Scale invariant scotogenic model: CDF-II W-boson mass and the 95 GeV excesses

Amine Ahriche ,1,* Mohamed Lamine Bellilet ,2,† Mohammed Omer Khojali ,3,4,‡

Mukesh Kumar ,3,§ and Anza-Tshildzi Mulaudzi3,∥
1Department of Applied Physics and Astronomy,

University of Sharjah, P.O. Box 27272 Sharjah, United Arab Emirates
2Laboratoire de Physique des Rayonnements, Badji Mokhtar University, B.P. 12, 23000 Annaba, Algeria

3School of Physics and Institute for Collider Particle Physics,
University of the Witwatersrand, Johannesburg, Wits 2050, South Africa

4Department of Physics, University of Khartoum, PO Box 321, Khartoum 11115, Sudan

(Received 19 April 2024; accepted 20 June 2024; published 22 July 2024)

The anomalies observed in the W mass measurements at the CDF-II experiments and the excesses seen
around 95 GeV at the Large Hadron Collider (LHC) motivate this work, in which we investigate and
constrain the parameter space of the Scale Invariant Scotogenic Model with a Majorana dark matter
candidate. The scanned parameters are chosen to be consistent with the dark matter relic density and the
observed excesses at ∼95 GeV signal strength rates in different channels. We found that significant part of
the viable space addresses the excess in the channel γγ, while a tight part can address the excess in both γγ

and bb̄ channels. Furthermore, the model’s viable parameters can be probed in both the LHC and future
eþe− colliders for di-Higgs production.

DOI: 10.1103/PhysRevD.110.015025

I. INTRODUCTION

In the Standard Model (SM), the mass of theW-boson is
a fundamental parameter, and precise measurements of this
mass are crucial for testing the model’s predictions. The
reported measurements at CDF-II [1] have shown a sig-
nificant discrepancy between the measured W-boson mass
(MCDF

W ¼ 80.4335� 0.0094 GeV) and the mass predicted
by the SM mW ¼ 80.357� 0.006 GeV [2]. This discrep-
ancy is said to be at the level of seven standard deviations.
The W-boson is a weak interaction carrier; and any
deviation from its SM-predicted properties, including its
mass, has important implications, potentially indicating the
presence of new physics beyond the Standard Model
(BSM). Note that a recent measurement from ATLAS [3]
(MATLAS

W ¼ 80.370� 0.019 GeV) shows no deviation from
the SM expectation. Excluding the recent measurements
from CDF-II [1], the current world average from

experiments yields Mavg
W ¼ 80.377� 0.012 GeV, based

on measurements at LEP-2 [4], Tevatron [5,6], and the
LHC [7,8].
In search of a light scalar Higgs boson, the CMS

and ATLAS experiments at the Large Hadron Collider
(LHC) reported a local excess of 2.9σ and 1.7σ at
95.4 GeV in the di-photon (γγ) invariant mass spectrum
in Run 2 dataset [9–12], respectively. The Higgs-boson
(H) production in the Higgsstrahlung process eþe− → ZH
with H → bb̄ an excess of 2.3σ has been observed in the
mass range 95 GeV < mH < 100 GeV at the LEP collider
experiments [13,14]. CMS also reported another local
excess in the light Higgs-boson searches in the τþτ− final
state with a significance of 3.1σ which is compatible with
the aforementioned excesses [15]. A recent study esti-
mates the global significance of the excesses at 95 GeV to
be 3.8σ [16].
The notable discovery of the Higgs boson at the

LHC [17,18] marks the completion of the SM’s founda-
tion. Nevertheless, the observed anomalies mentioned
above open new avenues for considering and constraining
BSM physics. Several such studies are being considered in
Refs. [16,19–39].
Despite its success, the SM has left many questions

unanswered, including the hierarchy problem, the nature of
dark matter (DM), and the smallness of neutrino masses.
Among the extensions of the SM that address these three
problems simultaneously is the Scale Invariant Scotogenic
Model (SI-SCM) [40]. In this framework, the SM is

*Contact author: ahriche@sharjah.ac.ae
†Contact author: medlamine.bellilet@outlook.com
‡Contact author: khogali11@gmail.com
§Contact author: mukesh.kumar@cern.ch
∥Contact author: anza-tshilidzi.mulaudzi@cern.ch

