# Observational bounds on a possible electron-to-proton mass ratio variation
and constraints in the lepton-specific 2HDM

R. G. Albuquerque1<EMAIL_ADDRESS>R. F. L. Holanda2,3,4
<EMAIL_ADDRESS>I. E. T. R. Mendonça2,5<EMAIL_ADDRESS>P.
S. Rodrigues da Silva1<EMAIL_ADDRESS>1Departamento de Física,
Universidade Federal da Paraíba, Caixa Postal 5008, 58051-970, João Pessoa,
PB, Brasil  2Departamento de Física, Universidade Federal do Rio Grande do
Norte,Natal - Rio Grande do Norte, 59072-970, Brasil  3Departamento de Física,
Universidade Federal de Campina Grande, 58429-900, Campina Grande - PB, Brasil
4Departamento de Física, Universidade Federal de Sergipe, 49100-000, Aracaju -
SE, Brazil  5Biofy.tech IA Department, 38407-180, Uberlândia-MG Brazil

###### Abstract

In this work, we test a possible redshift variation of the electron-to-proton
mass ratio, $\mu=m_{e}/m_{p}$, directly from galaxy cluster gas mass fraction
measurements and type Ia Supernovae observations. Our analysis is completely
independent of any cosmological model. Our result reveals no variation of
$\mu$ within 1 $\sigma$ confidence level. From the point of view of Particle
Physics, we can use the precision on these results to constrain the parameter
space of models beyond the Standard Model of electroweak interactions. We
exemplify this by focusing in a specific Two Higgs Doublet model (2HDM), where
the second scalar doublet couples exclusively to leptons. An important
parameter in the model concerns the ratio between its vacuum expectation
values, defined by $\tan\beta$. In our approach we can constrain the inverse
parameter (cot$\beta$) to an optimal value, (tan$\beta)^{-1}=$ 0.02127 $\pm$
0.0029, with the largest vacuum expectation value for 2HDM, $v_{2}$, estimated
at around 240.033 $\pm$ 0.21 GeV. Also, by taking into account the
$(g-2)_{\mu}$ discrepancy found between theory and experiment, we can reduce
the validity region for this model and establish bounds on the scalar masses,
in the light of our findings from galaxy clusters data for $\mu$. This study
contributes valuable insights to the understanding of Particle Physics and
Astrophysics interface, establishing a new interplay between data from large
scale structure of the Universe and subatomic Physics.

###### pacs:

98.62.Ra, 98.65.-r, 98.80.-k, 12.60.Fr, 14.80.Ec

##  I Introduction

Fundamental physical theories rely on certain parameters known as fundamental
constants, for instance the fine structure constant, gauge and Yukawa
couplings, and so on. Their values cannot be predicted from existing knowledge
and must be determined through experimentation. We assume these constants are
mathematical constructs and to be unchanging across both time and space,
according to the simpler successful physical theories. Dirac DIRAC (1937) was
among the first to question whether these constants are merely mathematical
constructs or if they hold deeper significance within a broader cosmological
framework. Current laboratory experiments have ruled out significant
variations in these constants over laboratory Dzuba et al. (1999), geological
Fujii et al. (2000); Flambaum and Tedesco (2006), and solar system time scales
Olive et al. (2004). Astrophysical methods, specifically, the spectroscopy of
quasar absorption lines which involve measuring transition wavelengths at high
redshifts over vast time intervals, probe a much wider span of time Fiorenzano
et al. (2004); Tzanavaris et al. (2005); Molaro et al. (2013); Murphy et al.
(2016).

In recent years, observational tests seeking possible variations in
fundamental constants have grown in the literature (see Uzan (2003); Barrow
(2005); Martins (2017); Bambi (2022) for a review). In this scenario, two
dimensionless constants, the fine structure constant, ($\alpha\equiv
e^{2}/4\pi\epsilon_{0}\hslash c$), and the electron-to-proton mass ratio,
($\mu\equiv m_{e}/m_{p}$), hold particular importance as these constants are
not only related to each other, but their possible variation is also linked to
new physics such as: models of extended gravityWei et al. (2011); Harko et al.
(2013); Hees et al. (2014), string theory Damour and Polyakov (1994); Martins
et al. (2015), and more recently, dark matter models Davoudiasl and Giardino
(2019); Brzeminski et al. (2021). Therefore, constraining the temporal and
spatial change of these parameters could profoundly impact both cosmology and
fundamental physics. The use of physics of large scale structure of the
universe, i.e., the physics of galaxy clusters, to constrain variations on
$\alpha$ is not new Galli (2013), nevertheless, it has been a field of intense
investigation recently de Martino et al. (2016); De Martino et al. (2016);
Colaço et al. (2019); Holanda et al. (2016); Colaço et al. (2019); Liu et al.
(2021). On the other hand, as far as we are aware, there have been no report
on the issue of $\mu$ variation using galaxy clusters until now, a path we are
going to follow in this work.

Concerning the physics of small scale, the Standard Model of Strong and
Electroweak interactions (SM) has been thoroughly confirmed along the past
decades by several collider data measurements, culminating with the last
important ingredient predicted by the model and discovered at the large Hadron
Collider (LHC), namely, the Higgs boson Aad et al. (2012); Chatrchyan et al.
(2012); Aad et al. (2022). It was an important step to definitely establish
the SM as a fiducial paradigm for describing the visible matter content in the
Universe and its interactions in a detailed and precise way. However, it is
not the most complete theory we can envisage. That is because we know there
are several issues, like the evidence for Cold Dark Matter Bertone and Hooper
(2018), neutrino’s mass and oscillation de Salas et al. (2021); Capozzi et al.
(2017), the experimental deviation from the theoretical prediction of the muon
anomalous magnetic moment Cotrozzi and Sorbara (2022); Keshavarzi et al.
(2022), etc, that cannot be addressed by the model without extending its
matter or interaction content (gauge group). In this sense, several models
have been proposed in literature to go beyond SM, but a class of such models
is being particularly of interest for phenomenologists as the scalar sector of
the SM is being scrutinized, the so called Two Higgs Doublet Models (2HDM)
Branco et al. (2012); Wang et al. (2022). An extra doublet scalar is the
simplest part of supersymmetric theories 111Supersymmetric theories (for a
review, see Ref. Martin (1998); Luty (2005); Mummidi et al. (2023) and
references therein) are thoroughly investigated for decades since it is a
fundamental piece for the mathematical consistency of string theory, although
no sign of it has shown up in accelerators till now. content that can be
theoretically studied without, necessarily, addressing specific details of the
intricate spectrum of such theories. Besides, 2HDM can add sources of CP
violation in the scalar sector Lee (1973), allowing for for Baryogenesis in
this class of models Bochkarev et al. (1990); Turok and Zadrozny (1991), and
the extra neutral scalar can offer a richer vacuum structure that could lead
to a first order phase transition Cline and Lemieux (1997); Fromme et al.
(2006); Cline et al. (2011); Dorsch et al. (2013), also required for
Baryogenesis. Finally, it is well known that additional scalar doublets do not
alter the SM prediction for the observable $\rho$-parameter222The
$\rho$-parameter is defined (at tree level) as
$\rho_{tree}=\frac{M_{W}^{2}}{M_{Z}^{2}\,\cos{\theta_{W}}^{2}}$, where $M_{W}$
and $M_{Z}$ are the W and Z weak gauge bosons’ masses, respectively, and
$\theta_{W}$ is the weak mixing angle. Its theoretical value in the SM is
$\rho_{tree}=1$. .

