# Exploring the dark sectors via the cooling of white dwarfs

Jaime Hoefken Zink<EMAIL_ADDRESS>Dipartimento di Fisica e
Astronomia, Università di Bologna, via Irnerio 46, 40126 Bologna, Italy  INFN,
Sezione di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy  Maura E.
Ramirez-Quezada<EMAIL_ADDRESS>Department of Physics,
University of Tokyo, Bunkyo-ku, Tokyo 113–0033, Japan,  Dual CP Institute of
High Energy Physics, C.P. 28045, Colima, México

###### Abstract

As dense and hot bodies with a well-understood equation of state, white dwarfs
offer a unique opportunity to investigate new physics. In this paper, we
examine the role of dark sectors, which are extensions of the Standard Model
of particle physics that are not directly observable, in the cooling process
of white dwarfs. Specifically, we examine the role of a dark photon, within
the framework of a three-portal Model, in enhancing the neutrino emission
during the cooling process of white dwarfs. We compare this scenario to the
energy release predicted by the Standard Model. By analyzing the parameter
space of dark sectors, our study aims to identify regions that could lead to
significant deviations from the expected energy release of white dwarfs.

##  I Introduction

Dark sectors (DS) have gained increasing importance in recent years as a
potential extension to the limitations of the Standard Model of particle
physics. In particular, several theoretical extensions have been proposed to
explain the origin of neutrino masses and mixing, with the Type-I seesaw
mechanism and its variations receiving significant attention. These models
only introduce heavy-neutral leptons (HNLs) as part of their minimal framework
[1]. However, a more compelling setup has recently been proposed as a
potential solution, where the HNLs are considered as part of a richer low-
energy dark sector [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. The non-minimal DS
introduces a new particle called the “dark photon”, which can interact with
both electromagnetic and dark currents. One interesting feature of this kind
of model is that it could also explain many of the experimental anomalies
which cannot be accounted for just by appealing to the SM, such as MiniBooNE
low energy excess [13, 9, 14, 15, 16, 17, 18] and the muon $(g-2)$ anomaly
[14, 16, 17, 18, 19], among other ones. To fully explore the potential and
test the predictions of the three-portal model, further theoretical
investigations are necessary. However, due to the introduction of several new
parameters and particles, experimental testing of the model may become more
complex and challenging.

White dwarfs (WD) provide unique opportunities to study new physics due to
their extreme conditions. By analyzing the luminosity emitted by white dwarfs,
it is feasible to probe the underlying physics in ways that are impossible in
Earth-based laboratories. One particular aspect is the cooling process of
young and hot white dwarfs, which is primarily driven by plasmon decay into
neutrinos originating from their core [20, 21]. The potential for enhanced
cooling through plasmon decay, mediated by new particles, has been explored in
the context of models with anomaly-free symmetry, such as U(1)B-L and
U(1)${}_{L_{\alpha}-L_{\beta}}$. These decay processes contribute to the
production of the SM neutrinos [22] and could also lead to the generation of
new light particles [23]. Furthermore some studies have shown that there could
be other important mechanisms under the assumption of a strong magnetic field
[24, 25, 26, 27, 28, 29], such as neutrino pair synchrotron emission from
electrons [30, 31, 32, 33, 34, 35].

The aim of this study is to investigate the emission of neutrinos from white
dwarfs, with a focus on beyond the SM (BSM) interactions in the neutrino
production rate. Specifically, we will use the DS model, known as the “three-
portal” model, proposed in Ref. [14]. Since the dark photon is capable of
interacting with neutrinos as well as electromagnetic-charged particles, its
presence could potentially alter the production of neutrinos through plasmon
decay at the early stages of a WD.

The structure of this paper is organized as follows. In Section II, we provide
a concise overview of the physics of white dwarfs. Additionally, we introduce
the three-portal model that we are considering in this study, along with its
key features. In Section III, we present the emission rate of neutrino
production, both within the context of SM interactions and with the modified
expressions that include the DS. Our results are discussed and presented in
Section IV, followed by our conclusions in Section V.

##  II Theoretical framework

###  II.1 White dwarfs

A white dwarf is a dense star that forms after a normal star has exhausted its
nuclear fuel and undergone the final stages of its evolution. This process
ejects the outer layers of the star, leaving behind a hot, dense core
primarily composed of carbon and oxygen. The core is supported by electron
degeneracy pressure, which prevents it from collapsing further and instead
causes it to contract and cool over billions of years. White dwarfs are
incredibly dense ($\sim 10^{6}\,\rm kg/m^{3}$), with a mass similar to the Sun
but a size similar to that of the Earth.

Electrons play a crucial role in determining the equation of state (EoS) and
the structure of white dwarfs. In Ref. [36], a theoretical EoS for WDs is
obtained by assuming a Wigner-Seitz (WS) cell, which is a uniformly negatively
charged spherical cell with a positively charged ion at its centre.
Corrections to the electron energy due to the electrostatic potential were
also introduced, which are the most significant correction to the EoS. Further
corrections, such as considering the non-rigidity of ions within the WS cell
and self-interactions, were also taken into account. However, it was
determined that these corrections do not significantly affect the behaviour of
high-density matter.

Initially, white dwarfs have extremely high temperatures before cooling down
to become faint objects. There are several stages of cooling that it goes
through, with each stage characterized by a different mechanism of energy
loss. The equation governing the temperature evolution of WDs depends on the
cooling mechanism and the physical properties,

$\frac{dT_{\star}}{dt}=-\frac{L_{\gamma}}{4\pi R_{\star}\sigma_{\rm
SB}T_{\star}}-\frac{L_{\nu}}{4\pi R_{\star}\sigma_{\rm SB}T_{\star}}.$ (1)

The left-hand side represents the temperature change rate with respect to
time, where $T_{\star}$ is the temperature of the white dwarf. The right-hand
side consists of two terms, representing the rates of energy loss due to
photon radiation ($L_{\gamma}$) and neutrino emission ($L_{\nu}$),
respectively. Here $R_{\star}$ is the WD radius, and $\sigma_{\rm SB}$ is the
Stefan-Boltzmann constant.

The dominant cooling mechanism for newborn WDs is through the emission of
neutrinos, produced primarily through plasmon decay [20, 21]. The produced
neutrinos easily escape the dense core of the WD, carrying away energy and
facilitating the loss of thermal energy. As the white dwarf continues to cool
down, its temperature eventually drops to approximately $10^{3}\ \rm K$, at
which point it enters the photon cooling stage. In this stage, the white dwarf
radiates energy primarily in the form of photons, and its luminosity is
dominated by photon radiation. Consequently, this stage is not of interest in
the discussion that follows.