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extended by a real scalar singlet, three Majorana singlet
fermions, and an inert scalar doublet. The real scalar singlet
develops a vacuum expectation value (VEV) to assist in the
radiatively induced electroweak symmetry breaking
(EWSB), à la Coleman [41]. Here, we have two CP-even
scalars whose tree-level eigenmasses are 0 and 125 GeV
which correspond to a dilaton and a SM-like Higgs,
respectively. When considering the radiative corrections
(RCs), two scenarios are possible: (1) the dilaton mass
squared acquires a positive nonzero value, mD < mH, and
the Higgs mass remains mH ¼ 125 GeV (light dilaton
case); and (2) the zero mass value shifts to 125 GeV due
to the RCs, and the 125 GeV tree-level eigenstate becomes a
heavy scalar, with mS > mH, referred as the Pure Radiative
Higgs Mass (PRHM) case [42].
In this setup, the new Yukawa interactions that couple the

Majorana singlet fermions and the inert scalar doublet to the
lepton doublets induce a neutrino mass at one-loop level
similar to the minimal scotogenic model [43]. The DM
candidate here could be either a scalar (the lightest neutral
inert scalar), resembling the case of the inert Higgs model
extended by a real scalar [44], or the lightest Majorana
singlet fermion [45]. The Majorana DM scenario in this
model differs from the minimal scotogenic model, as DM
annihilation occurs additionally into all SM fermions and
gauge bosons via processes mediated by the Higgs boson
and dilaton. This makes the new Yukawa coupling restricted
only by the requirements of neutrino oscillation data and
lepton flavor constraints. Here, in the setup, we investigate
whether the dilaton scalar field could address the 95 GeV
excess mentioned previously while considering theoretical
and experimental constraints and requirements, including
the DM relic density and direct detection, and the W-boson
mass values measured by CDF-II. In addition, we would
like to investigate the impact of all these assumptions and
constraints on the di-Higgs production at the LHC (and at
future eþe− colliders) with

ffiffiffi
s

p ¼ 14 TeV (500 GeV).
This work is organized as follows: Sec. II is dedicated to

presenting the SI-SCM model, describing EWSB, and
discussing various theoretical and experimental con-
straints. Next, in Sec. III, we delve into the discussion
and formulation of W mass corrections and the 95 GeV
signal strength modifiers in the SI-SCM model. The di-
Higgs production mechanism is detailed in Sec. IV, and
our numerical results are presented and discussed in
Sec. V. We conclude our work in Sec. VI.

II. MODEL AND FRAMEWORK

In the SI-SCM, the SM is extended by one inert doublet
scalar, S, three singlet Majorana fermions Ni (i ¼ 1, 2, 3),
and one real singlet scalar ϕ to assist the radiative EWSB, as
shown in Table I. The model is assigned by a global Z2 to
make the lightest Z2-odd field stable, which plays the DM
candidate role. The Lagrangian contains the following terms

L ⊃ −fgi;αNc
i S

†Lβ þ H:c:g − 1

2
yiϕNc

i Ni; ð1Þ

where gi;α and yi are new Yukawa couplings; Lβ are (lαR)
the left-handed lepton doublet (right-handed leptons); the
Greek letters label the SM flavors, α; β∈ fe; μ; τg; and the
SM Higgs and the inert scalar doublets are parametrized as:
HT ¼ ðχþ; ðhþ iχ0Þ= ffiffiffi

2
p Þ and ST ¼ ðSþ;ðS0þ iA0Þ= ffiffiffi

2
p Þ,

respectively (where χþ and χ0 are Goldstone bosons). The
most general SI scalar potential that obeys the Z2 symmetry
is given by

VðH; S;ϕÞ ¼ 1

6
λHðjHj2Þ2 þ λϕ

24
ϕ4 þ λS

2
jSj4

þ ω

2
jHj2ϕ2 þ κ

2
ϕ2jSj2 þ λ3jHj2jSj2

þ λ4jH†Sj2 þ
�
λ5
2
ðH†SÞ2 þ H:c:

