation to the potential up to corrections of
O(Mh41 =MS4) [92], which is reasonable provided one restricts attention to MS & 200 GeV.
Adopting this approximation, and writing the SM scalar in unitary gauge as
V1 l (h; ) =
4
H h4 +
4
H 2h2 +
4
4 +
X
i=all elds
ni G Mi2 (h; ) ;
(2.3)
where ni is a multiplicity factor,
is the renormalization scale, and the sum is over all
elds barring the light scalars (h and ) and the light SM fermions (all but the topquark).
The function G is given by
In the absence of bare dimensionful parameters, the elddependent masses can be written as
where i and i are constants.
Symmetry breaking is triggered via dimensional transmutation, introducing a
dimensionful parameter into the theory in exchange for one of the dimensionless couplings (which
is now
xed in terms of the other parameters). Analyzing the potential reveals a minimum
with both h i
x 6= 0 and hhi
v 6= 0 for
H < 0. If one considers the treelevel poten
tial, the desired VEV pattern is triggered at the scale
where the running couplings obey
H( ) = 0. Including loop corrections, subject to our approximation,
is also satis ed. Absent netuning, we observe that with
H; H = O(1) one obtains v
x
and the exotic scale is expected near the TeV scale. Eqs. (2.6) and (2.7) ensure that the
tadpoles vanish.
Oneloop vacuum stability requires that the couplings obey:
1H l; 1 l; 1Hl + 2
q 1 l 1 l > 0;
H
where the oneloop couplings are de ned as
1 l =
H
1 l =
1Hl =
:
Eq. (2.8) guarantees that the masses for the neutral scalars h and
are strictly positive,
forcing one of the beyondSM scalars in the doublet S to be the heaviest particle in the
spectrum, to overcome topquark contributions to the dilaton mass. Demanding
also ensures that the vacuum with v 6= 0 and x 6= 0 is preferred over the vacuum with a
1Hl < 0
single nonzero VEV.
2.2
The scalar spectrum
Writing the inertdoublet as S = (S+; (S0 + iA)=p2)T , the components have masses
MS2+ =
MS20;A =
2
2
S x2 +
2
two mass eigenstates, which we denote by h1;2,
5term splits the neutral scalar masses MS0 and MA, with the splitting becoming
1.2 After symmetry breaking, the scalars h and
mix to give
h1 = h cos h
sin h ;
h2 = h sin h +
cos h :
Due to the Z2 symmetry, the neutral components of S do not mix with these elds. At
treelevel the mixing angle is determined by the VEVS,
x
px2 + v2
;
ch
cos h =
sh
sin h =
v
px2 + v2
;
and the SMlike scalar mass is given by
massless at treelevel, though radiative corrections induce Mh2 6= 0. A useful approximation
for Mh2 is [95]
Mh22 ' 8 2(x2 + v2)
1
(
Mh41 + 6M W4 + 3MZ4
12Mt4 + 2MS4+ + MA4 + MS40
(2.11)
(2.12)
(2.13)
3
2 X
i=1
)
Mi4 :
(2.14)
Here the singlet fermion masses are given by Mi = yi x, and are ordered as M1 < M2 < M3.
Eq. (2.14) shows that viable symmetry breaking requires one of the scalars S+, S0 or A to
be the heaviest particle in the spectrum, to overcome negative loop contributions to Mh2
from the top quark and the fermions Ni.
Treelevel expressions for Mh1 and h are presented above for convenience, however, in
our numerical analysis (detailed below), we use the mass eigenvalues Mh1;2 and the mixing
angle h obtained by diagonalizing the oneloop corrected potential. We note that the SI
symmetry imposes nontrivial constraints on the model, with
and
H xed by eqs. (2.6)
and (2.7), and the Higgs mass Mh1 ' 125 GeV further xes H
.
3
Neutrino mass
The combined terms in eqs. (2.1) and (2.2) explicitly break lepton number symmetry, giving
rise to radiative neutrino mass at the oneloop level, as shown in
gure 1. Observe that
plays a key role in allowing the neutrino mass diagram, without which neutrinos would
remain massless.3 Calculating the mass diagram gives
(M )
=
i
X gi gi Mi
MA2
MA2
:
(3.1)
the limit 5 ! 0.
are possible [96].