###  II.2 Three portal model

In this section, we briefly describe the comprehensive three-portal model that
extends the SM of particle physics by introducing a new $U(1)$ symmetry known
as $U(1)_{X}$, which spontaneously breaks at the sub-GeV scale. The new
symmetry is accompanied by a Higgs singlet ($\Phi$), a gauge field mediator
($X^{\mu}$), sterile neutrinos ($N$), and dark neutrinos ($\nu_{D}$). The
interactions between the new particles and the SM particles are described by
the Lagrangian [14],

$\displaystyle\mathscr{L}$
$\displaystyle=\mathscr{L}_{\mathrm{SM}}+(D^{x}_{\mu}\Phi)^{\dagger}(D^{x\mu}\Phi)-V(\Phi,H)-\frac{1}{4}X_{\mu\nu}X^{\mu\nu}$
$\displaystyle-\frac{\sin\chi}{2}B_{\mu\nu}X^{\mu\nu}+\overline{N}i\not{\partial}N+\overline{\nu}_{D}i\not{D}^{x}\nu_{D}$
$\displaystyle-[y_{\nu}^{\alpha}(\overline{L}_{\alpha}\\!\cdot\\!\widetilde{H})N^{C}\\!+\frac{\mu^{\prime}}{2}\overline{N}N^{C}\\!+y_{N}\overline{N}\nu_{D}^{C}\Phi+\mathrm{h.c.}],$
(2)

where $\widetilde{H}\equiv i\sigma_{2}H^{*}$,
$D^{x}_{\mu}\equiv\partial_{\mu}-ig_{D}X_{\mu}$, $\sin\chi$ is a small
coupling for the kinetic mixing between the hypercharge and the gauge field
$X^{\mu}$. The terms in square brackets represent the neutrino mass terms and
interactions with the Higgs.

The three-portal model incorporates three possible communication channels
between the SM and the DS. These portals allow for interactions and exchanges
of particles between the two sectors. The first portal is the scalar portal,
which occurs through the mixing of the DS Higgs singlet with the SM Higgs
doublet. This mixing creates a scalar field that can interact with SM and DS
particles. The second portal is the neutrino portal, which occurs through the
mixing of SM neutrinos with dark neutrino states. Finally, we have a vector
portal, which appears thanks to the kinetic mixing of the $X$ field with the
hypercharge, $B$, and through a broken symmetry resulting in a dark photon
$Z^{\prime}$. This new particle can interact with both electromagnetic and
dark currents.

In the three-portal model, the mass of the dark photon is typically assumed to
be less than $\mathcal{O}(1\mathrm{GeV})$. When the mass of the dark photon is
much smaller than the mass of the $Z$ boson, $M_{Z^{\prime}}/M_{Z}\ll 1$, the
complicated interactions of the dark photon can be approximated by a
simplified form as follows,

$\begin{split}\mathcal{L}_{I}\simeq-\epsilon
eJ_{\mu}^{\mathrm{EM}}Z^{\prime\mu}-g_{\mathrm{D}}J_{\mu}^{\mathrm{D}}Z^{\prime\mu},\end{split}$
(3)

where $J_{\mu}^{\mathrm{EM}}$ is an electromagnetic current,
$J_{\mu}^{\mathrm{D}}$ is a dark current that consists of dark neutrino
states: $\bar{\nu}_{\mathrm{D}}\gamma_{\mu}\nu_{\mathrm{D}}$, and
$Z^{\prime\mu}$ is a dark photon. We consider $\epsilon$ a small number, while
$g_{\mathrm{D}}$ is not highly constrained.

Once the electroweak and dark symmetries are spontaneously broken, and
considering $\nu_{\alpha}$, $N$ and $\nu_{D}$ to be the SM-flavor, sterile and
dark states, respectively, the neutrino mass matrix takes a form similar to an
inverse [37, 38] or an extended seesaw [39, 40]:

$\begin{split}\mathscr{L}_{\mathrm{mass}}^{\nu}=-\frac{1}{2}\begin{pmatrix}\overline{\nu_{\alpha}}&\overline{N}&\overline{\nu_{D}}\end{pmatrix}\begin{pmatrix}0_{3\times
3}&m_{D}^{\mathrm{T}}&0\\\ m_{D}&\mu^{\prime}&\Lambda^{\mathrm{T}}\\\
0&\Lambda&0\end{pmatrix}\begin{pmatrix}\nu_{\alpha}\\\ N^{C}\\\
\nu_{D}^{C}\end{pmatrix}\end{split}$ (4)

where we define,

$m_{D}=\Bigg{[}\frac{y_{\nu}^{\alpha}}{\sqrt{2}}v_{H}\Bigg{]}^{\mathrm{T}},\,\,\
\Lambda=\Bigg{[}\frac{y_{N}}{\sqrt{2}}v_{\varphi}\Bigg{]}^{\mathrm{T}}$ (5)

$v_{H}$ and $v_{\varphi}$ correspond to the SM Higgs vacuum expectation value
(VEV) and the dark scalar VEV, respectively. Each vector runs over $\alpha$
(neutrino SM-flavor states) and $N$ (neutrino sterile states). The mass matrix
can be diagonalized using 111In order to account for the cancellation of the
chiral anomaly, the number of dark neutrino states should consider the pairing
of right-handed and left-handed dark states. One way to achieve this is by
introducing Dirac terms in the mass matrix, which would preserve the chiral
symmetry. If Majorana terms were included instead, they would explicitly break
the dark gauge symmetry. It is important to note that our calculations remain
independent of the precise approach taken in this regard

$\begin{split}\widehat{M}&=\begin{pmatrix}U_{\alpha}^{\mathrm{T}}&U_{N}^{\mathrm{T}}&U_{D}^{\mathrm{T}}\end{pmatrix}\begin{pmatrix}0_{3\times
3}&m_{D}^{\mathrm{T}}&0\\\ m_{D}&\mu^{\prime}&\Lambda^{\mathrm{T}}\\\
0&\Lambda&0\end{pmatrix}\begin{pmatrix}U_{\alpha}\\\ U_{N}\\\
U_{D}\end{pmatrix}\\\ &=U^{\mathrm{T}}MU.\end{split}$ (6)

The strength of the dark photon interaction with active neutrinos is
proportional to the components of the mixing matrix $U$ as $\propto
g_{D}U_{Di}^{\ast}U_{Dj}\gamma_{\nu}P_{\rm L}$. Notice that the standard PMNS
neutrino mixing matrix is typically obtained by examining the flavour and
light neutrino sectors.

##  III Neutrino emission rate

###  III.1 Review of emission rate

The release of neutrinos has a significant impact on the energy loss of stars
that are extremely hot or dense. The production rate of neutrinos can be
greatly affected by the combined effects of the stellar plasma. For instance,
photons may decay into pairs of neutrinos ($\gamma\to\nu\overline{\nu}$),
carrying away energy [20]. This is made possible by modifying the dispersion
relations of the photon due to thermal effects that allow it to decay.

