###### Abstract

Tensions in measurements of neutron and kaon weak decays, such as of the
neutron lifetime, may speak to the existence of new particles and dynamics not
present in the Standard Model (SM). In scenarios with dark sectors, particles
that couple feebly to those of the SM appear. We offer a focused overview of
such possibilities and describe how observations of neutron stars, that probe
either their structure or dynamics, limit them. In realizing these constraints
we highlight how the assessment of particle processes within dense baryonic
matter impacts the emerging picture — and we emphasize both the flavor
structure of the constraints and their broader connections to cogenesis models
of dark matter and baryogenesis.

###### keywords:

neutron stars; baryon number; neutron lifetime anomaly; dark matter; dark
sectors

1 1 0 https://doi.org/ 100 Probing Dark Sectors with Neutron Stars Probing
Dark Sectors with Neutron Stars Susan Gardner 1,‡ and Mohammadreza Zakeri 2,‡
Susan Gardner and Mohammadreza Zakeri Gardner, S.; Zakeri, M. These authors
contributed equally to this work.

##  1 Introduction

Various astrometric observations speak to the existence of dark matter and
dark energy Zwicky (1933, 1937); Peebles and Yu (1970); Press and Schechter
(1974); White and Rees (1978); Peebles (1982); Blumenthal et al. (1984); Davis
et al. (1985); Rubin et al. (1978); Faber and Gallagher (1979); Rubin et al.
(1980); Hu and Dodelson (2002); Aghanim et al. (2020); Eisenstein et al.
(2005); Riess et al. (1998); Perlmutter et al. (1999), and the detections of
neutrinos from extraterrestrial sources reveal nonzero neutrino masses Davis
et al. (1968); Fukuda et al. (1998); Ahmad et al. (2002). Also, the cosmic
excess of matter over antimatter Sakharov (1967); Aghanim et al. (2020), i.e.,
the baryon asymmetry of the universe (BAU), cannot be described within the
Standard Model (SM), because the electroweak phase transition is not of first
order Kajantie et al. (1996); Csikor et al. (1999) for the empirically
determined Higgs mass Aad et al. (2012); Chatrchyan et al. (2012). Thus
physics beyond the Standard Model (SM) has been established, even if its
origins continue to be mysterious. Although these may come from dynamics
disjoint from the SM, the possibility of identifying its nature mandates the
careful consideration of any experimental anomalies that appear, for they
could point to new dynamics. In this article, we consider the neutron lifetime
anomaly Serebrov et al. (2009); Wietfeldt and Greene (2011), the possibility
that it could, at least in part, derive from new physics Fornal and Grinstein
(2018); Berezhiani (2019); Barducci et al. (2018); Rajendran and Ramani
(2021), and how that possibility can be constrained by observations of neutron
stars Nelson et al. (2019); Baym et al. (2018); Motta et al. (2018);
Berezhiani et al. (2021); Berryman et al. (2022, 2023). To set the stage, in
this introduction we (i) place the neutron lifetime anomaly in the context of
other low-energy $\beta$-decay experiments, (ii) motivate why new physics in
neutron decay could be linked to models that explain both dark matter and the
BAU, and (iii) discuss how neutron dark decays connect to neutron star
structure and dynamics.

