# Superposition of CP-Even and CP-Odd Higgs Resonances:
Explaining the 95 GeV Excesses within a Two-Higgs Doublet Model

Rachid Benbrik1<EMAIL_ADDRESS>Mohammed Boukidi1
<EMAIL_ADDRESS>Stefano Moretti2,3<EMAIL_ADDRESS><EMAIL_ADDRESS>1Polydisciplinary Faculty, Laboratory of
Fundamental and Applied Physics, Cadi Ayyad University, Sidi Bouzid, B.P.
4162, Safi, Morocco.
2School of Physics and Astronomy, University of Southampton,
Southampton, SO17 1BJ, United Kingdom.
3Department of Physics and Astronomy, Uppsala University,
Box 516, SE-751 20 Uppsala, Sweden.

###### Abstract

We propose an explanation for the observed excesses around 95 GeV in the di-
photon and di-tau invariant mass distributions, as reported by the CMS
collaboration at the Large Hadron Collider (LHC). These findings are
complemented by a long-standing discrepancy in the $b\bar{b}$ invariant mass
at the Large Electron-Positron (LEP) Collider. Additionally, the ATLAS
collaboration has reported a corroborative excess in the di-photon final state
within the same mass range, albeit with slightly lower significance. Our
approach involves the superposition of CP-even and CP-odd Higgs bosons within
the Type-III Two-Higgs Doublet Model (2HDM) to simultaneously explain these
excesses at 1$\sigma$ Confidence Level (C.L.), while remaining consistent with
current theoretical and experimental constraints.

##  I Introduction

In the last decade, following the Higgs boson’s discovery at the Large Hadron
Collider (LHC) in 2012 Aad  _et al._ (2012); Chatrchyan  _et al._ (2012), the
scientific community has made significant strides towards the precise
characterization of its properties. These efforts have affirmed the SM (SM)
predictions with accuracies rarely exceeding 10%. Despite these achievements,
the quest for physics Beyond the SM (BSM) persists, encouraged by the
precision of current Higgs physics at the LHC. This has opened the door to
exploring additional Higgs states beyond the SM-like one, ranging in mass from
a few GeV to the TeV scale. Extended Higgs sectors, as anticipated in various
BSM scenarios including Supersymmetric models Moretti and Khalil (2019) and
2-Higgs Doublet Models (2HDMs) Gunion  _et al._ (1992); Branco  _et al._
(2012), suggest the presence of both light and heavy non-standard Higgs
bosons. These predictions have spurred searches for these (pseudo)scalar
states across lepton and hadron colliders.

The 2HDM is a particularly well-studied framework within BSM theories,
extending the Standard Model (SM) Higgs sector by an additional Higgs doublet.
Its general version allows non-diagonal Yukawa couplings, potentially leading
to Flavor Changing Neutral Currents (FCNCs) at tree level, contrary to
experimental evidence. To circumvent this issue, a $Z_{2}$ symmetry is
typically imposed to define the coupling structure of the two Higgs doublets
to SM fermions. This classification includes the so-called Type-I, Type-II,
lepton-specific and flipped scenarios Branco  _et al._ (2012), alongside the
2HDM Type-III, which allows direct couplings of both doublets to all SM
fermions. Its Yukawa structure is then refined by both theoretical consistency
requirements and experimental measurements of Higgs masses and couplings.

During ongoing searches for a low-mass Higgs boson, the CMS collaboration
reported an excess near 95 GeV in di-photon event invariant masses in 2018
Sirunyan  _et al._ (2019). In March 2023, CMS confirmed this excess with a
local significance of 2.9$\sigma$ at $m_{\gamma\gamma}=95.4$ GeV, employing
advanced analyses on data from Run 2’s first three years CMS (2023).
Similarly, ATLAS observed an excess at 95 GeV with a local significance of
1.7$\sigma$, aligning with CMS’ findings and showcasing enhanced sensitivity
over previous analyses Arcangeletti (2023); ATL (2023).

Moreover, CMS has reported an excess in the search for a light neutral
(pseudo)scalar boson $\phi$ decaying into $\tau\tau$ pairs, with local(global)
significance of $2.6\sigma(2.3\sigma)$ around the mass of 95 GeV. However,
attempts to attribute the di-tau excess to a CP-even resonance encounter
difficulties, notably due to CMS searches for a scalar resonance in
$t\bar{t}$-associated production decaying into tau pairs, which do not support
such a finding CMS (2022a).