Throughout this section and in subsequent discussions, we will follow the
convention of calling “photon” the transverse polarisation, while “plasmon” is
the longitudinal one [41].

####  III.1.1 Photon-self energy at finite temperature

To determine the neutrino emissivity due to plasmon decay, we must first
compute the self-energy of the photon at a finite temperature, as it plays a
crucial role in the emission rate computation. The most comprehensive
expression for the photon-self energy at a finite temperature is given as [41,
42],

$\begin{split}\Pi^{\mu\nu}&=4e^{2}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{f_{e}(E_{k})+f_{\overline{e}}(E_{k})}{2E_{k}}\\\
&\times\frac{Q\cdot
K(K^{\mu}Q^{\nu}+K^{\nu}Q^{\mu})-Q^{2}K^{\mu}K^{\nu}-(Q\cdot
K)^{2}g^{\mu\nu}}{(Q\cdot K)^{2}-Q^{4}/4}\end{split}$ (7)

here, $Q=(q_{0},\vec{q})$ and $K=(E_{k},\vec{k})$ represent the 4-momentum of
the photon and electron (or positron), respectively and $Q\cdot
K=q_{0}E_{k}-\vec{k}\cdot\vec{q}$. The thermal distribution of the electron
(or positron) is denoted by $f_{e}(E_{k})$ (or $f_{\overline{e}}(E_{k})$). The
self-energy tensor of a thermal photon can be divided into two components:
longitudinal and transverse components as follows,

$\Pi^{\mu\nu}=FP_{L}^{\mu\nu}+GP_{T}^{\mu\nu}$ (8)

where the projectors are:

$\begin{split}P^{\mu\nu}_{T}&=\big{(}\delta^{ij}-\hat{q}^{i}\hat{q}^{j}\big{)}\delta^{\mu}_{i}\delta^{\nu}_{j}\\\
P^{\mu\nu}_{L}&=\bigg{(}-g^{\mu\nu}+\frac{Q^{\mu}Q^{\nu}}{Q^{2}}\bigg{)}-P^{\mu\nu}_{T}.\end{split}$
(9)

Evaluating the $00$-elements of the longitudinal and transverse components of
the photon self-energy yields $P_{T}^{00}=0$ and
$P_{L}^{00}=\vec{q}^{\,2}/Q^{2}$, respectively. Consequently, we find that
$F=Q^{2}/\vec{q}^{\,2}\Pi^{00}$. Similarly, considering the $xx$-elements of
the transverse and longitudinal components gives $P_{T}^{xx}=1$ and
$P_{L}^{xx}=0$, respectively, leading to $G=\Pi^{xx}$. As a result, we can
express Eq. (8) as follows,

$\Pi^{\mu\nu}=\frac{Q^{2}}{\vec{q}^{\,2}}\Pi^{00}P_{L}^{\mu\nu}+\Pi^{xx}P_{T}^{\mu\nu}.$
(10)

We identify the first and second terms as $\Pi_{L}^{\mu\nu}$ and
$\Pi_{T}^{\mu\nu}$, respectively.

It is necessary to determine various factors to calculate the photon
polarization $4$-vectors at finite temperature. This includes the dispersion
functions of photons and plasmons as well as the corresponding residual
functions ($Z_{t}(q)$ and $Z_{l}(q)$), where $q\equiv|\vec{q}|$. In this
context, the dispersion relations for photons and plasmons provide the
relationship between the frequency $\omega_{t}$ and $\omega_{l}$ and the
momentum $q$ of the photon/plasmon. These relations provide information about
the energy of the photon/plasmon.

The dispersion function for the plasmon is determined using the longitudinal
propagator. This propagator is found by considering
$D^{00}=\frac{1}{q^{2}-\Pi_{L}(Q)}$. If $\omega_{l}(q)$ represents the energy
of the longitudinally polarized plasmon on-shell, in its vicinity it can be
deduced that [41],

$\lim_{q_{0}\to\omega_{l}(q)}D^{00}=\frac{\omega_{l}(q)^{2}}{q^{2}}\frac{Z_{l}(q)}{q_{0}^{2}-\omega_{l}(q)^{2}}$
(11)

and the dispersion relation for $\omega_{l}$ is obtained by setting the
denominator equal to zero, meaning that $\Pi_{L}(\omega_{l}(q),q)=q^{2}$.
Therefore, we obtain

$\omega_{l}(q)^{2}=\frac{\omega_{l}(q)^{2}}{q^{2}}\Pi_{L}(\omega_{l}(q),q).$
(12)

Furthermore, since $\omega_{l}(q)$ is also the pole of the propagator, the
value of $Z_{l}(q)$ can be easily found from Eq. (11)

$Z_{l}(q)=\frac{q^{2}}{\omega_{l}(q)^{2}}\bigg{[}-\frac{\partial\Pi_{L}}{\partial
q_{0}^{2}}(\omega_{l}(q),q)\bigg{]}^{-1}$ (13)

Similarly, for the transverse propagator, where $x$ is a transverse direction,
we have $D^{xx}=\frac{1}{q_{0}^{2}-q^{2}-\Pi_{T}(Q)}$. The pole of the
propagator is given by $\omega_{t}(q)$, and in its vicinity, the propagator
takes the form [41]

$\lim_{q_{0}\to\omega_{t}(q)}D^{xx}=\frac{Z_{t}(q)}{q_{0}^{2}-\omega_{t}(q)^{2}},$
(14)

the dispersion relation for $\omega_{t}$ is found to be

$\omega_{t}(q)^{2}=q^{2}+\Pi_{T}(\omega_{t}(q),q),$ (15)

and the residual function $Z_{t}(q)$ is given by

$Z_{t}(q)=\bigg{[}1-\frac{\partial\Pi_{T}}{\partial
q_{0}^{2}}(\omega_{t}(q),q)\bigg{]}^{-1}.$ (16)

Finally, we can define the photon/plasmon polarization $4$-vectors modified
due to the effects of the temperature,

$\begin{split}&\varepsilon^{\mu}(q,\lambda=0)=\frac{\omega_{l}(q)}{q}\sqrt{Z_{l}(q)}(1,0)^{\mu}\\\
&\varepsilon^{\mu}(q,\lambda=\pm
1)=\sqrt{Z_{t}(q)}(0,\varepsilon_{\pm}(q))^{\mu}\end{split}$ (17)

where $\varepsilon_{\pm}(q)$ are two unit mutually orthogonal vectors on the
plane transverse to $\vec{q}$.