Our main goal in this work is to impose constraints on a possible redshift
variation of electron to proton mass ratio, $\mu$, by using observational
data, more precisely, galaxy cluster gas mass fraction data and SNe Ia
luminosity distance measurements, and put constraints on particle physics
models beyond the SM. We do that for a specific version of 2HDM, where leptons
and quarks interact with different scalar doublets. Since the constraint on
$\mu$ from galaxy clusters is directly related to the ratio between the vacuum
expectation values (vev) of the model 333As we will see, in our approach we
restrict to the variation of electron mass only in the electron-to-proton mass
ratio, and that alone allows us to infer that the ratio of vevs of 2HDM,
namely, $\tan{\beta}$ is the parameter to be constrained by the $\mu$
variation., we can use the constraint on $\mu$, to get bounds on the
parameters of the 2HDM. In order to impose tighter limits in the parameter
space, including the scalar masses of the model, we also perform the
computation of the muon anomalous magnetic moment, restricting to the region
of parameters that can solve the $4.2\sigma$ discrepancy between experiment
and theory that presently seems a promising indication for physics beyond the
SM Abi et al. (2021).

The manuscript is organized as follows: we first present the methodology by
which we derive the relation between the electron-to-proton mass ratio and the
galaxy cluster gas mass fraction in Section II. Then, in Section III, we
briefly explain the cosmological data sample used in our analysis. We
describe, in Section IV, the theoretical structure of the 2HDM that we base
our investigation on. In Section V we describe the details of our analysis and
the results we get for the $\mu$ variation. After that, in Section VI, we
present constraints on $\tan{\beta}$, as well as the masses of the heavier
Higgs in the 2HDM, including the computation of the muon anomalous magnetic
moment contribution to solve the theoretical-experimental discrepancy. Finally
we draw our conclusions and share some perspectives in Section VII.

##  II Electron-to-proton mass ratio and galaxy cluster gas mass fraction

Galaxy clusters represent the largest virialized structures in the universe
Allen et al. (2011). Predominantly composed of dark matter and intergalactic
gas in a plasma state known as the Intra-Cluster Medium (ICM), which emits a
continuous X-ray spectrum. The spectrum is dominated by thermal bremsstrahlung
for temperatures from $2-10\,\text{keV}$Sarazin (1988). The ratio of ICM mass
to the total cluster mass, i.e., the gas mass fraction $f_{gas}$, holds
particular significance in cosmology, since it allows to constrain
cosmological parameters Allen et al. (2008); Ettori, S. et al. (2009); Mantz
et al. (2014). Moreover, it enables various tests on the fundamental physics,
including, for instance, the exploration of fundamental constant variations
Colaço et al. (2019); Mendonça et al. (2021a) and violations of Einstein’s
equivalence principle Mendonça et al. (2021b); Holanda et al. (2017a, b).
Consider the gas mass, $M_{gas}$, within a radius $R$, obtained through X-ray
spectrum measurements Sarazin (1988); Sasaki (1996):

$M_{gas}(<R)=\left(\frac{3\pi\hbar
m_{e}c^{2}}{2(1+X)e^{6}}\right)^{1/2}\left(\frac{3m_{e}c^{2}}{2\pi
k_{B}T_{e}}\right)^{1/4}m_{H}\\\
\times\frac{1}{[\overline{g_{B}}(T_{e})]^{1/2}}{r_{c}}^{3/2}\left[\frac{I_{M}(R/r_{c},\beta)}{I_{L}^{1/2}(R/r_{c},\beta)}\right][L_{X}(<R)]^{1/2}.$
(1)

Here, $X$ is the fraction of Hydrogen mass in the gas, $m_{e}$ and $m_{H}$ are
the masses of the electron and Hydrogen, respectively. $T_{e}$ is the gas
temperature, $\overline{g}_{B}$ is the Gaunt factor, multiplicative correction
to take quantum effects in consideration for the calculations of absorption or
emission of radiation Dopita and Sutherland (2003), and $r_{c}$ is the core
radius. $I_{M}$ and $I_{L}$ are defined as:

$I_{M}(y,\beta)\equiv\int_{0}^{y}(1+x^{2})^{-3\beta/2}x^{2}dx,$
$I_{L}(y,\beta)\equiv\int_{0}^{y}(1+x^{2})^{-3\beta}x^{2}dx,$

obtained using a spherical $\beta$ model, and the bolometric luminosity
$L_{X}(<R)$ is given by:

$L_{X}(<R)=4\pi[D_{L}(z,\Omega_{0},H_{0})]^{2}f_{X}(<\theta)\,\,\,,$

where $f_{X}(<\theta)$ is the total bolometric flux within the outer angular
radius $\theta$ Peebles (1993). $D_{L}$ is the luminosity distance, which
depends on redshift, $z$, and cosmological densities parameters, $\Omega$’s,
and the Hubble parameter, $H_{0}$. The total cluster’s mass $M_{tot}$,
obtained by using the Euler equation for a fluid in hydrostatic equilibrium,
with the additional assumption of isothermality White (2011), is given by:

$M_{tot}(<R)=-\left.\frac{k_{B}T_{e}R}{Gx_{H}m_{H}}\frac{d\ln n_{e}(r)}{d\ln
r}\right|_{r=R}.$ (2)

Here, $x_{H}$ is the molar fraction of hydrogen in the gas, $G$ is Newton’s
gravitational constant and $n_{e}$ is the electron gas density. Thus, from
Eq.(1), we observe that $M_{gas}\propto m_{H}$, and from Eq.(2),
$M_{tot}\propto 1/m_{H}$. Consequently, we can conclude that the gas mass
fraction, $f_{gas}$ within the intra-cluster medium which is the ratio between
Eq.(1) and Eq.(2),

$f_{gas}\propto m_{H}^{2},$ (3)

and since $m_{H}=m_{e}+m_{p}$, then

$f_{gas}\propto\left(1+\mu\right)^{2}m_{p}^{2},$ (4)

Which as we can see, explicitly depends on $\mu$, the ratio of electron to
proton masses.