Previously, the Large Electron Positron (LEP) collider collaborations Barate
_et al._ (2003) explored the low-mass domain extensively in the $e^{+}e^{-}\to
Z\phi$ production mode, with a generic Higgs boson state $\phi$ decaying via
the $\tau\tau$ and $b\overline{b}$ channels. Interestingly, an excess has been
reported in 2006 in the $e^{+}e^{-}\to Z{\phi}(\to b\overline{b})$ mode for
$m_{b\bar{b}}$ around 98 GeV Schael  _et al._ (2006). Given the limited mass
resolution of the di-jet invariant mass at LEP, this anomaly may well coincide
with the aforementioned excesses seen by CMS and/or ATLAS in the
$\gamma\gamma$ and $\tau\tau$ final state. Since the excesses appear in very
similar mass regions, several studies Cao  _et al._ (2017); Heinemeyer  _et
al._ (2022); Biekötter  _et al._ (2022a, 2020); Cao  _et al._ (2020);
Biekötter  _et al._ (2023a); Iguro  _et al._ (2022); Li  _et al._ (2022);
Cline and Toma (2019); Biekötter and Olea-Romacho (2021); Crivellin  _et al._
(2018); Cacciapaglia  _et al._ (2016); Abdelalim  _et al._ (2022); Biekötter
_et al._ (2022b, 2023b); Azevedo  _et al._ (2023); Biekötter  _et al._ (2024);
Cao  _et al._ (2024a); Wang and Zhu (2024); Li  _et al._ (2023); Dev  _et al._
(2024); Borah  _et al._ (2024); Cao  _et al._ (2024b); Ellwanger and Hugonie
(2023); Aguilar-Saavedra  _et al._ (2023); Ashanujjaman  _et al._ (2023);
Dutta  _et al._ (2023); Ellwanger and Hugonie (2024); Diaz  _et al._ (2024);
Ellwanger  _et al._ (2024); Ayazi  _et al._ (2024) have explored the
possibility of simultaneously explaining these anomalies within BSM frameworks
featuring a non-standard Higgs state lighter than 125 GeV, while being in
agreement with current measurements of the properties of the $\approx 125$ GeV
SM-like Higgs state observed at the LHC. In the attempt to explain the
excesses in the $\gamma\gamma$ and $b\overline{b}$ channels, it was found in
Refs. Benbrik  _et al._ (2022a, b); Belyaev  _et al._ (2023) that the 2HDM
Type-III with a particular Yukawa texture can successfully accommodate both
measurements simultaneously with the lightest CP-even Higgs boson of the
model, while being consistent with all relevant theoretical and experimental
constraints. Further recent studies have shown that actually all three
aforementioned signatures can be simultaneously explained in the 2HDM plus a
real (N2HDM) Biekötter  _et al._ (2022b) and complex (S2HDM) Biekötter  _et
al._ (2023b, 2024) singlet.

In this study, we demonstrate that a superposition of CP-even and CP-odd
resonances within the 2HDM Type-III offers a compelling explanation for these
excesses at 1$\sigma$ Confidence Level (C.L.) through a $\chi^{2}$ analysis
while, again, satisfying both theoretical requirements and up-to-date
experimental constraints.

The organization of the paper is as follows. Section II reviews the
theoretical framework of the 2HDM Type-III, emphasizing its potential in
explaining the observed excesses. Section III provides a detailed account of
the excesses, setting the stage for our analysis. In Section IV, we discuss
the theoretical and experimental constraints that shape our exploration of the
2HDM Type-III parameter space. Section V details our numerical approach and
the outcomes of scanning the 2HDM Type-III parameter space, with the aim of
finding plausible explanations for the observed anomalies. We conclude in
Section VI, underscoring the importance of our findings and their implications
for future LHC searches.

##  II General 2HDM

The 2HDM serves as one of the most straightforward extensions of the SM. It
comprises two complex doublets of Higgs fields, denoted as $\Phi_{i}$
($i=1,2$), each with a hypercharge of $Y=+1$. The scalar potential, invariant
under the SU(2)${}_{L}\otimes$U(1)Y gauge symmetry, can be expressed as Branco
_et al._ (2012):

$\displaystyle\mathcal{V}$
$\displaystyle=m_{11}^{2}\Phi_{1}^{\dagger}\Phi_{1}+m_{22}^{2}\Phi_{2}^{\dagger}\Phi_{2}-\left[m_{12}^{2}\Phi_{1}^{\dagger}\Phi_{2}+\rm{H.c.}\right]\leavevmode\nobreak\
$
$\displaystyle+\lambda_{1}(\Phi_{1}^{\dagger}\Phi_{1})^{2}+\lambda_{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}+\lambda_{3}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})\leavevmode\nobreak\
$
$\displaystyle+\lambda_{4}(\Phi_{1}^{\dagger}\Phi_{2})(\Phi_{2}^{\dagger}\Phi_{1})+\frac{1}{2}\left[\lambda_{5}(\Phi_{1}^{\dagger}\Phi_{2})^{2}+\rm{H.c.}\right]\leavevmode\nobreak\
$
$\displaystyle+\left\\{\left[\lambda_{6}(\Phi_{1}^{\dagger}\Phi_{1})+\lambda_{7}(\Phi_{2}^{\dagger}\Phi_{2})\right](\Phi_{1}^{\dagger}\Phi_{2})+\rm{H.c.}\right\\}$
(1)

The hermiticity of this potential implies that the parameters $m_{11}^{2}$,
$m_{22}^{2}$ and $\lambda_{1,2,3,4}$ are real. In contrast, $\lambda_{5,6,7}$
and $m_{12}^{2}$ can be complex, although they are considered real in the CP-
conserving versions of the 2HDM, which we do here as well. Notably, the
$\lambda_{6,7}$ terms have a minimal effect in this study and are thus set to
zero. This simplification leaves the model with seven independent parameters,
reduced to six in our analysis with the assumption of $H$ being the observed
SM-like Higgs boson with a mass of 125 GeV.