During the integration of the self-energy, the quantity $v\equiv k/E_{k}$,
where $k\equiv|\vec{k}|$, may lead to three different temperature regimes: a
non-relativistic, relativistic or degenerate regime. In the non-relativistic
limit ($T\ll m_{e}$) where particles are non-degenerate ($T\ll m_{e}-\mu$),
$v$ is equal to $0$. On the other hand, in the relativistic limit, where
electrons can be considered massless, the parameter $v$ takes on the value of
1. This limit is observed in either a high-density regime ($\mu\gg m_{e}$) or
a high-temperature regime ($T\gg m_{e}$). Finally, $v=v_{F}\equiv p_{F}/E_{F}$
in the degenerate limit, where $p_{F}\equiv\big{(}3\pi^{2}n_{e}\big{)}^{1/3}$
is the Fermi momentum, $E_{F}\equiv\mu(T=0)$ is the Fermi energy, and $n_{e}$
is the number density of electrons. In the degenerate limit, $df_{e}/dk$ peaks
at $v_{F}$. In such cases, we can approximate the integrals by extracting this
central value, denoted as $v_{*}$, from the integral so that [41],

$\displaystyle\Pi_{L}$
$\displaystyle=\omega_{p}^{2}\frac{3}{v^{2}_{*}}\Bigg{(}\frac{q_{0}}{2v_{*}q}\ln\frac{q_{0}+v_{*}q}{q_{0}-v_{*}q}-1\Bigg{)}$
(18) $\displaystyle\Pi_{T}$
$\displaystyle=\omega_{p}^{2}\frac{3}{2v^{2}_{*}}\Bigg{(}\frac{q_{0}^{2}}{q^{2}}-\frac{q_{0}^{2}-v_{*}^{2}q^{2}}{q^{2}}\frac{q_{0}}{2v_{*}q}\ln\frac{q_{0}+v_{*}q}{q_{0}-v_{*}q}\Bigg{)}$
(19) $\displaystyle\omega_{p}^{2}$
$\displaystyle=\frac{4\alpha}{\pi}\int_{0}^{\infty}dk\frac{k^{2}}{E_{k}}\bigg{(}1-\frac{1}{3}v^{2}\bigg{)}\big{(}f_{e}(E_{k})+f_{\overline{e}}(E_{k})\big{)},$
(20)

here, $\omega_{p}$ refers to the plasma frequency, which describes the
oscillation of the plasma itself due to changes in the separation of electric
charges caused by the motion of the charged particles. This type of
oscillation is also referred to as Langmuir waves [43]. By using the
aforementioned approximations in Eq. (13) and (16), we can express $Z_{l}$ and
$Z_{t}$ as follows,

$\displaystyle Z_{l}^{-1}$
$\displaystyle=\frac{3\omega_{p}^{2}}{2v^{2}_{*}q^{2}}\Bigg{(}\frac{\omega_{l}^{2}}{\omega_{l}^{2}-v_{*}^{2}q^{2}}-\frac{\omega_{l}}{2v_{*}q}\ln\frac{\omega_{l}+v_{*}q}{\omega_{l}-v_{*}q}-1\Bigg{)}$
(21) $\displaystyle Z_{t}^{-1}$
$\displaystyle=1-\frac{3\omega_{p}^{2}}{2v^{2}_{*}q^{2}}\Bigg{(}\frac{3}{2}-\frac{3\omega_{t}^{2}-v_{*}^{2}q^{2}}{2\omega_{t}^{2}}\frac{\omega_{t}}{2v_{*}q}\ln\frac{\omega_{t}+v_{*}q}{\omega_{t}-v_{*}q}\Bigg{)}$
(22)

In the computation of the photon/plasmon decay, an additional contribution to
the self-energy arises from a diagram containing a $\gamma^{5}$ matrix at one
of its vertices. This axial contribution is due to the electroweak sector of
neutrinos and must also be calculated. The magnitude of this purely transverse
tensor is given by [41]

$\displaystyle\Pi_{A}(Q)=$
$\displaystyle\frac{2\alpha}{\pi}\frac{Q^{2}}{q}\int
dk\frac{k^{2}}{E_{k}^{2}}\big{[}f_{e}(E_{k})-f_{\overline{e}}(E_{k})\big{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\bigg{(}\frac{q_{0}}{2qv}\ln\frac{q_{0}+vq}{q_{0}-vq}-\frac{Q^{2}}{q_{0}^{2}-v^{2}q^{2}}\bigg{)}$
(23)

Here $\alpha$ is the fine structure constant. By employing the same
approximation as previously mentioned, we can obtain the following expression

$\Pi_{A}(Q)=\omega_{A}\frac{Q^{2}}{q}\frac{3}{v_{*}^{2}}\bigg{(}\frac{q_{0}}{2qv_{*}}\ln\frac{q_{0}+v_{*}q}{q_{0}-v_{*}q}-1\bigg{)}.$
(24)

Here, $\omega_{A}$ represents an axial frequency, which is defined as,

$\begin{split}\omega_{A}&=\lim_{q\to 0}\frac{\Pi_{A}(\omega_{t}(q),q)}{q}\\\
&=-\frac{2\alpha}{3\pi}\int
dk\frac{k^{3}}{E_{k}^{2}}\frac{d}{dk}\big{[}f_{e}(E_{k})-f_{\overline{e}}(E_{k})\big{]}.\end{split}$
(25)

We now have all the necessary elements to compute the plasmon decay and the
neutrino emission rates.

####  III.1.2 Amplitude and decay width

Figure 1: Diagrams that contribute to the plasmon decay: trough the charged
and neutral currents.

Fig. 1 displays the two SM diagrams contributing to plasmon decay. These
diagrams can be expressed in a way that includes the previously computed self-
energy,

$\displaystyle\mathcal{M}$
$\displaystyle=\frac{G_{F}}{\sqrt{8\pi\alpha}}\Bigg{[}\varepsilon_{\mu}(\omega_{l},q)C_{V}^{\rm
SM}\bigg{(}\Pi_{L}(\omega_{l},q)\Big{(}1,\frac{\omega_{l}}{q}\hat{q}\Big{)}^{\mu}\Big{(}1,\frac{\omega_{l}}{q}\hat{q}\Big{)}^{\nu}\bigg{)}$
$\displaystyle+\varepsilon_{\mu}(\omega_{t},q)g^{\mu i}\bigg{(}C_{V}^{\rm
SM}\Pi_{T}(\omega_{t},q)\Big{(}\delta^{ij}-\hat{q}^{i}\hat{q}^{j}\Big{)}$
$\displaystyle+C_{A}\Pi_{A}(\omega_{t},q)(i\varepsilon^{ijm}\hat{q}^{m})\bigg{)}g^{\nu
j}\Bigg{]}\overline{u}(p_{1})\gamma_{\nu}(1-\gamma_{5})v(p_{2})$ (26)