###  1.1 Low-energy experimental anomalies in beta decay

The expected pattern of quark-flavor mixing under charged-current interactions
in the SM make the Cabibbo angle, i.e., the mixing of the first- and second-
generation of quarks, central to a unitarity test of the Cabibbo-Kobayashi-
Maskawa (CKM) matrix. This test probes the possibility of additional
contributions to quark mixing and/or whether the effective strength of the
weak interactions, as measured through the muon lifetime and thus $G_{F}$, is
really universal. The so-called Cabibbo angle anomaly (CAA) has different
components, and its resolution gains in importance because possible non-SM
effects need not be very small Coutinho et al. (2020); Crivellin and Mellado
(2023). There are disagreements in the assessment of $V_{ud}$, which is most
precisely probed through neutron decays or through superallowed ($0^{+}\to
0^{+}$) transitions in nuclei, and also in $V_{us}$, for which $K_{\ell 2}$
and $K_{\ell 3}$ decay studies are the most precise. Theoretical corrections
and inputs are needed in the analysis of the experiments to determine
$V_{ij}$. The determinations of $V_{us}$ from $K_{\mu\nu}$ and $K_{\ell 3}$
decays do not agree Moulson (2017); Seng et al. (2022); Cirigliano et al.
(2023), and the extracted values of $V_{ud}$ are also in slight tension with
each other Cirigliano et al. (2023). The combined results are also at odds
with CKM unitarity, with the differences predicated by the computed radiative
corrections Czarnecki et al. (2004); Marciano and Sirlin (2006); Seng et al.
(2019). In the superallowed case, the effect of the axial vector current
enters in the radiative corrections, which can be modified by nuclear
structure as well Towner and Hardy (2002). Recent developments in the theory
of the radiative corrections in neutron $\beta$ decay have led to refined
assessments of their value, impacting the value of $V_{ud}$ Seng et al.
(2019); we refer to Gorchtein and Seng (2023) in this volume for a summary. It
is worth noting that these determination also differ from the value of
$|V_{us}/V_{ud}|$ determined from the ratio of rates in $K_{\ell\nu}$ and
$\pi_{\ell\nu}$ decays Seng et al. (2022); Cirigliano et al. (2023). These
anomalies could have various resolutions, arising from either limitations in
the experiments themselves (there is only one high quality measurements of
$K\to\mu\nu$) or in the SM theoretical inputs, though particular new-physics
models, such as those with right-handed currents, could be at work Cirigliano
et al. (2023); Crivellin and Mellado (2023).

The mixed symmetry nature of neutron $\beta$-decay implies that at least two
different experiments are needed to extract $V_{ud}$, and our interests make
it important to flesh out the details. In the SM, the neutron lifetime
$\tau_{n}$ is

$|V_{ud}|^{2}\tau_{n}(1+3\lambda^{2})=\frac{2\pi^{3}}{G_{F}^{2}m_{e}^{5}(1+\delta_{\rm
RC})f}\equiv\eta\,,$ (1)

where $\lambda\equiv g_{A}/g_{V}$ and $\eta$ is $4908.6(1.9)\,{\rm s}$
Marciano and Sirlin (2006), or $4903.6(1.0)\,{\rm s}$ Seng et al. (2019),
$4905.7(1.5)\,{\rm s}$ Czarnecki et al. (2019), depending on the calculation
of the radiative correction $\delta_{\rm RC}$ used. Following common practice,
we use $\lambda$ and $g_{A}$ interchangeably, which is also strongly supported
by an analysis of the radiative corrections to $g_{A}$ Gorchtein and Seng
(2021). The uncertainties in $\delta_{\rm RC}$ dominate those in $\eta$ and
have been used to determine its reported error. Thus we see that the right-
hand side of Eq. (1) is strongly constrained, and the parameter $\lambda$ is
determined separately through measurements of the neutron $\beta$ decay
correlations, with coefficients Jackson et al. (1957) termed the “neutron
alphabet Abele (2008).” Thus to determine $V_{ud}$, to combine with $V_{us}$
for a test of CKM unitarity, both the lifetime and a measured decay
correlation are needed. The most precisely determined correlation coefficient
is “$A$,” the correlation between the neutron spin and the momentum of the
decay electron, and thus that determination has been commonly used. Recently,
precise determinations of “$a$,” the correlation of the decay electron and
antineutrino momenta, have been achieved. Thus we shall compare the outcomes
of combining measurements of either the $A$ or $a$ correlation coefficients
with the neutron lifetime for $V_{ud}$.

