compat=1.0.0

IFIC/23-20

A Scotogenic explanation for the 95 GeV excesses

Pablo<EMAIL_ADDRESS>Víctor Martín
<EMAIL_ADDRESS>Avelino
<EMAIL_ADDRESS>
(a) Instituto de Física Corpuscular, CSIC-Universitat de València, 46980
Paterna, Spain

(b) Departament de Física Teòrica, Universitat de València, 46100 Burjassot,
Spain

###### Abstract

Several hints of the presence of a new state at about $95$ GeV have been
observed recently. The CMS and ATLAS collaborations have reported excesses in
the diphoton channel at about this diphoton invariant mass with local
statistical significances of $2.9\,\sigma$ and $1.7\,\sigma$, respectively.
Furthermore, a $2\,\sigma$ excess in the $b\bar{b}$ final state was also
observed at LEP, again pointing at a similar mass value. We interpret these
intriguing hints of new physics in a variant of the Scotogenic model, an
economical scenario that induces Majorana neutrino masses at the loop level
and includes a viable dark matter candidate. We show that our model can
naturally explain the 95 GeV excesses while respecting all experimental
constraints and discuss other phenomenological predictions of our scenario.

##  1 Introduction

The Standard Model (SM) of particle physics has been an incredibly successful
framework for understanding the fundamental particles and forces that make up
our Universe. However, it faces significant challenges when it comes to
explaining two crucial phenomena: neutrino masses and dark matter (DM).
Neutrinos were long thought to be massless, as suggested by the original
formulation of the SM. However, experimental evidence has now firmly
established that neutrinos do have masses, albeit very small ones. Similarly,
the existence of DM, which is inferred from its gravitational effects on
visible matter, poses another major challenge and various theoretical
extensions of the SM have been proposed to account for it. Explaining the
origins and properties of neutrino masses and DM continues to be an active
area of research.

After the discovery of the Higgs boson, which proved the existence of a scalar
field and provided important insights into the mechanism of mass generation,
the LHC has continued to search for additional scalar particles. In fact, many
Beyond the Standard Model (BSM) scenarios include new scalar states. This is
also the case of models addressing the neutrino and DM problems, which
typically require extended scalar sectors. If these new states have masses and
couplings within the reach of the LHC, their signals may be hidden in the
currently existing searches or show up in the near future, and may appear at
lower or higher masses than the scalar found at the LHC at a mass of 125 GeV.
If lighter states exist, they may be produced given the high energies
available at colliders. In fact, different experiments have performed searches
for low mass scalars in different channels [1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11].

The diphoton channel plays a crucial role in the search for new scalar
particles at the LHC. This final state allows for precise measurements and
clean experimental signatures, making it easier to isolate potential signals
of new scalar particles amidst background noise. Interestingly, the CMS
collaboration has been consistently finding an excess over the SM prediction
in this channel at a diphoton invariant mass of $\sim 95$ GeV [5, 6]. The
statistical support for this excess has been reinforced by recent results
obtained after the analysis of the full Run 2 dataset [7]. The excess is
maximal for a mass of $95.4$ GeV and has a local (global) significance of
$2.9\,\sigma$ ($1.3\,\sigma$) and can be interpreted as the production via
gluon fusion and subsequent decay of a new scalar state, $h_{95}$. It can be
parametrized numerically in terms of the $\mu_{\gamma\gamma}$ signal strength,
which normalizes the cross section of the process to the analogous cross
section for a Higgs-like state $H$ at the same mass. The latest CMS result
points to [7, 12]

$\mu^{\rm CMS}_{\gamma\gamma}=\frac{\sigma^{\rm CMS}(gg\to
h_{95}\to\gamma\gamma)}{\sigma^{\rm SM}(gg\to
H\to\gamma\gamma)}=0.33^{+0.19}_{-0.12}\,.$ (1)

The ATLAS collaboration has also performed searches in the diphoton channel,
although with a lower sensitivity. A very mild excess in a mass region
compatible with that hinted by CMS was found in their Run 1 analysis [8].
Their update including 140 fb-1 of Run 2 data appeared recently [13].
Intriguingly, the statistical significance of the excess increases in the new
ATLAS results. This can be attributed to the addition of more statistics as
well as to several improvements in the analysis. In particular, the model-
dependent analysis presented in this update hints at an excess, curiously at
$95.4$ GeV too, with a local significance of $1.7\,\sigma$. We note that this
result is compatible with that of CMS, which ATLAS cannot exclude.

The 95 GeV region is particularly interesting due to the existence of other
excesses hinting at similar mass values. LEP has reported an excess in
$b\bar{b}$ production at about 95 GeV with a local significance of $2\,\sigma$
[2]. This excess can be interpreted in terms of a new scalar state
contributing to the process $e^{+}e^{-}\to Z\,h_{95}\to Z\,b\bar{b}$, with a
signal strength given by [14, 15]

$\mu^{\rm LEP}_{bb}=\frac{\sigma^{\rm LEP}(e^{+}e^{-}\to Z\,h_{95}\to
Z\,b\bar{b})}{\sigma^{\rm SM}(e^{+}e^{-}\to Z\,H\to Z\,b\bar{b})}=0.117\pm
0.057\,.$ (2)

Other searches for light scalars in CMS also gave a small excess in the ditau
channel $\mu_{\tau\tau}^{\rm CMS}=1.2\pm 0.5$ [10]. ATLAS has not published
any ditau search in this mass region, but has only provided results for scalar
masses above $200$ GeV [16].

The 95 GeV excesses have received some attention recently [15, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. We interpret them
in a variant of the Scotogenic model [36], a well-motivated and economical BSM
scenario that incorporates a mechanism for the generation of neutrino masses
and provides a testable DM candidate. We thus consider the possibility that
these excesses are the first collider hints of a new BSM sector addressing
some of the most important open questions in particle physics.

The rest of the manuscript is organized as follows. Section 2 introduces our
model, whereas Section 3 interprets the 95 GeV excesses in terms of a new
scalar state in the particle spectrum. The most relevant experimental
constraints are discussed in Section 4 and our numerical results, which prove
that our setup can accommodate the excesses, are presented in Section 5. Other
aspects of our scenario, such as neutrino masses, DM and additional collider
signatures, are discussed in Section 6. Finally, we summarize our work in
Section 7. An Appendix is also included with some technical details.