On the other hand, the expected constancy of the $f_{gas}$ within massive, hot
and relaxed galaxy clusters with redshifts in the interval of interest has
been used to constrain cosmological parameters via the following equation
Allen et al. (2008); Ettori, S. et al. (2009); Mantz et al. (2014):

$f_{gas}^{th}(z)=\gamma(z)\,K(z)\,A(z)\frac{\Omega_{b}}{\Omega_{m}}\left[\frac{D^{*}_{L}}{D_{L}}\right]^{3/2}.$
(5)

In the previous Eq.(5), $D_{L}^{*}$ indicates the luminosity distance
calculated using a fiducial concordance model, $D_{L}$ is the distance of
interest, and $\gamma(z)$ is the baryon depletion factor, which for our
purposes and conservatively, can be taken as the best value
$\gamma_{0}=0.799\pm 0.11$ independent of $z$ Battaglia et al. (2013);
Planelles et al. (2013). $\Omega_{b}$ and $\Omega_{m}$ are the baryonic and
matter abundances density parameters, respectively, $K(z)$ is the mean ratio
of lensing to X-ray mass, which too, can be taken at best value:
$K_{0}=0.96\pm 0.12$ independent of $z$ Mantz et al. (2014) and $A(z)$
represents the angular correction factor, which is close to unity for all
cosmologies and redshifts of interest, and it can be neglected without
significant loss of accuracy Allen et al. (2008). Thus, considering a possible
variation of $\mu$ with redshift, we modify Eq.(5) accordingly, arriving at
the expression:

$f_{gas}^{obs}=\left(1+\mu(z)\right)^{2}\gamma\,K\,A\frac{\Omega_{b}}{\Omega_{m}}\left[\frac{D^{*}_{L}}{D_{L}}\right]^{3/2}.$
(6)

Finally, if the luminosity distance is known for each galaxy cluster, $\mu(z)$
can be inferred through the deviation from zero of $(1+\mu)^{2}$ with respect
to the ratio $\frac{f_{\text{gas}}^{\text{obs}}}{f_{\text{gas}}^{\text{th}}}$,

$\left(1+\mu\right)^{2}-\frac{f_{\text{gas}}^{\text{obs}}}{f_{\text{gas}}^{\text{th}}}=0\,\,\,,$
(7)

where $f_{\text{gas}}^{\text{obs}}$ is the gas mass fraction dataset444Which
would be equivalent to the modified equation for $f_{gas}$. presented in
Section III and $f_{\text{gas}}^{\text{th}}$ comes from Eq (5). This will
provide 44 values of $\mu$, which will be used to reconstruct the function
$\mu(z)$ via Gaussian process regression, as we describe next.

##  III Cosmological data

###  III.1 Gas Mass Fraction

We utilize the most new sample of $f_{\text{gas}}$ measurements spanning the
redshift range $0.018\leq z\leq 1.160$, compiled in Ref. Mantz et al. (2021).
This dataset comprises 44 massive, hot, and morphologically relaxed galaxy
clusters observed by the Chandra telescope (see Fig.1). The selection of
relaxed systems aims to minimize systematic biases in the hydrostatic masses.
The gas mass fraction is computed within the spherical shell defined by
$0.25\leq r/r_{2500}\leq 0.8$ rather than integrated at all radii. Further
details about the data can be found in Ref. Mantz et al. (2014, 2021). In
order to perform a cosmological model independent analysis, the
$\Omega_{b}/\Omega_{m}$ ratio utilized is from galaxy clustering observations
as given by Krolewski and Percival (2024), such as:
$\Omega_{b}/\Omega_{m}=0.173\pm 0.027$. Conversely, the majority of
cosmological constraints derived from observations of gas mass fraction in
X-ray emitting sources are dependent on hydrodynamic simulations Zheng et al.
(2019); Applegate et al. (2014).

Figure 1: The 44 Chandra X-ray gas mass fraction as a function of redshift
compiled by Mantz et al. Mantz et al. (2021).

###  III.2 SNe Ia sample

As formerly discussed, in order to perform our test, we need the luminosity
distance for each galaxy cluster. Then, we consider the PANTHEON dataset of
Type Ia supernovae, compiled in 2018 by Scolnic et al. Scolnic et al. (2018),
which is a combination of observations from three subsets: 279 Type Ia
supernovae from Pan-STARRS ($0.03<z<0.68$), samples from SDSS, SNLS, and
various low-z samples from HST. This forms one of the largest combined set of
Type Ia supernovae, totaling 1048 Type Ia supernovae with redshift intervals
ranging from 0.01 to 2.26. The luminosity distance ($d_{L}$) calculated to
this sample is obtained by considering the SH0ES (Supernovae and H0 for the
Equation of State) calibration, where the Hubble Constant, $H_{0}=73\pm 1$
km/s/Mpc Riess et al. (2022). To obtain the luminosity distance for each
galaxy cluster, we employ the Gaussian process methodology to determine the
central value along with its associated variance. The results for the gaussian
process regression for $d_{L}$ are shown in Fig.(2). In the following sections
we present the main content of the 2HDM to be addressed in this work, and
translate the above analysis to constrain the $\mu$ variation with redshift so
as to exemplify how our approach is able to put bounds on Particle Physics
models.

Figure 2: Luminosity distances for the 44 galaxy clusters obtained through
gaussian process regression using the value for the Hubble constant, $H_{0}$,
as given by Ref. Riess et al. (2022).