In Figure 1 we show the data set which constitutes the neutron lifetime
anomaly — we see that the measured lifetime $\tau_{n}$ is significantly
different if the neutrons are stored in a bottle, or trap, with the surviving
neutrons counted after some storage time, or if the neutrons in a beam decay
within a device that counts the protons from neutron decay. The determined
neutron lifetimes differ by some $9.5\,\rm s$, a significant variation. One
prominent impact this shift could have is on big-bang nucleosynthesis (BBN),
because the neutron lifetime determines, to a good approximation, the
temperature at which the weak-interactions freeze out in the early universe —
and a shorter lifetime reduces the expected yield of ${}^{4}{\rm He}$ Mathews
et al. (2005). We refer to Yeh et al. (2023) for a review of the neutron
lifetime’s impact on element yields from BBN. Since the bottle lifetime is
shorter, the difference between the two determinations can be reasonably
attributed to the possibility of neutron dark decays Fornal and Grinstein
(2018), coming from physics beyond the SM, and we refer to the review Fornal
(2023) in this volume for further discussion.

Figure 1:  Measured values of the neutron lifetime from beam (blue) and
material trap (red) and magnetic trap (green) experiments since 1990, from LRP
. In all neutron trap experiments, the neutron lifetime is determined by
counting the neutrons that persist, whereas in the beam experiments it is
determined by counting the decay protons. The central values of the two sets
of experiments disagree by $9.5\,\rm s$, with a significance in excess of
$4\,\sigma$ — this is the neutron lifetime anomaly. The PDG does not include
the beam measurements in their fits Workman et al. (2022).

Testing the $V-A$ structure of neutron (and nuclear) $\beta$ decay in the SM
has been a focus of intense research for decades Abele (2008); Dubbers and
Schmidt (2011); Cirigliano et al. (2013); Dubbers and Märkisch (2021). Here
the various decay observables, including the lifetimes and $\beta$-decay
correlations, can constrain the possibility that the neutron lifetime anomaly
comes from new physics Czarnecki et al. (2018); Dubbers et al. (2019);
Berryman et al. (2022). That is, the agreement of the value of $V_{ud}$ with
CKM unitarity and the value of $\lambda$ in a neutron lifetime and in a decay
correlation limit the possibility of non-$V-A$ contributions, which could
include neutron dark decays or the possibility of $n-n^{\prime}$ oscillations,
to the lifetime. In that context, the PDG gives an average of
$|\lambda|=1.2754(13)$ with a scale factor of 2.7 from the $A$ and $a$
correlation coefficients Workman et al. (2022), where the most precise
determination, from an $A$ measurement, is from PERKEO III
$|\lambda|=1.27641(56)$ Märkisch et al. (2019). The other two most precise
determinations of $\lambda$, which are also determined from $A$ Mund et al.
(2013); Brown et al. (2018), are consistent with that result. Recent, precise
$a$ determinations Beck et al. (2020, 2023); Wietfeldt et al. (2023) have
emerged, however, that differ significantly from the $A$ results. We note,
e.g., $|\lambda|=1.2677(28)$ Beck et al. (2020) — and the latest status of the
various measurements is shown in Fig. 2. This discrepancy cannot yet be
challenged by direct calculation in lattice QCD. As for the lattice QCD
results, we note the FLAG average from simulations with $N_{f}=2+1+1$ flavors,
$g_{A}=1.246(28)$ Aoki et al. (2022), as well as CalLat-18 result
$g_{A}=1.271(13)$ Chang et al. (2018), which is compatible with the PERKEO III
determination, though it has been updated to $g_{A}=1.2642(93)$ Walker-Loud et
al. (2020). Placed in the context of a test of CKM unitarity, as shown in
Figure 2, using the radiative corrections of Seng et al. (2019), we see that
the value of $|V_{ud}|$ from the bottle lifetime and the PERKEO III $A$
measurement is entirely consistent with the value determined in superallowed
decays. Interestingly, moreover, the value of $|V_{ud}|$ from the measured
beam lifetime combined with $\lambda$ from an $a$ measurement is very
comparable to its value determined using the bottle lifetime and the $\lambda$
from $A$. In the presence of two possible solutions for $\lambda$ and $V_{ud}$
that are similarly compatible with CKM unitarity, it would seem we cannot
(yet) limit a dark-decay solution to the neutron lifetime anomaly on the basis
of this comparison. This is perspective differs from that of earlier work
Dubbers et al. (2019); Czarnecki et al. (2018), and it is driven by the recent
$a$ measurements. Ultimately, any solution to the neutron lifetime anomaly
with new physics is required to be compatible with the measured $V-A$
structure of the SM currents.