##  2 The Model

We consider a variant of the Scotogenic model [36] that extends the SM
particle content with $n_{N}$ generations of singlet fermions $N_{n}$,
$n=1,\dots,n_{N}$, and $n_{\eta}$ doublet scalars $\eta_{a}$,
$a=1,\dots,n_{\eta}$. These fields are assumed to be odd under a new
$\mathbb{Z}_{2}$ symmetry, under which all the SM states are even. This
generalizes the original Scotogenic model [37], which corresponds to $n_{N}=3$
and $n_{\eta}=1$. The $\eta_{a}$ doublets can be decomposed in terms of their
$\rm SU(2)_{L}$ components as

$\eta_{a}=\left(\begin{array}[]{c}\eta_{a}^{+}\\\
\eta_{a}^{0}\end{array}\right)\,.$ (3)

In addition, we include a real singlet scalar $S$. The lepton and scalar
particle content of the model is summarized in Tab. 1.

Field | Generations | $\boldsymbol{\rm SU(3)_{c}}$ | $\boldsymbol{\rm SU(2)_{L}}$ | $\boldsymbol{\rm U(1)_{Y}}$ | $\boldsymbol{\mathbb{Z}_{2}}$
---|---|---|---|---|---
$\ell_{L}$ | $3$ | 1 | 2 | -1/2 | +
$e_{R}$ | $3$ | 1 | 1 | -1 | +
$N$ | $n_{N}$ | 1 | 1 | 0 | -
$H$ | $1$ | 1 | 2 | 1/2 | +
$\eta$ | $n_{\eta}$ | 1 | 2 | 1/2 | -
$S$ | $1$ | 1 | 1 | 0 | +
Table 1: Lepton and scalar particle content of the model and their
representations under gauge and global symmetries.

The Yukawa Lagrangian of the model includes the terms

$\displaystyle\mathcal{L}\supset
y_{na\alpha}\,\overline{N}_{n}\widetilde{\eta}_{a}^{\dagger}\ell_{L}^{\alpha}-\kappa_{nm}\,S\,\overline{N^{c}}_{n}N_{m}-\frac{1}{2}\left(M_{N}\right)_{nn}\overline{N^{c}}_{n}N_{n}+\text{h.c.}\,,$
(4)

where $n,m=1,\dots,n_{N}$, $a=1,\dots,n_{\eta}$ and $\alpha=1,2,3$ are
generation indices. The Yukawa coupling $y$ is an $n_{N}\times n_{\eta}\times
3$ object while $\kappa$ and $M_{N}$ are $n_{N}\times n_{N}$ symmetric
matrices. $M_{N}$ has been chosen diagonal without loss of generality.
Finally, we define $\widetilde{\eta}=i\sigma_{2}\eta^{*}$. The scalar
potential of the model can be written as

$\mathcal{V}=\mathcal{V}_{H}+\mathcal{V}_{\eta}+\mathcal{V}_{S}+\mathcal{V}_{\rm
mix}\,,$ (5)

with

$\displaystyle\mathcal{V}_{H}=$ $\displaystyle
m_{H}^{2}\,H^{\dagger}H+\frac{1}{2}\lambda_{1}\,\left(H^{\dagger}H\right)^{2}\,,$
(6) $\displaystyle\mathcal{V}_{\eta}=$
$\displaystyle\left(m_{\eta}^{2}\right)_{aa}\eta_{a}^{\dagger}\eta_{a}+\frac{1}{2}\lambda_{2}^{abcd}\left(\eta_{a}^{\dagger}\eta_{b}\right)\left(\eta_{c}^{\dagger}\eta_{d}\right)\,,$
(7) $\displaystyle\mathcal{V}_{S}=$
$\displaystyle\frac{1}{2}m_{S}^{2}\,S^{2}+\frac{1}{3}\mu_{S}\,S^{3}+\frac{1}{4}\lambda_{S}\,S^{4}\,,$
(8) $\displaystyle\mathcal{V}_{\rm mix}=$
$\displaystyle\lambda_{3}^{ab}\left(H^{\dagger}H\right)\left(\eta_{a}^{\dagger}\eta_{b}\right)+\lambda_{4}^{ab}\left(H^{\dagger}\eta_{a}\right)\left(\eta_{b}^{\dagger}H\right)+\frac{1}{2}\left[\lambda_{5}^{ab}\left(H^{\dagger}\eta_{a}\right)\left(H^{\dagger}\eta_{b}\right)+\text{h.c.}\right]$
$\displaystyle+\mu_{H}\,H^{\dagger}HS+\frac{1}{2}\lambda_{HS}\,H^{\dagger}HS^{2}+\mu_{\eta}^{ab}\,\eta_{a}^{\dagger}\eta_{b}\,S+\frac{1}{2}\lambda_{\eta
S}^{ab}\,\eta_{a}^{\dagger}\eta_{b}\,S^{2}\,.$ (9)

Here all the indices are $\eta$ generation indices and then $m_{\eta}^{2}$,
$\lambda_{3,4,5}$, $\mu_{\eta}$ and $\lambda_{\eta S}$ are $n_{\eta}\times
n_{\eta}$ matrices, while $\lambda_{2}$ is an $n_{\eta}\times n_{\eta}\times
n_{\eta}\times n_{\eta}$ object. We also note that $\lambda_{5}$ must be
symmetric whereas $\lambda_{3,4}$, $\mu_{\eta}$ and $\lambda_{\eta S}$ are
Hermitian. Again, $m_{\eta}^{2}$ will be assumed to be diagonal without any
loss of generality.

###  2.1 Symmetry breaking, scalar masses and mixings

We will assume that the vacuum of our model is given by

$\langle
H^{0}\rangle=\frac{v}{\sqrt{2}}\,,\quad\langle\eta^{0}_{a}\rangle=0\,,\quad\langle
S\rangle=v_{S}\,.$ (10)

The vacuum expectation value (VEV) $v$ breaks the electroweak symmetry in the
usual way. In contrast, the $\mathbb{Z}_{2}$ Scotogenic parity remains exactly
conserved due to $\langle\eta_{a}^{0}\rangle=0$. The VEV configuration in Eq.
(10) imposes some conditions on the scalar potential parameters due to the
minimization equations

$\displaystyle\left.\frac{\partial V}{\partial H}\right|_{\langle
H\rangle=\frac{v}{\sqrt{2}}\,,\,\langle S\rangle=v_{S}}=$ $\displaystyle
m_{H}^{2}v+\mu_{H}v\,v_{S}+\frac{1}{2}\lambda_{1}v^{3}+\frac{1}{2}\lambda_{HS}v_{S}^{2}v=0\,,$
(11) $\displaystyle\left.\frac{\partial V}{\partial S}\right|_{\langle
H\rangle=\frac{v}{\sqrt{2}}\,,\,\langle S\rangle=v_{S}}=$ $\displaystyle
m_{S}^{2}v_{S}+\mu_{S}v_{S}^{2}+\frac{1}{2}\mu_{H}v^{2}+\lambda_{S}v_{S}^{3}+\frac{1}{2}\lambda_{HS}v^{2}\,v_{S}=0\,.$
(12)