The Yukawa sector of the 2HDM involves general scalar-to-fermion couplings,
expressed as:

$\displaystyle-{\cal L}_{Y}$
$\displaystyle=\bar{Q}_{L}Y^{u}_{1}U_{R}\tilde{\Phi}_{1}+\bar{Q}_{L}Y^{u}_{2}U_{R}\tilde{\Phi}_{2}+\bar{Q}_{L}Y^{d}_{1}D_{R}\Phi_{1}$
$\displaystyle+\bar{Q}_{L}Y^{d}_{2}D_{R}\Phi_{2}+\bar{L}Y^{\ell}_{1}\ell_{R}\Phi_{1}+\bar{L}Y^{\ell}_{2}\ell_{R}\Phi_{2}+H.c.$
(2)

Before Electro-Weak Symmetry Breaking (EWSB), the Yukawa matrices
$Y^{f}_{1,2}$, which govern the interactions between the Higgs fields and
fermions, are arbitrary $3\times 3$ matrices. In this state, fermions do not
yet represent physical eigenstates. This allows us the flexibility to choose
diagonal forms for the matrices $Y^{u}_{1}$, $Y^{d}_{2}$ and $Y^{\ell}_{2}$.
Specifically, we can set
$Y^{u}_{1}=\mathrm{diag}(y^{u}_{1},y^{u}_{2},y^{u}_{3})$ and
$Y^{d,\ell}_{2}=\mathrm{diag}(y^{d,\ell}_{1},y^{d,\ell}_{2},y^{d,\ell}_{3})$.

In our study, we focus on the 2HDM Type-III. This variant does not impose a
global symmetry on the Yukawa sector nor enforces alignment in flavor space.
Instead, we adopt the Cheng-Sher ansatz Cheng and Sher (1987); Diaz-Cruz  _et
al._ (2004), which posits a specific flavor symmetry in the Yukawa matrices.
Under this assumption, FCNC effects are proportional to the masses of the
fermions and dimensionless real parameters Hernandez-Sanchez  _et al._ (2013)
$\chi_{ij}^{f}$ ($\propto\sqrt{m_{i}m_{j}}/v\leavevmode\nobreak\
\chi_{ij}^{f}$), where $i,j=1-3$. After EWSB, the Yukawa Lagrangian is
expressed in terms of the mass eigenstates of the Higgs bosons. It can be
represented as follows:

$\displaystyle-{\cal L}^{III}_{Y}$
$\displaystyle=\sum_{f=u,d,\ell}\frac{m^{f}_{j}}{v}\times\left((\xi^{f}_{h})_{ij}\bar{f}_{Li}f_{Rj}h+(\xi^{f}_{H})_{ij}\bar{f}_{Li}f_{Rj}H-i(\xi^{f}_{A})_{ij}\bar{f}_{Li}f_{Rj}A\right)$
$\displaystyle+\frac{\sqrt{2}}{v}\sum_{k=1}^{3}\bar{u}_{i}\left[\left(m^{u}_{i}(\xi^{u*}_{A})_{ki}V_{kj}P_{L}+V_{ik}(\xi^{d}_{A})_{kj}m^{d}_{j}P_{R}\right)\right]d_{j}H^{+}+\frac{\sqrt{2}}{v}\bar{\nu}_{i}(\xi^{\ell}_{A})_{ij}m^{\ell}_{j}P_{R}\ell_{j}H^{+}+H.c.\,$
(3)

Here, $V_{kj}$ represents the Cabibbo-Kobayashi-Maskawa (CKM) matrix, while
the specific reduced Yukawa couplings are elaborated in Tab. 1, with
expressions defined in relation to the mixing angle $\alpha$, $\tan\beta$ and
the independent parameters $\chi_{ij}^{f}$.

---
$\phi$ | $(\xi^{u}_{\phi})_{ij}$ | $(\xi^{d}_{\phi})_{ij}$ | $(\xi^{\ell}_{\phi})_{ij}$
$h$ |  $\frac{c_{\alpha}}{s_{\beta}}\delta_{ij}-\frac{c_{\beta-\alpha}}{\sqrt{2}s_{\beta}}\sqrt{\frac{m^{u}_{i}}{m^{u}_{j}}}\chi^{u}_{ij}$ |  $-\frac{s_{\alpha}}{c_{\beta}}\delta_{ij}+\frac{c_{\beta-\alpha}}{\sqrt{2}c_{\beta}}\sqrt{\frac{m^{d}_{i}}{m^{d}_{j}}}\chi^{d}_{ij}$ |  $-\frac{s_{\alpha}}{c_{\beta}}\delta_{ij}+\frac{c_{\beta-\alpha}}{\sqrt{2}c_{\beta}}\sqrt{\frac{m^{\ell}_{i}}{m^{\ell}_{j}}}\chi^{\ell}_{ij}$
$H$ | $\frac{s_{\alpha}}{s_{\beta}}\delta_{ij}+\frac{s_{\beta-\alpha}}{\sqrt{2}s_{\beta}}\sqrt{\frac{m^{u}_{i}}{m^{u}_{j}}}\chi^{u}_{ij}$ | $\frac{c_{\alpha}}{c_{\beta}}\delta_{ij}-\frac{s_{\beta-\alpha}}{\sqrt{2}c_{\beta}}\sqrt{\frac{m^{d}_{i}}{m^{d}_{j}}}\chi^{d}_{ij}$ | $\frac{c_{\alpha}}{c_{\beta}}\delta_{ij}-\frac{s_{\beta-\alpha}}{\sqrt{2}c_{\beta}}\sqrt{\frac{m^{\ell}_{i}}{m^{\ell}_{j}}}\chi^{\ell}_{ij}$
$A$ | $\frac{1}{t_{\beta}}\delta_{ij}-\frac{1}{\sqrt{2}s_{\beta}}\sqrt{\frac{m^{u}_{i}}{m^{u}_{j}}}\chi^{u}_{ij}$ | $t_{\beta}\delta_{ij}-\frac{1}{\sqrt{2}c_{\beta}}\sqrt{\frac{m^{d}_{i}}{m^{d}_{j}}}\chi^{d}_{ij}$ | $t_{\beta}\delta_{ij}-\frac{1}{\sqrt{2}c_{\beta}}\sqrt{\frac{m^{\ell}_{i}}{m^{\ell}_{j}}}\chi^{\ell}_{ij}$
Table 1: Yukawa couplings of the neutral Higgs bosons $h$, $H$, and $A$ to the
quarks and leptons in the 2HDM Type-III.