Figure 2:  The SM correlation between the CKM matrix element $|V_{ud}|$ and
the axial-vector to vector coefficient ratio $|\lambda|=|g_{A}/g_{V}|$ is
presented, utilizing the averaged neutron lifetime values from bottle (blue)
and beam (green) methods given in Figure 1. This figure includes radiative
corrections as per Ref. Seng et al. (2019). Vertical bands illustrate
$|\lambda|$ measurements from aSPECT Beck et al. (2020) (light green), aCORN
Wietfeldt et al. (2023) (light blue), and PERKEO III Märkisch et al. (2019)
(light purple). The measured value of $V_{ud}$ via superallowed nuclear
transitions is shown by a red band Workman et al. (2022). Additionally,
lattice values of $g_{A}$ are shown, comprising the 2021 FLAG average for
$N_{f}=2+1+1$ flavors Aoki et al. (2022) and the 2019 results from the CalLat
collaboration Walker-Loud et al. (2020).

###  1.2 From anomalies to dark matter and the cosmic baryon asymmetry

Although astrometric observations speak to the existence of dark matter that
is (i) stable or nearly so on Gyr time scales Aghanim et al. (2020);
Eisenstein et al. (2005), (ii) possessive of no substantial strong or
electromagnetic charge Clowe et al. (2006), and (iii) engenders an “inside-
out” formation of large-scale structure Blumenthal et al. (1984), many
different sorts of dark-matter candidates are possible Battaglieri et al.
(2017); Chou et al. (2022), and they can impact astrophysical observations at
small scales differently Gardner et al. (2021). It is intriguing that the
contribution of baryonic matter and of the dark matter to the present cosmic
density parameter $\Omega_{0}$ are not too dissimilar, with the fraction in
baryonic matter some 20% of the dark matter one Aghanim et al. (2020).
Perhaps, then, dark matter is asymmetric Nussinov (1985); Barr et al. (1990),
much as baryonic matter is, suggesting that the two effects could share a
common origin. That possibility could be realized through a dark co-genesis
model Davoudiasl and Mohapatra (2012); Elor et al. (2019); Alonso-Álvarez et
al. (2021), and interestingly there is also connection between such models and
a dark-decay model of the neutron lifetime anomaly Fornal and Grinstein
(2018), yielding the dark decay $n\to\chi\gamma$, where $\chi$ is a fermion
with baryon number ${\cal B}=1$. This connection is concrete even if the
ultimate numerical impact of the $n\to\chi\gamma$ decay rate on the neutron
lifetime anomaly finally turns out to be small. Dark co-genesis models can act
in counterpoint to other, popular mechanisms, such as via weak-scale
supersymmetry — in the latter the severity of permanent EDM searches for the
electron and neutron limit the possible phase space severely Morrissey and
Ramsey-Musolf (2012). In contrast, the $B$-mesogenesis Elor et al. (2019)
mechanism and its variants Nelson et al. (2019); Elahi and Mohammadi
Najafabadi (2020); Elor and McGehee (2021); Alonso-Álvarez et al. (2021, 2021)
do not require sources of CP-violation beyond the SM to operate, and it is,
moreover, a testable mechanism of baryogenesis Barrow et al. (2020). The
branching ratio for $B\to{\cal B}\bar{\chi}$ decay cannot be too small Elor
and Guerrera (2023), and experimental constraints exist from searches with
BaBar data, with improvements expected from Belle-2 Lees et al. (2023). To
generate the noted decays, new scalars ought exist as well, as they can be
probed through collider searches and through constraints from flavor physics
Fajfer and Susič (2021); Alonso-Álvarez et al. (2021). Here we discuss what
further constraints come from precision studies of neutron stars, and their
dynamics Berryman et al. (2022, 2023).