After symmetry breaking, the real component of the neutral $H^{0}$ field mixes
with the real $S$ field. In the basis
$\mathcal{H}=\left\\{S,\text{Re}(H^{0})\right\\}$ their mass matrix reads

$\displaystyle\mathcal{M}_{\mathcal{H}}^{2}=\left(\begin{matrix}m_{S}^{2}+2\mu_{S}v_{S}+\frac{1}{2}\lambda_{HS}\,v^{2}+3\lambda_{S}v_{S}^{2}&\mu_{H}v+\lambda_{HS}\,v\,v_{S}\\\
\mu_{H}v+\lambda_{HS}\,v\,v_{S}&m_{H}^{2}+\mu_{H}v_{S}+\frac{3}{2}\lambda_{1}v^{2}+\frac{1}{2}\lambda_{HS}\,v_{S}^{2}\end{matrix}\right)\,.$
(13)

After application of Eqs. (11) and (12), solved for $m_{H}^{2}$ and
$m_{S}^{2}$, this matrix becomes

$\displaystyle\mathcal{M}_{\mathcal{H}}^{2}=\left(\begin{matrix}v_{S}\left(\mu_{S}+2\lambda_{S}v_{S}\right)-\frac{\mu_{H}v^{2}}{2v_{S}}&\mu_{H}v+\lambda_{HS}\,v\,v_{S}\\\
\mu_{H}v+\lambda_{HS}\,v\,v_{S}&\lambda_{1}v^{2}\end{matrix}\right)\,.$ (14)

It can be brought to diagonal form as
$V_{\mathcal{H}}\,\mathcal{M}^{2}_{\mathcal{H}}\,V_{\mathcal{H}}^{T}=\widehat{\mathcal{M}}^{2}_{\mathcal{H}}=\text{diag}\left(m^{2}_{h_{1}},m^{2}_{h_{2}}\right)$,
where $h_{1}$ and $h_{2}$ are mass eigenstates and

$V_{\mathcal{H}}=\left(\begin{matrix}\cos\alpha&\sin\alpha\\\
-\sin\alpha&\cos\alpha\end{matrix}\right)\,,$ (15)

with

$\tan
2\alpha=\frac{2\,\left(\mathcal{M}^{2}_{\mathcal{H}}\right)_{12}}{\left(\mathcal{M}^{2}_{\mathcal{H}}\right)_{11}-\left(\mathcal{M}^{2}_{\mathcal{H}}\right)_{22}}\,.$
(16)

This mixing angle between the singlet and doublet scalars plays a central role
in the phenomenology of our model, as we will explain in the following
Subsection. We focus now on the $\mathbb{Z}_{2}$-odd scalars $\eta^{+}_{a}$
and $\eta^{0}_{a}$. We decompose the neutral components of the $\eta_{a}$
doublets as

$\eta_{a}^{0}=\frac{1}{\sqrt{2}}(\eta_{R_{a}}+i\eta_{I_{a}})\,,$ (17)

and they do not mix if we assume that CP is conserved in the scalar sector.
This can be easily achieved if all the parameters in the scalar potential are
real. Again, after electroweak symmetry breaking, the $n_{\eta}\times
n_{\eta}$ mass matrices are given by

$\displaystyle\bigl{(}\mathcal{M}_{\eta_{R}}^{2}\bigr{)}_{ab}$
$\displaystyle=\left(m_{\eta}^{2}\right)_{ab}+\left(\lambda_{3}^{ab}+\lambda_{4}^{ab}+\lambda_{5}^{ab}\right)\frac{v^{2}}{2}+\frac{v_{S}^{2}}{2}\lambda_{\eta
S}^{ab}+\mu_{\eta}^{ab}v_{S}\,,$ (18)
$\displaystyle\bigl{(}\mathcal{M}_{\eta_{I}}^{2}\bigr{)}_{ab}$
$\displaystyle=\left(m_{\eta}^{2}\right)_{ab}+\left(\lambda_{3}^{ab}+\lambda_{4}^{ab}-\lambda_{5}^{ab}\right)\frac{v^{2}}{2}+\frac{v_{S}^{2}}{2}\lambda_{\eta
S}^{ab}+\mu_{\eta}^{ab}v_{S}\,,$ (19)
$\displaystyle\bigl{(}\mathcal{M}_{\eta^{+}}^{2}\bigr{)}_{ab}$
$\displaystyle=\left(m_{\eta}^{2}\right)_{ab}+\lambda_{3}^{ab}\frac{v^{2}}{2}+\frac{v_{S}^{2}}{2}\lambda_{\eta
S}^{ab}+\mu_{\eta}^{ab}v_{S}\,.$ (20)

Notice that the mass matrices for the real and imaginary components are the
same in the limit in which all the elements of $\lambda_{5}$ vanish.

###  2.2 Scalar couplings

$h_{i}$$\eta^{\pm}$$\eta^{\mp}$$\eta^{\pm}$$\gamma$$\gamma$$h_{i}$$W^{\pm}$$W^{\mp}$$W^{\pm}$$\gamma$$\gamma$$h_{i}$$f$$f$$f$$\gamma$$\gamma$

Figure 1: 1-loop contributions to $h_{i}\to\gamma\gamma$.

The couplings of the $h_{1}$ and $h_{2}$ scalars to the SM fermions are
determined by the $\alpha$ mixing angle which, as discussed below, will be
constrained to be small. We note that the singlet $S$ does not have a Yukawa
term with the SM fermions. Hence, it can only couple to them via mixing with
the SM Higgs $H$. As a result of this, one of the mass eigenstates ($h_{1}$,
the mostly-singlet one) couples to the SM fermions proportional to
$\sin\alpha$ while the other mass eigenstate ($h_{2}$, the SM-like one)
couples proportional to $\cos\alpha$. The same happens for the coupling to
$WW$ and $ZZ$ bosons and the loop coupling to gluons. However, the 1-loop
couplings to $\gamma\gamma$ and $\gamma Z$ will be affected by the new
particle content present in our model, in particular the $\eta$ doublets. In
Fig. 1 we can see the different contributions to the loop decays
$h_{i}\to\gamma\gamma$. Similar diagrams can be drawn for the $\gamma Z$ final
state. On the one hand the Higgses will couple to $\gamma\gamma/Z$ through the
SM loops with $W$ bosons and fermions, with the largest contribution among the
fermions given by the top quark. On the other hand, the charged $\eta$ states
can also run into the loop and contribute to the decay through the
$g_{h_{i}\eta\eta}$ coupling, given by

