#  When Standard Model Higgs Meets Its Lighter 95 GeV Higgs

Abdesslam Arhrib<EMAIL_ADDRESS>Abdelmalek Essaadi University, Faculty of
Sciences and Techniques, Tanger, Morocco  Khiem Hong Phan
<EMAIL_ADDRESS>Institute of Fundamental and Applied Sciences, Duy
Tan University, Ho Chi Minh City 70000, Vietnam Faculty of Natural Sciences,
Duy Tan University, Da Nang City 50000, Vietnam
Van Que Tran  0000-0003-4643-4050<EMAIL_ADDRESS>Tsung-Dao Lee Institute &
School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai
200240, China  Phenikaa Institute for Advanced Study, Phenikaa University, Yen
Nghia, Ha Dong, Hanoi 100000, Vietnam Tzu-Chiang Yuan  0000-0001-8546-5031
<EMAIL_ADDRESS>Institute of Physics, Academia Sinica, Nangang,
Taipei 11529, Taiwan

###### Abstract

Two excesses reported recently at the LHC in the lighter Higgs mass region
around 95 GeV and in the rare $Z\gamma$ final state of the Standard Model (SM)
125 GeV Higgs decay are simultaneously scrutinized within the framework of
minimal gauged two-Higgs-doublet model (G2HDM). Viable parameter space in
G2HDM is obtained to account for both excesses. We find a strong correlation
between the signal strengths of SM 125 GeV Higgs decays into $\gamma\gamma$
and $Z\gamma$ modes, whereas this correlation does not extend to its lighter
95 GeV cousin.

##  I Introduction

Facing the mounting several hundreds petabytes of physics data collected by
the Large Hadron Collider (LHC) since the first collision at 2010, Standard
Model (SM) is holding up amazingly well, surviving all the challenges posted
by our experimental colleagues. Albeit there were reports from the LHC over
the years about some middling 2$\sim$3$\sigma$ excesses and created some
excitements in the community, the joys did not last long as the excesses faded
away quickly when more statistics was accumulated. Nevertheless, few people
would doubt there must be new physics beyond the SM (BSM). From the bottom-up
viewpoint, right-handed neutrinos are seem to be required as suggested by the
experiments of neutrino oscillations. Dark matter and dark energy are also
missing in SM but they are essential ingredients for the evolution and history
of our observed universe. From the top-down viewpoint, we also have good
arguments for new physics from grand unification, gauge hierarchy problem,
theory of everything like string theory, etc.

Recently, CMS has reported an excess in the light Higgs-boson search in the
di-photon ($\gamma\gamma$) decay mode at about 95.4 GeV based on the 8 TeV
data and the full Run 2 data set at 13 TeV with the local significance is
$2.9\sigma$ CMS (2023). The corresponding signal strength is given as

$\mu_{\gamma\gamma}^{\mathrm{CMS}}=\frac{\sigma^{\exp}(gg\rightarrow
h\rightarrow\gamma\gamma)}{\sigma^{\mathrm{SM}}(gg\rightarrow
H\rightarrow\gamma\gamma)}=0.33_{-0.12}^{+0.19}\;,$ (1)

where $\sigma^{\rm SM}$ denotes the cross section for a hypothetical scalar
boson $h$ with the mass is 95.4 GeV, and $H$ is the SM Higgs.

ATLAS recently also presented the result of the search for new neutral scalars
in the di-photon final state with mass window from 66 GeV to 110 GeV, using
full Run 2 data collected at 13 TeV ATLAS ; ATL (2023). ATLAS observed an
excess at the same mass value as reported by the CMS with the local
significance is $1.7\sigma$. The corresponding signal strength is given as
Biekötter  _et al._ (2023a)

$\mu_{\gamma\gamma}^{\mathrm{ATLAS}}=\frac{\sigma^{\exp}(gg\rightarrow
h\rightarrow\gamma\gamma)}{\sigma^{\mathrm{SM}}(gg\rightarrow
H\rightarrow\gamma\gamma)}=0.18_{-0.1}^{+0.1}\;.$ (2)

The combined local significance from ATLAS and CMS is $3.1\sigma$ and the
signal strength is Biekötter  _et al._ (2023a)

$\mu_{\gamma\gamma}^{\exp}=\mu_{\gamma\gamma}^{\text{ATLAS
}+\mathrm{CMS}}=0.24_{-0.08}^{+0.09}\;.$ (3)

Using the full Run 2 data set, CMS reported another local excess with a
significance of $3.1\sigma$ ($2.6\sigma$) for light Higgs with mass $\sim 100$
GeV ($95$ GeV) produces from gluon-gluon fusion and subsequently decays to di-
tau ($\tau^{+}\tau^{-}$) final state CMS (2022). The signal strength for the
scalar mass at $95$ GeV is given as

$\mu_{\tau\tau}^{\mathrm{exp}}=\frac{\sigma^{\exp}(gg\rightarrow
h\rightarrow\tau^{+}\tau^{-})}{\sigma^{\mathrm{SM}}(gg\rightarrow
H\rightarrow\tau^{+}\tau^{-})}=1.2\pm{0.5}\;.$ (4)

Note that ATLAS has not yet reported a search in the di-tau final state that
covers the mass range around 95 GeV.

In additional, searches for a low-mass scalar boson were previously carried
out at LEP. A local significance excess of $2.3\sigma$ for the light scalar
mass of about 98 GeV in the process $e^{+}e^{-}\rightarrow Z(H\rightarrow
b\overline{b})$ and the corresponding signal strength of Barate  _et al._
(2003)

$\mu^{\rm exp}_{bb}=0.117\pm 0.057\;.$ (5)

At face values, the signal strengths in (3), (4) and (5) indicate that the 95
GeV Higgs, if its existence is confirmed, would be very much SM-like for the
di-tau mode but rather non-SM like for both the di-photon and $b\bar{b}$
modes. Excesses in these channels are strong motivation for BSM. Indeed in
light of these excesses, many BSMs have been studied in recent years, e.g. Dev
_et al._ (2023); Li  _et al._ (2023); Chen  _et al._ (2023); Ahriche (2023);
Arcadi  _et al._ (2023); Ahriche  _et al._ (2023); Borah  _et al._ (2023); Cao
_et al._ (2023); Ellwanger and Hugonie (2023); Dutta  _et al._ (2023);
Aguilar-Saavedra  _et al._ (2023); Ashanujjaman  _et al._ (2023); Belyaev  _et
al._ (2023); Escribano  _et al._ (2023); Azevedo  _et al._ (2023); Biekötter
_et al._ (2023b); Iguro  _et al._ (2022); Biekötter  _et al._ (2022); Sachdeva
and Sadhukhan (2020); He  _et al._ (2024).

More recently, ATLAS and CMS Collaborations Aad  _et al._ (2024) reported an
analysis for the first evidence of the rare decay mode $H\rightarrow Z\gamma$,
where the $Z$ boson decays into a $e^{+}e^{-}$ or $\mu^{+}\mu^{-}$ pair. The
number of events is found twice as many as predicted by the SM. To be more
precise, the combined observed signal yield is

$\mu_{Z\gamma}^{\rm exp}=\frac{\sigma^{\rm exp}(gg\rightarrow H\rightarrow
Z\gamma)}{\sigma^{\rm SM}(gg\rightarrow H\rightarrow Z\gamma)}=2.2\pm 0.7\;,$
(6)

with a 3.4$\sigma$ statistical significance.

Currently, the data are insufficient to rule out the possibility that the
above discrepancies are merely statistical fluke. Nevertheless, they open up
the opportunity window for new physics.

Is this paper, we study the excesses (3), (4), (5) and (6) in the framework of
minimal gauged two-Higgs-doublet model (G2HDM) which contains a predominantly
$SU(2)_{L}$ singlet scalar $h_{1}$, a mixture of a hidden scalar with the SM
Higgs boson, can become the 95 GeV Higgs boson candidate. The orthogonal
combination $h_{2}$ will be identified as the observed 125 GeV Higgs boson.
Besides the SM $W^{\pm}$ boson and top quark $t$ contributions, the presence
of charged heavy hidden fermions $(f^{H})$ and charged Higgs $H^{\pm}$ can
provide additional contributions to the production and decay through one-loop
diagrams for both Higgs bosons. Thus it is natural that the signal strengths
for these two Higgs bosons deviate from their SM expectations.

This paper is organized as follows. We will give a brief review of the minimal
G2HDM in Section II, followed by a discussion in Section III for the
computation of signal strengths in the model. Numerical studies of scanning
the parameter space in G2HDM are presented in Section IV. We conclude in
Section V. For convenience, Appendix A collects the detailed decay rates for
$\gamma\gamma$, $Z\gamma$ and $gg$ modes of the Higgs bosons. We also discuss
our numerical study of the two loop functions entered in the Higgs decay
amplitudes.