Figure 3:  A schematic illustration of $B$-mesogenesis Elor et al. (2019), a
late-scale dark co-genesis model. Here a scalar $\Phi$, produced out of
equilibrium, decays and hadronizes to a $B_{q}^{0}{\bar{B}}_{q}^{0}$ pair,
with $q\in d,s$. The pair mixes, in the presence CP violation, and, finally, a
$B$ meson decays preferentially to ${B}\to{{\cal B}}\bar{\chi}$, a baryon
${\cal B}$ \- dark anti-baryon ${\bar{\chi}}$ pair. Since ${\bar{\chi}}$
carries ${\cal B}=-1$ the ${\cal B}\bar{\chi}$ final states do not break
baryon number, yet this scenario produces a baryon asymmetry in the visible
sector. The produced ${\bar{\chi}}(\chi)$ decays to other dark sector states,
preventing washout of the baryon asymmetry, with the particles of the dark-
sector final states serving as dark matter candidates. Different dark-sector
choices, that we note in text, differ in their impact on the dynamics of a
neutron star Berryman et al. (2023).

###  1.3 Constraints from neutron star structure and dynamics

Taken at face value, the difference in the beam and bottle lifetimes reflects
roughly a $1\%$ change in the neutron decay rate — and if the neutron does
decay via $n\to\chi\gamma$ or $n\to\chi\phi$, where $\phi$ is scalar that
could also decay via $\phi\to e^{+}e^{-}$ Fornal and Grinstein (2018), then
the ability to form a proto-neutron star is impacted McKeen et al. (2018);
Baym et al. (2018); Motta et al. (2018). In particular, its maximum mass is
only some $0.7M_{\odot}$, in conflict with observations of neutron stars of
much larger mass. This constraint can be side-stepped in different ways, such
as if $\chi$ self-interactions are included in the model, as also studied in
Cline and Cornell (2018), as an example. We note Husain et al. (2023) in this
volume for a discussion of neutron star constraints on dark scalar boson
production in this context. Here, rather, we focus on the new-physics channels
with dark fermions, particularly with a baryon-number-carrying particle
$\chi$, and the connections of this possibility to models that describe dark
matter and the baryon asymmetry. In this context $\chi$ must decay, though not
to anti-baryons, to avoid wash-out of the baryon asymmetry, and the size of
realized baryon asymmetry relies on the rate of $B\to{\bar{\chi}}{\cal B}$.
Here, then, the neutron dark decay rate can be smaller, and thus be in accord
with neutron-star observations. It thus would not explain the full size of the
neutron lifetime anomaly, but constraints on its size are useful in that they
can be interpreted in terms of limits on the flavor structure of the quark-
dark-baryon couplings. Here we note that although $\chi$ must decay to other
dark particles, different dark sector choices can be made. The precise energy-
loss constraints we are able to make from the study of existing neutron stars
are sensitive to those choices, and we will explain how in Section 2. There
are also dark decay models, such as the $n\to 3\chi$ model of Strumia (2022),
that we cannot constrain with the methods we review in this paper, though that
model is constrained by the would-be agreement between the decay correlations
and the bottle lifetime, should the anomaly in the $a$ and $A$ determinations
of $\lambda$ (or $g_{A}$) be resolved.

In what follows, we discuss, in turn, the effects of existence of dark sectors
on stars and, particularly, how neutron stars can constrain models of neutron
dark decays, how computations of particle processes in dense matter are
important to determining those constraints, and finally what constraints
emerge from neutron stars. We then conclude with a perspective on the neutron
lifetime anomaly, and a final summary.

##  2 Stellar constraints on dark sectors

Neutron stars, remnants of stellar evolution, present a unique opportunity to
study the intricate interactions of dark matter within extreme astrophysical
environments. This section delves into the multifaceted role of dark matter
across different stages of stellar evolution, from the early Universe’s first
stars to the dynamic phenomena observed in supernovae. We explore how dark
matter not only influences the lifecycle of stars but also plays a crucial
role in the formation, structure, and evolution of neutron stars.