2Note that the limit 5
1 is technically natural due to the restoration of lepton number symmetry in
3The Feynman diagram in gure 1 is an example of the SI type T3 oneloop topology. Related variants
{ 5 {
(M )
'
X gi gi 5v2
i
1
i
:
(3.2)
Note that the Z2 symmetry prevents mixing between SM neutrinos and the exotics Ni.
One can relate the entries in the neutrino mass matrix to the elements of the
PontecorvoMakiNakawagaSakata (PMNS) mixing matrix [97, 98] elements. We
parameterize the latter as
0
(3.3)
j m213j = 2:55+0:06
0:09
with d being the Dirac phase and Um = diag(1; ei =2; ei =2) giving the dependence on
the Majorana phases
; . We use the shorthand sij
sin ij and cij
cos ij to refer to
s23 = 0:43+00::0033, and the masssquared di erences:
2
m221 = 7:62+0:19
0:19
the mixing angles. In our numerical scans of the parameter space in the model, we t to
the best t experimental values for the mixing angles: s213 = 0:025+00::000033, s212 = 0:320+00::001167,
10 5 eV2 and
To determine the parameter space that generates viable neutrino masses, we use the
(M )
=
X gi gi i = g
T g
;
i
i =
Mi
MA2
MA2
M 2 ln
i
MA2
:
According to the CasasIbarra parameterization, the coupling g can be written as
g = Dp
1 RDpm U y;
where Dp
1 = diag
q
1 ;
1 q
2 ;
1 q
3
1 , Dpm
= diag
pm1; pm2; pm3 , and R is
an orthogonal rotation matrix (m1;2;3 are the neutrino eigenmasses).
(3.4)
(3.5)
(3.6)
{ 6 {
Invisible Higgs decays
The model is subject to constraints on the branching fraction for invisible Higgs decays,
B(h ! inv) < 17% [101]. One should use inv
available, with corresponding decay widths given by
fh2h2g; fNDMNDMg, when kinematically
122 is de ned in eq. (6.11) below. As a result of the SI
symmetry, the coupling
122 vanishes at treelevel, and the nonzero looplevel coupling is
su ciently small to ensure that decay to h2 pairs is highly suppressed.4
5
avor violating decays
The new
elds give rise to oneloop contributions to
Br(
! e
e), the corresponding branching fraction is
! e + . Normalized relative to
where AD is the dipole form factor:
with the loop function given by
Br(
Br(
! e )
! e
e)
=
4G2F
3(4 )3 em jADj2 ;
AD =
X geigi
1
i
A simple change of labels allows one to use the above formulae for the related decay
+ . In our analysis we also include the constraint from neutrinoless double
Note that, in general, the scotogenic model is subject to strong LFV constraints,
relating to the fact that the DM annihilates via the same Yukawa couplings that mediate
LFV processes. Consequently one cannot decouple the two e ects and there can be tension
between the demands of suppressed LFV processes and the attainment of a viable DM
abundance (actually, in the scotogenic model, constraints from other LFV processes, like
e conversion, can be more severe than the above LFV decays; see the 3rd and 4th
4Note that h2 decays to SM states, similar to a light SM Higgs boson but with suppression by the mixing
angle, s2 . However, dedicated ATLAS or CMS searches for such light scalars, in the channels 2b, 2
h
or 2 ,
do not currently exist, so we classify the decay h1 ! h2h2 as invisible. In practice, however, the suppression
of (h1 ! h2h2) due to SI symmetry renders this point moot.
(5.1)
(5.2)
(5.3)
{ 7 {
papers in refs. [2{8]). However, we shall see that the situation di ers in the SI model,
due to additional annihilation processes mediated by the dilaton. This provides a degree
of decoupling between the LFV processes and DM annihilations, such that LFV bounds
are more readily satis ed. Thus, for our purposes, it is su cient to consider the above
LFV decays (we shall see that the viable parameter space includes regions wellbelow the
LFV bounds, so slightly stronger bounds do not have a large e ect). We note that the
correlation between
! e and the DM relic abundance, for the case of fermionic DM in
the scotogenic model, was rst noted in ref. [102], while ref. [103] noted that models with
a singlet scalar allow one to decouple these issues.
6
6.1
Dark matter
Relic density
As the universe cools, the temperature eventually drops below the DM mass. Consequently
the DM number density becomes Boltzmann suppressed and the DM annihilation rate
can become comparable to the Hubble parameter.