##  III The Excesses in the $\gamma\gamma$, $\tau\tau$ and $b\bar{b}$ Channels

In this section, we investigate whether the 2HDM Type-III can describe
consistently the excesses observed by both LEP and the LHC in the 94–100 GeV
mass windows in the $b\bar{b}$ as well as $\gamma\gamma$ and $\tau\tau$
channels, respectively. Starting with the LHC excesses, the parametrization
used to access possible BSM signals invokes the so-called ‘signal strength’
(defined in terms of ratios of production cross sections $\sigma$ and decay
Branching Ratios ${\cal BR}$s), which, for these excesses, are as follows:

$\displaystyle\mu_{\mathrm{\tau\tau}}$ $\displaystyle=$
$\displaystyle\frac{\sigma_{\rm 2HDM}(gg\to\phi)}{\sigma_{\rm SM}(gg\to h_{\rm
SM})}\times\frac{{\cal BR}_{\rm 2HDM}(\phi\to\tau\tau)}{{\cal BR}_{\rm
SM}(h_{\rm SM}\to\tau\tau)},$ $\displaystyle\mu_{\mathrm{\gamma\gamma}}$
$\displaystyle=$ $\displaystyle\frac{\sigma_{\rm 2HDM}(gg\to\phi)}{\sigma_{\rm
SM}(gg\to h_{\rm SM})}\times\frac{{\cal BR}_{\rm
2HDM}(\phi\to\gamma\gamma)}{{\cal BR}_{\rm SM}(h_{\rm SM}\to\gamma\gamma)}.$
(4)

The experimental measurements for these two signal strengths are expressed as
Biekötter  _et al._ (2024, 2022b, 2023b):

$\displaystyle\mu_{\gamma\gamma}^{\mathrm{exp}}$ $\displaystyle=$
$\displaystyle\mu_{\gamma\gamma}^{\mathrm{ATLAS+CMS}}=0.24^{+0.09}_{-0.08},$
(5) $\displaystyle\mu_{\tau\tau}^{\mathrm{exp}}$ $\displaystyle=$
$\displaystyle 1.2\pm 0.5,$ (6)

where $h_{\rm SM}$ corresponds to a SM-like Higgs boson with a mass of $95$
GeV.

In our analysis, we have combined the di-photon measurements from the ATLAS
and CMS experiments, denoted as $\mu_{\gamma\gamma}^{\mathrm{ATLAS}}$ and
$\mu_{\gamma\gamma}^{\mathrm{CMS}}$, respectively. The ATLAS measurement
yields a central value of $0.18{\pm 0.1}$ Biekötter  _et al._ (2024) while the
CMS measurement yields a central value of $0.33^{+0.19}_{-0.12}$Biekötter  _et
al._ (2023b). By doing so, we aimed to leverage the strengths of both
experiments and improve the precision of our analysis. The combined
measurement, denoted as $\mu_{\gamma\gamma}^{\mathrm{ATLAS+CMS}}$, is
determined by taking the average of the central values without assuming any
correlation between them. To evaluate the combined uncertainty we sum ATLAS
and CMS uncertainties in quadrature.

The signal strength for the $b\bar{b}$ channel from LEP data is defined as:

$\displaystyle\mu_{{b\bar{b}}}$ $\displaystyle=$
$\displaystyle\frac{\sigma_{\mathrm{2HDM}}(e^{+}e^{-}\to
Zh)}{\sigma_{\mathrm{SM}}(e^{+}e^{-}\to
Zh_{\mathrm{SM}})}\times\frac{\mathcal{BR}_{\mathrm{2HDM}}(h\to
b\bar{b})}{\mathcal{BR}_{\mathrm{SM}}(h_{\mathrm{SM}}\to b\bar{b})}.$ (7)

Here, the expected experimental value of $\mu_{b\bar{b}}$ is reported as
Barate  _et al._ (2003):

$\displaystyle\mu_{b\bar{b}}^{\mathrm{exp}}=0.117\pm 0.057.$ (8)

To determine whether a simultaneous fit to the observed excesses is possible,
a $\chi^{2}$ analysis is performed using the measured central values
$\mu^{\mathrm{exp}}$ and the 1$\sigma$ uncertainties
$\Delta\mu^{\mathrm{exp}}$ of the signal rates related to the three excesses
as defined in Eqs. (4) and (7). The contribution to the $\chi^{2}$ value for
each channel is calculated using the formula:

$\displaystyle\chi^{2}_{\gamma\gamma,\tau\tau(+b\bar{b})}=\frac{\left(\mu_{\gamma\gamma,\tau\tau(+b\bar{b})}-\mu_{\gamma\gamma,\tau\tau(+b\bar{b})}^{\mathrm{exp}}\right)^{2}}{\left(\Delta\mu^{\mathrm{exp}}_{\gamma\gamma,\tau\tau(+b\bar{b})}\right)^{2}}.$
(9)

So, the resulting $\chi^{2}$, which we will use to judge whether the points
from the model describe the excesses, reads as:

$\chi^{2}_{\gamma\gamma+\tau\tau(+b\bar{b})}=\chi^{2}_{\gamma\gamma}+\chi^{2}_{\tau\tau}\leavevmode\nobreak\
(+\chi^{2}_{b\bar{b}}),$ (10)

where the inclusion of the $b\bar{b}$ data depends on the solution that we
will attempt. Specifically, having tested the $h$-only explanation in Refs.
Benbrik  _et al._ (2022a, b); Belyaev  _et al._ (2023), here, we are concerned
with the $A$-only one (which will then necessarily not capture the LEP data as
there is no $AZZ$ coupling) as well as with the superposition of the two
(which can potentially capture both LHC and LEP anomalies). However, before
using the above measure to test the viability of 2HDM Type-III against the
anomalous LHC data, we describe the aforementioned theoretical and
experimental constraints adopted here.