##  IV The Two Higgs Doublet Model

The 2HDM Branco et al. (2012); Wang et al. (2022) consists of introducing an
additional scalar to the SM spectrum, transforming as a dublet under the weak
isospin group and carrying hypercharge $Y=+1$. As usual, we choose the vev of
both doublets to be along the direction of weak isospin $T^{3}=-1/2$, the
electrically neutral field components, so that expansion around the minimum
energy configuration $\left<\Phi_{i}\right>_{0}$ reads,

$\Phi_{i}=\left(\begin{array}[]{c}\phi_{i}^{+}\\\
\phi_{i}^{0}\end{array}\right)=\left(\begin{array}[]{c}\phi_{i}^{+}\\\
\frac{1}{\sqrt{2}}(v_{i}+\rho_{i}+i\eta_{i})\end{array}\right)\
,\quad\left<\Phi_{i}\right>_{0}=\left(\begin{array}[]{c}0\\\
v_{i}/\sqrt{2}\end{array}\right)\ ,\quad i=1,2\,.$ (8)

In order to obtain a simplified scalar potential, we introduce a discrete
symmetry for the scalar doublets (which will be justified later), $Z_{2}:\
\Phi_{1}\rightarrow-\Phi_{1}\ ,\Phi_{2}\rightarrow\Phi_{2}$. Then, allowing
only a bilinear, $\Phi_{1}^{\dagger}\Phi_{2}+\text{h.c.}$ term, which softly
breaks the $Z_{2}$ symmetry so as to avoid the domain wall problem Zeldovich
et al. (1974), the most general renormalizable, gauge, Lorentz and CP
invariant scalar potential is written as,

$\displaystyle V(\Phi_{1},\Phi_{2})$ $\displaystyle=$
$\displaystyle\displaystyle
m_{1}^{2}\Phi_{1}^{\dagger}\Phi_{1}+m_{2}^{2}\Phi_{2}^{\dagger}\Phi_{2}-m_{12}^{2}(\Phi_{1}^{\dagger}\Phi_{2}+\Phi_{2}^{\dagger}\Phi_{1})+\frac{\lambda_{1}}{2}(\Phi_{1}^{\dagger}\Phi_{1})^{2}+\displaystyle\frac{\lambda_{2}}{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}$
(9) $\displaystyle+$
$\displaystyle\lambda_{3}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{4}(\Phi_{1}^{\dagger}\Phi_{2})(\Phi_{2}^{\dagger}\Phi_{1})+\displaystyle\frac{\lambda_{5}}{2}\left[(\Phi_{1}^{\dagger}\Phi_{2})^{2}+(\Phi_{2}^{\dagger}\Phi_{1})^{2}\right]\
.$

After the electroweak spontaneous symmetry breaking (SSB), the bilinear terms
in the scalar potential in Eq. (9) must be diagonalized, yielding the scalar
mass spectrum of the model. From the original eight scalars, three are
absorbed by the massive gauge bosons, $W^{\pm}$ and $Z^{0}$, while the five
remaining get mixed to provide the mass eigenstates. The physical basis (mass
basis) is achieved through the following rotations Melo (2020),

$\begin{array}[]{c}\left(\begin{array}[]{c}{h}\\\
{H}\end{array}\right)=R(\alpha)\left(\begin{array}[]{c}\rho_{1}\\\
\rho_{2}\end{array}\right)\ ,\quad\left(\begin{array}[]{c}{G}\\\
{A}\end{array}\right)=R(\beta)\left(\begin{array}[]{c}\eta_{1}\\\
\eta_{2}\end{array}\right)\ ,\quad\left(\begin{array}[]{c}G^{\pm}\\\
H^{\pm}\end{array}\right)=R(\beta)\left(\begin{array}[]{c}\phi_{1}^{\pm}\\\
\phi_{2}^{\pm}\end{array}\right)\,.\end{array}$ (10)

Where, each of the rotation matrices, $R(\theta)$, with
$\theta=\alpha\,,\beta$ can be written as,

$\displaystyle\begin{array}[]{c}R(\theta)\equiv\left(\begin{array}[]{rr}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{array}\right)\,,\end{array}$ (14)

with

$tan(2\alpha)\equiv\displaystyle\frac{-2m_{12}^{2}v_{1}v_{2}+2\lambda_{345}v_{1}^{2}v_{2}^{2}}{m_{12}^{2}(v_{2}^{2}-v_{1}^{2})+\lambda_{1}v_{1}^{3}v_{2}-\lambda_{2}v_{1}v_{2}^{3}}\,\,\,\,\,;\,\,\,\tan\beta\equiv\frac{v_{2}}{v_{1}}\,,$
(15)

where we have defined
$\lambda_{345}\equiv\lambda_{3}+\lambda_{4}+\lambda_{5}$.

The scalar’s mass spectrum then becomes:

$\displaystyle m^{2}_{h}$ $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{2}(\Delta_{\rho}-\delta_{\rho})\,;\,\,\,\,\,\,\,\,m^{2}_{H}=\displaystyle\frac{1}{2}(\Delta_{\rho}+\delta_{\rho})\,;\,\,\,\,\,\,\,\,m^{2}_{A}=\displaystyle\left[\frac{m_{12}^{2}}{v_{1}v_{2}}-\lambda_{5}\right](v_{1}^{2}+v_{2}^{2})\,;$
(16) $\displaystyle m^{2}_{H^{\pm}}$ $\displaystyle=$
$\displaystyle\displaystyle\left[\frac{m_{12}^{2}}{v_{1}v_{2}}-\frac{1}{2}(\lambda_{4}+\lambda_{5})\right](v_{1}^{2}+v_{2}^{2})\,,$
(17)

where we have defined,

$\Delta_{\rho}=\displaystyle\frac{m_{12}^{2}(v_{1}^{2}+v_{2}^{2})}{v_{1}v_{2}}+\lambda_{1}v_{1}^{2}+\lambda_{2}v_{2}^{2}\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\delta_{\rho}=\displaystyle\left[\left(\frac{m_{12}^{2}(v_{1}^{2}-v_{2}^{2})}{v_{1}v_{2}}-\lambda_{1}v_{1}^{2}+\lambda_{2}v_{2}^{2}\right)^{2}+4(m_{12}^{2}-\lambda_{345}v_{1}v_{2})^{2}\right]^{\frac{1}{2}}\,.$