##  II The Minimal G2HDM

The crucial idea of the original G2HDM Huang  _et al._ (2016a) was to embed
the two Higgs doublets $H_{1}$ and $H_{2}$ in the popular scalar dark matter
model, the inert two-Higgs-doublet model (I2HDM), into a 2-dimensional
irreducible representation of a hidden gauge group $SU(2)_{H}\times U(1)_{X}$,
a dark replica of the SM electroweak group $SU(2)_{L}\times U(1)_{Y}$. Besides
the two Higgs doublets in I2HDM, the original G2HDM also introduced two hidden
scalar multiplets, one doublet ($\Phi_{H}$) and one triplet ($\Delta_{H}$), to
generate the hidden particle mass spectra. The hidden gauge group acts
horizontally on the two Higgs doublets in I2HDM.

Various refinements Arhrib  _et al._ (2018); Huang  _et al._ (2019); Chen  _et
al._ (2020) and collider phenomenology Huang  _et al._ (2016b); Chen  _et al._
(2019); Huang  _et al._ (2018) were pursued subsequently with the same
particle content as in the original model. Recently Ramos  _et al._ (2021a,
b), it has been realized that removing the hidden triplet scalar field
$\Delta_{H}$ in the model without jeopardizing a realistic hidden particle
mass spectra. Interpretation of the $W$ boson mass measurement at the CDF II
Tran  _et al._ (2023a) and FCNC processes $l_{i}\rightarrow l_{j}\gamma$ Tran
and Yuan (2023) and $b\rightarrow s\gamma$ Liu  _et al._ (2024) have been
recently studied within the framework of this minimal G2HDM. We will again
focus on the minimal G2HDM in this work.

The gauge group of G2HDM is

$\mathcal{G}=SU(3)_{C}\times SU(2)_{L}\times SU(2)_{H}\times U(1)_{Y}\times
U(1)_{X}\;.$

The minimal particle representations under $\mathcal{G}$ are as follows:

Spin 0 Bosons:

$\mathcal{H}=\left(H_{1}\;\;H_{2}\right)^{\rm T}\sim\left({\bf 1},{\bf 2},{\bf
2},\frac{1}{2},{\frac{1}{2}}\right)\;,\;$ $\Phi_{H}\sim\left({\bf 1},{\bf
1},{\bf 2},0,{\frac{1}{2}}\right)\;;$

Spin 1/2 Fermions:

Quarks

$Q_{L}=\left(u_{L}\;\;d_{L}\right)^{\rm T}\sim\left({\bf 3},{\bf 2},{\bf
1},\frac{1}{6},0\right)\;,\;$ $U_{R}=\left(u_{R}\;\;u^{H}_{R}\right)^{\rm
T}\sim\left({\bf 3},{\bf 1},{\bf 2},\frac{2}{3},\frac{1}{2}\right)\;,$
$D_{R}=\left(d^{H}_{R}\;\;d_{R}\right)^{\rm T}\sim\left({\bf 3},{\bf 1},{\bf
2},-\frac{1}{3},-\frac{1}{2}\right)\;;$ $u_{L}^{H}\sim\left({\bf 3},{\bf
1},{\bf 1},\frac{2}{3},0\right)\;,\;d_{L}^{H}\sim\left({\bf 3},{\bf 1},{\bf
1},-\frac{1}{3},0\right)\;;$

Leptons

$L_{L}=\left(\nu_{L}\;\;e_{L}\right)^{\rm T}\sim\left({\bf 1},{\bf 2},{\bf
1},-\frac{1}{2},0\right)\;,\;$ $N_{R}=\left(\nu_{R}\;\;\nu^{H}_{R}\right)^{\rm
T}\sim\left({\bf 1},{\bf 1},{\bf 2},0,\frac{1}{2}\right)\;,$
$E_{R}=\left(e^{H}_{R}\;\;e_{R}\right)^{\rm T}\sim\left({\bf 1},{\bf 1},{\bf
2},-1,-\frac{1}{2}\right)\;;$ $\nu_{L}^{H}\sim\left({\bf 1},{\bf 1},{\bf
1},0,0\right)\;,\;e_{L}^{H}\sim\left({\bf 1},{\bf 1},{\bf 1},-1,0\right)\;.$

We assume three families of matter fermions in minimal G2HDM and the family
indices are often omitted. In addition to the SM gauge fields $W_{i}$
($i=1,2,3$) of $SU(2)_{L}$ and $B$ of $U(1)_{Y}$, the hidden gauge fields of
$SU(2)_{H}$ and $U(1)_{X}$ are denoted as $W^{\prime}_{i}$ ($i=1,2,3)$ and $X$
respectively.

One of the nice features of G2HDM is the presence of the accidental $h$-parity
Chen  _et al._ (2020) such that all the SM particles can be assigned to be
even. The $h$-parity odd particles in G2HDM are
$W^{\prime\,(p,m)}=(W^{\prime}_{1}\mp iW^{\prime}_{2})/\sqrt{2}$, $D^{(*)}$
and ${\tilde{G}}^{(*)}$ (the two orthogonal combinations of the complex
neutral component in $H_{2}$ and the hidden complex Goldstone field in
$\Phi_{H}$), $H^{\pm}$, and all new heavy fermions collectively denoted as
$f^{H}$. Among them, $W^{\prime\,(p,m)}$, $D^{(*)}$, and $\nu^{H}$ are
electrically neutral and hence any one of them can be a dark matter (DM)
candidate. Phenomenology of a complex scalar $D^{(*)}$ as DM was studied in
details in Chen  _et al._ (2020); Dirgantara and Nugroho (2022) and for low
mass $W^{\prime\,(p,m)}$ as DM, see Ramos  _et al._ (2021a, b). A pure gauge-
Higgs sector with $W^{\prime\,(p,m)}$ as self-interacting dark matter was also
studied recently in Tran  _et al._ (2023b). For further details of G2HDM, we
refer our readers to the earlier works Huang  _et al._ (2019); Arhrib  _et
al._ (2018); Ramos  _et al._ (2021a, b). Phenomenology of a new heavy neutrino
$\nu^{H}$ as DM in the model, which is necessarily implying both DM and
neutrino physics, has yet to be explored.

The most general renormalizable Higgs potential invariant under both the SM
$SU(2)_{L}\times U(1)_{Y}$ and the hidden $SU(2)_{H}\times U(1)_{X}$ is given
by

$\displaystyle V={}$ $\displaystyle-\mu^{2}_{H}\left({\mathcal{H}}^{\alpha
i}{\mathcal{H}}_{\alpha i}\right)+\lambda_{H}\left({\mathcal{H}}^{\alpha
i}{\mathcal{H}}_{\alpha
i}\right)^{2}+\frac{1}{2}\lambda^{\prime}_{H}\epsilon_{\alpha\beta}\epsilon^{\gamma\delta}\left({\mathcal{H}}^{\alpha
i}{\mathcal{H}}_{\gamma i}\right)\left({\mathcal{H}}^{\beta
j}{\mathcal{H}}_{\delta j}\right)$
$\displaystyle-\mu^{2}_{\Phi}\Phi_{H}^{\dagger}\Phi_{H}+\lambda_{\Phi}\left(\Phi_{H}^{\dagger}\Phi_{H}\right)^{2}$
(7)
$\displaystyle+\lambda_{H\Phi}\left({\mathcal{H}}^{\dagger}{\mathcal{H}}\right)\left(\Phi_{H}^{\dagger}\Phi_{H}\right)+\lambda^{\prime}_{H\Phi}\left({\mathcal{H}}^{\dagger}\Phi_{H}\right)\left(\Phi_{H}^{\dagger}{\mathcal{H}}\right),$

where ($\alpha$, $\beta$, $\gamma$, $\delta$) and ($i$, $j$) refer to the
$SU(2)_{H}$ and $SU(2)_{L}$ indices respectively, all of which run from 1 to
2, and ${\mathcal{H}}^{\alpha i}={\mathcal{H}}^{*}_{\alpha i}$. We note that
every term in the above potential $V$ is self-Hermitian. Therefore all
parameters in (II) are real and no CP violation can be arise from the scalar
potential.

To achieve spontaneous symmetry breaking (SSB) in the model, we follow
standard lore to parameterize the Higgs fields in the doublets linearly as

$\displaystyle H_{1}=\begin{pmatrix}G^{+}\\\ \frac{v+h_{\rm
SM}}{\sqrt{2}}+i\frac{G^{0}}{\sqrt{2}}\end{pmatrix},\;H_{2}=\begin{pmatrix}H^{+}\\\
H_{2}^{0}\end{pmatrix},\;\Phi_{H}=\begin{pmatrix}G_{H}^{p}\\\
\frac{v_{\Phi}+\phi_{H}}{\sqrt{2}}+i\frac{G_{H}^{0}}{\sqrt{2}}\end{pmatrix}\;\;\;$
(8)

where $v$ and $v_{\Phi}$ are the only non-vanishing vacuum expectation values
(VEVs) in the $H_{1}$ and $\Phi_{H}$ doublets respectively. $v=246$ GeV is the
SM VEV, and $v_{\Phi}$ is an unknown hidden VEV. We note that $h$-parity would
be broken spontaneously should $\langle H_{2}\rangle$ develops a VEV. This is
undesirable since we want a DM candidate to address DM physics at low energy.