At a certain temperature the DM particles freeze out of equilibrium, such that the DM number density in a comoving volume henceforth remains constant. The cold DM relic abundance therefore depends on the total thermally averaged annihilation cross section
HJEP06(21)8
h (NDM NDM)vri =
h (NDM NDM ! X)vri
=
X
X
X
X Z 1
4MD2M
ds N DM NDM!X (s)
s
where vr is the relative velocity, s is the Mandelstam variable, K1;2 are the modi ed Bessel
functions and
at the CM energy ps. At freezeout, the thermal relic density can be given in terms of the
NDM NDM!X (s) is the annihilation cross due to the channel NDM NDM ! X,
thermally averaged annihilation cross section by
DMh2 ' pg Mpl(GeV) h (NDM NDM)vri
;
(1:07
109)xF
where Mpl is the Plank mass and g counts the e ective degrees of freedom of the
relativistic elds in equilibrium. The inverse freezeout temperature, xF = MDM=TF , can be
determined iteratively from the equation
xF = log
8
3pg xF
r 45 MDMMpl h (NDMNDM)vri
!
:
In the present model, the classes of DM annihilation channels are shown in gure 2.
The DM can annihilate into: (1) charged leptons and neutrinos, ` `
+ and
, including
LFV
nal states with
6
=
, (2) SM fermions and gauge bosons bb, tt, W +W , ZZ and the
scalars SS, and (3) nal states comprised of the Higgs and/or dilaton, hihk. The rst class
of channels are h1;2mediated schannel processes, the second class are Smediated tchannel
processes while the third class contains both s and tchannels processes mediated by h1;2.
p
s
T
(6.1)
;
(6.2)
(6.3)
{ 8 {
N1
N1
N1
(a)
hj
(d)
S
Lβ
hi
hk
N1
N1
N1
(b)
(e)
S
Lβ
hi
hk
N1
N1
N1
hj
(c)
(f)
hi
hk
XSM,S
The cross section for the annihilation channel into charged
leptons5 is given by [104]
HJEP06(21)8
(6.4)
:
(6.5)
(6.6)
p
s ;
(6.7)
(NDMNDM ! ` `+)vr =
1
8 s(MS2+
+ s
2
3
s
4
jg1 g1 j
2
MD2M
MD2M + 2s )2
" m`2 + m2
`
2
s
2
MD2M
(MS2+
MD2M)2 + 2s (MS2+
MD2M) + s82 #
(MS2+
MD2M + 2s )2
The cross section for annihilation into neutrinos can be obtained from eq. (6.4) by replacing
MS2+ ! MS20 and sending the charged lepton masses to zero, i.e.,
(NDMNDM !
)vr =
jg1 g1 j
2
12
s
4
MD2M
(MS20
MD2M)2 + 2s (MS20
(MS20
MD2M + 2s )4
2
MD2M) + s8 :
(2) schannel processes.
The processes NDMNDM ! bb, tt, W +W
and ZZ can occur
as shown in gure 2c. The corresponding amplitude can be written as
M = ichshy1u (k2) u (k1)
s
i
Mh21
i
Mh22
!
Mh!SM mh !
p
s ;
with Mh!SM (mh !
Higgs mass replaced as mh !
ps. This leads to the cross section
ps) being the amplitude of the Higgs decay h ! XSMXSM, with the
(NDMNDM ! XSMXSM) r = 8pss2hc2hy12 s
s
1
Mh22
2
h!XSMXSM
mh !
where h!XSMXSM (mh !
ps) is the total decay width, with mh !
ps.
5For same avor charged leptons ( = ), there are also schannel processes mediated by h1;2. However,
these are proportional to their Yukawa couplings and may therefore be ignored.
s
1
Mh21
{ 9 {
(3) Higgs channel.
The DM can selfannihilate into hihk, as seen in gure 2d, e and f.
The amplitude squared is given by
jMj2 = 2y~D2Ms
" ch 1ik
s
Mh21 +
s
sh 2ik
Mh22
" ch 1ik
s
#2
+ 4cicky~D3MMDM
+
(t
2ci2c2ky~D4M
MD2M)
2
+ a2 2ci2c2ky~D4M
+ a
(u
(t
MD2M)2
2ci2c2ky~D4M
MD2M) (u
MD2M)
Mh21 +
s
sh 2ik
Mh22
#
s
Mh2i + Mh2k + a
MD2M
s + Mh2i
u
4MD2MMh2k +
MD2M + Mh2i
MD2M + Mh2i
u
Mh2k !