In the above description, the three Goldstone bosons are denoted by, $G$ and
$G^{\pm}$, which constitute the longitudinal polarization of the massive gauge
bosons, whose masses are the same as in the SM, with the same vev but now
written as, $v\equiv\sqrt{v_{1}^{2}+v_{2}^{2}}=246$ GeV. The remaining scalar
fields, $h$ and $H$, are electrically neutral CP even, with $m_{H}>m_{h}$,
while $A$ is a pseudoscalar (CP odd), which together with the two charged
scalars, $H^{\pm}$, are the physical scalars in the spectrum of the theory.
For our purposes, the relevant interactions for future computations come from
the Yukawa Lagrangian, responsible for fermion mass generation,

$-\pazocal{L}^{\text{2HDM}}_{\text{yukawa}}=y_{ij}^{1e}(\bar{L}^{i}\Phi_{1})e_{R}^{j}+y_{ij}^{1d}(\bar{Q}^{i}\Phi_{1})d_{R}^{j}+y_{ij}^{1u}(\bar{Q}^{i}\tilde{\Phi}_{1})u_{R}^{j}+y_{ij}^{2e}(\bar{L}^{i}\Phi_{2})e_{R}^{j}+y_{ij}^{2d}(\bar{Q}^{i}\Phi_{2})d_{R}^{j}+y_{ij}^{2u}(\bar{Q}^{i}\tilde{\Phi}_{2})u_{R}^{j}+\text{h.c.}\,,$
(18)

therefore,

$M^{f}_{ij}=\displaystyle\frac{v_{1}}{\sqrt{2}}y_{ij}^{1f}+\frac{v_{2}}{\sqrt{2}}y_{ij}^{2f}\,,$
(19)

where, $L^{i}$ and $Q^{i}$ represent the usual SM left-handed lepton and quark
doublets, respectively, while $e_{R}$, $u_{R}$ and $d_{R}$, their
corresponding singlet right-handed partners. $y_{ij}$ denotes the matrix
elements of dimensionless Yukawa couplings and $M$ refers to the fermionic
mass matrix (not necessarily diagonal). Each term is summed over the family
indices $i$ and $j$ and we defined, $\tilde{\Phi_{i}}\equiv
i\sigma_{2}\Phi_{i}^{*}$.

Observe that, differently from the SM, here we have two unrelated sources of
contributions to fermion mass terms, one from the matrix $v_{1}y^{1f}$ and
another from $v_{2}y^{2f}$. The bold implication is that there is no change of
basis that diagonalizes both terms simultaneously, leading to Flavor Changing
Neutral Currents (FCNC) in the fermion-scalar couplings. The fact that such
interactions are experimentally unobserved, calls for some arrangement to
strongly suppress them. This can be achieved if we demand that each fermion
mass matrix come from only one of the scalar doublets, so that diagonalizing
all fermion mass matrices, $M^{f}$, implies the automatic diagonalization of
$y^{1,2}$ (with the corresponding change from fermion interaction eigenstates
to mass eigenstates) and, consequently, the disappearance of FCNC. This
mechanism is known as Natural Flavor Conservation (NFC) Paschos (1977);
Glashow and Weinberg (1977). The easiest way to implement that is through the
introduction of an extra symmetry, under which the scalar doublets transform
differently, so that each fermion (also transforming under this symmetry) can
couple to only one of the doublets. Our choice for that is a discrete $Z_{2}$
symmetry such that, $\Phi_{1}\rightarrow-\Phi_{1}\
,\Phi_{2}\rightarrow\Phi_{2}$. This is the reason we omitted odd terms in
either of the scalar doublets in the scalar potential, Eq. (9), except for the
soft $Z_{2}$ breaking term as we mentioned before.

The convention is to take all the fermion doublets, as well as singlet up-type
quarks ($u_{R}$), to transform trivially, so that the up-type quarks always
couple to $\Phi_{2}$ and we are left to decide which of the scalar doublets
couples to the singlet down-type fermions (right-handed down-type quarks
$d_{R}$ and charged leptons $e_{R}$). In the scenario we choose to work here,
called lepton-specific 2HDM (or LS2HDM for short), $d_{R}$ is assigned
positive parity under $Z_{2}$ (so that both, $d_{R}$ and $u_{R}$ couple to
$\Phi_{2}$, only) and $e_{R}$ is assigned negative parity (couples to
$\Phi_{1}$, only). We can then rewrite the Yukawa Lagrangian, Eq. (18), in the
mass basis for both, scalars and fermions Melo (2020),

$\displaystyle-\pazocal{L}^{\text{LS2HDM}}_{\text{yukawa}}$ $\displaystyle=$
$\displaystyle\sum_{f=u,d,e}m_{f}\bar{f}f+\sum_{f=u,d,e}\frac{m_{f}}{v}(\xi_{h}^{f}\bar{f}fh+\xi_{H}^{f}\bar{f}fH-i\xi_{A}^{f}\bar{f}\gamma_{5}fA)$
(20) $\displaystyle-$
$\displaystyle\left[\frac{\sqrt{2}}{v}\bar{u}V_{ud}(m_{u}\xi_{A}^{u}P_{L}+m_{d}\xi_{A}^{d}P_{R})dH^{+}+\frac{\sqrt{2}m_{e}\xi_{A}^{e}}{v}\bar{\nu}_{L}e_{R}H^{+}+\text{h.c.}\right]\,,$

with,

$\xi_{h}^{u}=\xi_{h}^{d}=s_{\alpha}/s_{\beta}\ ,\
\xi_{h}^{e}=c_{\alpha}/c_{\beta}\ ,\
\xi_{H}^{u}=\xi_{H}^{d}=c_{\alpha}/s_{\beta}\ ,\
\xi_{H}^{e}=-s_{\alpha}/c_{\beta}\ ,\
\xi_{A}^{u}=-\xi_{A}^{d}=(t_{\beta})^{-1}\ ,\ \xi_{A}^{e}=t_{\beta}\,,$

where we defined $c_{\theta}\equiv\cos\theta$, $s_{\theta}\equiv\sin\theta$
and $t_{\theta}\equiv\tan\theta$ for shorthand notation, the sum is over all
up-type and down-type fermions (leptons and quarks) and the coefficient
$V_{ud}$ denotes the appropriate element of the CKM matrix. There is a linear
combination of the neutral scalars that corresponds to the SM Higgs boson Melo
(2020),

$H^{\text{SM}}\sim h\cos(\beta-\alpha)+H\sin(\beta-\alpha)\,$ (21)

which allows us to choose between $h$ and $H$ to represent the observed Higgs
boson, $H^{\text{SM}}$, with mass around $125$ GeV, and consistent with SM
predictions for its interactions with fermions, implying a constraint on the
mixing angles $\alpha$ and $\beta$. Here we make the choice of $h$, the
lighter scalar, to be the observed Higgs boson, so that $\alpha=\beta$, from
now on. We are now, prepared to obtain a constraint on this model from the
bounds on $\mu$ varying with the redshift as inferred from the gas mass
fraction in galaxy clusters, as we present next.