In G2HDM, the SM charged gauge boson $W^{\pm}$ does not mix with
$W^{\prime\,(p,m)}$ and its mass is the same as in SM: $m_{W}=gv/2$. However
the SM neutral gauge boson $Z_{\rm SM}$ will in general mix further with the
gauge field $W^{\prime}_{3}$ associated with the third generator of
$SU(2)_{H}$ and the $U(1)_{X}$ gauge field $X$ via the following mass matrix

$\mathcal{M}_{Z}^{2}=\begin{pmatrix}m^{2}_{Z}&-\frac{1}{2}g_{H}vm_{Z}&-{\frac{1}{2}}g_{X}vm_{Z}\\\
-\frac{1}{2}g_{H}vm_{Z}&m^{2}_{W^{\prime}}&{\frac{1}{4}}g_{H}g_{X}v_{-}^{2}\\\
-{\frac{1}{2}}g_{X}vm_{Z}&{\frac{1}{4}}g_{H}g_{X}v_{-}^{2}&{\frac{1}{4}}g_{X}^{2}v_{+}^{2}+M_{X}^{2}\end{pmatrix}\;,$
(9)

where

$\displaystyle v_{\pm}^{2}$ $\displaystyle=$ $\displaystyle\left(v^{2}\pm
v^{2}_{\Phi}\right)\;,$ (10) $\displaystyle m_{Z}$ $\displaystyle=$
$\displaystyle\frac{1}{2}v\sqrt{g^{2}+g^{\prime\,2}}\;,$ (11) $\displaystyle
m_{W^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}g_{H}\sqrt{v^{2}+v_{\Phi}^{2}}\;,$ (12)

and $M_{X}$ is the Stueckelberg mass for the $U(1)_{X}$.

The real and symmetric mass matrix $\mathcal{M}_{Z}^{2}$ in (9) can be
diagonalized by a 3 by 3 orthogonal matrix ${\mathcal{O}}^{G}$, i.e.
$(\mathcal{O}^{G})^{\rm T}\mathcal{M}_{Z}^{2}\mathcal{O}^{G}={\rm
Diag}(m^{2}_{Z_{1}},m^{2}_{Z_{2}},m^{2}_{Z_{3}})$, where $m_{Z_{i}}$ is the
mass of the physical fields $Z_{i}$ for $i=1,2,3$. We will identify
$Z_{1}\equiv Z$ to be the neutral gauge boson resonance with a mass of 91.1876
GeV observed at LEP Zyla  _et al._ (2020). The lighter/heavier of the other
two states is the dark photon ($\gamma^{\prime}$)/dark $Z$ ($Z^{\prime}$).
These neutral gauge bosons are $h$-parity even in the model, despite the
adjective ‘dark’ are used for the other two states. We note that these neutral
gauge bosons can decay into SM particles and thus they can be constrained by
experimental data, including the electroweak precision measurement at the $Z$
pole physics from LEP, searches for dark $Z$ and dark photon at colliders,
beam-dump experiments, and astrophysical observations. The DM candidate
considered in this work is $W^{\prime\,(p,m)}$, which is electrically neutral
but carries one unit of dark charge and chosen to be the lightest $h$-parity
odd particle in the parameter space.

In G2HDM there are mixings effects of the two doublets $H_{1}$ and $H_{2}$
with the hidden doublet $\Phi_{H}$. The neutral components $h_{\rm SM}$ and
$\phi_{H}$ in $H_{1}$ and $\Phi_{H}$ respectively are both $h$-parity even.
They mix to form two physical Higgs fields $h_{1}$ and $h_{2}$

$\left(\begin{matrix}h_{\rm SM}\\\
\phi_{H}\end{matrix}\right)={\mathcal{O}}^{S}\cdot\left(\begin{matrix}h_{1}\\\
h_{2}\end{matrix}\right)=\left(\begin{matrix}\cos\theta_{1}&\sin\theta_{1}\\\
-\sin\theta_{1}&\cos\theta_{1}\end{matrix}\right)\cdot\left(\begin{matrix}h_{1}\\\
h_{2}\end{matrix}\right)\;.$ (13)

The mixing angle $\theta_{1}$ is given by

$\tan
2\theta_{1}=\frac{\lambda_{H\Phi}vv_{\Phi}}{\lambda_{\Phi}v^{2}_{\Phi}-\lambda_{H}v^{2}}\;.$
(14)

The masses of $h_{1}$ and $h_{2}$ are given by

$\displaystyle m_{h_{1},h_{2}}^{2}$ $\displaystyle=$
$\displaystyle\lambda_{H}v^{2}+\lambda_{\Phi}v_{\Phi}^{2}$ (15)
$\displaystyle\mp$
$\displaystyle\sqrt{\lambda_{H}^{2}v^{4}+\lambda_{\Phi}^{2}v_{\Phi}^{4}+\left(\lambda^{2}_{H\Phi}-2\lambda_{H}\lambda_{\Phi}\right)v^{2}v_{\Phi}^{2}}\,.\hskip
17.07182pt$

Depending on its mass, either $h_{1}$ or $h_{2}$ is designated as the observed
Higgs boson at the LHC. Currently, the most precise measurement of the Higgs
boson mass is $125.38\pm 0.14$ GeV Sirunyan  _et al._ (2020). In this
analysis, $h_{1}$ and $h_{2}$ are identified as the lighter and SM Higgs
bosons with masses approximately $95$ GeV and $125$ GeV, respectively, to
address the LHC excesses. For clarity in denoting the masses of the scalars,
we use the notation $h_{95}\equiv h_{1}$ and $h_{125}\equiv h_{2}$ in the
following.

The complex fields $H_{2}^{0\,*}$ and $G^{p}_{H}$ in $H_{2}$ and $\Phi_{H}$
respectively are both $h$-parity odd. They mix to form a physical dark Higgs
$D^{*}$ and an unphysical Goldstone field $\tilde{G}^{*}$ absorbed by the
$W^{\prime\,p}$

$\begin{pmatrix}G^{m}_{H}\\\
H_{2}^{0}\end{pmatrix}=\mathcal{O}^{D}\cdot\begin{pmatrix}\tilde{G}\\\
D\end{pmatrix}=\begin{pmatrix}\cos\theta_{2}&\sin\theta_{2}\\\
-\sin\theta_{2}&\cos\theta_{2}\end{pmatrix}\cdot\begin{pmatrix}\tilde{G}\\\
D\end{pmatrix}\;.$ (16)

The mixing angle $\theta_{2}$ satisfies

$\tan 2\theta_{2}=\frac{2vv_{\Phi}}{v^{2}_{\Phi}-v^{2}}\;,$ (17)

and the mass of $D$ is

$m_{D}^{2}=\frac{1}{2}\lambda^{\prime}_{H\Phi}\left(v^{2}+v_{\Phi}^{2}\right)\;.$
(18)

In the Feynman-’t Hooft gauge the Goldstone field $\tilde{G}^{*}$
$(\tilde{G})$ has the same mass as the $W^{\prime\,p}$ ($W^{\prime\,m}$) which
is given by (12). Finally the charged Higgs $H^{\pm}$ is also $h$-parity odd
and has a mass

$m^{2}_{H^{\pm}}=\frac{1}{2}\left(\lambda^{\prime}_{H\Phi}v^{2}_{\Phi}-\lambda^{\prime}_{H}v^{2}\right)\;.$
(19)

One can do the inversion to express the fundamental parameters in the scalar
potential in terms of the particle masses and mixing angle $\theta_{1}$ Ramos
_et al._ (2021b, a):

$\displaystyle\lambda_{H}$ $\displaystyle=$
$\displaystyle\frac{1}{2v^{2}}\left(m^{2}_{h_{95}}\cos^{2}\theta_{1}+m^{2}_{h_{125}}\sin^{2}\theta_{1}\right)\;,$
(20) $\displaystyle\lambda_{\Phi}$ $\displaystyle=$
$\displaystyle\frac{1}{2v_{\Phi}^{2}}\left(m^{2}_{h_{95}}\sin^{2}\theta_{1}+m^{2}_{h_{125}}\cos^{2}\theta_{1}\right)\;,$
(21) $\displaystyle\lambda_{H\Phi}$ $\displaystyle=$
$\displaystyle\frac{1}{2vv_{\Phi}}\left(m^{2}_{h_{125}}-m^{2}_{h_{95}}\right)\sin\left(2\theta_{1}\right)\;,$
(22) $\displaystyle\lambda^{\prime}_{H\Phi}$ $\displaystyle=$
$\displaystyle\frac{2m_{D}^{2}}{v^{2}+v_{\Phi}^{2}}\;,$ (23)
$\displaystyle\lambda^{\prime}_{H}$ $\displaystyle=$
$\displaystyle\frac{2}{v^{2}}\left(\frac{m_{D}^{2}v_{\Phi}^{2}}{v^{2}+v_{\Phi}^{2}}-m^{2}_{H^{\pm}}\right)\;.$
(24)

From (12), we also have

$g_{H}=\frac{2m_{W^{\prime}}}{\sqrt{v^{2}+v^{2}_{\Phi}}}\;.$ (25)

Thus one can elegantly use the masses $m_{h_{95}}$, $m_{W^{\prime}}$, $m_{D}$
and $m_{H^{\pm}}$, mixing angle $\theta_{1}$ and VEV $v_{\Phi}$ as input in
our numerical scan.