MD2M
sMh2i
4MD2MMh2i +
MD2M + Mh2k
u
MD2M + Mh2k
t
sMh2k
MD2M + Mh2i
t
MD2M + Mh2k
t
Similarly, the SS annihilation cross section can written as
(NDMNDM ! SS)vr = S h h 1 s
where S0 = A = 1; S+ = 2, and 1SS and 2SS are the triple couplings of a scalar h1;2
with two S elds, given by
1S+S
= 3chv
Sshx; 2S+S
= 3shv +
Schx;
1
2
1
2
1S0S0;1AA =
2S0S0;2AA =
(e )
LN1 q = aq qq NDcMNDM;
+ MD2M +Mh2k
u
MD2M +Mh2i
u
s
s
with s, t and u being the Mandelstam variables, and the Yukawa couplings are de ned as
y~DM
y1, c1
ch and c2
sh. Here, we integrate the phase space numerically to obtain
the cross section for a given value of s. At treelevel the e ective cubic scalar couplings
( 1ik and 2ik) are given by
3
111 = 6 H chv
3
Hc2hshv + 3
Hchs2hx
6
s3hx;
though for completeness we employ the oneloop results, obtained from the loopcorrected
potential following ref. [105]. We note that the (leading order) absence of the cubic
interactions h1h22 and h32, is a general feature of SI models.
6.3
Direct detection
With regard to directdetection experiments, interactions between the DM and quarks are
described by an e ective lowenergy Lagrangian:
with
where
Consequently, the e ective nucleonDM interaction is written as
aq =
shchMqMDM
2 h i hH0i
"
1
B the baryon mass in the chiral limit [106].
This leads to the following nucleonDM elastic cross section in the chiral limit
det =
s4hM 2
N
MN
The analysis below will show that the upper bound reported by LUX experiment [109, 110]
provides a stringent constraint on det.
7
Analysis and results
Next we turn to our numerical analysis and results.
We perform a numerical scan
of the parameter space to determine whether radiative electroweak symmetry breaking
is compatible with oneloop radiative neutrino mass and singlet neutrino DM. In the
scans, we enforce the minimization conditions, eqs. (2.6) and (2.7), vacuum stability via
eq. (2.8), and demand that the SMlike Higgs mass is in the experimentally allowed range,
Mh1 = 125:09
0:21 GeV. Compatibility with constraints from LEP (OPAL) on a light
Higgs [107] are enforced, and we consider the constraint from the Higgs invisible decay,
(however, we only nd viable benchmark points for h i & 150 GeV).6
B(h ! inv) < 17%, [101]. Dimensionless couplings are restricted to the perturbative range
throughout, and we consider values of 100 GeV < h i < 5 TeV for the beyondSM VEV
The scan reveals a spread of viable values for the dilaton mass Mh2 , consistent with
OPAL, as plotted in
gure 3. In the scan we tend to
nd Mh2 in the range O(1) GeV .
Mh2 . 90 GeV. Lighter values of Mh2 seemingly require an amount of engineered
cancellation among the radiative masscorrections from fermions and bosons, or larger values for
h i; see eq. (2.14). We noticed that regions with h i & 500 GeV tend to be preferred.
We further scan for parameter space giving viable neutrino masses and mixing,
subject to the LFV and muon anomalous magnetic moment constraints, while simultaneously
generating a viable DM relic density. Figure 4 shows viable benchmark sets for the Yukawa
couplings gi , along with the corresponding LFV branching ratios and a contributions.
The couplings gi are typically wellbelow the perturbative bound. Note that the range for
6In principle, one can consider larger values for h i. However, these require hierarchically small couplings
in the scalar potential [108], which we do not consider here.
M(e103
S
dashed line denotes the degenerate case, i.e, min jgj = max jgj. Right: the LFV branching ratios
versus the muon anomalous magnetic moment, both scaled by the experimental bounds.
the Yukawa couplings varies over several orders of magnitude. This re ects the freedom to
take the leptonnumber violating quartic coupling 5 to be small, and accordingly transfer
some of the neutrino mass suppression between the Yukawa and quartic coupling sectors.