##  V Analysis and Results

Now we are going to obtain bounds on the variation of the electron to proton
mass ratio by comparing the observed gas mass fraction within galaxy clusters
to the theoretical gas mass fraction which is predicted by cosmological models
as explained earlier in Sec. II. From the 44 data-points we inferred a
variation of $\mu$ ($\Delta\mu$), such as,

$\Delta\mu=1.77\times 10^{-2}\pm 9.8\times 10^{-2}.$ (22)

Although the constraints are not as restrictive as those obtained through
quasar absorption spectra, the measurements are in agreement within 1 $\sigma$
confidence level. It is worth noting that the physics involved here is
completely different. The techniques used to measure $\mu(z)$ in quasar
spectroscopy rely on the hyperfine transition of molecules such as
$\text{H}_{2}$, $\text{Fe}_{\text{II}}$, and $\text{Si}_{\text{II}}$ Ubachs et
al. (2016); Barrow and Magueijo (2005), with this transition being
proportional to the ratio $\frac{\alpha_{EM}^{2}g_{p}}{\mu}$ and the effective
radius of the nucleus. In this scenario, the ratio between the mass of the
electron and the mass of the proton is not directly measured; rather, it is
the ratio between the mass of the electron and the effective mass of the
nucleus Martins (2017). A measurement using the Eq. (4) obtained through X-ray
emission from ICM plasma, which consists predominantly of hydrogen ions, is a
more direct measurement.

Figure 3: Reconstruction of $\mu(z)$ via gaussian processes regression for the
GMF data using Riess’ prior for luminosity distance. Best fit curve and 1 and
2 $\sigma$ interval of confidence.

Gaussian Processes (GPs) offer a method for reconstructing a function from
observational data without assuming a specific parametrization. The GP
reconstruction process involves selecting a prior mean function and a
covariance function that quantifies the correlation between the values of the
dependent variable in the reconstruction, which are characterized by a set of
hyperparameters such as the signal variance $\sigma$ and the characteristic
length scale $l$. Unlike regular parameters, hyperparameters do not dictate
the shape of a function, rather, they characterize typical variations in the
reconstructed function. $l$ roughly represents the scale over which
significant changes occur in the function values across $z$, while $\sigma$
indicates the typical magnitude of these alterations Seikel and Clarkson
(2013). While more complex covariance functions may involve additional
hyperparameters, in this work we used a covariance function that solely
incorporate $\sigma$ and $l$. To reconstruct $\mu(z)$, we choose zero as the
prior mean function to avoid biased results and also require that $\mu(z)$
matches the laboratory value (at redshift 0), we show the plot of our
reconstruction in Fig. (3) along with a Gaussian kernel serving as the
covariance function, as follows,

$k(z,z^{\prime})=\sigma^{2}\exp{\left(-\frac{(z-z^{\prime})^{2}}{2l^{2}}\right)}.$
(23)

To perform the GP we used the algorithm scikit-learn Pedregosa et al. (2011).
Finally, we defined a likelihood function that compares the values of $\mu(z)$
obtained through GPR with the experimentally measured value $\mu_{0}$,

$L(y_{e},v_{2},cot\beta(z))=\prod_{i=1}^{n}\frac{1}{\sqrt{2\pi\sigma_{i}^{2}}}\exp\left(-\frac{(\mu_{\text{GPR}}(z_{i})-\mu_{\text{0}}(z_{i}))^{2}}{2\sigma_{i}^{2}}\right).$
(24)

We then chose an initial distribution for the electron Yukawa coupling,
$y_{e}$, the vev of $\Phi_{1}$, $v_{1}$, and the ratio of vevs in the 2HDM,
$\tan\beta(z)$, and then implemented a MCMC. Using the diagonalized Yukawa
coupling for electrons in the LS2HDM, obtained from Eqs. (18,19), and assuming
that any variation in $\mu$ with $z$ arises from $v_{1}(z)$, that is, the
proton mass remains constant, we obtain,

$\mu(z)=\frac{y_{e}v_{2}}{\sqrt{2}m_{p}(\Lambda_{\text{QCD}})}\cot\beta(z).$
(25)

This expression, evaluated at $z=0$, provides constraints for the Yukawa
coupling of the electron, $y_{e}$, for the cot$\beta_{0}$, and the vev,
$v_{2}$. The samples generated provide us with the best estimates for the
parameters as given in Fig. (4), along with the conditional 2D regions for the
confidence intervals. The results for the constrained parameters are
summarized in Table 1.

Figure 4: Constraints on $y_{e}$ , $v_{2}$ , and cot$\beta$ from $\mu(z)$ obtained using Riess’ prior. The diagonal plots represent 1-D marginalized likelihood distributions of each parameter present and off-diagonal contours are 68%, 95%, and 99% 2-D marginalized confidence regions. Parameter |  Value
---|---
$y_{e}(10^{-4})$ |  $1.31\cdot 10^{-4}\pm 0.00095$
$v_{2}(\text{GeV})$ |  $240.033\pm 0.21$
$\cot\beta$ |  $0.02127\pm 0.0029$
Table 1: Summary of the best values for the model’s parameters obtained from
our MCMC sampling.

This constitutes the main result of our work. Next we compute the contribution
of LS2HDM to the muon anomalous magnetic moment and gather additional
constraints to the model so as to find the favorable parameter space that is
constrained by $\mu(z)$.

##  VI LS2HDM and the Muon anomalous magnetic moment

There is, currently, a discrepancy between the muon anomalous magnetic moment,
$a_{\mu}\equiv(g-2)_{\mu}$, predicted by the SM and its experimental value,
$\Delta a_{\mu}^{obs}\equiv a_{\mu}^{\text{EXP}}-a_{\mu}^{\text{SM}}=(25.1\pm
5.9)\cdot 10^{-10}$ Cotrozzi and Sorbara (2022); Keshavarzi et al. (2022). We
show here that our model is capable of accounting for that difference at
$2\sigma$.

