When Standard Model Higgs Meets Its Lighter 95 GeV Higgs

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.

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:

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.

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.

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.

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.