The capacity to obtain viable neutrino masses, with Yukawa couplings that vary over a
considerable range, in uences the strength of the signal from LFV decays. Figure 4 shows
that the bound from
! e gives important constraints in parameter space with larger
gi , while smaller values of gi allow the model to easily evade the bound. Constraints
from the weaker
!
bound are readily satis ed. Also, we veri ed that constraints
from neutrinoless doublebeta decay searches are satis ed by the benchmark points.
With regards to the DM relic density, recall that there are multiple classes of
annihilation channels, namely NDMNDM ! X (X = ` ` ,
, bb, tt, W W , ZZ, SS, h1;2h1;2).
200
100
0
1
102
104
lepton pairs, gauge bosons, heavy quarks and scalars. Right: the charged scalar masses MS+ versus
the DM mass. The palette shows the DM Yukawa coupling yDM
y1.
Depending on the speci c value of the DM mass, a given channel may be signi cant or
suppressed. To probe the role of the distinct channels, in gure 5left we plot the contribution
of each channel relative to the total cross section at freezeout, X = tot, versus the DM
mass. Annihilations into lepton pairs typically play a subdominant role. These are
mediated by the couplings gi , whose values should be su ciently small to ensure viable neutrino
masses and consistency with LFV constraints. For lighter values of MDM . 75 GeV, the
cross section tends to be dominated by annihilations into b quarks, while annihilations
into Z2even neutral scalar nal states (X = hh with h
h1;2) are dominant for heavier
values of MDM & 125 GeV. In the intermediate range, annihilations into gauge bosons can
also be important. For completeness, we include the
nal states X = 2S in the plot, for
components of the doublet S. Although the doublet scalars are typically heavier than the
DM, thermal uctuations can allow a contribution from these modes (though the e ect is
clearly subdominant, as seen in the gure). Figure 5right shows the mass of the charged
scalar, MS+ , versus the DM mass. In the lighter DM mass range, MDM . O(100) GeV,
one notices that the charged scalar mass should not exceed 450 GeV, while for larger
values of MDM one can have MS+ at the TeV scale. Such light charged scalars may be of
phenomenological interest as they can be within reach of collider experiments.
We note that gure 5right contains disconnected regions for viable DM, with the
region 31 GeV . MDM . 48 GeV not returning viable benchmark points. This \missing
region" results from an overabundance of DM, due to an insu ciently large,
thermallyaveraged annihilation cross section. In the small MDM region, the annihilation cross section
is dominated by bb nal states, with an important subcontribution from annihilations into
dilatons. However, below MDM
48 GeV, we nd that the dilaton contribution is too small
to allow the observed relic abundance. The allowed island at MDM . 31 GeV corresponds
to parameter space that approaches the h2 resonance, such that 2MDM is around, or just
below, the dilaton mass, namely MDM . Mh2 =2 (the dilaton mass is shown in
gure 6).
This enhances annihilations into SM
nal states. The corresponding enhancement to the
schannel process NDMNDM ! h2h2, via an intermediate h2, is not su cient to overcome
1040
1041
1042
90
80
70
60
50
40
30
20
10
0
LUX(2015)
1
10
100
1000
MDM (GeV)
recent constraints from LUX, while the palette gives the mass for the neutral beyondSM scalar
(dilaton), Mh2 , in units of GeV.
the small cubic coupling
222, as shown in eq. (6.11). Note also that points in the region
MDM . Mh1 =2
60 GeV experience some enhancement from the h1 resonance. Such
enhancements do not occur in heavier MDM regions, as both the dilaton and Higgs are
much lighter than the DM. Throughout the lighter MDM regions, the Higgs may decay into
NDM and h2 nal states, though the bound on invisible Higgs decays is readily satis ed.
The decay h1 ! NDMNDM is su ciently small due to Yukawa suppression (in addition to
small h mixing), as seen from the palette in gure 5right, while the decay h1 ! h2h2 is
suppressed by the small cubic scalar coupling 122.
Next we consider the constraints from directdetection experiments. We plot the
directdetection cross section versus the DM mass for the benchmark parameter sets in
gure 6.