�
: ð2Þ

The first term in Eq. (1) and the last term in Eq. (2) are
responsible for generating neutrino mass via the one-loop
diagrams as illustrated in Fig. 1.
The neutrino mass matrix element [46] can be written as

mðνÞ
αβ ¼ P

i gi;αgi;βΛi ¼ ðgT · Λ · gÞαβ, which permits us to
estimate the new Yukawa couplings using the Casas–Ibarra
parametrization [47], where lepton flavor violating (LFV)
bounds on the branching ratios of lα → lβγ and lα →
lβlβlβ should be fulfilled.
Here, the EWSB is triggered by the RCs where the

counterterm δλH; δλϕ; δω, corresponding to terms in
Eq. (2), are chosen to fulfill the tadpole conditions and
one of the CP-even eigenmasses matches the 125 SM-like

FIG. 1. The neutrino mass is generated in the SI-scotogenic
model at one-loop level.

TABLE I. The field charges under the symmetry Z2, where XSM
denotes all SM fields.

Gauge group S Ni ϕ XSM

SUð2ÞL 2 1 1
Uð1ÞY −1 0 0
Z2 −1 −1 1 1

AMINE AHRICHE et al. PHYS. REV. D 110, 015025 (2024)

015025-2



Higgs and the other corresponds to light Higgs (PRHM
case) or a heavy Higgs (light dilaton case). After the EWSB
(hhi ¼ υ; hϕi ¼ x), we obtain two CP-even eigenstates as
H ¼ cαh − sαϕ and D ¼ sαhþ cαϕ, where H denotes the
125 GeV Higgs, D is the dilaton scalar whose mass should
be around mD ¼ 95.4 GeV in this setup, and α is the
Higgs-dilaton mixing angle. Here, the RCs in both PRHM
and light dilaton cases ensure the mixing angle α to be in
the experimental range dictated by the Higgs gauge
couplings measurements. Detailed discussions on these
conditions can be found in [42].
The vacuum stability must be ensured by imposing the

coefficients of the term ϕ4 logϕ to be positive, which
represents the leading term in the scalar; instead the term
ϕ4, where ϕ refers to any direction in the h − ϕ plane. Since
all field dependent squared masses can be written as
m2

i ðh;ϕÞ ¼ 1
2
ðαih2 þ βiϕ

2Þ, the vacuum stability condi-
tions can be written as

P
i niα

2
i > 0 and

P
i niβ

2
i > 0, with

ni to be the multiplicity of the field “i.” In addition to these
conditions, the quartic couplings in Eq. (2) must fulfill the
perturbative unitarity conditions [42].
In this model, the DM candidate could be fermionic (the

lightest Majorana fermion, N1) or a scalar (the lightest
among S0 and A0). In the case of a scalar DM, the situation
matches the singlet extended inert doublet model case [44],
where the coannihilation effect should be considered in
order to have viable parameters space. In the minimal
scotogenic model with Majorana DM, the DM annihilation
occurs via t-channel diagrams mediated by the inert fields,
which makes the Yukawa couplings gi;α values constrained
by the relic density, and therefore the neutrino mass
smallness can be achieved only in extreme S0 − A0 mass
degeneracy, i.e., imposing a very small value for λ5 ∼
Oð10−10Þ [48]. However, in the scale-invariant version,
new s-channels mediated by the Higgs boson or dilaton
exist, which allows all the perturbative range for the gi;α
Yukawa couplings. Also, it is worth noting that, in contrast
to many Majorana dark matter models, in this model, the
dark matter couples to quarks at the tree level. This feature
underscores the significance of direct detection constraints
on the parameter space [45].

III. MW MEASUREMENTS & 95 GeV EXCESSES

The mass of theW-boson can be calculated as a function
of the oblique parametersΔS,ΔT, andΔU, and is given by:

MW ¼mW

�
1þ α

c2W − s2W

×

�
−
1

2
ΔSþ c2WΔT þ c2W − s2W

4s2W
ΔU

��1
2

; ð3Þ

where cW ¼ cos θW and sW ¼ sin θW , with θW being the
weak mixing angle. The oblique parameters in SI-SCM
model are given by [49]