The connector sector linking the SM particles and the DM $W^{\prime(p,m)}$
consists of the $h$-parity even or odd particles. Specifically, we have
$\gamma$, $Z_{i}(i=1,2,3)$, $h_{125}$ and $h_{95}$ in the $s$-channel, and new
heavy fermions $f^{H}$s or even the DM $W^{\prime(p,m)}$ itself in the
$t$-channel and/or $u$-channel.

##  III The signal strengths

In this section we show the signal strengths of the lighter scalar boson
$h_{95}$ decays into di-photon and $\tau^{+}\tau^{-}$ from gluon-gluon fusion,
and into $b\bar{b}$ from Higgs-strahlung process. The signal strengths from
the gluon-gluon fusion process can be given as

$\displaystyle\mu_{\gamma\gamma/\tau\tau}$ $\displaystyle=$
$\displaystyle\frac{\sigma(gg\rightarrow h_{95})\times{\rm
BR}(h_{95}\rightarrow\gamma\gamma/\tau^{+}\tau^{-})}{\sigma^{\rm
SM}(gg\rightarrow H)\times{\rm BR}^{\rm
SM}(H\rightarrow\gamma\gamma/\tau^{+}\tau^{-})|{{}_{m_{H}=m_{h_{95}}}}}\,,$
(26) $\displaystyle=$ $\displaystyle\left.\frac{\Gamma^{\rm
SM}_{H}}{\Gamma_{h_{95}}}\cdot\frac{\Gamma_{h_{95}\rightarrow gg}}{\Gamma^{\rm
SM}_{H\rightarrow
gg}}\cdot\frac{\Gamma_{h_{95}\rightarrow\gamma\gamma/\tau^{+}\tau^{-}}}{\Gamma^{\rm
SM}_{H\rightarrow\gamma\gamma/\tau^{+}\tau^{-}}}\right|_{m_{H}=m_{h_{95}}}\,,$
(27)

where $\Gamma^{\rm SM}$ is the SM total decay with of a hypothetical scalar
with mass $m_{H}=m_{h_{95}}\simeq 95$ GeV, while $\Gamma_{h_{95}}$ is total
decay width of $h_{95}$ boson. We have the decay width ratio of
$\Gamma_{h_{95}\rightarrow\tau^{+}\tau^{-}}/\Gamma^{\rm
SM}_{H\rightarrow\tau^{+}\tau^{-}}=\cos^{2}\theta_{1}$. The decay width of
$\Gamma_{h_{95}\rightarrow gg}$ and $\Gamma_{h_{95}\rightarrow\gamma\gamma}$
are given in the Appendix A.

The signal strength of the Higgs-strahlung process ($e^{+}e^{-}\rightarrow
Zh_{95}(h_{95}\rightarrow b\overline{b})$) is given by

$\mu_{bb}=\frac{\sigma(e^{+}e^{-}\rightarrow Zh_{95})}{\sigma^{\rm
SM}(e^{+}e^{-}\rightarrow ZH)|_{{m_{H}=m_{h_{95}}}}}\times\frac{{\rm
BR}(h_{95}\rightarrow b\overline{b})}{{\rm BR^{SM}}(H\rightarrow
b\overline{b})|_{{m_{H}=m_{h_{95}}}}}\,.$ (28)

The LO cross section for the Higgs-strahlung process can be given as Kilian
_et al._ (1996)

$\sigma\left(e^{+}e^{-}\rightarrow
Zh_{95}\right)=\cos^{2}\theta_{1}\frac{G_{F}^{2}m_{Z}^{4}}{96\pi
s}\left(v_{e}^{2}+a_{e}^{2}\right)\lambda^{\frac{1}{2}}\frac{\lambda+12m_{Z}^{2}/s}{\left(1-m_{Z}^{2}/s\right)^{2}}\,,$
(29)

where $\sqrt{s}$ is the center-of-mass energy,
$\lambda=(1-(m_{h_{95}}+m_{Z})^{2}/s)(1-(m_{h_{95}}-m_{Z})^{2}/s)$ is the two-
particle phase space function, and $v_{e}$ ($a_{e}$) is the vector (axial-
vector) coupling of $Ze^{+}e^{-}$ vertex. Ignoring the mixing effects between
the neutral gauge bosons given in (9), the coupling $Ze^{+}e^{-}$ is the same
as SM, i.e. $a_{e}=-1$ and $v_{e}=-1+4\sin^{2}\theta_{W}$, and hence one can
obtain

$\mu_{bb,{\rm LO}}\simeq\cos^{2}\theta_{1}\times\frac{{\rm
BR}(h_{95}\rightarrow b\overline{b})}{{\rm BR^{SM}}(H\rightarrow
b\overline{b})|_{{m_{H}=m_{h_{95}}}}}\,.$ (30)

We emphasize that one-loop electroweak radiative corrections to
$e^{-}e^{+}\rightarrow ZH$ have been computed within the standard model
Fleischer and Jegerlehner (1983); Denner  _et al._ (1992); Kniehl (1992); Sun
_et al._ (2017); Bondarenko  _et al._ (2019) and several beyond standard
models Heng  _et al._ (2015); Abouabid  _et al._ (2021); Aiko  _et al._
(2021). At LEP center-of-mass energy regions ($\sqrt{s}\sim 200$ GeV), full
one-loop electroweak corrections are about $\sim-20\%$ contributions. However,
the corrections are mainly from initial-state radiative (ISR) corrections
(about $\sim-18\%$ contributions as indicated in Greco  _et al._ (2018);
Bondarenko  _et al._ (2019)). We know that the ISR corrections are universal
for many processes. Therefore, the corrections are cancelled out in the signal
strength given in (28). For the reasons explained above, it is enough to take
tree-level cross sections for the processes $e^{-}e^{+}\rightarrow ZH,Zh_{95}$
in our analysis.

##  IV Numerical results

Figure 1:  The decay rate and branching ratio of $h_{95}$ boson as a function
of DM mass with $v_{\Phi}>0$ (top panels) and $v_{\Phi}<0$ (bottom panels).
Here we fixed $m_{h_{95}}=95$ GeV, $m_{H^{\pm}}=600$ GeV, $m_{D}=550$ GeV,
$g_{H}=10^{-3}$, $g_{X}=10^{-6}$, $m_{X}=1$ TeV and $y_{f^{H}}=1$. Left
panels: dashed lines in black, blue, and purple denote decay rates
corresponding to $\cos^{2}\theta_{1}$ values of 0.2, 0.1, and 0.05,
respectively. Right panels: We fixed $\cos^{2}\theta_{1}=0.1$. Solid lines in
blue, orange, green, red, and purple indicate the branching ratios of the
$h_{95}$ boson decaying into pairs of SM particles, including gluons, photons,
tau leptons, bottom quarks, and $W$ bosons, respectively. Meanwhile, dashed
lines in brown and pink represent pairs of BSM particles, $W^{\prime}$ and
$Z^{\prime}$ respectively.

###  IV.1 Decay Rates and Branching Ratios of the Lighter Scalar $h_{95}$

If kinematically allowed, the lighter scalar $h_{95}$ may decay into a pair of
SM particles, including fermions and gauge bosons, and a pair of dark sector
particles such as DMs, dark photons, and dark Z bosons.

Fig. 1 illustrates decay rate and branching ratio of $h_{95}$ boson as a
function of DM mass. Here we fixed $m_{h_{95}}=95$ GeV, $m_{H^{\pm}}=600$ GeV,
$m_{D}=550$ GeV, $g_{H}=10^{-3}$, $g_{X}=10^{-6}$, $m_{X}=1$ TeV and
$y_{f^{H}}=1$. In the top panels, we fix the VEV $v_{\Phi}$ to be positive,
whereas in the bottom panels, it is set to be negative. From the left panels
of Fig. 1, we observe that the decay rate of the $h_{95}$ boson is notably
depended on the DM mass, particularly in the low mass region
($m_{W^{\prime}}<m_{h_{95}}/2$), where decaying into pairs of DMs and
$Z^{\prime}$ bosons becomes kinematically feasible. Furthermore, the
dependence of the decay rate on the mixing angle $\theta_{1}$ is significant
in the high DM mass region but less pronounced in the low DM mass region.