##  IV Theoretical and experimental constraints

In our work, we employ a diverse set of theoretical and experimental
constraints that must be met to establish a viable model.

  * •

Unitarity The scattering processes involving (pseudo)scalar-(pseudo)scalar,
gauge-gauge and/or (pseudo)scalar-gauge initial and/or final states must
satisfy unitarity constraints. The eigenvalues $e_{i}$ of the tree-level
2-to-2 body scattering matrix should meet the following criteria:
$|e_{i}|<8\pi$ Kanemura  _et al._ (1993); Akeroyd  _et al._ (2000).

  * •

Perturbativity Adherence to perturbativity constraints imposes an upper limit
on the quartic couplings of the Higgs potential: $|\lambda_{i}|<8\pi$ Branco
_et al._ (2012).

  * •

Vacuum Stability The scalar potential must be positive and bounded from below
in any direction of the fields $\Phi_{i}$ to ensure vacuum stability. This
requires that $\lambda_{1}>0$, $\lambda_{2}>0$,
$\lambda_{3}>-\sqrt{\lambda_{1}\lambda_{2}}$, and
$\lambda_{3}+\lambda_{4}-|\lambda_{5}|>-\sqrt{\lambda_{1}\lambda_{2}}$ Barroso
_et al._ (2013); Deshpande and Ma (1978).

$\displaystyle\lambda_{1}>0,\quad\lambda_{2}>0,\quad\lambda_{3}>-\sqrt{\lambda_{1}\lambda_{2}},$
$\displaystyle\lambda_{3}+\lambda_{4}-|\lambda_{5}|>-\sqrt{\lambda_{1}\lambda_{2}}.$
(11)

  * •

SM-like Higgs Boson Discovery The compatibility of the SM-like scalar with the
observed Higgs boson is tested. The relevant quantities calculated with
HiggsSignals-3 Bechtle  _et al._ (2020, 2021) via HiggsTools Bahl  _et al._
(2023) must satisfy the measurements at 95% confidence level (C.L.).

  * •

BSM Higgs Boson Exclusions Exclusion limits at 95% C.L. from direct searches
for Higgs bosons at LEP, Tevatron, and LHC are taken into account using
HiggsBounds-6 Bechtle  _et al._ (2010, 2011, 2014, 2015) via HiggsTools.

  * •

$B$-physics observables The constraints from $B$-physics observables are
implemented using the code SuperIso_v4.1 Mahmoudi (2009) as described in Ref.
Benbrik  _et al._ (2022a). The relevant experimental measurements used are as
follows:

    1. 1.

${\cal BR}(\overline{B}\to
X_{s}\gamma)|_{E_{\gamma}<1.6\mathrm{\leavevmode\nobreak\ GeV}}$
$\left(3.32\pm 0.15\right)\times 10^{-4}$ Amhis  _et al._ (2017),

    2. 2.

${\cal BR}(B^{+}\to\tau^{+}\nu_{\tau})$ $\left(1.06\pm 0.19\right)\times
10^{-4}$ Amhis  _et al._ (2017),

    3. 3.

${\cal BR}(D_{s}\to\tau\nu_{\tau})$ $\left(5.51\pm 0.18\right)\times 10^{-2}$
Amhis  _et al._ (2017),

    4. 4.

${\cal BR}(B_{s}\to\mu^{+}\mu^{-})$ (LHCb)
$\left(3.09^{+0.46}_{-0.43}\right)\times 10^{-9}$ Aaij  _et al._ (2022a, b),

    5. 5.

${\cal BR}(B_{s}\to\mu^{+}\mu^{-})$ (CMS)
$\left(3.83^{+0.38}_{-0.36}\right)\times 10^{-9}$ Tumasyan  _et al._ (2023),

    6. 6.

${\cal BR}(B^{0}\to\mu^{+}\mu^{-})$ (LHCb)
$\left(1.2^{+0.8}_{-0.7}\right)\times 10^{-10}$ Aaij  _et al._ (2022a, b),

    7. 7.

${\cal BR}(B^{0}\to\mu^{+}\mu^{-})$ (CMS)
$\left(0.37^{+0.75}_{-0.67}\right)\times 10^{-10}$ Tumasyan  _et al._ (2023).

##  V Explanation of the Excesses

In this section, we present our numerical analysis of the 2HDM Type-III
parameter space. For the 2HDM Type-III spectrum generation, we have employed
2HDMC Eriksson  _et al._ (2010), which considers the theoretical constraints
discussed in the previous section, along with the Electro-Weak Precision
Observables (EWPOs). Subsequently, we validate our results by comparing them
to Higgs data, utilizing HiggsTools Bahl  _et al._ (2023), which includes the
most recent versions of both HiggsBounds and HiggsSignals. In accordance with
the above discussions, we consider the scenario where the heavier CP-even
Higgs boson $H$ is the SM-like Higgs particle $H_{\rm SM}$ discovered at the
LHC with $m_{H_{\rm SM}}\approx$ 125 GeV. In this scenario the CP-odd Higgs,
$A$, is the source of the observed LHC excess in $\gamma\gamma$ and $\tau\tau$
channels around 95 GeV, which we previously labeled as $h_{\rm SM}$. To
explore this scenario, we conducted a systematic random scan across the
parameter ranges specified in Tab. 2.