ΔS ¼ 1

24π

�
ð2s2W − 1Þ2Gðm2

S� ; m
2
S� ; m

2
ZÞ

þ Gðm2
S0 ; m

2
A0 ; m2

ZÞ þ log

�
m2

S0m
2
A0

m4
S�

�

þ s2α

�
log

m2
D

m2
H
− Ĝðm2

H;m
2
ZÞ þ Ĝðm2

D;m
2
ZÞ
��

; ð4Þ

ΔT ¼ 1

16πs2Wm
2
W
fFðm2

S� ; m
2
S0Þ

þ Fðm2
S� ; m

2
A0Þ − Fðm2

S0 ; m
2
A0Þ

þ 3s2α½Fðm2
W;m

2
HÞ − Fðm2

Z;m
2
HÞ − Fðm2

W;m
2
DÞ

þ Fðm2
Z;m

2
DÞ�g; ð5Þ

ΔU ¼ 1

24π
fGðm2

S� ; m
2
S0 ; m

2
WÞ þGðm2

S� ; m
2
A0 ; m2

WÞ
− ½2s2W − 1�2Gðm2

S� ; m
2
S� ; m

2
ZÞ −Gðm2

S0 ; m
2
A0 ; m2

ZÞ
þ s2a½Ĝðm2

D;m
2
WÞ − Ĝðm2

D;m
2
ZÞ − Ĝðm2

H;m
2
WÞ

þ Ĝðm2
H;m

2
ZÞ�g; ð6Þ

where the one-loop functions G, F, and Ĝ can be found
in [49].1 The oblique parameter ΔT quantifies the con-
tribution of new physics at low energies and ΔS at different
energy scales.
In order to analyze whether the SI-SCM model can yield

a shift in the prediction for MW that is compatible with the
measurements at experiments and simultaneously provides
a possible explanation of the observed excesses, namely:
(1) γγ & bb or (2) γγ, bb̄ & τþτ−, we perform a χ2 analysis.
This analysis quantifies the agreement between the theo-
retically predicted signal rates μXX̄ [where X ¼ γ, b for case
(1) and X ¼ γ, b, τ for case (2)] and the experimentally
observed values μexpXX̄ . Experimentally, it was determined
that the excesses at ∼95 GeV were best described assuming
signal rates of a scalar resonance as [10,14,15]:

μexpγγ ¼ 0.27þ0.10
−0.09 ;

μexp
bb̄

¼ 0.117� 0.057;

μexpττ ¼ 1.2� 0.5

9>>>=
>>>;
; ð7Þ

where the signal strengths are defined as the cross section
times the branching ratios divided by the corresponding
predictions for the hypothetical SM Higgs boson at the
same mass and the experimental uncertainties are given as
1σ uncertainties.

1Note: Subsequent to the CDF-II results, several research groups
have adjusted their fits for the oblique parametersΔS,ΔT, andΔU
in the context of electroweak precision measurements [50–52],
examining their potential effects on BSM physics.

SCALE INVARIANT SCOTOGENIC MODEL: CDF-II W- … PHYS. REV. D 110, 015025 (2024)

015025-3



The theoretically predicted values for μXX̄ can be
expressed as

μXX̄ ¼ σðgg → DÞ · BðD → XX̄Þ
σSMðgg → HÞ · BSMðH → XX̄Þ

¼ ρXð1 − BðD → XBSMÞÞ; ð8Þ

where ρX is defined for X ¼ γ, b, τ as:

ργ ¼
						1þ

υ

2

λDS�S∓

m2
Sþ

Aγγ
0



m2

D
4m2

Sþ

�

Aγγ
1



m2

D
4m2

W

�
þ 4

3
Aγγ
1=2



m2

D
4m2

t

�
						
2

;

ρb ¼ ρτ ¼ s2α; ð9Þ

with XBSM ¼ NiNk; S0S0; A0A0. Here, σðgg → DÞ and
BðD → XX̄Þ represent the ggF dilaton production and
the final state XX̄ branching ratio, respectively. The scalar
triple coupling of the dilaton with charged scalars, λDS�S∓ ,
is given by λDS�S∓ ¼ sαλ3υþ cακx, and the loop functions
Aγγ
0;1;1=2 are provided in [53]. Additionally, we have

ΓD
tot ¼ s2αΓD;SM

tot þ ΓðD → XBSMÞ, and σSMðgg → HÞ and
BSMðH → XX̄Þ are the corresponding SM quantities evalu-
ated at the Higgs-boson mass mh → mD [54].
To assess the combined description of the three excesses,

we define a total χ2ðNÞ (N ¼ 2, 3) function as

χ2ð2Þ ¼ χ2γγ þ χ2
bb̄
;