The most important contributions to $(g-2)_{\mu}$ from the LS2HDM are given by
the diagrams in figure 5, with $\phi$ representing the neutral scalars $h,\
H,\ A$. Here, $h$ is chosen to be the SM Higgs, with mass, $m_{h}=125$ GeV,
whose contribution is already considered in the theoretical prediction so it
does not enter in the computation of the deviation. Furthermore, the charged
scalar contribution is insignificant, so it will not be considered here,
meaning that we only take $H$ and $A$ into account. The first one loop diagram
in Fig (5) results in Botella et al. (2022),

$\Delta a_{\mu}^{\text{1
loop}}=\frac{1}{8\pi^{2}}\left(\frac{m_{\mu}}{v}\right)^{2}\left\\{(\xi_{H}^{\mu})^{2}\left[2I_{1}\left(\frac{m_{\mu}^{2}}{m_{H}^{2}}\right)-I_{2}\left(\frac{m_{\mu}^{2}}{m_{H}^{2}}\right)\right]-(\xi^{\mu}_{A})^{2}I_{2}\left(\frac{m_{\mu}^{2}}{m_{A}^{2}}\right)\right\\}\,,$
(26)

with,

$I_{1}(x)=1+\frac{1-2x}{2x\sqrt{1-4x}}\ln\left(\frac{1+\sqrt{1-4x}}{1-\sqrt{1-4x}}\right)+\frac{1}{2x}\ln
x\approx x\left(-\frac{3}{2}-\ln x\right)+x^{2}\left(-\frac{16}{3}-4\ln
x\right)+\pazocal{O}(x^{3})\,,$ (27)
$I_{2}(x)=\frac{1}{2}+\frac{1}{x}+\frac{1-3x}{2x^{2}\sqrt{1-4x}}\ln\left(\frac{1+\sqrt{1-4x}}{1-\sqrt{1-4x}}\right)+\frac{1-2x}{2x^{2}}\ln
x\approx x\left(-\frac{11}{6}-\ln x\right)+x^{2}\left(-\frac{89}{12}-5\ln
x\right)+\pazocal{O}(x^{3})\,.$ (28)

At one loop, $H$ contributes positively and $A$, negatively Kim (2022), so
that the overall contribution is not enough to explain the anomaly. For this
reason we include the second diagram in Fig. (5), a special type of two loop
contribution called Barr-Zee diagram, which yields Botella et al. (2022),

$\Delta a_{\mu}^{\text{Barr-
Zee}}=-\frac{2\alpha}{\pi}\frac{1}{8\pi^{2}}\left(\frac{m_{\mu}}{v}\right)^{2}\sum_{f}N_{c}^{f}Q_{f}^{2}\left\\{\xi_{H}^{\mu}\xi_{H}^{f}f\left(\frac{m_{f}^{2}}{m_{H}^{2}}\right)-\xi_{A}^{\mu}\xi_{A}^{f}g\left(\frac{m_{f}^{2}}{m_{A}^{2}}\right)\right\\}$
(29)
$f(z)=\frac{z}{2}\int_{0}^{1}dx\frac{1-2x(1-x)}{x(1-x)-z}\ln\left(\frac{x(1-x)}{z}\right)$
(30)
$g(z)=\frac{z}{2}\int_{0}^{1}dx\frac{1}{x(1-x)-z}\ln\left(\frac{x(1-x)}{z}\right)$
(31)

Here $H$ contributes negatively and $A$, positively Kim (2022). We sum over
all the fermions, $f$, in the loop, each one characterized by its color
number, $N_{c}^{f}$ (which is equal to 3 for quarks and zero for leptons), and
its electric charge (multiple of unit positron charge), $Q_{f}$.

Figure 5: Neutral scalar contributions to muon, $(g-2)_{\mu}$ (the wiggy lines
are photon ones).

The above expressions depend on $m_{H}$, $m_{A}$ and $\tan{\beta}$, which are
in principle hard to constrain because (considering $\beta=\alpha$) the quark
couplings from Eq. (20) are proportional to $\cot\beta\ll 1$. This means that
hadronic cross sections are attenuated and thus incapable of providing bounds
for the 2HDM masses. So it becomes necessary to establish model-dependent
bounds on these parameters. We follow here the model suggested by Ref. Kim
(2022), which uses lepton universality constraints, the CDF W boson mass
results and muonic $g-2$ measurements to probe the allowed parameter space (as
well as the requirement of vacuum stability, perturbativity and unitarity in
the wrong sign scenario). The $h\rightarrow AA$ decay channel is considered,
adding the constraint $m_{h}\geq 2m_{A}$.

The regions in parameter space that meet these requirements and keep $\Delta
a_{\mu}^{\text{2HDM}}(m_{H},m_{A},t_{\beta})=\Delta a_{\mu}^{\text{1
loop}}+\Delta a_{\mu}^{\text{Barr-Zee}}$ inside $1\sigma$, $2\sigma$ or
$3\sigma$ deviation from $\Delta a_{\mu}^{obs}=(25.1\pm 5.9)\cdot 10^{-10}$
are shown in Figs. (6) and (7), displayed as green bands. In Fig. (6), we
present the predictions of $\tan{\beta}$ against the pseudoscalar mass,
$m_{A}$, keeping the mass of the heavier scalar, $m_{H}$, fixed at $m_{H}=300$
GeV. We take into account the bounds stemming from our astrophysical analysis
of $\mu(z)$ in Table 1, representing $\cot\beta=0.0213\pm 0.0029$ (light gray
band). We see that a light pseudoscalar is favored, with mas ranging from
around 10 to 50 GeV. As for the Fig. (7), our prediction for $\Delta
a_{\mu}^{\text{2HDM}}$ is plotted as a function of $m_{A}$ with $m_{H}=300$
GeV. We kept $\tan{\beta}$ fixed with central value $\tan{\beta}=47$ (solid
purple curve) and the limiting values at $\pm 1\sigma_{\tan\beta}=\pm 6.4$
(dashed purple curves). The green bands are the observed $(g-2)_{\mu}$
discrepancy at $1\sigma$, $2\sigma$ and $3\sigma$. From this figure we
conclude that the neutral scalar contributions can solve the muon anomaly at
$1\sigma$ when $m_{A}\gtrapprox 10$ GeV. The critical point at $m_{A}\approx
10.6$ GeV is due to the competition between the one loop (negative) and Barr-
Zee (positive) contributions.