###  2.1 Dark matter’s role in stellar evolution

The role of dark matter extends beyond the confines of neutron stars. Within
the CDM cosmological framework, cold dark matter is a key driver of the
Universe’s “inside-out” structural growth, crucially shaping the formation of
galaxies and stars from the earliest density fluctuations Peebles (1982). Its
influence is not limited to gravitational interactions: the dark sector
potentially engages in a variety of interactions and phenomena that have a
lasting impact on stellar evolution. This section explores these interactions,
starting with the pivotal role of dark matter in the lifecycle of the first
stars.

####  2.1.1 Impact on First Stars and Population-I Stars

Formation of Population III Stars. In the earliest chapters of cosmic history,
Population III (Pop III) stars, the Universe’s very first luminous bodies,
emerged from the gravitational collapse of pockets of gas in small halos
driven by dark matter Ciardi and Ferrara (2005). These stars are predicted to
form at redshifts $z<20-30$, in halos with masses ranging from
$10^{6}\,M_{\odot}$ to $10^{8}\,M_{\odot}$. Recent reports suggest the
possible identification of a Population III star cluster in the remote fringes
of a galaxy, highlighting ongoing efforts to deepen our understanding of these
primordial objects Maiolino et al. (2023). Beyond its gravitational influence,
dark matter, through its annihilation channels, might have significantly
impacted the formation of Pop III stars. Such annihilation could act as an
additional heat source, potentially inhibiting proto-star formation by
offsetting essential cooling mechanisms Iocco et al. (2008).

End of Pop III Stars. Pop III stars, luminous yet transient features of the
early universe, concluded their short lifecycles in supernovae, enriching the
cosmos with heavy elements. However, the influence of dark matter could extend
beyond the inception of these stars, significantly shaping the endpoint of
their evolution. This influence is epitomized by the black hole mass gap
(BHMG), a range delineated by $45\,M_{\odot}\leq M_{\rm BH}\leq
120\,M_{\odot}$, where no black holes emerge from the direct collapse of
massive stars due to pair-instability.

New Physics Implications for the Black Hole Mass Gap. The upper boundary of
the BHMG results from supermassive stars bypassing pair-instability supernovae
to directly form black holes, while the lower threshold arises from less
massive stars that enter a phase of pulsational-pair instability supernovae,
shedding significant mass through pulsations and leading to lower-mass black
hole remnants than their progenitors. Modifications to Pop III stellar
physics, potentially through new physics such as light particle emissions, can
shift the BHMG boundaries. Such particles can expedite helium burning and
diminish stellar wind mass loss, consequently reducing the available oxygen
for burning and suppressing the pulsational pair-instability mechanisms in Pop
III progenitors Croon et al. (2021).

Transitioning from dark-matter considerations in the stellar evolution in the
early universe, we now turn our focus to the influence of dark sector physics
on more contemporary stars.

Effects on Stellar Luminosities. Luminosity measurements from the Sun and
horizontal branch (HB) stars are instrumental in constraining dark sector
particles, known for their potential to alter stellar cooling rates. The Sun’s
emission of energy through a “dark channel,” such as dark photons, is
particularly restricted by the solar boron (${}^{8}B$) neutrino flux
measurements Haxton et al. (2013). These observations establish a conservative
upper limit on the Sun’s dark luminosity, which is mandated to be less than
10% of the Sun’s total photon luminosity Redondo and Raffelt (2013).
Furthermore, the emission of light particles such as neutrinos and axions from
the cores of red giants can delay helium ignition. This results in the
development of more massive cores, leading to a brighter phase prior to their
evolution into HB stars Raffelt (1996). The tip of the red-giant branch (TRGB)
is especially sensitive to new energy loss mechanisms, including those
prompted by axions Capozzi and Raffelt (2020), dark photons An et al. (2013),
or millicharged particles Davidson et al. (1991), which can cause noticeable
deviations in TRGB brightness from their standard values. Conversely, dark
matter annihilation within a star’s core might trigger an earlier helium
ignition, which could potentially lead to a dimming of the TRGB Lopes and
Lopes (2021), offering a contrasting perspective on the impact of dark matter
within stellar environments.