From the right panels of Fig. 1, one sees that the branching ratios of
$h_{95}$ to pairs of DMs and to $Z^{\prime}$ bosons are dominant in the low DM
mass region while higher DM mass region the branching ratios to SM particles
becomes predominant. This implies that in a heavy DM mass region, there is a
relatively large signal strength for the decay of $h_{95}$ into SM particles,
such as di-photon and di-tau, as observed at the LHC. Notably, the branching
ratios to pairs of gluons and photons undergo significant alterations in the
low DM mass region, depending on the sign of $v_{\Phi}$. This effect arises
from the interference between contributions from charged Higgs, SM quarks and
hidden quarks for the process of $h_{95}$ decaying into di-photon. Whereas for
the process of $h_{95}$ decaying into gluons, it is due to the interference
between contributions from SM quarks and hidden quarks. Note that these
alterations in the decays into pairs of gluons and photons result in slight
changes in the total decay width of $h_{95}$ in the low mass region for
different signs of $v_{\Phi}$ as shown in the left panels of Fig. 1.

###  IV.2 Correlations between $h_{95,125}\rightarrow\gamma\gamma$ and
$h_{95,125}\rightarrow Z\gamma$

In this section, we present an analysis of the correlations between the modes
$h_{95,125}\rightarrow\gamma\gamma$ and $h_{95,125}\rightarrow Z\gamma$. These
processes occur via one-loop induced with the $W^{\pm}$ boson, SM fermions,
hidden fermions, and charged Higgs running in the loop. We find that the
contribution from the hidden fermion loop is negligible for our parameters of
interest.

Figure 2:  The signal strengths of lighter scalar and SM 125 GeV Higgs bosons
as a function of charged Higgs mass. Here we set $m_{D}=m_{H^{\pm}}-10$ GeV in
the left panel and $m_{D}=m_{H^{\pm}}+70$ GeV in the right panel. The remain
parameters are set to be $m_{h_{1}}=95$ GeV, $\cos^{2}\theta_{1}=0.1$,
$m_{W^{\prime}}=50$ GeV, $g_{H}=10^{-3}$, $g_{X}=10^{-6}$, $m_{X}=1$ TeV and
$y_{f^{H}}=1$ in both panels. The dashed red, blue and green lines represent
the signal strength of lighter scalar boson decays into di-photon, $Z\gamma$
and di-tau, respectively. Whereas the solid red and blue lines indicate the
signal strength of SM 125 GeV Higgs boson decays into di-photon and $Z\gamma$
, respectively.

In Fig. 2, we illustrate the signal strength of both lighter scalar $h_{95}$
and Higgs bosons $h_{125}$ decaying into di-photon, $Z\gamma$, and di-tau as a
function of the charged Higgs mass. Here, for both scalar bosons, the
production mode considered is gluon-gluon fusion. With parameters set as
follows: $m_{h_{1}}=95$ GeV, $\cos^{2}\theta_{1}=0.1$, $m_{W^{\prime}}=50$
GeV, $g_{H}=10^{-3}$, $g_{X}=10^{-6}$, $m_{X}=1$ TeV, and $y_{f^{H}}=1$, the
signal strength of $h_{95}\rightarrow\tau^{+}\tau^{-}$ is approximately 0.1.
The signal strengths of the di-photon and $Z\gamma$ final states undergo
significant alterations in the low mass range of the charged Higgs, where the
contribution from charged Higgs loop becomes important. Particularly
noteworthy is when the charged Higgs becomes on-shell
$(m_{H^{\pm}}<m_{h_{95,125}}/2)$, the amplitude from the charged Higgs loop
process acquires an imaginary part, while its real part peaks at the mass
threshold $(m_{H^{\pm}}=m_{h_{95,125}}/2)$.

On the other hand, while the sum of loop form factors from $W^{\pm}$ boson and
SM fermions diagrams yields a negative value in the decay channels
$h_{125}\rightarrow\gamma\gamma$ and $h_{125}\rightarrow Z\gamma$, as well as
$h_{95}\rightarrow\gamma\gamma$ processes, it exhibits a positive value in the
$h_{95}\rightarrow Z\gamma$ process. This is mainly due to the sign changing
of the $W^{\pm}$ boson loop form factor in the $h\rightarrow Z\gamma$ process
(shown in (49)) at scalar mass $m_{h}\sim 100$ GeV. Depending on the relative
sign between the loop form factors of the charged Higgs and the total
$W^{\pm}$ and SM fermions contributions, the decay rate will either be
enhanced or suppressed.

Here, we note that the bound on the charged Higgs mass from LEP Achard  _et
al._ (2003) ($m_{H^{\pm}}>80$ GeV) within the framework of the well-known two-
Higgs-doublet models may not be directly applicable to the charged Higgs in
this model. This is because of the different production and decay modes of the
charged Higgs boson in this model compared to the conventional models.
Specifically, the charged Higgs boson in this model is odd under $h$-parity,
necessitating its decay into a $h$-parity odd particle and a $h$-parity even
particle. For example, charged Higgs can decay into $W^{\pm}W^{\prime}$ and/or
$W^{\pm}D$ followed by $D\rightarrow W^{\prime}h_{i}$ and/or $D\rightarrow
W^{\prime}Z_{i}$. Nevertheless, searches for multilepton or multijet plus
missing energy events at the LHC can establish constraints on the charged
Higgs mass, resembling signatures similar to searches for charginos and
neutralinos in supersymmetry Aad  _et al._ (2021); Tumasyan  _et al._ (2022a).
Moreover, one can put a lower bound on the charged Higgs mass in minimal G2HDM
depending upon the hidden up-type quark mass using the data from rare $B$
meson decays and oblique parameters Liu  _et al._ (2024). Taking into account
these considerations, in scanning results presented in following subsections,
we set $m_{H^{\pm}}>100$ GeV.

In the left panel of Fig. 2, we set $m_{D}=m_{H^{\pm}}-10$ GeV, ensuring that
the coupling between $h_{95,125}$ and charged Higgs (as shown in 54) is
positive. Consequently, a cancellation between the loop contributions causes
dips at the mass thresholds for the signal strength of
$h_{125}\rightarrow\gamma\gamma$ and $Z\gamma$, as well as
$h_{95}\rightarrow\gamma\gamma$, while an enhancement between the loop
contributions results in a peak shape for the signal strength of
$h_{95}\rightarrow Z\gamma$.

On the other hand, when $m_{D}=m_{H^{\pm}}+70$ GeV is assumed, the coupling
between $h_{95,125}$ and charged Higgs becomes negative, resulting in peaks at
the mass thresholds for the signal strength of
$h_{125}\rightarrow\gamma\gamma$, $h_{125}\rightarrow Z\gamma$ and
$h_{95}\rightarrow\gamma\gamma$ as shown in the right panel of Fig. 2. In the
same panel, for the $h_{95}\rightarrow Z\gamma$ signal strength, there is
enhancement at charged Higgs masses below the mass threshold due to the
dominance of the imaginary part from the charged Higgs loop contribution,
while there is suppression above the mass threshold due to the cancellation
between real part of the charged Higgs, $W^{\pm}$, and SM fermion loop
contributions.

From Fig. 2, we find a strong correlation between the signal strengths of
$h_{125}\rightarrow\gamma\gamma$ and $h_{125}\rightarrow Z\gamma$, whereas
this correlation does not extend to the lighter scalar boson $h_{95}$. This
feature persists even after conducting a comprehensive parameter scan, as
presented in the following subsection. Moreover, with
$\cos^{2}\theta_{1}=0.1$, the signal strength of
$h_{95}\rightarrow\gamma\gamma$ is approximately one order of magnitude
smaller than the same channel for $h_{125}$. A larger value of
$\cos^{2}\theta_{1}$ results in an increase in signal strengths of $h_{95}$
and a decrease in $h_{125}$.