χ2ð3Þ ¼ χ2γγ þ χ2
bb̄

þ χ2ττ;

9=
;; ð10Þ

and

χ2i ¼
�
μi − μexpi

Δμexpi

�
2

; ð11Þ

where i ¼ γγ, bb̄, or ττ. These functions are useful for
checking whether the excess can be addressed simulta-
neously in the channels: (i) γγ; bb̄ and (ii) γγ; bb̄; ττ.
In the parameter space region corresponding to a small

charged scalar effect on the effective coupling Dγγ (i.e.,
ργ ∼ 1), as indicated by the experimental values in Eq. (7),
we found the following results for χ2N < 1σ:

(i) χ2ð2Þjmin ¼ 2.262: s2α ¼ 0.11, and BðD → XBSMÞ ¼
70.15%.

(ii) χ2ð3Þjmin ¼ 7.708: s2α ¼ 0.11, and BðD → XBSMÞ ¼
69.7%.

This scenario becomes plausible only if the channel
D → inv is accessible. Here, “inv.” denotes invisible
channels such as NiNk;H0H0; A0A0. Given that the inert
neutral masses are expected to be large, the Majorana
singlet fermions should be light, i.e., mDM ¼ M1 < mD=2,
as will be confirmed later. In the subsequent numerical
analysis (Sec. V), we will consider parameter points to

provide a good description of the excesses if they account
for the combined effect of the excess in the channels γγ and
bb̄ at the 1σ level, since the ττ channel appears to be
hopeless.

IV. THE DI-HIGGS PRODUCTION

In the SM, the measurement of di-Higgs (HH) produc-
tion is intriguing not only because it enables the determi-
nation of Higgs-boson self-interaction but also because it
contributes to understanding EWSB. In cases where EWSB
proceeds via a single scalar (the SM Higgs boson), the di-
Higgs signal occurs through two Feynman diagrams: box
and triangle diagrams. The triangle diagram involves a triple
Higgs-boson vertex that could be modified by new physics

effects, represented as λhhh ¼ λðSMÞ
hhh ð1þ ΔhhhÞ. Therefore,

the di-Higgs measurement can precisely determine the new
physics effect by measuring Δhhh. However, when EWSB
involves more than one scalar, as in the model considered in
this study, the di-Higgs signal occurs through a box and two
(or more) triangle diagrams that involve additional triple
couplings λhhS (with S ¼ h, D) and new CP scalar (dilaton
in our case). Consequently, the determination of λhhh is not
straightforward since σðHHÞ depends on several model
parameters. Nonetheless, the experimental bounds on
σðHHÞ through different channels, either via resonant or
nonresonant production, are very useful for constraining
the scalar sector, especially if σðHHÞ exhibits values
greater than the SM predictions. Nonresonant HH pro-
duction at the LHC occurs primarily through the dominant
gluon fusion (ggF) mode and the subdominant vector-
boson fusion (VBF) mode. The cross section for HH
production at next-to-next-to-leading order, including
finite top-quark-mass effects in the ggF mode, is σSMggF ¼
31.05þ2.1

−7.2 fb [55–62]. However, in the VBF mode, at next-
to-next-to-next-to-leading order, the cross section is
σSMVBF ¼ 1.73� 0.04 fb [63–67] for mh ¼ 125 GeV atffiffiffi
s

p ¼ 13 TeV. The smallness of the HH production cross
section in ggF mode at leading order (LO) results from the
negative interference between the box and triangle
Feynman diagrams, determined by three contributing
factors [42,68,69]:

σSMðHHÞ ¼ σB þ σT þ σBT; ð12Þ

where σB ¼ 70.1 fb represents the box contribution,
σT ¼ 9.66 fb corresponds to the triangle contribution,
and σBT ¼ −49.9 fb accounts for their interference [70].
In the SI-SCM framework, nonresonant HH production

through ggF mode includes an additional triangle Feynman
diagram mediated through the dilaton field D. The pro-
duction cross section of di-Higgs production either at the
LHC or at e−eþ colliders has three distinct contributions
that come from: (1) the Feynman diagrams involving only
the triple scalar couplings that have one scalar propagator,