$g_{h_{i}\eta\eta}=\left(V_{\mathcal{H}}\right)_{i1}(\lambda_{\eta
S}\,v_{S}+\mu_{\eta})+\left(V_{\mathcal{H}}\right)_{i2}\,\lambda_{3}\,v\,,$
(21)

or, equivalently,

$\displaystyle g_{h_{1}\eta\eta}=$ $\displaystyle\cos\alpha\left(\lambda_{\eta
S}\,v_{S}+\mu_{\eta}\right)+\sin\alpha\,\lambda_{3}\,v\,,$ (22) $\displaystyle
g_{h_{2}\eta\eta}=$ $\displaystyle-\sin\alpha\left(\lambda_{\eta
S}\,v_{S}+\mu_{\eta}\right)+\cos\alpha\,\lambda_{3}\,v\,.$ (23)

###  2.3 Neutrino masses

After symmetry breaking, the $n_{N}\times n_{N}$ mass matrix of the singlet
fermions is given by

$\mathcal{M}_{N}=M_{N}+2\,\kappa\,v_{S}\,.$ (24)

As already explained, one can take the matrix $M_{N}$ to be diagonal without
loss of generality. We will further assume that $\kappa$ is diagonal too.
Then, the singlet fermion masses are simply given by
$m_{N_{n}}=\left(\mathcal{M}_{N}\right)_{nn}=\left(M_{N}\right)_{nn}+2\,\kappa_{nn}\,v_{S}$.

Figure 2: Neutrino mass generation in our model.

The simultaneous presence of the $y$ and $\lambda_{5}$ couplings and the
$M_{N}$ Majorana mass term (or the $\kappa$ coupling) leads to explicit lepton
number violation. Neutrino masses vanish at tree-level due to the
$\mathbb{Z}_{2}$ symmetry of the model, that forbids a neutrino Yukawa
interaction with the SM Higgs doublet. However, neutrinos acquire non-zero
Majorana masses at the 1-loop, as shown in Fig. 2. This mechanism is exactly
the same as in the original Scotogenic model [36], although our scenario
includes a variable number of $N$ and $\eta$ fields. The general expression
for the light neutrinos Majorana mass matrix can be found in [37] where it is
particularized for specific $(n_{N},n_{\eta})$ cases.

##  3 Interpretation of the 95 GeV excesses

In the following we will assume that the lightest $\mathbb{Z}_{2}$-even scalar
in our model, $h_{1}$, has a mass of 95 GeV and is thus identified with the
scalar resonance hinted by CMS, ATLAS and LEP precisely this energy scale.
Therefore, $h_{2}$ is identified with the 125 GeV Higgs discovered at the LHC.
In summary,

$h_{1}\equiv h_{95}\,,\quad h_{2}\equiv h_{125}\,.$ (25)

We should then study whether our model can accommodate the experimental hints
at 95 GeV. In other words, we must determine the regions in the parameter
space leading to a $h_{1}$ diphoton signal strength in agreement with Eq. (1)
that also comply with the existing experimental constraints. We will also
explore the possibility to simultaneously explain the other anomalies at 95
GeV, in the $b\bar{b}$ and ditau channels.

The signal strength for the diphoton channel is given in our model by

$\mu_{\gamma\gamma}=\frac{\sigma(gg\to h_{1})}{\sigma_{\rm SM}(gg\to
H)}\times\frac{{\rm BR}(h_{1}\to\gamma\gamma)}{{\rm BR}_{\rm
SM}(H\to\gamma\gamma)}=\sin^{2}\alpha\,\frac{{\rm
BR}(h_{1}\to\gamma\gamma)}{{\rm BR}_{\rm SM}(H\to\gamma\gamma)}\,,$ (26)

where we normalize again to the SM values, the usual suppression by the
$\alpha$ mixing angle has been taken into account and ${\rm
BR}(h_{1}\to\gamma\gamma)$ is the $h_{1}\to\gamma\gamma$ branching ratio in
our model. This is modified with respect to the predicted value for a Higgs-
like state with a mass of 95 GeV due to the presence of the $\eta$ doublets.
The decay width of a CP-even scalar to two photons has been studied in great
detail [38, 39, 40]. With $n_{\eta}$ generations of $\eta$ doublets and
assuming diagonal $g_{h_{i}\eta\eta}$ couplings, it is given by

$\Gamma(h_{i}\to\gamma\gamma)=\frac{G_{F}\alpha^{2}m_{h_{i}}^{3}}{128\sqrt{2}\pi^{3}}\left|\sum_{f}N_{c}Q_{f}^{2}g_{h_{i}ff}\,A_{1/2}(\tau_{f})+g_{h_{i}WW}\,A_{1}(\tau_{W})+\sum_{a}\frac{v}{2m^{2}_{\eta_{a}}}g_{h_{i}\eta\eta}^{aa}\,A_{0}(\tau_{\eta})\right|^{2}\,,$
(27)

where $\tau_{k}=m^{2}_{h_{i}}/4m^{2}_{k}$ and the $A_{i}(\tau)$ functions are
defined in Appendix A. As we can see, the presence of the $\eta$ doublets not
only modifies the diphoton decay of the mostly-singlet state, but it also
affects to the diphoton decay of the SM-like Higgs. As discussed below in Sec.
4, this feature constrains the parameter space from the existing measurements
of the 125 GeV Higgs.

For the $b\bar{b}$ excess in LEP we must consider the signal strength

$\mu_{bb}=\frac{\sigma(e^{+}e^{-}\to Z\,h_{1})}{\sigma_{\rm SM}(e^{+}e^{-}\to
Z\,H)}\times\frac{{\rm BR}(h_{1}\to b\bar{b})}{{\rm BR}_{\rm SM}(H\to
b\bar{b})}=\sin^{2}\alpha\,\frac{\Gamma(h_{1}\to b\bar{b})/\Gamma^{\rm
tot}}{\Gamma_{\rm SM}(H\to b\bar{b})/\Gamma^{\rm tot}_{\rm
SM}}=\sin^{4}\alpha\,\frac{\Gamma^{\rm tot}_{\rm SM}}{\Gamma^{\rm tot}}\,.$
(28)

In this case, the mixing angle not only suppresses the production cross
section, but also the decay width of $h_{1}\to b\bar{b}$, which can only take
place via singlet-doublet mixing.