---
Parameters |  |  Scanned ranges
$m_{h}$ |  |  [$94$, $97$]
$m_{H}$ |  |  129.05
$m_{A}$ |  |  [$94$, $97$]
$m_{H^{\pm}}$ |  |  [$160$, $300$]
$\tan\beta$ |  |  [$1$, $10$]
$s_{\beta-\alpha}$ |  |  [-$0.5$, $0$]
$\chi_{ij}^{f,\ell}$ |  |  [$-3$, $3$]
Table 2: Scan ranges of the 2HDM Type-III input parameters. Masses are given
in GeV.

###  V.1 The $A$ Solution

Here, we investigate parameter spaces that satisfy the condition
$\chi^{2}_{125}\leq 189.4$, corresponding to a 95% C.L. for 159 degrees of
freedom, where $\chi^{2}_{125}$ corresponds to the $\chi^{2}$ evaluated by
HiggsSignals for the 125 GeV Higgs signal strength measurements. Subsequently,
we examine 2-dimensional (2D) planes of the signal strength parameters:
$(\mu_{\gamma\gamma}-\mu_{\tau\tau})$.

Figure 1: Di-photon signal rate $\mu(A)_{\gamma\gamma}$ versus di-tau signal
rate $\mu(A)_{\tau\tau}$ for a 95 GeV pseudo-scalar within the 2HDM Type-III.
The bands represent the experimentally observed signal rates along with their
1$\sigma$ uncertainty intervals. The minimum value of
$\chi^{2}_{\gamma\gamma+\tau\tau}$ is marked by a green star.

In Fig. 1, we present the results for $\chi^{2}_{\gamma\gamma+\tau\tau}$ in
the form of a color map projected onto the
($\mu_{\tau\tau}-\mu_{\gamma\gamma}$) plane, representing the signal strength
parameters. The dashed ellipse delineates the regions consistent with the
excess observed at the 1$\sigma$ C.L., as described by the equation
$\chi^{2}_{\gamma\gamma}+\chi^{2}_{\tau\tau}=2.30$. The value of
$\chi^{2}_{\gamma\gamma+\tau\tau}$ is represented by the vertical color map.
The gray(green) dashed line represents the central value for
$\mu_{\gamma\gamma}$($\mu_{\tau\tau}$) the gray(green) band showing the
1$\sigma$ range. The green star indicates the position of
$\chi^{2}_{\gamma\gamma+\tau\tau,\mathrm{min}}$, which is the minimum value of
$\chi^{2}_{\gamma\gamma+\tau\tau}$, noted at 0.12. Furthermore, numerous
points surrounding $\chi^{2}_{\gamma\gamma+\tau\tau,\mathrm{min}}$ are
depicted in dark orange, demonstrating the capability of the 2HDM Type-III
model with a CP-odd resonance to perfectly explain the observed excess across
both channels simultaneously, as well as individually, at the 1$\sigma$ level.

Fig. 2 depicts the values of the branching ratios, $\mathcal{BR}(A)$, for our
best-fit point through various possible decay channels.

In Figs. 3 and 4, we directly compare our allowed parameter points with the
experimental data by superimposing them onto the CMS 13 TeV low-mass
$\tau\tau$ CMS (2022b) and $\gamma\gamma$ CMS (2023) analysis data,
respectively. The light green colour represents the parameter points that fit
the excesses within a two-dimensional confidence level (C.L.) of 1$\sigma$,
while the points fitting the excesses at 2$\sigma$ or more are shown in
orange. It can be clearly observed from the plots that our parameter points
are well-suited to satisfy the LHC excesses.

Figure 2: $\mathcal{BR}s$ of the CP-odd Higgs boson $A$ at our best fit point.

Figure 3: Allowed points, following the discussed theoretical and experimental
constraints, superimposed onto the results of the CMS 13 TeV low-mass
$\gamma\gamma$ CMS (2023) analysis. (Notably, the plot further includes the
depiction of the ATLAS expected and observed limits from Arcangeletti (2023),
showcased in blue.) The light green colour represents the parameter points
that fit the excesses within a three-dimensional C.L. of 1$\sigma$, whereas
the points that fit the excesses at 2$\sigma$ or more are shown in orange.

Figure 4: The same points as in Fig. 3, superimposed onto the results of the
CMS 13 TeV low-mass $\tau\tau$ CMS (2023) analysis.

###  V.2 The $h$ \+ $A$ Solution

In this section, we conduct a combined analysis of the CP-even ($h$)111It
should be noted that the CMS CMS (2022a) limit for the production of a Higgs
boson in association with either a top-quark pair or a $Z$ boson, subsequently
decaying into a tau pair, is taken into account in our analysis for the scalar
$h$ resonance. and CP-odd ($A$) resonances. We now explore also the $b\bar{b}$
excess, potentially attributable to the $h$ resonance, by incorporating the
$\chi^{2}_{b\bar{b}}$ into our total $\chi^{2}$ analysis.