AMINE AHRICHE et al. PHYS. REV. D 110, 015025 (2024)

015025-4



(2) the diagrams with only pure gauge couplings; and (3) the
interference contribution [42,69]. Therefore, the di-Higgs
production cross section can be expressed as follows:

σðHHÞ ¼ ζ1σB þ ζ2σT þ ζ3σBT; ð13Þ

where the coefficients ζi in this model are modified with
respect to the SM as [69]

ζ1 ¼ c4α;

ζ2 ¼
				cα λHHH

λSMHHH
þ sα

λHHD
λSMHHH

s−m2
HþimHΓH

s−m2
DþimDΓD

				
2

;

ζ3 ¼ c2αℜ

�
cα

λHHH
λSMHHH

þ sα
λHHD
λSMHHH

s−m2
HþimHΓH

s−m2
DþimDΓD

�

9>>>>>>=
>>>>>>;
; ð14Þ

where λSMHHH is the Higgs triple coupling in the SM; and
ffiffiffi
s

p
is the center-of-mass collision energy, which we will
consider to be

ffiffiffi
s

p ¼ 14 TeV at LHC. The expression for
the one-loop triple Higgs coupling in the SM is [71]:

λSMHHH ≃
3m2

H

υ

�
1 −

m4
t

π2υ2m2
H

�
; ð15Þ

with mt as the top quark mass.
Interestingly, a direct measurement of the triple

Higgs-boson self-coupling is achievable through reso-
nant HH production at a future eþe− collider. This
involves double Higgsstrahlung processes with W or
Z bosons, as well as through WW or ZZ fusion. In
the case of double Higgsstrahlung (eþe− → HHZ)
production at

ffiffiffi
s

p ¼ 500 GeV, the production cross sec-
tion can be expressed as Eq. (13) using the same
coefficients in Eq. (14) and the cross-section contributions
given as σB ¼ 0.0837 fb, σT ¼ 0.01565 fb, and σBT ¼
0.05685 fb [42].

V. NUMERICAL RESULTS AND DISCUSSION

In this section, our attention is directed towards the
parameter space corresponding to the dilaton mass window
of 94 GeV < mD < 97 GeV. We systematically consider
various theoretical and experimental constraints, including
perturbativity, perturbative unitarity, Higgs-boson di-photon
and invisible decays, LEP negative searches, and the
electroweak precision tests. In addition, we require the
DM relic density to match the observed values, and the DM
direct detection constraints to be satisfied [45]. In our
analysis, instead of relying on the bounds on the total
Higgs strength modifier (μtot ¼ c2α × ð1 − BinvÞ ≥ 0.89 at
95% confidence level (CL) [72]),2 we perform a detailed
analysis by considering the bounds on the Higgs total decay

width (Γh ¼ 4.6þ2.6
−2.5 MeV); and the partial Higgs strength

signal modifiers μhXX for X ¼ μ; τ; b; γ;W; Z [2]. For this
reason, we define the SM χ2SM function

χ2SM ¼
X
O

χ2O ¼
X
O

�
O −Oexp

ΔOexp

�
2

; ð16Þ

with the observables O denotes the Higgs total decay width
(Γh) and the Higgs signal strength modifiers (μhXX). In our
analysis, we consider only benchmark points (BPs) with a
precision of 95% CL, i.e., χ2SM < 11.07.
Additionally, we ensure that the considered values for

the model’s free parameters, including the inert masses
(mS0 ,mA0 ,mS�), Majorana masses (Mi), scalar coupling λ3,
and the singlet VEV x, correspond to values of the new
Yukawa couplings gi;α that satisfy the neutrino oscillation
data and LFV constraints.
Through a random numerical scan adhering to the

aforementioned constraints and conditions, we consider
15,000 BPs that satisfy the 95 GeVexcess in the γγ channel
(μexpγγ ¼ 0.27þ0.10

−0.09 ). The viable parameter space fulfilling the
various conditions and constraints is illustrated in Fig. 2.
Figure 2(a) reveals that the assumptions of mD ≈