As we can see from both signal strengths, the main features of this model to
explain the signals are, on the one hand, the reduced couplings to fermions
and vector bosons given by the mixing of the singlet state with the doublet,
that introduce powers of $\sin\alpha$ in the observables of interest. This
allows (i) to evade the existing limits from LEP and the LHC on light scalars
with masses below 125 GeV decaying into SM states and, (ii) to easily
accomodate the correct range of values to explain the $b\bar{b}$ excess at
LEP. This occurs easily because a singlet of a mass of 95 GeV with a small
admixture with the doublet will predominantly decay into a $b\bar{b}$ pair. On
the other hand, the singlet couples directly to the $\eta$ doublets. This
induces, via loops, the decay into a pair of photons. This feature allows to
explain in a natural way the diphoton rate at CMS. We can see that the
features for both excesses have different origins, the $b\bar{b}$ signal comes
through the mixing with the doublet state while the diphoton signal is mainly
driven by the singlet couplings. In order to accomodate both signals, the
interplay between the singlet and doublet components of $h_{1}$ must be looked
for.

Finally, the signal strength for $\tau^{+}\tau^{-}$ is exactly the same as
that for $b\bar{b}$ in Eq. (28). In fact, our model predicts
$\mu_{bb}=\mu_{\tau\tau}$. This obviously precludes our model from explaining
the CMS ditau excess in the same region of parameter space that explains the
LEP $b\bar{b}$ excess. In fact, as we will see below, the value of the mixing
angle $\alpha$ required to explain the ditau excess is too large, already
excluded by Higgs data. Thefore, the CMS ditau excess will be interpreted as
an upper limit instead.

##  4 Constraints

The presence of a 95 GeV scalar that could lead to an explanation to the
anomalies in the data is subject to different constraints.

First of all there are some theoretical constraints that affect the parameters
of our model. Such constraints involve mainly the parameters of the scalar
potential. First of all, we demand all the quartic couplings in the potential
to be below $\sqrt{4\pi}$ in order to ensure perturbativity. Furthermore, we
demand the potential to be bounded from below to ensure that we have a stable
global minimum. This requirement is rather complicated in the presence of many
scalar fields so we apply a copositivity requirement as described in the
Appendix of Ref. [37]. Although being overconstraining, once the potential
passes this requirement, it is guaranteed to be bounded from below.

Furthermore, several experimental searches are sensitive to the spectrum of
our singlet extension of the Scotogenic model. In that sense, the most
important searches are the ones provided by colliders. Since the Higgs sector
gets modified with respect to the one of the SM, one must ensure that all our
predictions are in agreement with the existing collider measurements. We
remind the reader that we have adopted a setup characterized by a light scalar
around 95 GeV and a SM-like Higgs boson at 125 GeV. In order to take into
account the bounds on the 95 GeV scalar we make use of the public code
HiggsBounds-v.6 [41, 42, 43, 44, 45, 46, 47], integrated now in the public
code HiggsTools [48]. This code compares the potential signatures of such a
scalar against BSM scalar searches performed at the LHC. A point in the
parameter space of our model would be excluded if its signal rate for the most
sensitive channel excedes the observed experimental limit at the 95%
confidence level.

Moreover, as a 125 GeV SM-like Higgs boson has been observed at the LHC, we
ask the second scalar state to be in agreement with the experimental
measurements of its signal rates using the public code HiggsSignals-v.3 [49,
50, 51, 52] that is also part of HiggsTools [48]. This code constructs a
$\chi^{2}$ function using the different data from the measured cross sections
at the LHC involving the measured 125 GeV Higgs boson. For that purpose we
provide the code with the rescaled effective couplings for the different
sensitive channels in our model. Once the $\chi^{2}$ function is built for a
point of the parameter space we compare it with the result of the fit for a
$125.09$ GeV SM-like Higgs boson with HiggsSignals-v.3,
$\chi^{2}_{\rm{SM},125}=152.49$, and impose that the difference between the
calculated $\chi^{2}_{125}$ and $\chi^{2}_{\rm{SM},125}$ is less than
$2\sigma$ away from the LHC measurements in order to consider a point as
experimentally allowed. Since we perform a two-dimensional analysis of the
parameter space, our consideration for a point to be allowed becomes
$\Delta\chi^{2}_{125}=\chi^{2}_{125}-\chi^{2}_{\rm{SM},125}\leq 6.18$.

Another point to have into consideration is the fact that the presence of an
$\rm SU(2)_{L}$ scalar doublet can induce sizeable contributions to the
electroweak precision observables. In particular, the oblique parameters $S$,
$T$ and $U$ are generally affected by the presence of these particles, but
these strongly depend on the scalar masses [53, 54]. When the CP-even and CP-
odd $\mathbb{Z}_{2}$-odd neutral states are mostly degenerate, or equivalently
when the entries of the $\lambda_{5}$ matrix are small, the $T$ parameter
imposes a restrictive bound over the difference in masses between the charged
and neutral states, $\Delta
m(\eta^{+},\eta^{0})=|m_{\eta^{+}}-m_{\eta^{0}}|\lesssim 140$ GeV [54].
Charged particles are also heavily constrained by different searches at
colliders. However, these searches assume specific decay modes. In our case
the decay of the charged $\eta^{\pm}$ scalar takes place via electroweak
couplings as $\eta^{\pm}\to\eta^{0}W^{\pm}$. Since all decay chains must
include the DM state, this eventually leads to missing transverse energy and
leptons or jets in the final state. For that purpose we impose a conservative
limit on the charged particles given by the LEP experiment of about
$m_{\eta^{\pm}}\gtrsim 100$ GeV. We impose this bound even if a detailed
analysis could show that the limits might be weaker in some specific
configurations, due to the decay modes and mass differences. Such a detailed
analysis is out of the scope of this paper, that just aims at showing that our
model can accommodate the 95 GeV excesses. The LHC has also performed searches
looking for charged particles that decay into a neutral one and different
objects [55, 56]. Although the current limits on charged particles can reach
high values of the mass, they are again strongly dependent on the mass
splitting between the charged and neutral states, making the searches almost
not sensitive for differences lower than $\Delta m(\eta^{+},\eta^{0})\lesssim
60$ GeV. Furthermore, there are searches that look for charged particles that
decay into neutral states that are close in mass, producing soft objects as
final state [57, 58]. These searches aim to cover the gap in mass values of
the previous analyses for charged particles. Their sensitivity is maximized
for mass differences of order $\Delta m(\eta^{+},\eta^{0})\sim$ 10 GeV,
decreasing for increased values of $\Delta m(\eta^{+},\eta^{0})$, until it
reaches $\Delta m(\eta^{+},\eta^{0})\sim$ 60 GeV, where the searches from
Refs. [55, 56] are sensitive. For that reason, we take the masses of the
$\eta$ doublet in such a way that fulfil both $S$, $T$ and $U$ parameters and
the collider constraints. This requirement can be achieved naturally in this
model according to Eq (20) since $\Delta m(\eta^{+},\eta^{0})$ is driven by
the couplings $\lambda^{ab}_{4}$ and $\lambda^{ab}_{5}$. The first of these
matrices has entries tipically smaller than 1, while the second one is usually
very small due to its link with neutrino masses.