Then, we calculate the combined contributions to the signal strengths from
both resonances for the $\gamma\gamma$ and $\tau\tau$ channels, as follows:

$\displaystyle\mu_{\gamma\gamma}(h+A)$ $\displaystyle=$
$\displaystyle\mu_{\gamma\gamma}(h)+\mu_{\gamma\gamma}(A),$
$\displaystyle\mu_{\tau\tau}(h+A)$ $\displaystyle=$
$\displaystyle\mu_{\tau\tau}(h)+\mu_{\tau\tau}(A),$ (12)

since there is no interference between the $h$ and $A$ states, given that in
our 2HDM Type-III we have assumed CP conservation.

Figure 5: Di-photon signal rate $\mu(h+A)_{\gamma\gamma}$ versus di-tau signal
rate $\mu(h+A)_{\tau\tau}$ for a 95 GeV pseudo-scalar within the 2HDM Type-
III. The bands represent the experimentally observed signal rates along with
their 1$\sigma$ uncertainty intervals. The minimum value of
$\chi^{2}_{\gamma\gamma+\tau\tau+b\bar{b}}$ is marked by a magenta star.

Integrating data from both resonances, $h$ and $A$, we demonstrate the 2HDM
Type-III ability to account for observed excesses through their superposition,
achieving a 1$\sigma$ C.L. This is clearly illustrated in Fig. 5, which shows
the combined $\chi^{2}_{\gamma\gamma+\tau\tau+b\bar{b}}$ in the signal
strengths $(\mu_{\gamma\gamma},\mu_{\tau\tau})$ plane. Points that explain the
three excesses at 1$\sigma$ ($\chi^{2}\leq 3.53$) are marked in light green.
Notably, the minimum value of $\chi^{2}_{\gamma\gamma+\tau\tau+b\bar{b}}$ is
2.35, which is highlighted by a magenta star.

Additional insights are provided in Fig. 6, displaying allowed parameter
points in the $(\chi^{u}_{33}-\chi^{\ell}_{33})$,
$(\chi^{u}_{33}-\chi^{d}_{33})$, and $(\chi^{d}_{33}-\chi^{\ell}_{33})$
planes, highlighted in orange. Areas meeting the criterion
$\chi^{2}_{\gamma\gamma+\tau\tau+b\bar{b}}\leq 3.53$ (1$\sigma$ C.L.) are
indicated in light green. The analysis confirms that to accommodate the three
excesses at the 1$\sigma$ level, the necessary parameter intervals are:
$\chi^{u}_{33}\in[-0.71,0.53]$, $\chi^{d}_{33}\in[0.75,1.57]$, and
$\chi^{\ell}_{33}\in[0.52,1.70]$.

Figure 6: Allowed points, following the discussed theoretical and experimental
constraints (orange), while the area indicating
$\chi^{2}_{\gamma\gamma+\tau\tau+b\bar{b}}\leq 3.53$ (1$\sigma$ C.L) is
illustrated in light green. The magenta star marks the minimum of
$\chi^{2}_{\gamma\gamma+\tau\tau+b\bar{b}}$.

Figure 7: Correlation between the normalized couplings $|c_{h_{125}VV}|$ and
$|c_{h_{125}t\bar{t}}|$, with colours corresponding to those in Fig. 6, are
illustrated. The green solid (dashed) ellipses indicate the projected
uncertainties at the HL-LHC Cepeda  _et al._ (2019) at $1\sigma$($2\sigma$).

Fig. 7 illustrates the correlation between the normalized couplings of the
$\approx 125$ GeV Higgs, using the same color scheme as detailed in Figure 5.
The plot features green solid and dashed lines, representing the projected
experimental precision for these couplings at the High Luminosity LHC (HL-LHC)
at the 1$\sigma$ and 2$\sigma$ levels, respectively, based on an expected
integrated luminosity of 3000 fb-1. The center of these projections,
corresponding to the SM values, is marked by a red diamond. Our analysis
reveals that explaining the observed excesses in the $b\overline{b}$,
$\tau\tau$, and $\gamma\gamma$ channels requires an enhancement of the
$h_{125}t\overline{t}$ couplings, which deviate by approximately 12% from the
SM predictions for $t\bar{t}$, as discussed in Belyaev  _et al._ (2023).
Additionally, it is evident that each point that successfully accounts for the
three excesses consistently lies outside the 1$\sigma$ ellipse, though some
points may fit within the 2$\sigma$ level. Given these deviations, the
expected precision of HL-LHC experiments will enable a clear differentiation
between the SM-like properties of $h_{125}$ and those of the $H$ boson from
the 2HDM Type-III model within the parameter ranges consistent with these
observed excesses.

In summary, this section provides a comprehensive overview of our best fit
points, as presented in Tab. 3. The first point corresponds to the best fit
for the CP-odd state, explaining the LHC excesses in the $\gamma\gamma$ and
$\tau\tau$ channels. The second point, indicated by a magenta star, represents
the best fit point for the superposition solution of both CP-even and CP-odd
resonances, addressing the LHC excesses along with the LEP excess in the
$b\bar{b}$ channel.