95 GeV and μexpγγ ¼ 0.27þ0.10
−0.09 lead to inert masses exceed-

ing 300 GeV. Additionally, the new Yukawa couplings are
an order of magnitude smaller than the perturbative limit,
contrasting with general cases in the SI-SCM [40,45]. From
Fig. 2(a), one can observe that the couplings gi;α have
values ranging from 0.001 to 0.4 for inert masses across
different ranges. These values are dictated by the require-
ments for DM relic density [45,48,76], where the DM
annihilation channels N1N1 → lαlβ; νανβ play a key role
in the majority of the viable parameter space.
In Fig. 2(b), we show the singlet scalar VEV x as a

function of the DM mass mDM and the freeze-out param-
eter xf. It is noteworthy that mDM must be smaller than
mD=2 to ensure the branching ratio BðD → invÞ lies
between 50% and 85%, satisfying the assumption μexpγγ ¼
0.27þ0.10

−0.09 [Fig. 2(b)]. It’s important to mention that the
values of the DM freeze-out observable (the annihilation
cross section and freeze-out parameter xf ¼ mDM=Tf,
where Tf is the freeze-out temperature) exhibit typical
values for a Weakly Interacting Massive Particle DM
candidate. One notices that the singlet scalar VEV, x, has
lower bounds arising from various constraints. Perturb-
ativity and unitarity together require x > 57 GeV, which
is the dominant constraint for M1 ≡mDM ≲ 1 GeV.
For M1 ≡mDM > 1 GeV, the dominant constraint comes
from the Higgs-dilaton mixing, which is equivalent to
χ2SM < 11.07. Despite the tight range of the dilaton mass,
94 GeV < mD < 97 GeV, the singlet scalar VEV spans a
broad interval, x∈ ½57 GeV; 100 TeV�. This can be under-
stood by considering that the dilaton mass is a one-loop
effect sensitive to multiple parameters, including x, the

2Here, the Higgs invisible branching ratio is constrained
by ATLAS to be Binv ¼ BðH → NiNkÞ < 0.11 [73].

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multiplicity (ni), and the couplings of the new fields to the
Higgs doublet (αi) and the singlet scalar (βi). Thus, any
value for the dilaton mass can be achieved by selecting
appropriate values for the couplings (αi, βi), making the
range of x indistinguishable for cases of restricted and
nonrestricted dilaton mass [40,42,45].
In Fig. 2(c), we depict the MW prediction within the

framework of the SI-SCM model as a function of the scalar
mixing angle s2α and mS� . Additionally, we compare these
predictions with various measurements at their respective
2σ limits. Importantly, in this model, the correction ΔmW is
strictly positive, driven by the fact that ΔT is always
positive and dominates over the values of ΔS and ΔU. For
instance, without considering the MW bounds, the relative
mass difference between CP-even and CP-odd scalars
could reach 20%, whereas incorporating the MW measure-
ments reduces this difference to below 5%.
In Fig. 3, we present the signal strength modifiers and

the relevant observable for the 95 GeV scalar candidate.

Notably, the excesses μγγ;bb̄ can be simultaneously
addressed, whereas the excess μττ shows suppressed
values. As illustrated in the μγγ − μbb̄ panels in Fig. 3,
the excesses in the channels (γγ; bb̄) can be accommo-
dated simultaneously at the 1σ level for a very
small region of the parameter space (the BPs point inside
the 1σ contour in Figs. 3(a)–3(c) with χ2ð2Þ < 2.53); and

at the 2σ level for a significant region of the parameter
space. However, the ττ channel can only be accommo-
dated at the 99% CL, i.e., 8.02 < χ2ð3Þ < 11.34. The

preference for maximal values in both scalar mixing,
s2α ≈ 0.11, and BðD → invÞ ≈ 70%, becomes evident
when matching the excess in channels other than γγ as
shown in Fig. 3(d). If the di-tau excess is analyzed by
ATLAS and/or reanalyzed by CMS with additional data,
and the measured μexpττ is relaxed to around 0.6-0.7, then
addressing the three excesses within this model becomes
feasible.

FIG. 2. (a) Maximum Yukawa couplings maxðjgi;αjÞ plotted against the masses of the charged inert doublet, mS� , and the neutral
scalar,mS0 . (b) The singlet scalar VEV xas a function ofmDM and the freeze-out parameter, xf . (c)MW prediction in the SI-SCM model
shown with respect to the scalar mixing angle, s2α, and mS� . The horizontal color bands represent MW measurements at a 2σ level from
various experiments: green for PDG [74], blue for LEP [75], red for CDF [1], gray for ATLAS [3], cyan for the world average [38], and
yellow for the SM value.