##  5 Numerical results

Figure 3: Branching ratios of the $h_{1}$ scalar into several final states for
$n_{\eta}=2$, $\overline{\lambda}_{3}=0.55$, $\overline{\lambda}_{\eta
S}\,v_{S}=500$ GeV and $\overline{\mu}_{\eta}=1500$ GeV. Left: contours of
BR($h_{1}\to\gamma\gamma$) in the $m_{\eta}-\sin\alpha$ plane. Right:
BR($h_{1}\to$ XX) as a function of $\sin\alpha$ for a fixed $m_{\eta}=200$
GeV.

We now show our numerical results. For the sake of simplicity, we will assume
in the following that $\lambda_{3}$, $\mu_{\eta}$ and $\lambda_{\eta S}$ are
proportional to the $n_{\eta}\times n_{\eta}$ identity matrix
$\mathcal{I}_{n_{\eta}}$, that is, $X=\overline{X}\,\mathcal{I}_{n_{\eta}}$,
with $X=\lambda_{3}$, $\mu_{\eta}$, $\lambda_{\eta S}$. Fig. 3 shows our
results for the $h_{1}$ decay width and branching ratios into several final
states. This figure has been made with the specific choice $n_{\eta}=2$,
fixing also $\overline{\lambda}_{3}=0.55$, $\overline{\lambda}_{\eta
S}\,v_{S}=500$ GeV and $\overline{\mu}_{\eta}=1500$ GeV. The left-hand side of
this figure shows contours of BR($h_{1}\to\gamma\gamma$) in the
$m_{\eta}-\sin\alpha$ plane. One can see that BR($h_{1}\to\gamma\gamma$)
decreases with $m_{\eta}$, as expected, and gets enhanced for low values of
$\sin\alpha$. In fact, the branching ratio into the diphoton final state can
be of order $1$ for very low values of $\sin\alpha$. This behavior is also
illustrated on the right-hand panel of the figure, which shows the dependence
of the different BR($h_{1}\to$ XX) on $\sin\alpha$ for a fixed $m_{\eta}=200$
GeV. The enhancement is caused by the strong suppression of all the other
channels, which have negligible branching ratios for low values of
$\sin\alpha$. This is simply due to the fact that the $h_{1}\approx S$ decay
into SM states can only take place via singlet-doublet mixing. It is important
to notice that for values of $\sin\alpha\gtrsim 0.05$ the branching ratio to a
$b\bar{b}$ pair becomes predominant favouring the LEP signal as was explained
in Sec. 3.

Figure 4: Regions of the $m_{\eta}-\sin\alpha$ plane that explain the CMS
diphoton excess for fixed values of $\overline{\lambda}_{\eta S}\,v_{S}=500$
GeV and $\overline{\lambda}_{3}=0.6$. The colored regions accommodate
$\mu^{\rm CMS}_{\gamma\gamma}$ at $1\,\sigma$, while the solid lines
correspond to the CMS central value $\mu^{\rm CMS}_{\gamma\gamma}=0.33$. The
different colors are associated to different values of
$\overline{\mu}_{\eta}=$ 500 GeV (purple), 1000 GeV (yellow), 2000 GeV (red).
The gray band on the left is (in principle) excluded by direct searches at
LEP, although it would be allowed for sufficiently compressed spectra.

Fig. 4 displays different examples that prove that our model can easily fit
the CMS diphoton excess. In order to obtain these solutions we have assumed
the coupling of the $\eta$ scalars to the $h_{1}$ and $h_{2}$ scalars to be
$\overline{\lambda}_{\eta S}\,v_{S}=500$ GeV and $\overline{\lambda}_{3}=0.6$,
respectively, and we also fixed the value of $\overline{\mu}_{\eta}$ in the
three figures to 500 GeV (purple regions), 1 TeV (yellow regions), and 2 TeV
(red regions). Then, we found the $(m_{\eta},\sin\alpha)$ pairs that can
reproduce $\mu_{\gamma\gamma}^{\rm CMS}$. Furthermore, we also vary the number
of $\eta$ generations in the model and assume degenerate $\eta$ doublets. The
upper left plot of Fig. 4 represents the solution for only one generation of
$\eta$ doublets. We can see that the $m_{\eta}$ range in which our model
explains the CMS diphoton excess depends on the values of $\mu_{\eta}$. For
example, for the lowest value of the coupling, $\overline{\mu}_{\eta}=500$
GeV, the mass of the charged $\eta$ scalars must be around $m_{\eta}\sim 100$
GeV to compensate the low value of the parameter, whereas for larger values of
the coupling the $\eta$ masses can reach $\sim 180$ GeV. In the upper right
plot of Fig. 4 we can see the case of 2 generations of $\eta$ that have the
same mass $m_{\eta_{1}}=m_{\eta_{2}}=m_{\eta}$. With two generations, the
diphoton rate increases and, for this reason, the CMS diphoton excess is
explained for greater values of the $\eta$ mass. Something similar happens in
the case of 3 generations, shown in the lower pannel of Fig. 4. In this case,
the $m_{\eta}$ value required to accommodate the CMS excess can be as high as
300 GeV when the $\mu_{\eta}$ couplings are of the order of 2 TeV.

It is important to note that an explanation for the CMS diphoton excess can be
found for small values of $\sin\alpha$, in the $\sim 0.02$ ballpark. This may
be surprising at first, since such low values of the singlet-doublet mixing
angle strongly suppress the production of $h_{1}$ at the LHC. However, the
existence of the low $\alpha$ region is due to the abovementioned increase in
BR($h_{1}\to\gamma\gamma$), see Fig. 3, which compensates for the reduction in
the production cross section. In fact, one can estimate a lower limit on
$\sin\alpha$, below which $h_{1}$ cannot fit the CMS diphoton signal strength
because the required branching ratio into $\gamma\gamma$ would be larger than
1. The production cross section for a SM Higgs that decays into a pair of
photons at $\sqrt{s}=$ 13 TeV is approximately
$\sigma_{H}\times\text{BR}(H\to\gamma\gamma)\sim 0.125\,{\rm pb}$ [7]. Then,
assuming the hypothetical scenario with $\text{BR}(h_{1}\to\gamma\gamma)\to
1$111It is important to note that this limit is just hypothetical. Once it is
reached then the singlet cannot be produced in the LHC. one finds the limit
$\sin\alpha\gtrsim 0.0215$ for the central value of $\mu^{\rm
CMS}_{\gamma\gamma}$ and the range $\sin\alpha\gtrsim[0.027-0.017]$ taking the
$1\,\sigma$ region. This is precisely what determines the low $\alpha$ region
observed in Fig. 4.