---
Parameters | $m_{h}$ | $m_{H}$ | $m_{A}$ | $m_{H^{\pm}}$ | $t_{\beta}$ | $s_{\beta-\alpha}$ | $\chi^{u}_{11}$ | $\chi^{u}_{22}$ | $\chi^{u}_{33}$ | $\chi^{d}_{11}$ | $\chi^{d}_{22}$ | $\chi^{d}_{33}$ | $\chi^{\ell}_{11}$ | $\chi^{\ell}_{22}$ | $\chi^{\ell}_{33}$
★☆ | 94.62 | 125.09 | 94.96 | 162.95 | 1.82 | -0.16 | 1.55 | 0.33 | -0.04 | -0.10 | 1.56 | 1.14 | 0.67 | -0.44 | 1.62
★☆ | 95.61 | 125.09 | 94.36 | 162.92 | 2.93 | -0.19 | 0.56 | 0.36 | -0.31 | -0.01 | -0.16 | 1.24 | 1.27 | 0.44 | 1.08

Signal strengths |  |  | $\mu_{\gamma\gamma}(h)$ |  | $\mu_{\gamma\gamma}(A)$ |  | $\mu_{\gamma\gamma}(h+A)$ |  | $\mu_{\tau\tau}(h)$ |  | $\mu_{\tau\tau}(A)$ |  | $\mu_{\tau\tau}(h+A)$ |  | $\mu_{b\bar{b}}(h)$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
★☆ |  |  | 0.16 |  | 0.24 |  | 0.40 |  | 0.22 |  | 1.02 |  | 1.25 |  | 0.02
★☆ |  |  | 0.08 |  | 0.18 |  | 0.26 |  | 0.38 |  | 0.89 |  | 1.27 |  | 0.03

Effective couplings |  $c_{h_{95}t\bar{t}}$ |  $c_{h_{95}b\bar{b}}$ |  $c_{h_{95}VV}$ |  $c_{h_{125}t\bar{t}}$ |  $c_{h_{125}b\bar{b}}$ |  $c_{h_{125}VV}$ |  $c_{At\bar{t}}$ |  $c_{Ab\bar{b}}$
---|---|---|---|---|---|---|---|---
★☆ |  0.41 |  -0.31 |  -0.16 |  1.08 |  0.96 |  0.99 |  0.14 |  0.58
★☆ |  0.37 |  -0.40 |  -0.19 |  1.09 |  0.94 |  0.98 |  0.21 |  0.57

Branching ratios in % |  |  |  |  |  |
---|---|---|---|---|---|---
$\bm{h}$ |  $gg$ |  $b\bar{b}$ |  $\tau^{+}\tau^{-}$ |  $\gamma\gamma$ |  $W^{+}W^{-}$ |  $ZZ$
★☆ |  12.29 |  74.32 |  10.79 |  0.13 |  0.10 |  0.01
★☆ |  5.97 |  68.26 |  22.85 |  0.08 |  0.09 |  0.01
$\bm{H}$ |  $gg$ |  $b\bar{b}$ |  $\tau^{+}\tau^{-}$ |  $\gamma\gamma$ |  $W^{+}W^{-}$ |  $ZZ$
★☆ |  8.78 |  58.66 |  6.98 |  0.18 |  19.27 |  2.42
★☆ |  9.46 |  59.23 |  4.83 |  0.19 |  20.13 |  2.52
$\boldsymbol{A}$ |  $gg$ |  $b\bar{b}$ |  $\tau^{+}\tau^{-}$ |  $\gamma\gamma$ |  |
★☆ |  51.80 |  18.97 |  23.72 |  0.10 |  |
★☆ |  41.23 |  33.14 |  21.66 |  0.07 |  |
$\boldsymbol{H^{\pm}}$ |  $tb$ |  $\tau^{\pm}\nu$ |  $W^{\pm}h$ |  $W^{\pm}H$ |  $W^{\pm}A$ |
★☆ |  23.92 |  2.27 |  36.69 |  0.05 |  36.58 |
★☆ |  23.44 |  2.52 |  34.05 |  0.07 |  39.48 |

Table 3: The full description of our best fit points.

##  VI Conclusion

Extensive data samples collected by LHC experiments have facilitated detailed
analyses of the reported 95 GeV excesses following their initial observations.
A rigorous examination of this data, coupled with in-depth simulations and
advanced computational techniques, has been conducted. In this context, we
have introduced the 2HDM Type-III with a specific Yukawa texture as a
theoretical framework for potentially the observed $\gamma\gamma$, $\tau\tau$
and $b\bar{b}$ anomalies. This model focuses on a Higgs boson with a mass of
approximately 95 GeV, produced via gluon-gluon fusion at the 13 TeV LHC and
decaying into $\tau\tau$ and $\gamma\gamma$, as well as being produced through
Higgs-strahlung at LEP and decaying into $b\bar{b}$.

Assuming that the heavy CP-even $H$ state in our model is the 125 GeV Higgs
boson discovered at the LHC, we have explored parameter spaces where the 95
GeV CP-odd state, $A$, comprehensively explains the LHC excesses at a
1$\sigma$ level, consistent with current theoretical and experimental
constraints. We have also demonstrated that the superposition of the light CP-
even state $h$ and the CP-odd state $A$ can account for the anomalies observed
at both the LHC and LEP, through a $\chi^{2}$ analysis at the 1$\sigma$ level.

Further analysis confirms that effectively addressing these discrepancies
requires an enhancement of the ${t\bar{t}}H$ coupling, deviating from SM
predictions. With upcoming advancements at the HL-LHC, precise measurements
are expected to clearly differentiate between the SM-like properties of the
$H$ state and the predictions of the 2HDM Type-III. This differentiation is
critical for data points showing significant deviations, particularly with the
enhanced ${t\bar{t}}H$ coupling parameter space. Such measurements will be
crucial in conclusively confirming or refuting our model. We have provided
detailed descriptions of our best fit points to aid further phenomenological
studies.

##  VII Acknowledgments

SM is supported in part through the NExT Institute and the STFC Consolidated
Grant ST/L000296/1. We thank A. Belyaev, M. Chakraborti and S. Semlali for
useful discussions.

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