$G_{F}$ is the Fermi constant, which appears because $M_{W}^{2}(M_{Z}^{2})\gg
p^{2}$. The coefficient $C_{V}^{\rm SM}$ takes on a value of
$2\sin^{2}\theta_{W}+1/2$ for $\nu_{e}$, and $2\sin^{2}\theta_{W}-1/2$ for
other neutrino species. Meanwhile, $C_{A}$ is equal to $1/2$ for $\nu_{e}$ and
$-1/2$ for all others. We are explicitly stating that $C_{V}$ belongs to the
SM to distinguish it from the DS contribution. This is not the case for the
axial component, as it remains unaffected by the inclusion of new physics. It
should be noted that we assume the masses of the weak bosons to be much larger
than the momenta involved in the process. Finally, if $\lambda=l,\,t$
represents the longitudinal or transverse polarization, respectively, the
above equation can be expressed using shorthand notation

$\begin{split}\mathcal{M}=\frac{G_{F}}{\sqrt{2}}\Big{(}\Gamma^{\mu\nu}_{\lambda}\varepsilon_{\nu}(\vec{q},\lambda)\Big{)}\overline{u}(p_{1})\gamma_{\nu}(1-\gamma_{5})v(p_{2})\end{split}$
(27)

where a sum over polarisation is assumed.

The decay width of the process given a specific polarization $\lambda$ is,

$\begin{split}\Gamma_{\lambda}(q)=\frac{1}{2\omega_{\lambda}(q)}&\int\frac{d^{3}p_{1}}{(2\pi)^{3}}\frac{1}{2p_{1}}\int\frac{d^{3}p_{2}}{(2\pi)^{3}}\frac{1}{2p_{2}}\\\
&(2\pi)^{4}\delta^{(4)}(P_{1}+P_{2}-Q)|\mathcal{M}|^{2}.\end{split}$ (28)

The integral can be performed easily since the final states only involve
neutrinos, which are independent of the self-energy

$\Gamma_{\lambda}(q)=-\frac{G_{F}^{2}}{12\pi}\frac{\omega_{\lambda}(q)^{2}-q^{2}}{\omega_{\lambda}(q)}\Big{(}\Gamma^{\alpha\mu}_{\lambda}\varepsilon_{\mu}(q,\lambda)\Big{)}\Big{(}\Gamma_{\alpha\rho}^{\lambda}\varepsilon^{\rho}(q,\lambda)\Big{)}^{*}$
(29)

Since the expression is evaluated at $q^{0}=\omega_{\lambda}(q)$, it is
possible to use the dispersion relations to derive more explicit relations for
each polarisation,

$\displaystyle\Gamma_{l}(q)$ $\displaystyle=(C_{V}^{\rm
SM})^{2}\frac{G_{F}^{2}}{48\pi^{2}\alpha}Z_{l}(q)\Big{(}\omega_{l}(q)^{2}-q^{2}\Big{)}^{2}\omega_{l}(q)$
(30) $\displaystyle\Gamma_{t}(q)$
$\displaystyle=\frac{G_{F}^{2}}{48\pi^{2}\alpha}Z_{t}(q)\frac{\omega_{t}(q)^{2}-q^{2}}{\omega_{t}(q)}$
$\displaystyle\times\bigg{(}(C_{V}^{\rm
SM})^{2}\Big{(}\omega_{t}(q)^{2}-q^{2}\Big{)}^{2}+C_{A}^{2}\Pi_{A}(\omega_{t}(q),q)^{2}\bigg{)}$
(31)

####  III.1.3 Emissivity of the plasma

To calculate the Emissivity $\mathcal{Q}$ of the plasma, which represents the
rate of energy loss per unit volume, we must integrate the decay rate over the
phase space of the photon, with weighting by number density and energy, and
sum over the polarization states of the photon as well as the different
species of neutrinos. Upon substitution of the previously derived expressions,
we obtain the following Emissivity for each type of polarization

$\displaystyle\mathcal{Q}_{T}=$ $\displaystyle\sum_{\nu}(C_{V}^{\rm
SM})^{2}\frac{G_{F}^{2}}{48\pi^{4}\alpha}\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\\!\\!\\!dqq^{2}Z_{t}(q)\Big{(}\omega_{t}(q)^{2}-q^{2}\Big{)}^{3}\\!\\!\\!n_{B}(\omega_{t}(q))$
(32) $\displaystyle\mathcal{Q}_{A}=$
$\displaystyle\sum_{\nu}C_{A}^{2}\frac{G_{F}^{2}}{48\pi^{4}\alpha}\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\\!\\!\\!dqq^{2}Z_{t}(q)\Big{(}\omega_{t}(q)^{2}-q^{2}\Big{)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\Pi_{A}(\omega_{t}(q),q)^{2}n_{B}(\omega_{t}(q))$
(33) $\displaystyle\mathcal{Q}_{L}=$ $\displaystyle\sum_{\nu}(C_{V}^{\rm
SM})^{2}\frac{G_{F}^{2}}{96\pi^{4}\alpha}\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\\!\\!\\!dqq^{2}Z_{l}(q)\Big{(}\omega_{l}(q)^{2}-q^{2}\Big{)}^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\omega_{l}(q)^{2}n_{B}(\omega_{l}(q))$
(34)

the sum of the squares of the vector and axial-vector couplings, represented
by $C_{V}^{\rm SM}$ and $C_{A}$ respectively, overall neutrino species, is
given by $\sum_{\nu}(C_{V}^{\rm
SM})^{2}=3/4-2\sin^{2}\theta_{W}+12\sin^{4}\theta_{W}\approx 0.911$, while
$\sum_{\nu}C_{A}^{2}=3/4$. The variable $n_{B}$ represents the distribution
function of the photons within the system.

To calculate the neutrino luminosity, the Emissivity must be integrated over
the entire volume of the star. Assuming spherical symmetry, this can be
expressed as

$\mathcal{L}_{\Lambda}=4\pi\int_{0}^{R_{\star}}\mathcal{Q}_{\Lambda}(r)r^{2}dr$
(35)

where $\Lambda$ stands for $L$, $T$ or $A$. $R_{\star}$ is the radius of the
WD core, which is the volume considered for the plasmon decay. The quantities
used to compute the emissivities depend simultaneously on the radius of the WD
through its density or $p_{F}$, depending on the regime. For high chemical
potential, the high degenerate approximation can be used. As we approach the
surface of the core, the non-relativistic approximation is used instead.