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In Fig. 4, we present the di-Higgs production cross section
for both (a) the LHCwith

ffiffiffi
s

p ¼ 14 TeV and (b) future e−eþ

colliders with
ffiffiffi
s

p ¼ 500 GeV plotted against the scalar
mixing angle s2α. As shown in Fig. 4(a), di-Higgs production
at the LHC exhibits no enhancement; however, a reduction
of 73% is possible for benchmark points with a smaller
singlet VEV x and nonsuppressed scalar mixing. In contrast,
at eþe− colliders, the double Higgsstrahlung cross section
ranges from a reduction of 24% to an enhancement of 46%
compared to the SM cross section, as illustrated in Fig. 4b.
Current constraints from the LHC [77–86], particularly

those from the HH → γγbb̄ and other di-Higgs decay
channels, place stringent limits on the di-Higgs production
cross section and the Higgs cubic self-coupling. Our model,
which predicts a nonzero branching ratio to invisible
particles, must be evaluated in light of these constraints.
The di-Higgs cross section predicted by our model is
smaller than the SM values, inherently satisfying all the

bounds from negative searches at the LHC. Given the
smaller cross section, the predicted rates for HH → γγbb̄
and other di-Higgs decay channels remain below the upper
limits set by the LHC. This implies that the model does not
predict an excess of di-Higgs events that would have
already been excluded by current experimental data.
While our results suggest that the SI-SCM model is

consistent with current studies on collider anomalies,
including the CDF-II W-mass anomaly and the 95 GeV
excess [34,35], it is important to note that the muon’s
magnetic moment, ðg − 2Þμ, anomaly cannot be addressed
in this model. This limitation arises from its single negative
contribution, which is insufficient to match the measured
value. To reconcile this discrepancy in ðg − 2Þμ, an exten-
sion of the SI-SCM by incorporating additional scalar
components may be required. Such an extension could
potentially enable the model to successfully account for the
experimental values associated with ðg − 2Þμ.

FIG. 3. (a) The plot illustrates the signal strengths for the three excesses: γγ, ττ, and bb̄. (b) This plot presents the signal strengths of γγ
and bb̄ as functions of the total χ2ð2Þ. (c) Signal strengths of γγ and bb̄ are plotted against the total χ2ð3Þ. (d) This plot shows the dilaton’s
invisible branching ratio versus the scalar mixing angle, s2α, presented alongside the total χ2ð2Þ. In panels (a)–(c), all the benchmark points

(BPs) fall within the 2σ contour, though it is visible only in a restricted region due to the chosen μγγ values.

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VI. CONCLUSION

In response to anomalies observed in the measurement of
the W-mass at CDF-II and an excess around ∼95 GeV
reported by LEP, CMS, and ATLAS, we conducted a study
within the framework of the SI-SCM model to address
these issues. This model, characterized by radiatively
induced EWSB, not only accommodates light neutrino
masses but also proposes a Majorana dark matter candidate.
Its predictions are well within the reach of collider experi-
ments. We identified a viable parameter space where inert
scalar masses can explain the W-mass anomaly, and the
light dilaton with mD ∼ 95 GeV aligns with the observa-
tions in that region. Under these assumptions, the dilaton
might need to decay into invisible Majorana singlet
fermions to address the excess (7) in the channels
γγ & bb̄. For this reason, the excesses considered in this

study can be accommodated simultaneously at the 1σ level
within a very small region of the parameter space, and at the
2σ level within a significant region. Within this parameter
space, di-Higgs production at the LHC shows no enhance-
ment compared to the SM. However, at eþe− colliders, the
double Higgsstrahlung cross section varies from a reduc-
tion of 24% to an enhancement of 46% compared to the SM
cross section.

ACKNOWLEDGMENTS

A. A. and M. L. B. were funded by the University of
Sharjah under the research Projects No. 21021430107
“Hunting for New Physics at Colliders” and
No. 23021430135 “Terascale Physics: Colliders vs
Cosmology.”

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