###  IV.3 Scanning Results

We scan over the parameter space in the model through all the theoretical
constraints for the Higgs potential and experimental constraints from the
Higgs signal strengths data from CMS CMS (2020) measurements, the invisible
Higgs decay data Tumasyan  _et al._ (2022b) and constraints from electroweak
precision measurements at the $Z$ pole Zyla  _et al._ (2020), as well as from
$Z^{\prime}$ Aad  _et al._ (2019) and dark photon physics (see Fabbrichesi
_et al._ (2020) for a recent review). The dark matter constraints on relic
density from the Planck Collaboration Aghanim  _et al._ (2020) and DM direct
detection from various experiments Angloher  _et al._ (2017); Agnes  _et al._
(2018, 2023); Aprile  _et al._ (2018, 2019, 2023); Meng  _et al._ (2021);
Aalbers  _et al._ (2023) are taken into account as well. The detailed
discussion on these constraints on the model have been studied in previous
works Ramos  _et al._ (2021b, a)

To sample the parameter space in the model, we employ MCMC scans using emcee
Foreman-Mackey  _et al._ (2013). The scan range is given as,

$\displaystyle m_{h_{95}}/{\rm GeV}$ $\displaystyle\in$
$\displaystyle(94,\,97)\,,$ (31) $\displaystyle m_{H^{\pm}}/{\rm GeV}$
$\displaystyle\in$ $\displaystyle(100,\,1000)\,,$ (32) $\displaystyle
m_{D}/{\rm GeV}$ $\displaystyle\in$ $\displaystyle(10,\,1000)\,,$ (33)
$\displaystyle m_{W^{\prime}}/{\rm GeV}$ $\displaystyle\in$
$\displaystyle(0.1,\,1000)\,,$ (34) $\displaystyle\theta_{1}/{\rm rad}$
$\displaystyle\in$
$\displaystyle\left(-\frac{\pi}{2},\,\frac{\pi}{2}\right)\,,$ (35)
$\displaystyle v_{\Phi}/{\rm GeV}$ $\displaystyle\in$
$\displaystyle(10^{2},\,10^{5})\,,$ (36) $\displaystyle g_{X}$
$\displaystyle\in$ $\displaystyle(10^{-6},\,10^{-2})\,,$ (37) $\displaystyle
m_{X}/{\rm GeV}$ $\displaystyle\in$ $\displaystyle(10^{-2},\,10^{3})\,,$ (38)

and we fix the Yukawa couplings of the hidden fermions to be $y_{f^{H}}=1$.
The parameters $m_{W^{\prime}}$, $v_{\Phi}$, $g_{X}$ and $m_{X}$ are sampling
in log prior while the remain ones are in linear.

Figure 3:  The signal strength of lighter scalar boson $h_{95}$ and SM Higgs
boson $h_{125}$. The red and blue lines represent $1\sigma$ and $2\sigma$
experimental contours, respectively. From left to right and top to bottom, the
viable parameter space projected on ($\mu_{\gamma\gamma}^{h_{95}}$,
$\mu_{bb}^{h_{95}}$), ($\mu_{\gamma\gamma}^{h_{95}}$,
$\mu_{\tau\tau}^{h_{95}}$), ($\mu_{bb}^{h_{95}}$, $\mu_{\gamma Z}^{h_{125}}$),
($\mu_{\gamma\gamma}^{h_{95}}$, $\mu_{\gamma Z}^{h_{125}}$) planes,
respectively. The color indicates the values of $|\cos(\theta_{1})|$.

Fig. 3 illustrates the viable parameter space within the model, projected onto
the signal strength of the 95 GeV scalar boson and the SM-like scalar boson.
We find that a portion of the viable data points can explain the signal
strength of the 95 GeV scalar boson decaying into di-photon events, as
measured at the LHC, and into $b\bar{b}$ pairs, as measured at LEP, as
depicted in the top-left panel of Fig. 3. However, the signal strength of the
95 GeV scalar boson decaying into di-tau pairs is found to be too small, thus
conflicting with current measurements at CMS, as shown in the top-right panel
of Fig. 3. Notably, a larger value of $|\cos(\theta_{1})|$ can result in a
higher value of $\mu_{\tau\tau}^{h_{95}}$, however the current constraints
from Higgs data measurements at the CMS CMS (2020) require
$|\cos(\theta_{1})|\lesssim 0.36$.

The bottom panels in Fig. 3 present the viable data points projected on planes
of the signal strengths of $h_{125}\rightarrow\gamma Z$ and $h_{95}\rightarrow
b\bar{b}$ (left panel), and $h_{95}\rightarrow\gamma\gamma$ (right panel). The
results indicate that, within $2\sigma$ region, the model can simultaneously
accommodate the experimental data for 95 GeV scalar boson decaying into
$b\bar{b}$ from LEP and its decaying to di-photon from the LHC as well as the
SM-like Higgs boson decaying into $Z\gamma$ channel from recent measurements
at the LHC.

Figure 4: Viable parameter space projected on the signal strengths of
$h_{125}\rightarrow\gamma\gamma$ and $h_{125}\rightarrow Z\gamma$ plane
(left), and $h_{95}\rightarrow\gamma\gamma$ and $h_{95}\rightarrow Z\gamma$
plane (right).

In Figure 4, we present the detailed correlation between signal strengths of
$h_{95,125}\rightarrow\gamma\gamma$ and $h_{95,125}\rightarrow Z\gamma$ within
the viable parameter space. In the left panel of Fig. 4, we show the viable
parameter space spanned on the signal strengths of
$h_{125}\rightarrow\gamma\gamma$ and $h_{125}\rightarrow Z\gamma$ plane. The
signal strengths of $h_{95}\rightarrow\gamma\gamma$ and $h_{95}\rightarrow
Z\gamma$ is presented in the right panel. Here, for both $h_{125}$ and
$h_{95}$, the production mode considered is gluon-gluon fusion. It is evident
that the signal strengths of $h_{125}\rightarrow\gamma\gamma$ and
$h_{125}\rightarrow Z\gamma$ exhibit a strong correlation, whereas this
correlation is not observed for the $h_{95}$ boson.

##  V Conclusion

Recent results from low-mass Higgs boson searches at the LHC reveal excesses
around $95$ GeV in the di-photon final state, with a local significance of
$3.1\sigma$ combining both CMS and ATLAS data, and the di-tau final state,
with a local significance of $2.6\sigma$ from CMS. An excess at similar mass
with a local significance of $2.3\sigma$ was previously observed at the LEP
experiment. Furthermore, the LHC has reported the first evidence of a rare
decay mode of the 125 GeV Higgs boson into $Z\gamma$.

In this study, we have investigated the simultaneously interpretation of the
excesses at $95$ GeV arising from the production of a lighter scalar boson and
the rare decay mode of the $125$ Higgs boson into $Z\gamma$ in the framework
of gauged two-Higgs-doublet model.

We have presented an analysis of the decay properties of the lighter scalar
boson $h_{95}$. In addition to its decays into SM particles due to the mixing
with SM Higgs boson, $h_{95}$ can decay into particles within the dark sector
via its major hidden component, including DM, dark photons, and dark $Z$
bosons, where kinematically feasible. The decay rate of $h_{95}$ to SM
particles is significantly influenced by the mixing angle $\theta_{1}$, with a
larger value of $|\cos(\theta_{1})|$ correlating to higher decay rates.
Notably, the $h_{95}\rightarrow b\bar{b}$ channel emerges as the predominant
decay mode among SM particle final states. On the other hand, the decay rate
of $h_{95}$ into dark sector particles strongly depends on the new gauge
couplings and the mass of the final state particles. For a substantial gauge
coupling and a low mass region of DM, the branching ratios of $h_{95}$
decaying into dark sector particles can dominate, as illustrated in Fig. 1.

We focused our investigation on the di-photon and $Z\gamma$ final states from
the decays of both 95 GeV and 125 GeV Higgs bosons. In addition to the
anticipated contributions from $W^{\pm}$ boson and SM fermions loops, our
analysis found significant effects from charged Higgs loop, especially in low
charged Higgs mass region. The impact of the hidden fermions loop remains
negligible within our parameter range of interest. The signal strengths of
$h_{95,125}\rightarrow\gamma\gamma$ and $Z\gamma$ can either be enhanced or
suppressed depending on the constructive or destructive interference between
these contributions, as depicted in Fig. 2.

Employing an exhaustive parameter space scan, constrained by the theoretical
conditions and experimental data, we present our main results in Fig. 3. We
found that the viable parameter space in the model can simultaneously address
the excesses observed around 95 GeV in the $b\bar{b}$ final state channel at
LEP and the di-photon final state channel at the LHC as well as the recent
evidence for the 125 GeV Higgs boson decay into $Z\gamma$ at the LHC. On the
other hand, the signal strength of $h_{95}\rightarrow\tau^{+}\tau^{-}$ is
insufficient to account for the excess reported by CMS. Moreover, we found a
strong correlation between the signal strengths of
$h_{125}\rightarrow\gamma\gamma$ and $h_{125}\rightarrow Z\gamma$, although
this correlation doesn’t extend to the lighter scalar $h_{95}$, as shown in
Fig. 4.

Forthcoming results for the low-mass searches in di-tau final state channel
from the ATLAS experiment, along with upcoming Run 3 results from both ATLAS
and CMS, particularly for the di-photon final state channel, hold promise in
shedding light on the potential presence of an addition scalar boson around 95
GeV.

## Acknowledgments

This work was supported in part by the National Natural Science Foundation of
China, grant Nos. 19Z103010239 and 12350410369 (VQT), the NSTC grant No.
111-2112-M-001-035 (TCY), and the Moroccan Ministry of Higher Education and
Scientific Research MESRSFC and CNRST Project PPR/2015/6 (AB). VQT would like
to thank the Medium and High Energy Physics group at the Institute of Physics,
Academia Sinica, Taiwan for their hospitality during the course of this work.