Figure 6: Constraints on 2HDM parameters $m_{A}$ and $\tan{\beta}$ using the
$(g-2)_{\mu}$ prediction of LS2HDM ($m_{H}$ is fixed at 300 GeV). The green
regions represent, from darker to lighter, $1\sigma$, $2\sigma$ and $3\sigma$,
where a $\chi^{2}$ distribution is adopted. The region in light gray
represents $\cot\beta=0.0213\pm 0.0029$ from Table 1. Figure 7: $\Delta
a_{\mu}^{\text{2HDM}}$ as a function of $m_{A}$ with $m_{H}=300$ GeV. We took
the central value $\tan{\beta}=47$ (solid purple curve) and the limiting
values at $1\sigma_{\tan\beta}=6.4$ (dashed purple curves). The green bands
are the observed $(g-2)_{\mu}$ discrepancy at $1\sigma$, $2\sigma$ and
$3\sigma$.

Just for completeness, we use some additional constraints from Ref. Kim (2022)
that give $m_{H^{\pm}}\gtrapprox m_{H}$, besides solving the $(g-2)_{\mu}$
discrepancy and the W mass anomaly simultaneously. For that, we superpose our
results from the $\mu(z)$ bounds on $\tan{\beta}$ in the LS2HDM (table 1) to
their results on $(g-2)_{\mu}$, universality tests through $\tau$-decay and
$Z\rightarrow\bar{l}l$ (Z decay into a pair of leptons), as well as the scalar
mass difference, $\Delta\equiv m_{H^{\pm}}-m_{H}$, constrained by the
requirement that it solves the $W$ mass anomaly Aaltonen et al. (2022). We see
from Fig. (8) that these extra bounds are compatible with our constraints on
$\tan{\beta}$ in the $1\sigma$ range $40.6\leq\tan{\beta}\leq 53.4$, covering
the central value obtained before with a small variation depending on the
heavier neutral scalar mass. The first panel in that figure is for a heavier
Higgs, $m_{H}=300$ GeV, while the second one is for $m_{H}=400$ GeV. The upper
bounds on $\tan{\beta}$ from $\tau$-decay and $Z\rightarrow\bar{l}l$ are shown
as full and dashed lines for different choices of the scalar mass difference,
$\Delta$. Observe that their results for the $(g-2)_{\mu}$, sketched in brown,
is a little bit different from ours because they use only the lepton $\tau$ in
the Barr-Zee loop, while we summed over all the SM fermions.

Figure 8: Constraints given by Kim (2022). The $2\sigma$ region allowed by the
muon $g-2$ is sketched in brown. The coloured lines give upper bounds on
$\tan\beta$ from lepton universality tests, exploring both $\tau$ decay and
$Z$ decay in any pair of leptons. The full and dashed lines are associated
with the lower and upper bounds on $\Delta\equiv m_{H^{\pm}}-m_{H}$,
respectively, and we superimposed the region in light gray to represent our
astrophysical results.

##  VII Conclusions

In this work, we analyze the expression for the gas mass fraction of a galaxy
cluster and found that $f_{\text{gas}}\propto\left(1+\mu\right)^{2}$. Thus, by
combining type Ia Supernovae observations with gas mass fraction measurement
data from 44 galaxy clusters, we put constraints on a possible variation of
the electron-to-proton mass ratio, $\Delta\mu$ given in Eq. (22).
Additionally, using a Gaussian regression process, we reconstructed a $\mu(z)$
curve, given in Eq. (25), which allows us to impose limits on models beyond
the electroweak SM of particle physics. In order to expose our approach, we
employed a particular extension of the scalar sector of the SM by an extra
Higgs doublet and an additional discrete $Z_{2}$ symmetry, resulting in the
two-Higgs doublet model in the scenario of lepton-specific coupling, which we
called LS2HDM, presented in Sec. IV. This choice is made not only for
simplicity but to attach some useful meaning to a class of well studied models
in literature. We were able to translate the results on $\mu(z)$ into
constraints on the parameters of the model, namely the $\cot{\beta}$ (reverted
to $\tan{\beta}$), $v_{2}$ and $y_{e}$ (see table 1, finding their optimal
values from a MCMC sampling).

These constraints, when further contrasted with the possibility of the model
to account for the discrepancy between experimental and SM theoretical values
of the anomalous muon magnetic moment, $(g-2)_{\mu}$, resulted in a robust set
of bounds on the parameters, favoring a rather light pseudoscalar, with
$m_{A}\gtrapprox 10$ GeV and the heavier scalar with mass around $300\leq
m_{H}\leq 400$ GeV. This last conclusions we obtained by gathering existing
bounds Kim (2022) on the model when $\tau$-decay and $Z\rightarrow\bar{l}l$
are considered together with the accommodation of the W-mass anomaly as seen
by CDF-collaboration Aaltonen et al. (2022).

Finally, we stress that the main point of this work was to probe a
complimentary approach to Particle Physics models from an interface with
Astrophysics and Cosmology, which, in our case, derived from the gas
properties in galactic clusters. Our conclusions add to unveil the importance
of seeking new constraints for non-standard physics in places beyond Particle
Physics accelerators only, and confirm that the large-scale structures of the
universe might contain more information than we have previously thought. In
the years ahead, there will be a substantial rise in both the quantity and
quality of observations made possible by instruments like the eROSITA
telescope, a Russian-German collaboration. This progress will enable more
comprehensive statistical analyses. Consequently, it will be feasible to
achieve more restrictive limits by using the methodology outlined in this
article.

## ACKNOWLEDGEMENT

PSRS would like to thank the financial support from Conselho Nacional de
Desenvolvimento Cientifico e Tecnologico (CNPq) under the project No.
309650/2019-4 and the support of UFPB through an internal project
PIA16656-2023; RFLH thanks the financial support from the Conselho Nacional de
Desenvolvimento Cientıfico e Tecnologico (CNPq) under the project No.
308550/2023-47; RGA thanks the financial support from UFPB through PIBIC-CNPq
under the project No. PIA16656-2023.

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