The mass of the dilaton, Mh2 , in units of GeV, is shown in the corresponding palette. One
immediately observes that directdetection limits from LUX [109, 110] impose very serious
constraints on the model, with a large number of benchmark sets already excluded. The
plot shows that the surviving benchmark points mostly occur for MDM . 10 GeV, with a
smaller number of viable points found for MDM & 200 GeV. Benchmarks with intermediate
MDM values are excluded. The viable parameter space typically requires a lighter dilaton
mass, Mh2 . 10 GeV, as all benchmarks with Mh2 & 50 GeV are excluded. It is clear
from the
gure that the surviving benchmark sets can be probed in forthcoming
directdetection experiments.
In gure 7 we consider the oblique parameters. The variation with respect to the mixing
parameter sin2 h is shown in the left panel. One notices that the sin2 h dependence is
not the dominant source of variation. There is some sensitivity to sin2 h, primarily in
S.
However, for a given
xed value of sin2 h, benchmark points occur along the majority of
the Vshaped curve traced out in the plot. Thus, the sin2 h dependence is not driving the
variation. The dependence of the oblique parameters on the dimensionless massdi erence
for components of S, namely
MS0 ) =2MS+ , is shown in the right panel
0.4
0, as seen in the plot, while larger masssplittings can con ict with
ellipsoids show the 68%, 95% and 99% CL., respectively. In the Left frame, the palette shows the
mixing sin2 h between the Higgs and the dilaton; in the Right frame it shows the relative mass
splitting,
of gure 7. The plot shows that the majority of the variation in
T is due to the mass
splitting encoded in
. This is expected. The T parameter is sensitive to isospin violation
and thus constrains the splitting for SU(2)L multiplets. Viable benchmark points occur in
The benchmark points include a range of values for the masssplitting parameter
giving rise to the variation in
gure 7. However, in general, one can take the couplings
4;5 in the scalar potential su ciently small to ensure the masssplitting for S+, S0 and
A is consistent with oblique constraints. From the (technical) naturalness perspective,
arbitrarily small values of 5 are allowed, due to the enhanced lepton number symmetry for
5 ! 0.7 Natural values of 4 are bounded from below by oneloop gauge contributions
to the operator jHySj2. Consequently the mass splitting for components of S is not
expected to be smaller than the oneloop induced splitting, which is safely within the bounds.
Thus, although the oblique parameters can exclude some regions of parameter space, the
constraints are readily evaded.
The exotics in the model can also give new contributions to the Higgs decays h !
Z
. The ratio of the corresponding widths, relative to the SM values, is plotted
in gure 8. One sees that the overwhelming majority of the benchmark points are
consistent with constraints from ATLAS and CMS. Importantly, moreprecise measurements by
ATLAS and CMS during Run II of the LHC will provide further probes of the model.
Before concluding, we note that our analysis reveals considerable di erences between
the SI scotogenic model and the standard (nonSI) scotogenic model. These relate primarily
to the presence of the dilaton. The coupling between
and the DM provides new
annihilation channels for the sterile neutrino DM. This alleviates the need for larger Yukawa
couplings gi , normally required in the scotogenic model to generate the relic density, and
7In practice, the demand of viable neutrino masses gives a Yukawa couplingdependent lower bound
on 5.
1.5
1.4
reduces the tension with LFV constraints. However, the dilaton also permits new channels
at directdetection experiments making these constraints more severe for the SI model. As
a rough guide, one expects stronger LFV signals for the scotogenic model, and stronger
directdetection signals for the SI scotogenic model.
8
In this work, we performed a detailed study of the minimal SI scotogenic model. Our
analysis demonstrates the existence of viable parameter space in which one obtains radiative
electroweak symmetry breaking, oneloop neutrino masses and a good DM candidate. The
model predicts a new scalar with O(GeV) mass. This eld plays the dual roles of
triggering electroweak symmetry breaking and sourcing lepton number symmetry violation.
The model can give observable signals in LFV searches, directdetection experiments, and
precision searches for the Higgs decays h !
and h !
Z. It also predicts a scalar
doublet S, whose mass is expected to be . TeV, within reach of collider experiments. The
model is subject to strong constraints from directdetection experiments; viable parameter
space was found for MDM . 10 GeV and MDM & 200 GeV, while intermediate values for
MDM appear excluded.
Acknowledgments
AA is supported by the Algerian Ministry of Higher Education and Scienti c Research
under the CNEPRU Project No D01720130042.
KM is supported by the Australian
Research Council.
Open Access.
This article is distributed under the terms of the Creative Commons Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. { 16 {
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