###  III.2 Emission rate and the dark sector

If dark photons are present and interact with both an electromagnetic current
and a current of dark neutrinos, it introduces the possibility of an
additional diagram in the system that we sum to the SM ones. The new
interaction is analogous to the diagram involving the $Z$ particle, but
instead involving the $Z^{\prime}$ particle, as depicted in Fig. 2. In this
scenario, the dark final states would be mixed with the light states, which
can be expressed in terms of the SM neutrinos.

It is important to note that the photon/plasmon could not produce heavier mass
states at the energies characteristic of a WD if we consider those heavy
neutral leptons at the MeV scale. The plasma frequencies of a WD are well
below those energies.

Figure 2: Diagram that contributes to the plasmon decay through the dark
photon.

In this case, we will not use the effective propagator of the dark photo,
$(Q^{2}-M_{Z^{\prime}}^{2})^{-1}\sim-M_{Z^{\prime}}^{-2}$, just to consider a
general approach. The amplitude to neutrino mass states $i$ and $j$, coming
from the diagram in Fig. 2 is,

$\begin{split}\mathcal{M}^{ij}_{Z^{\prime}}&=-\varepsilon_{\mu}(Q)\int\frac{d^{4}k}{(2\pi)^{4}}\mathrm{tr}\big{[}\gamma^{\mu}S_{\beta}^{F}(K)\gamma^{\nu}S_{\beta}^{F}(K-Q)\big{]}\\\
&\times\epsilon
e^{2}\frac{1}{Q^{2}-M^{2}_{Z^{\prime}}}\frac{g_{\mathrm{D}}}{2}U^{*}_{i\mathrm{D}}U_{j\mathrm{D}}[\overline{u}_{j}(p_{1})\gamma_{\nu}(1-\gamma_{5})v_{i}(p_{2})]\\\
&=\frac{G_{F}}{2\sqrt{4\pi\alpha}}\Big{[}C_{\nu}^{\mathrm{D}}U^{*}_{i\mathrm{D}}U_{j\mathrm{D}}\Pi^{\mu\nu}\Big{]}\overline{u}_{j}(p_{1})\gamma_{\nu}(1-\gamma_{5})v_{i}(p_{2})\end{split}$
(36)

where $U$ represents the mixing matrix for neutrino states, with
$U_{i\mathrm{D}}$ denoting the mixing between a dark state $D$ and a light
mass state $i$. Here $S_{\beta}^{F}(P)$ is the thermal fermion propagator for
the electron/positron at temperature $T\equiv 1/\beta$. The constant
$C_{\nu}^{\mathrm{D}}$ is defined as

$C_{\nu}^{\mathrm{D}}=\frac{\sqrt{2\pi\alpha}}{G_{F}}\frac{\epsilon
g_{\mathrm{D}}}{M^{2}_{Z^{\prime}}-Q^{2}}.$ (37)

When computing the neutrino emission rates, it is convenient to express each
contribution in terms of neutrino mass states, $i$ and $j$, rather than
flavour states in order to observe the interference with the DS. These mass
states may not necessarily be the same, and hence, the summation must be
performed over all possible combinations,

$\displaystyle C_{V,ij}^{\mathrm{SM+D}}=$
$\displaystyle\sum_{\alpha}C_{V,\alpha}^{\mathrm{SM}}U_{\alpha i}^{*}U_{\alpha
j}+C_{\nu}^{\mathrm{D}}U_{\mathrm{D}i}^{*}U_{\mathrm{D}j}$
$\displaystyle\big{(}C_{V,ij}^{\mathrm{SM+D}}\big{)}^{2}$
$\displaystyle=\big{|}\sum_{\alpha}C_{V,\alpha}^{\rm SM}U_{\alpha
i}^{*}U_{\alpha j}\big{|}^{2}+|C_{V}^{\rm
D}\,U_{\mathrm{D}i}^{*}U_{\mathrm{D}j}|^{2}$
$\displaystyle+\frac{\sqrt{8\pi\alpha}}{G_{F}}\frac{\epsilon
g_{\mathrm{D}}U_{\mathrm{D}i}^{*}U_{\mathrm{D}j}}{M^{2}_{Z^{\prime}}-Q^{2}}\real[\sum_{\alpha}C_{V,\alpha}^{\rm
SM}U_{\alpha i}^{*}U_{\alpha j}\Bigg{]}$ (38)

where $C_{V,\alpha}^{\rm SM}$ is for a particular flavor $\alpha$. Performing
the sums over all possible final mass eigenstates, we compute the decay width,

$\displaystyle\big{(}C_{V}^{\mathrm{SM+D}}\big{)}^{2}$
$\displaystyle=\sum_{\alpha}\big{(}C_{V}^{\rm
SM}\big{)}^{2}+\frac{18\pi\alpha}{G_{F}^{2}}\frac{\epsilon^{2}g_{\mathrm{D}}^{2}\big{|}U_{\mathrm{D}}\big{|}^{4}}{(M^{2}_{Z^{\prime}}-Q^{2})^{2}}$
$\displaystyle+\frac{\sqrt{8\pi\alpha}}{G_{F}}\frac{\epsilon
g_{\mathrm{D}}\big{|}U_{\mathrm{D}}\big{|}^{2}}{M^{2}_{Z^{\prime}}-Q^{2}}\real[\sum_{\alpha,i,j}C_{V,\alpha}^{\rm
SM}U_{\alpha i}^{*}U_{\alpha j}\Bigg{]}.$ (39)

Here, $(C_{V}^{\mathrm{SM+D}})^{2}\equiv\sum_{\small
ij}(C_{V,\,ij}^{\mathrm{SM+D}})^{2}$ has contributions from three terms. The
first term represents the SM contribution, while the second term is solely
from the DS. The third term represents the interference between the SM and DS
amplitudes.

To simplify the computation, the assumption has been made that
$U_{\mathrm{D}i}$ is equal for every $i$, denoted as $U_{\mathrm{D}}$. This
implies that the mixing between each light mass state and the dark states is
real and equal without any loss of generality. With the inclusion of the dark
photon in the photon/plasmon decay, the transverse and longitudinal
emissivities are also altered as a consequence,

$\displaystyle\mathcal{Q}_{T}=2\frac{G_{F}^{2}}{96\pi^{4}\alpha}$
$\displaystyle\int_{0}^{\infty}dqq^{2}Z_{t}(q)\sum_{\alpha\beta}\big{(}C_{V}^{\rm
SM+D}(\omega_{t}(q),q)\big{)}^{2}$
$\displaystyle\times\Big{(}\omega_{t}(q)^{2}-q^{2}\Big{)}^{3}n_{B}(\omega_{t}(q)),$
(40) $\displaystyle\mathcal{Q}_{L}=\frac{G_{F}^{2}}{96\pi^{4}\alpha}$
$\displaystyle\int_{0}^{\infty}dqq^{2}Z_{l}(q)\sum_{\alpha\beta}\big{(}C_{V}^{\rm
SM+D}(\omega_{l}(q),q)\big{)}^{2}$
$\displaystyle\times\omega_{l}(q)^{2}\Big{(}\omega_{l}(q)^{2}-q^{2}\Big{)}^{2}n_{B}(\omega_{l}(q)).$
(41)

It is evident from Eq. (39), that the dependence on $G_{F}^{2}/\alpha$
vanishes for the purely DS contribution. Notice that $\mathcal{Q}_{A}$ remains
unchanged because there is no axial term next to the thermal loop in the dark
photon diagram (see Fig. 2).