Stellar Structural Changes. Significant dark matter accumulation within stars
can impact their structure by altering convective and radiative regions due to
energy transfer by dark matter particles Martins et al. (2017). These
structural changes influence stellar pulsations, with implications for
asteroseismology Rato et al. (2021); Ayala (2022). In asymmetric dark matter
(ADM) scenarios, where a primordial asymmetry determines the dark matter relic
abundance instead of annihilation processes Zurek (2014), ADM can accumulate
without annihilation counterbalance. This accumulation leads to notable
effects on stellar evolution, such as altered thermal transport in main
sequence stars Spergel and Press (1985); Iocco et al. (2012); Banks et al.
(2022). Specifically, ADM interactions, depending on their nature, can modify
the temperature gradient in a star’s core by evacuating nuclear energy, with
spin-dependent interactions playing a significant role during core hydrogen
burning Raen et al. (2021).

####  2.1.2 Influence on Supernovae Dynamics

The death of stars is also not free from potential dark sector influences.
Specifically, the study of core-collapse supernovae, which mark the
cataclysmic end of massive stars (with masses exceeding approximately
$8\,M_{\odot}$) offers a unique window for probing possible hidden sector
interactions with ordinary matter. The emission of light particles from
supernova remnants introduces an additional cooling mechanism, which operates
by draining energy from the proto-neutron star (PNS). This process is subject
to constraints, as it could lead to a reduction in the duration of the
neutrino burst observed from SN 1987A, a pivotal discriminant among
astrophysical observations Hirata et al. (1987); Bionta et al. (1987); Raffelt
(1990). To maintain consistency with these observations, a conservative upper
limit on the luminosity of dark particles ($L_{d}$) is established, ensuring
it remains below the neutrino luminosity observed during the Kelvin-Helmholtz
cooling phase, which takes place approximately one second after the core
bounce. This constraint results in the formulation of a specific condition:
$L_{d}\lesssim 3\times 10^{52}\,{\rm erg}\,s^{-1}$, imposed one second after
the bounce Raffelt (1996). Supernova 1987A has been instrumental in
establishing limits on various extensions of the dark sector. These include
constraints on axions Raffelt and Seckel (1988), dark photons Dent et al.
(2012), and notably on muonic dark forces Croon et al. (2021); Manzari et al.
(2023). The latter, particularly the gauged $L_{\mu}-L_{\tau}$ model He et al.
(1991); Ma et al. (2002), has garnered interest for potentially resolving both
the muon $(g-2)_{\mu}$ anomaly Aoyama et al. (2020); Abi et al. (2021), and
the Hubble constant tension Aghanim et al. (2020); Escudero et al. (2019); Di
Valentino et al. (2021). Moreover, SN 1987A has been critical in constraining
hyperon decay processes, encompassing both decays into light bosons Camalich
et al. (2021) and those leading to GeV-scale dark sector particles Alonso-
Álvarez et al. (2021).

The exploration of supernovae, crucial for understanding the dynamics of dark
sectors, culminates in the birth of compact star remnants.

###  2.2 Neutron stars as dark matter laboratories

Among these remnants, neutron stars stand out due to their extreme densities
and strong gravitational fields, making them ideal laboratories for probing
dark matter effects in dense matter Baryakhtar et al. (2022); Bramante and Raj
(2023). In this discussion, we categorize the effects of the dark sector on
neutron stars into two distinct types: static and dynamic.

Static Effects and Neutron Star EoS. Static effects pertain to modifications
in the equation of state (EoS) that governs the dense matter within neutron
stars. These alterations reshape the neutron stars’ configuration space in
hydrostatic equilibrium, as dictated by the EoS. Such effects reach their
complete manifestation well within the age (its observation time) of the star,
acting in a time-independent manner during our observations.

Dynamic Effects and Neutron Star Evolution. Dynamic effects, in contrast,
involve changes that occur over time, impacting the neutron star’s structure,
such as its temperature and spin. These changes evolve with the star,
reflecting ongoing transformations.

It is important to recognize that a model can induce both static and dynamic
effects on neutron stars. There are models that exclusively yield dynamic
effects, while still preserving the original EoS. In such cases, these models
facilitate a quasi-equilibrium transformation of the star, unfolding within a
configuration space that is derived from, and remains consistent with, the
unaltered EoS Berryman et al. (2022).