## Appendix A

The general analytical expressions for the one-loop amplitudes of the two
processes $h_{i}\rightarrow\gamma\gamma$ and $h_{i}\rightarrow Z_{j}\gamma$
111See for example Refs. Gunion  _et al._ (2000); Gamberini  _et al._ (1987);
Weiler and Yuan (1989); Chen  _et al._ (2013); Hue  _et al._ (2018) for the
computation of this process in a variety of BSM. in G2HDM were given in the
Appendix in Tran  _et al._ (2023a). To make this paper self-contained, we
briefly summarize these formulas here. As mentioned in the text, $h_{1}\equiv
h_{95}$, $h_{2}\equiv h_{125}$ and $Z_{1}$ are identified as the lighter
scalar with mass $\sim 95$ GeV, the observed Higgs with mass $125.38\pm 0.14$
GeV Sirunyan  _et al._ (2020) and $Z$ boson with mass $91.1876\pm 0.0021$ GeV
Zyla  _et al._ (2020), respectively.

Define the following two well-known loop functions $I_{1}(\tau,\lambda)$ and
$I_{2}(\tau,\lambda)$ Gunion  _et al._ (2000)

$\displaystyle I_{1}(\tau,\lambda)$ $\displaystyle=$
$\displaystyle\frac{\tau\lambda}{2(\tau-\lambda)}+\frac{\tau^{2}\lambda^{2}}{2(\tau-\lambda)^{2}}\left[f(\tau)-f(\lambda)\right]$
(39) $\displaystyle+$
$\displaystyle\frac{\tau^{2}\lambda}{2(\tau-\lambda)^{2}}\left[g(\tau)-g(\lambda)\right]\;,$
$\displaystyle I_{2}(\tau,\lambda)$ $\displaystyle=$
$\displaystyle-\frac{\tau\lambda}{2(\tau-\lambda)}\left[f(\tau)-f(\lambda)\right]\;,$
(40)

with

$\displaystyle f(x)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cr}\left[{\rm
arcsin}(1/\sqrt{x})\right]^{2}\,,&(x\geq 1)\,,\\\
-\frac{1}{4}\left[\ln\left(\eta_{+}/\eta_{-}\right)-i\pi\right]^{2}\,,&(x<1)\,;\end{array}\right.$
(43) $\displaystyle g(x)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cr}\sqrt{x-1}\,{\rm
arcsin}(1/\sqrt{x})\,,&(x\geq 1)\,,\\\
\frac{1}{2}\sqrt{1-x}\left[\ln\left(\eta_{+}/\eta_{-}\right)-i\pi\right]\,,&(x<1)\,;\end{array}\right.$
(46)

where

$\eta_{\pm}\equiv 1\pm\sqrt{1-x}\;.$ (47)

Fig. 5 illustrates the loop functions for the $h\rightarrow Z\gamma$ process,
multiplied by the phase space factor $(1-m_{Z}^{2}/m_{h}^{2})$, as a function
of $m_{h}$. In this case, $\tau=4m_{l}^{2}/m_{h}^{2}$ and
$\lambda=4m_{l}^{2}/m_{Z}^{2}$, where $m_{l}$ represents the mass of the
particle running inside the loop. We note that the loop function $I_{1}$
exhibits a singularity at $m_{h}=m_{Z}$; however, this will be canceled out by
the phase space factor. Furthermore, when the particle running inside the loop
is on-shell ($m_{l}<m_{h}/2$), the loop functions acquire imaginary parts,
with the real parts peaking at the mass threshold ($m_{l}=m_{h}/2$).

Figure 5: Loop functions $I_{1}^{\prime}\equiv(1-m_{Z}^{2}/m_{h}^{2})I_{1}$
(left panel) and $I_{2}^{\prime}\equiv(1-m_{Z}^{2}/m_{h}^{2})I_{2}$ (right
panel) for $h\rightarrow Z\gamma$ process as a function of $m_{h}$. The dashed
and dotted blue (red) represent the real and imaginary part of the loop
function with mass of the particle running inside the loop $m_{l}=m_{W}$
($m_{l}=m_{t}$).

###  A.1 Decay Rate of $h_{i}\rightarrow Z_{j}\gamma$

The partial decay rate for $h_{i}\rightarrow Z_{j}\gamma$ is

$\Gamma(h_{i}\rightarrow
Z_{j}\gamma)=\frac{1}{32\pi}m_{h_{i}}^{3}\left(1-\frac{m_{Z_{j}}^{2}}{m_{h_{i}}^{2}}\right)^{3}\biggl{|}F^{1}_{ij}+F^{1/2}_{ij}+F^{0}_{ij}\biggr{|}^{2}\;,$
(48)

where $F^{s}_{ij}$ with $s=0,1/2,1$ denotes the loop form factor for charged
particle with spin equals $0,1/2,1$ respectively running inside the loop.

In G2HDM, the only charged spin 1 particle is the SM $W^{\pm}$, thus
$F^{1}_{ij}=F_{ij}(W^{\pm})$,

$\displaystyle F_{ij}(W^{\pm})$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\cdot e\cdot gm_{W}\cdot
gc_{W}\cdot\frac{1}{m_{W}^{2}}\cdot{\mathcal{O}}^{S}_{1i}{\mathcal{O}}^{G}_{1j}$
(49)
$\displaystyle\times\left\\{\left[5+\frac{2}{\tau_{iW}}+\left(1+\frac{2}{\tau_{iW}}\right){\left(1-\frac{4}{\lambda_{jW}}\right)}\right]I_{1}\left(\tau_{iW},\lambda_{jW}\right)\right.$
$\displaystyle\left.-16{\left(1-\frac{1}{\lambda_{jW}}\right)}I_{2}\left(\tau_{iW},\lambda_{jW}\right)\right\\}\;.$

Here and below, we denote $\tau_{il}=4m^{2}_{l}/m^{2}_{h_{i}}$ and
$\lambda_{jl}=4m^{2}_{l}/m^{2}_{Z_{j}}$.

All the charged fermions in G2HDM, including both the SM fermions $f^{\rm SM}$
and the new heavy fermions $f^{H}$ contribute to $F^{1/2}_{ij}$. Thus

$F^{1/2}_{ij}=\sum_{f^{\rm SM}}F_{ij}(f^{\rm
SM})+\sum_{f^{H}}F_{ij}(f^{H})\;,$ (50)

where

$\displaystyle F_{ij}(f^{\rm SM})$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\cdot N^{c}_{f^{\rm SM}}\cdot eQ_{f^{\rm
SM}}\cdot\frac{m_{f^{\rm SM}}}{v}\cdot C^{f^{\rm
SM}}_{Vj}\cdot\frac{-2}{m_{f^{\rm SM}}}\cdot{\mathcal{O}}^{S}_{1i}$ (51)
$\displaystyle\times\biggl{[}I_{1}\left(\tau_{if^{\rm SM}},\lambda_{jf^{\rm
SM}}\right)-I_{2}\left(\tau_{if^{\rm SM}},\lambda_{jf^{\rm
SM}}\right)\biggr{]}\;,$

and

$\displaystyle F_{ij}(f^{\rm H})$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\cdot N^{c}_{f^{\rm H}}\cdot eQ_{f^{\rm
H}}\cdot\frac{m_{f^{\rm H}}}{v_{\Phi}}\cdot C^{f^{\rm
H}}_{Vj}\cdot\frac{-2}{m_{f^{\rm H}}}\cdot{{\mathcal{O}}^{S}_{2i}}$ (52)
$\displaystyle\times\biggl{[}I_{1}\left(\tau_{if^{\rm H}},\lambda_{jf^{\rm
H}}\right)-I_{2}\left(\tau_{if^{\rm H}},\lambda_{jf^{\rm
H}}\right)\biggr{]}\;,$

with $N^{c}_{f}$ being the color factor and $Q_{f}$ the electric charge of $f$
in unit of $e>0$; the vector couplings $C^{f}_{Vj}$ of quarks and leptons are
listed in Table 1 and Table 2 respectively.