Finally, using Eq. (35), (40) and (41), we can compute the WD luminosity due
to photon/plasmon decay into neutrinos.

##  IV Results and discussion

To obtain the total luminosity of a WD due to photon/plasmon decay is
necessary to obtain the WD radial profiles. This is achieved through the use
of the Tolman-Oppenheimer-Volkoff (TOV) equations [44], which describe the
hydrostatic equilibrium of a spherically symmetric, non-rotating star coupled
to the Salpeter EOS [36]. It is important to note that the following analysis
is performed on a hypothetical young WD with 1 solar mass. We fix its
temperature to be of the order of $10^{8}$ K, and using the evolutionary
sequences given in Ref. [45], this temperature corresponds to a WD of
$\sim\mathcal{O}(10^{6})$ s old.

It is necessary to establish some limits to determine the parameter space of
the three-portal model that will be explored. One such limit concerns the
quantity $\epsilon g_{\mathrm{D}}|U_{\mathrm{D}}|^{2}$. Here,
$|U_{\mathrm{D}}|$ refers to $|U_{\mathrm{Di}}|$, where $i$ denotes a massive
light neutrino state. Current limits on this element depend on the mass scale
of the heavy states, and we want to avoid the heavy states being so light that
they can be directly produced by the plasmon.222 Although this effect might
increase the energy lost by plasmon decay. Based on Ref. [46], we can assume
that $|U_{\mathrm{D}}|\lesssim 10^{-1}$. This would imply high masses for the
heavy neutrino states. We are not particularly interested in the specific
realization of this as long as the values are not fully excluded. The
parameter $g_{\mathrm{D}}$ is not a coupling of the dark photon to SM
fermionic currents, so limits such as those found in [47] are not relevant for
our model. There are no direct constraints on $g_{\mathrm{D}}$ neither if we
do not consider the dark photon or the HNLs to be dark matter, we still need
to keep the theory perturbative on $\alpha_{\mathrm{D}}\equiv
g_{\mathrm{D}}^{2}/4\pi$. Therefore, we can assume that
$g_{\mathrm{D}}\sim\mathcal{O}(1)$. The excluded regions for $\epsilon$ depend
heavily on the mass of the dark photon and the number of extra neutrino
states. We can assume $\epsilon\lesssim 10^{-2}$ based on [48, 19] which is
safe, especially for heavy neutrino mass states of the order of $1-10$ GeV
[49]. Hence, we can consider $\epsilon
g_{\mathrm{D}}|U_{\mathrm{D}}|^{2}\lesssim 10^{-4}$.

On the other hand, for the mass of the dark photon, the limits depend on
several conditions: whether the dark photon or the heavy neutral leptons
constitute dark matter, the value of $\epsilon$ or $g_{\mathrm{D}}$ and the
mixing matrix of the neutrino sector, $U$. Here, we will also consider the
same parameter space of [48, 19], such that $10\,\mathrm{MeV}\leq
M_{Z\prime}\leq 10\,\mathrm{GeV}$. The Lagrangian in Eq. (3) is valid as long
as $\big{(}M_{Z\prime}/M_{Z}\big{)}^{2}$ is negligible. Since for
$M_{Z\prime}=10\,\mathrm{GeV}$, this is of the order of $10^{-2}$, and its
contribution is still negligible.

###  IV.1 Luminosity

Figure 3: Luminosity of a $1M_{\odot}$ young WD with respect to its
temperature for different BSM scenarios. The solid (dashed) line represents
the longitudinal (transverse) components. The green (pink) bandwidth of the DS
longitudinal (transverse) contribution considers the values: $\epsilon
g_{D}|U_{D}|^{2}=[10^{-8}-10^{-4}]$. The axial contribution is not just shown
since no contribution comes from the new physics.

Fig. 3 illustrates the total luminosity of the WD attributed solely to plasmon
decay. The plot depicts different scenarios. The blue solid (dashed) lines
correspond to the longitudinal (transverse) contributions of the SM case. In
contrast, the green and pink regions show the longitudinal and transverse
contributions of the DS scenario, which includes a dark photon with a mass of
$M_{Z^{\prime}}=1$ GeV. Both bands demonstrate the luminosity for two
different coupling strengths: $\epsilon g_{D}|U_{D}|^{2}=10^{-8}$ and
$\epsilon g_{D}|U_{D}|^{2}=10^{-4}$. Note that when the coupling is too small,
the curves correspond to the SM scenario. The maximum luminosity from new
interactions in the neutrino emission rate is about one order of magnitude
greater than the SM. This suggests that dark photons, like the one proposed in
the DS scenario, may contribute to the evolution and behaviour of WDs in ways
not accounted for by the SM. As the WD cools down, the contributions from
plasmons decrease until they essentially disappear. Before reaching a
temperature of $10^{8}$ K, the main source of energy loss is from the
longitudinal contribution (solid line). At lower temperatures, the transverse
contribution (dashed line) dominates. Here, we do not show the axial
contribution in our analysis since it is suppressed several orders of
magnitude compared to the longitudinal and transverse contributions.

Figure 4: Luminosity of a $1M_{\odot}$ young WD with a temperature of
$T=10^{8}$ K, as a function of the dark photon mass for different BSM
scenarios. The solid (dashed) line represents the longitudinal (transverse)
components. The axial contribution is not shown since there is no contribution
from new physics. The bandwidth of the DS contribution considers the values:
$\epsilon g_{D}|U_{D}|^{2}=[10^{-8}-10^{-4}]$.