Figure 5: Contours of $\Gamma_{h_{1}}$ (in GeV) in the $m_{\eta}-\sin\alpha$
plane for a variant of the model with 2 generations of $\eta$ doublets. The
colored region explains the CMS diphoton excess at $1\,\sigma$ with 2
generations of $\eta$, $\overline{\lambda}_{\eta S}v_{s}=500$ GeV,
$\overline{\mu}_{\eta}=2000$ GeV and $\overline{\lambda}_{3}=0.6$.

Given that some regions of the $m_{\eta}-\sin\alpha$ plane considered in Fig.
4 have small $\sin\alpha$ values, one may wonder about the decay width of
$h_{1}$. We note that $h_{1}$ must decay promptly for our explanation of the
CMS diphoton excess to work. We explore this in Fig. 5, which shows contours
of $c\tau_{h_{1}}$ in the $m_{\eta}-\sin\alpha$ plane for a variant of our
model featuring 2 generations of $\eta$ doublets. We see that $h_{1}$ has a
short decay length, well below that regarded as prompt, even for small
$\alpha$ angles. Again, the reason is the enhancement of the diphoton decay
channel.

Figure 6: Regions of the $m_{\eta}-\sin\alpha$ plane that explain the CMS
diphoton and LEP $b\bar{b}$ excesses. The region that explains the CMS
diphoton excess at $1\,\sigma$ is shown in light red for $n_{\eta}=2$ (top
panels) and $3$ (bottom panels). We have fixed
$\overline{\lambda}_{S\eta}\,v_{S}$= 500 GeV and $\overline{\lambda}_{3}=0.6$,
while the trilinear $\overline{\mu}_{\eta}$ takes the values 500 GeV (left
panels) and 1000 GeV (right panels). The area that fits the LEP excess in
$b\bar{b}$ is shown as light orange while we show as a dark gray shaded
contour the area that is disfavoured at $95\%$ C.L. by Higgs constraints.

Let us now consider the LEP $b\bar{b}$ excess in combination with the
previously discussed CMS diphoton excess. One can see in Fig. 6 the region of
the $(m_{\eta},\sin\alpha)$ plane where both excesses can be explained. As
discussed in Sec. 3, the charged $\eta$ scalars not only affect the $h_{1}$
diphoton rate, but also modify the one for $h_{2}$, already measured at the
LHC. This implies limits from Higgs data, displayed in this figure by the dark
gray area, which is excluded at $95\%$ C.L.. One should notice that the LEP
excess can be explained in a region of parameter space that lies on a high
value of $\sin\alpha$ and is mostly excluded by Higgs data. However, there is
still a portion of the allowed parameter space where both excesses are
explained. As expected, this portion involves lighter charged $\eta$ scalars
when fewer generations are considered. With the specific values chosen in this
figure for the $\overline{\lambda}_{S\eta}\,v_{S}$, $\overline{\lambda}_{3}$
and $\overline{\mu}_{\eta}$ parameters, the required $\eta$ masses range from
$\sim 130$ GeV to $\sim 220$ GeV. Finally, we note that the dark gray area in
Fig. 6 depends very strongly on the $\overline{\lambda}_{3}$ value. This can
be easily understood by inspecting Eq. (23). For some values of
$\overline{\lambda}_{3}$, the $g_{h_{2}\eta\eta}$ coupling becomes
$\mathcal{O}(1)$ and excludes most of the parameter space due to Higgs data.
However, one can choose specific values of $\overline{\lambda}_{3}$ that
induce a cancellation in the $g_{h_{2}\eta\eta}$ coupling and make the
constraints from Higgs data less stringent. Our choice
$\overline{\lambda}_{3}=0.6$ is an example of this.

Alternatively, since the LEP $b\bar{b}$ excess is not very significant from a
statistical point of view, one can interpret it as an upper limit on the
$e^{+}e^{-}\to Z\,h_{95}\to Zb\bar{b}$ cross section. In this case, we
conclude that the restrictions imposed by the LEP search in this channel are
compatible with the areas where our model can explain the diphoton excess in
CMS. The case of the CMS ditau excess is similar. The value of $\sin\alpha$
that would be required to explain this excess is quite large, above the
current limit. The minimum value of the mixing angle in order to explain this
signal would be $\sin\alpha_{\rm{min}}\sim 0.7$. However, as there is only one
search by CMS at still low luminosity and there are no more searches, we
consider this excess as still not significant. We can again interpret it as an
upper bound on the $pp\to\,h_{95}\to\tau^{+}\tau^{-}$ cross section. In this
case, we conclude again that the region of parameter space where the CMS
diphoton excess is explained respects this bound.

##  6 Discussion

Once shown that our scenario can accommodate the 95 GeV anomalies, let us
comment on some other aspects of the model that have been ignored in our
previous discussion. This is the case of neutrino oscillation data. As already
explained in Sec. 2.3, neutrinos acquire non-zero masses via loops involving
the $\mathbb{Z}_{2}$-odd states $N_{n}$ and $\eta_{a}$. Therefore, the
resulting neutrino mass matrix depends on their masses, as well as on the
$\lambda_{5}$ quartics and the $y$ Yukawa couplings [37]. For specific values
of $n_{N}$ and $n_{\eta}$, the $y$ Yukawa couplings can be readily written in
terms of the parameters measured in oscillation experiments using a Casas-
Ibarra parametrization [59] adapted to the Scotogenic model [60, 61, 62].
However, we note that the $y$ Yukawa couplings do not play any role in the 95
GeV collider phenomenology.

The Scotogenic variant discussed here also contains a DM candidate. In this
family of models one usually has two options: fermion ($N_{1}$) or scalar
($\eta_{1}^{0}$) DM. However, in order to accommodate the 95 GeV anomalies we
require relatively light $\eta$ doublets, with masses $m_{\eta}\lesssim 300$
GeV for $n_{\eta}\leq 3$. 222The explanation of the diphoton excess involves
the charged $\eta$ states, not the neutral ones considered here. However, the
mass splitting between charged and neutral components is small, since it is
controlled by electroweak symmetry breaking effects. This is too light to
accommodate the DM relic density determined by the Planck collaboration [63].
In fact, the scalar DM scenario resembles the Inert Doublet Model [64], which
is known to fully account for the observed relic density for DM masses in the
$500-700$ GeV range [53, 65, 66, 67]. In contrast, $m_{\eta}\lesssim 300$ GeV
leads to underabundant DM, hence requiring an additional DM component. We
should also note that we consider more than one generation of $\eta$ doublets.
This variation of the Scotogenic model deserves further investigation, since
it may lead to novel possibilities in scenarios with scalar DM. Alternatively,
we may consider scenarios with fermion DM. This candidate is known to be
potentially problematic due to existing tension between the DM relic density
(which requires large $y$ Yukawas) and contraints from lepton flavor violating
observables (which require small $y$ Yukawas), see for instance [68]. Two
interesting scenarios emerge:

  * •

$m_{N_{1}}\ll m_{\eta_{1}^{0}}$. If the DM particle $N_{1}$ is much lighter
than the $\eta$ states (for instance, $m_{N_{1}}\sim 100$ GeV and
$m_{\eta_{1}}\sim 300$ GeV), the $y$ Yukawa parameters must be fine-tuned to
suppress the contributions to $\mu-e$ flavor violating processes, such as
$\mu\to e\gamma$, while being compatible with neutrino oscillation data.
Although tuned, this scenario is possible. It would be characterized at the
LHC by the pair production (due to the $\mathbb{Z}_{2}$ symmetry) of $\eta$
states which subsequently decay into the invisible $N_{1}$ and leptons:
$\eta_{1}^{0}\to N_{1}\,\nu$ and $\eta_{1}^{\pm}\to N_{1}\,\ell^{\pm}$. This
scenario is constrained by existing searches for sleptons, which would have a
very similar phenomenology in both R-parity conserving and violating
supersymmetry (see for instance [69, 70, 71, 72]). Furthermore, a light
$N_{1}$ may also contribute to the invisible decay of $h_{1}$ (if
$m_{N_{1}}\leq m_{h_{1}}/2$) and/or $h_{2}$ (if $m_{N_{1}}\leq m_{h_{2}}/2$).
In fact, the $h_{1}\to N_{1}N_{1}$ invisible channel may easily dominate the
$h_{1}$ decay width and preclude an explanation of the 95 GeV excess.

  * •

$m_{N_{1}}\lesssim m_{\eta_{1}^{0}}$. If the $N_{1}$ singlet is almost
degenerate with the lightest $\eta$ states, coannihilations become efficient
and the DM relic density is more easily obtained. This enlarges the viable
parameter space of the model and leads to novel signatures at the LHC. If the
mass splitting $\Delta m=m_{\eta_{1}^{\pm}}-m_{N_{1}}$ is small enough, the
decay $\eta_{1}^{\pm}\to N_{1}\,\ell^{\pm}$ may involve a long decay length,
hence producing charged tracks at the detector.

Finally, in parameter points with $m_{N_{1}}\approx m_{h_{i}}/2$ the
annihilation cross section in the early universe gets enhanced due to resonant
effects. In such cases one can achieve the correct DM relic density without
invoking large couplings. This is a generic feature that does not affect the
previous discussion.

##  7 Summary and conclusion

We have shown that a theoretically well-motivated and economical model can
accommodate the diphoton excess hinted by CMS and ATLAS at 95 GeV as well as
the hint for a $b\bar{b}$ excess at similar energies by LEP. Our model is a
minimal extension of the Scotogenic model and, besides addressing these
collider anomalies, also provides a mechanism for the generation of neutrino
masses and a testable dark matter candidate. We have allowed for variable
numbers of generations of the Scotogenic states $N$ and $\eta$ and discussed
our results for several choices of interest.

Two CP-even (and $\mathbb{Z}_{2}$-even) scalars: $h_{1}$ and $h_{2}$. The
lightest of these states, $h_{1}$, is identified with $h_{95}$, the
hypothetical scalar that is responsible for the $\gamma\gamma$ and $b\bar{b}$
excesses at 95 GeV, while $h_{2}$ is the Higgs-like state discovered by the
CMS and ATLAS collaborations in 2012. Our numerical analysis shows that the
excesses can be accommodated in our model in a large fraction of the parameter
space. The viable region is characterized by sizable $\mu_{\eta}$ trilinear
couplings and leads to $\eta$ scalars with masses below $\sim 300$ GeV ($\sim
180$ GeV) for $n_{\eta}=3$ (for $n_{\eta}=1$). As expected, a larger number of
$\eta$ generations implies larger contributions to the $h_{1}$ diphoton
coupling and enlarges the viable parameter space.

Our scenario has a rich phenomenology, both at colliders and at low-energy
experiments. The nature of the dark matter candidate and the particle spectrum
determines the phenomenology at colliders. Depending on the mass differences
between the lightest $\mathbb{Z}_{2}$-odd fermion and $\mathbb{Z}_{2}$-odd
scalar, one expects monolepton events including missing energy or charged
tracks at the LHC detectors. In addition, the usual lepton flavor violating
signatures, common to most low-energy neutrino mass models, are expected too.
Therefore, our setup not only is well motivated from a theoretical point of
view, but also has interesting phenomenological implications.

We conclude with a note of caution. Although the coincidence of several
excesses, hinted by independent experiments at the same invariant mass, is
highly intriguing, their relatively low statistical significance implies that
more data is required to fully assess their relevance. We eagerly look forward
to future updates on the 95 GeV excesses.

## Acknowledgments

The authors are thankful to Thomas Biekötter and Sven Heinemeyer for useful
discussions about the 95 GeV excesses and HiggsTools. This work has been
supported by the Spanish grants PID2020-113775GB-I00
(AEI/10.13039/501100011033) and CIPROM/2021/054 (Generalitat Valenciana). VML
is funded by grant María Zambrano UP2021-044 (ZA2021-081) funded by Ministerio
de Universidades and “European Union - Next Generation EU/PRTR”. AV
acknowledges financial support from MINECO through the Ramón y Cajal contract
RYC2018-025795-I.

##  Appendix A Loop functions

The loop functions involved in the calculation of
$\Gamma(h_{i}\to\gamma\gamma)$ are given by [38, 39, 40]

$\displaystyle A_{0}$ $\displaystyle=-[\tau-f(\tau)]/\tau^{2}\,,$
$\displaystyle A_{1/2}$ $\displaystyle=2[\tau+(\tau-1)f(\tau)]/\tau^{2}\,,$
(29) $\displaystyle A_{1}$
$\displaystyle=-[2\tau^{2}+3\tau+3(2\tau-1)f(\tau)]/\tau^{2}\,,$

where the function $f(\tau)$ is defined as

$\displaystyle
f(\tau)=\left\\{\begin{array}[]{lc}\arcsin^{2}\sqrt{\tau};&\tau\leq 1\\\ \\\
-\frac{1}{4}\left[\log\frac{1+\sqrt{1-\tau^{-1}}}{1-\sqrt{1-\tau^{-1}}}-i\pi\right]^{2};&\tau>1\end{array}\right.$
(33)

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