$C^{u}_{Vj}$ | $\frac{1}{2}\left[\frac{g}{c_{W}}\left(\frac{1}{2}-\frac{4}{3}s^{2}_{W}\right){\mathcal{O}}^{G}_{1j}+g_{H}\left(+\frac{1}{2}\right){\mathcal{O}}^{G}_{2j}+{\frac{1}{2}}g_{X}{\mathcal{O}}^{G}_{3j}\right]$
---|---
$C^{d}_{Vj}$ | $\frac{1}{2}\left[\frac{g}{c_{W}}\left(-\frac{1}{2}+\frac{2}{3}s^{2}_{W}\right){\mathcal{O}}^{G}_{1j}+g_{H}\left(-\frac{1}{2}\right){\mathcal{O}}^{G}_{2j}+g_{X}\left(-{\frac{1}{2}}\right){\mathcal{O}}^{G}_{3j}\right]$
$C^{u^{H}}_{Vj}$ | $\frac{1}{2}\left[-\frac{g}{c_{W}}\left(\frac{4}{3}\right)s^{2}_{W}{\mathcal{O}}^{G}_{1j}+g_{H}\left(-\frac{1}{2}\right){\mathcal{O}}^{G}_{2j}+{\frac{1}{2}}g_{X}{\mathcal{O}}^{G}_{3j}\right]$
$C^{d^{H}}_{Vj}$ | $\frac{1}{2}\left[\frac{g}{c_{W}}\left(\frac{2}{3}\right)s^{2}_{W}{\mathcal{O}}^{G}_{1j}+g_{H}\left(+\frac{1}{2}\right){\mathcal{O}}^{G}_{2j}+g_{X}\left(-{\frac{1}{2}}\right){\mathcal{O}}^{G}_{3j}\right]$
Table 1: Coupling coefficients $C^{f}_{Vj}$ for quarks. $C^{\nu}_{Vj}$ | $\frac{1}{2}\left[\frac{g}{c_{W}}\left(+\frac{1}{2}\right){\mathcal{O}}^{G}_{1j}+g_{H}\left(+\frac{1}{2}\right){\mathcal{O}}^{G}_{2j}+{\frac{1}{2}}g_{X}{\mathcal{O}}^{G}_{3j}\right]$
---|---
$C^{e}_{Vj}$ | $\frac{1}{2}\left[\frac{g}{c_{W}}\left(-\frac{1}{2}+2s^{2}_{W}\right){\mathcal{O}}^{G}_{1j}+g_{H}\left(-\frac{1}{2}\right){\mathcal{O}}^{G}_{2j}+g_{X}\left(-{\frac{1}{2}}\right){\mathcal{O}}^{G}_{3j}\right]$
$C^{\nu^{H}}_{Vj}$ | $\frac{1}{2}\left[g_{H}\left(-\frac{1}{2}\right){\mathcal{O}}^{G}_{2j}+{\frac{1}{2}}g_{X}{\mathcal{O}}^{G}_{3j}\right]$
$C^{e^{H}}_{Vj}$ | $\frac{1}{2}\left[\frac{g}{c_{W}}\left(2\right)s^{2}_{W}{\mathcal{O}}^{G}_{1j}+g_{H}\left(+\frac{1}{2}\right){\mathcal{O}}^{G}_{2j}+g_{X}\left(-{\frac{1}{2}}\right){\mathcal{O}}^{G}_{3j}\right]$
Table 2: Coupling coefficients $C^{f}_{Vj}$ for leptons.

There is only one charged Higgs $H^{\pm}$ in G2HDM. Thus
$F^{0}_{ij}=F_{ij}(H^{\pm})$ with

$\displaystyle F_{ij}(H^{\pm})$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\cdot eQ_{H^{+}}\cdot
g_{h_{i}H^{+}H^{-}}\cdot g_{Z_{j}H^{+}H^{-}}\cdot\frac{2}{m^{2}_{H^{\pm}}}$
(53) $\displaystyle\cdot$ $\displaystyle
I_{1}\left(\tau_{iH^{\pm}},\lambda_{jH^{\pm}}\right)\;,$

where $Q_{H^{+}}=+1$, and $g_{h_{i}H^{+}H^{-}}$ and $g_{Z_{j}H^{+}H^{-}}$ are
the $h_{i}H^{+}H^{-}$ and $Z_{j}H^{+}H^{-}$ couplings in the G2HDM
respectively. Explicitly they are

$\displaystyle g_{h_{i}H^{+}H^{-}}$ $\displaystyle=$
$\displaystyle\left(2\lambda_{H}-\lambda^{\prime}_{H}\right)v{\mathcal{O}}^{S}_{1i}+\left(\lambda_{H\Phi}+\lambda^{\prime}_{H\Phi}\right)v_{\Phi}\mathcal{O}^{S}_{2i}\,,\;\;\;\;\;\;$
(54) $\displaystyle g_{Z_{j}H^{+}H^{-}}$ $\displaystyle=$
$\displaystyle\;\frac{1}{2}(g\,c_{W}-g^{\prime}s_{W}){\mathcal{O}}^{G}_{1j}-\frac{1}{2}g_{H}{\mathcal{O}}^{G}_{2j}$
(55) $\displaystyle+$
$\displaystyle{\frac{1}{2}}g_{X}{\mathcal{O}}^{G}_{3j}\;.$

###  A.2 Decay Rates of $h_{i}\rightarrow\gamma\gamma$ and $h_{i}\rightarrow
gg$

The partial decay rate for $h_{i}\rightarrow\gamma\gamma$ is

$\Gamma(h_{i}\rightarrow\gamma\gamma)=\frac{1}{64\pi}m_{h_{i}}^{3}\biggl{|}F^{1}_{i}+F^{1/2}_{i}+F^{0}_{i}\biggr{|}^{2}\;,$
(56)

where $F^{1}_{i}=F_{i}(W^{\pm})$, $F^{1/2}_{i}=\sum_{f^{\rm SM}}F_{i}(f^{\rm
SM})+\sum_{f^{H}}F_{i}(f^{H})$, where $\sum_{f^{\rm SM}}$ and $\sum_{f^{H}}$
denote summation over all charged SM and heavy hidden fermions, and
$F^{0}_{i}=F_{i}(H^{\pm})$ with

$\displaystyle F_{i}(W^{\pm})$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\cdot e^{2}\cdot
gm_{W}\cdot\frac{-1}{m_{W}^{2}}\cdot{\mathcal{O}}^{S}_{1i}$ (57)
$\displaystyle\times\left[2+3\tau_{iW}+3\tau_{iW}\left(2-\tau_{iW}\right)f(\tau_{iW})\right]\,,$
$\displaystyle F_{i}(f^{\rm SM})$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\cdot N^{c}_{f^{\rm SM}}\cdot
e^{2}Q^{2}_{f^{\rm SM}}\cdot\frac{m_{f^{\rm SM}}}{v}\cdot\frac{4}{m_{f^{\rm
SM}}}\cdot{\mathcal{O}}^{S}_{1i}$ (58)
$\displaystyle\times\left\\{\tau_{if^{\rm SM}}\left[1+\left(1-\tau_{if^{\rm
SM}}\right)f(\tau_{if^{\rm SM}})\right]\right\\}\;,$ $\displaystyle
F_{i}(f^{H})$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi^{2}}\cdot
N^{c}_{f^{\rm H}}\cdot e^{2}Q^{2}_{f^{\rm H}}\cdot\frac{m_{f^{\rm
H}}}{v_{\Phi}}\cdot\frac{4}{m_{f^{\rm H}}}\cdot{\mathcal{O}}^{S}_{2i}$ (59)
$\displaystyle\times\left\\{\tau_{if^{\rm H}}\left[1+\left(1-\tau_{if^{\rm
H}}\right)f(\tau_{if^{\rm H}})\right]\right\\}\;,$ $\displaystyle
F_{i}(H^{\pm})$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi^{2}}\cdot
e^{2}\cdot g_{h_{i}H^{+}H^{-}}\cdot\frac{-1}{m^{2}_{H^{\pm}}}$ (60)
$\displaystyle\times\left\\{\tau_{iH^{\pm}}\left[1-\tau_{iH^{\pm}}f(\tau_{iH^{\pm}})\right]\right\\}\;.$

The partial decay rate for $h_{i}\rightarrow gg$ is

$\Gamma(h_{i}\rightarrow
gg)=\frac{1}{8\pi}m_{h_{i}}^{3}\biggl{|}G^{1/2}_{i}\biggr{|}^{2}\;,$ (61)

where $G^{1/2}_{i}=\sum_{q^{\rm SM}}G_{i}(q^{\rm
SM})+\sum_{q^{H}}G_{i}(q^{H})$ and

$\displaystyle G_{i}(q^{\rm SM})$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\cdot\frac{1}{2}\cdot
g_{s}^{2}\cdot\frac{m_{q^{\rm SM}}}{v}\cdot\frac{4}{m_{q^{\rm
SM}}}\cdot{\mathcal{O}}^{S}_{1i}$ (62)
$\displaystyle\times\left\\{\tau_{iq^{\rm SM}}\left[1+\left(1-\tau_{iq^{\rm
SM}}\right)f(\tau_{iq^{\rm SM}})\right]\right\\}\,,$ $\displaystyle
G_{i}(q^{\rm H})$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\cdot\frac{1}{2}\cdot
g_{s}^{2}\cdot\frac{m_{q^{\rm H}}}{v_{\Phi}}\cdot\frac{4}{m_{q^{\rm
H}}}\cdot{\mathcal{O}}^{S}_{2i}$ (63)
$\displaystyle\times\left\\{\tau_{iq^{\rm H}}\left[1+\left(1-\tau_{iq^{\rm
H}}\right)f(\tau_{iq^{\rm H}})\right]\right\\}\,.$

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