Fig. 4 shows the luminosity of the WD, at a temperature of $T=10^{8}$ K, as a
function of the dark photon mass $M_{Z^{\prime}}$. Again, the solid (dashed)
blue line represents the SM longitudinal (transverse) contribution. The green
and pink areas correspond to the emission rates, including the dark photon,
and the bands represent various fixed values of $\epsilon
g_{D}|U_{D}|^{2}=[10^{-8}-10^{-4}]$. As expected, for each set of couplings,
there is an upper bound on the dark photon mass above, which the new physics
does not visibly contribute, and the luminosity curve is just a horizontal
line approaching the SM. As the dark photon mass decreases, the contribution
to the luminosity grows exponentially, with the dominant contribution coming
from the longitudinal photon states (solid lines). For instance, when the
product of couplings is $\epsilon g_{D}|U_{D}|^{2}=10^{-8}$, and the dark
photon has a mass of $\sim 10$ MeV, the luminosity of the WD can be up to
$\sim 2$ orders of magnitude higher than the SM case. This effect becomes
insignificant for $M_{Z^{\prime}}\sim 100$ MeV and above. On the other hand,
in the case of $\epsilon g_{D}|U_{D}|^{2}=10^{-4}$, the effect of the new
interaction in the WD luminosity becomes irrelevant around $10$ GeV. Notice
that as the product $\epsilon g_{D}|U_{D}|^{2}$ increases, the impact of the
new physics in the luminosity is visible for heavier dark photon states. This
can be seen from the green solid and pink dashed lines showing, from left to
right, $\epsilon g_{D}|U_{D}|^{2}=10^{-7},10^{-6}$ and $10^{-5}$.

###  IV.2 Results

Figure 5: Limits on $\epsilon$ for dark photons decaying into neutrinos,
obtained with a young WD assuming $T=10^{8}$ K and a dark sector parameter of
$g_{D}|U_{D}|^{2}=10^{-5}$. The luminosity due to dark photons constitutes a
maximum of $F_{DS}=1\%$ (lightest green shaded area), $10\%$, and $50\%$
(darkest green shaded area) of the total SM luminosity. For comparison, we
also show the bounds from DIS [50], electron ($g-2$) (LKB) [51] and [52]
proportioned in Ref. [19].

Finally, we present a comprehensive investigation of the parameter space
associated with the dark photon in the context of the cooling of WDs. Here, we
express the contribution of dark photons as a fraction of the total SM
luminosity,

$F_{\mathrm{DS}}=\frac{\mathcal{L}_{\mathrm{DS+SM}}-\mathcal{L}_{\mathrm{SM}}}{\mathcal{L}_{\mathrm{SM}}}\times
100\%.$ (42)

Therefore, we can perform estimations and projections of the allowed parameter
space regarding the cooling of WD mediated by dark photons.

By imposing the condition that the luminosity generated through the presence
of dark photons constitutes a maximum of $1\%$, $10\%$, and $50\%$ of the
total SM luminosity, we can determine the excluded parameter space. In Fig. 5,
we set reasonable dark sector parameters $g_{D}|U_{D}|^{2}=10^{-5}$, and show
the estimated bounds on $\epsilon$. The bounds are depicted as shaded regions,
ranging from the lightest to the darkest shade of green, corresponding to
$1\%$, $10\%$, and $50\%$ maximum values of $F_{DS}$, respectively. For
comparison, we also show the bounds given by Deep-inelastic scattering (DIS)
[50], electron ($g-2$) (LKB) [51] and electroweak precision observables (EWPO)
[52], revised and presented in [19] as model-independent limits.

We notice that in all cases, the estimated bounds significantly surpass the
existing constraints. As the contribution of dark photons decreases, the bound
becomes progressively more stringent. This behaviour can be attributed to the
fact that, in order to achieve a smaller DS contribution, the coupling
parameter $\epsilon$ must decrease accordingly. The obtained bounds are up to
one and two orders of magnitude, larger than those given by precision
measurements of the electron anomalous magnetic moment when considering a
maximum fraction of $F_{DS}=50\%$ and $10\%$, respectively. However, the most
compelling result emerges when we restrict the luminosity to only $1\%$ of the
SM luminosity. In this scenario, the range where the bound exhibits its
greatest strength extends from 10 MeV to 1 GeV. It is important to note that
we are unable to extend our bounds beyond 10 GeV since the validity of the
dark photon interactions taken into account relies on the assumption that
$\big{(}M_{Z\prime}/M_{Z}\big{)}^{2}<<1$. Hence, for $M_{Z^{\prime}}>10^{4}$
MeV, the strongest bounds are still given by EWPO and DIS.

There are certainly additional constraints on this parameter space [53, 54].
However, in order to translate these bounds, we would need more specific
considerations on the model, especially those that affect the visible, semi-
visible and invisible decays of the dark photon. Since our computation is
independent of those regards, we are not showing them here. Finally, it is
worth noting that electron-neutrino scattering can be mediated through the
exchange of a dark photon. In the context of B-L models, the constraints
imposed on this process are more stringent compared to those derived from
white dwarf cooling. This is primarily due to the fact that in B-L models, the
vector boson couples with the same strength to the SM and the DS. On the other
hand, in the case of the three-portal model, the couplings are independent,
resulting in weaker constraints from the electron-neutrino scattering process.

##  V Conclusions

In the present paper, we have computed the total luminosity of a white dwarf
due to photon/plasmon decay and present various scenarios involving a dark
photon with different masses and couplings. We found that for a
$M_{Z^{\prime}}=1$ GeV, the maximum luminosity from new interactions,
corresponding to $\epsilon g_{D}|U_{D}|^{2}=10^{-4}$, exceeds that of the SM
by approximately one order of magnitude. Additionally, we have examined the
upper bound on the dark photon mass for each set of couplings. Above this
mass, the new physics has a minimal visible contribution, and the luminosity
from new interactions aligns with the SM one. We found that in the case of
$\epsilon g_{D}|U_{D}|^{2}=10^{-4}$, the dark photon mass at which the
luminosity becomes only SM luminosity is around 10 GeV.

Finally, by imposing the condition that the luminosity attributed to dark
photons should not exceed $1\%$, $10\%$, and $50\%$ of the total Standard
Model luminosity, we estimated bounds on the coupling parameter $\epsilon$ for
reasonable dark sector parameters $g_{D}|U_{D}|^{2}=10^{-5}$. Remarkably, our
estimated bounds consistently surpass the existing constraints, becoming
increasingly stringent as the contribution of dark photons decreases. This is
due to the need for smaller coupling values of $\epsilon$ to achieve a reduced
dark sector contribution. Notably, when restricting the luminosity to only
$1\%$ of the SM luminosity, our bounds exhibit their strongest range from 10
MeV to 1 GeV. Therefore, the cooling behaviour of WDs presents a promising
avenue to probe the existence and properties of dark photons.

###### Acknowledgements.

We thank Matheus Hostert for the helpful discussion on the dark sector model
and current bounds. We also thank Daniele Massaro for providing the revised
bounds from EWPO, LKB and DIS. MRQ would like to thank Koichi Hamaguchi for
the useful discussion and feedback. The research of JHZ has received support
from the European Union’s Horizon 2020 research and innovation programme under
the Marie Skłodowska-Curie grant agreement No 860881-HIDDeN. MRQ is supported
by the JSPS KAKENHI Grant Number 20H01897.

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