####  2.2.1 Static Effects of Dark Matter on Neutron Stars

Neutron Star Maximum Mass. Building on the concept of static dark effects, a
notable instance is observed through the introduction of a dark baryon. This
particle, comparable to baryons in terms of mass but markedly weaker in its
interactions, results in a decrease in the pressure within neutron stars. Such
a phenomenon effectively “softens” the EoS. This softening consequently
reduces the maximum stable mass for neutron stars ($M_{\rm TOV}$) as
determined by the Tolman-Oppenheimer-Volkoff (TOV) equations for hydrostatic
equilibrium Oppenheimer and Volkoff (1939); Tolman (1939). Such modifications
present significant challenges to the simplest models aimed at resolving the
neutron lifetime anomaly Fornal (2023). They reduce $M_{\rm TOV}$ to about
$0.7\,M_{\odot}$ McKeen et al. (2018); Baym et al. (2018); Motta et al.
(2018), markedly below the mass of the most massive observed neutron stars,
which exceed $2\,M_{\odot}$. To reconcile these discrepancies, the
introduction of repulsive interactions either within the dark sector or
between dark particles and baryons becomes necessary Baym et al. (2018);
McKeen et al. (2018); Motta et al. (2018); Grinstein et al. (2019); Husain et
al. (2023); Zhou (2023). Alternatively, a dark decay channel leading to a
significantly lower equilibrium density of dark particles could circumvent
this stringent bound Strumia (2022).

####  2.2.2 Dark Dynamics of Neutron Stars

Dynamic effects on neutron stars can manifest in various forms, ranging from
abrupt, dramatic events to more gradual changes. A prime example of a sudden
occurrence is the collapse of a neutron star into a black hole. Such extreme
transformations can be triggered by specific interactions with dark matter, as
seen in two distinct scenarios.

WIMP-Induced Collapse. In one case, certain WIMP variants can accumulate
within neutron stars. Over time, these WIMPs may gravitate towards the star’s
center, eventually forming a self-gravitating core. This group includes
massive bosons lacking vectorial repulsive interactions and both bosons and
fermions with strong attractive scalar interactions. Such a concentration of
WIMPs could precipitate the rapid destruction of neutron stars, in a period
shorter than their observed lifespans Goldman and Nussinov (1989).

Collapse by Asymmetric Dark Matter. In a parallel scenario, the accretion of
non-annihilating dark matter within a neutron star can result in the formation
of a degenerate dark core. If this core expands beyond its Chandrasekhar limit
Chandrasekhar (1931), the neutron star hosting it might undergo collapse de
Lavallaz and Fairbairn (2010); McDermott et al. (2012).

Thermal Effects - Heating. A gradual dynamic effect influencing the thermal
evolution of neutron stars is the deviation of their thermal history from the
predictions of standard cooling theory Yakovlev et al. (2005). This deviation
can occur due to excess heating, potentially caused by the capture or
annihilation of dark matter. Kinetic heating, resulting from dark matter
capture, involves dark matter particles being gravitationally accelerated to
velocities exceeding half the speed of light. As these particles scatter
within neutron stars, they deposit significant kinetic energy, possibly
heating the stars to infrared blackbody temperatures Baryakhtar et al. (2017).
Additionally, the annihilation of WIMPs inside neutron stars can impact their
thermal properties. This is particularly evident in stars older than 10
million years, where such interactions may result in an asymptotic internal
temperature of around $10^{4}\,K$ Kouvaris (2008).

Thermal Effects - Cooling. Temperature observations of NS1987A, the remnant of
SN1987A Cigan et al. (2019); Page et al. (2020), and of Cassiopeia A (Cas A),
the second-youngest known neutron star with an approximate age of 340 years,
impose constraints on anomalous cooling processes in neutron stars. These
observations play a crucial role in constraining the emissions of novel
bosonic particles that could serve as additional cooling mechanisms, thereby
altering the thermal evolution of stars Hong et al. (2021); Shin and Yun
(2022).

Effects on Binary Pulsar Dynamics. Binary pulsar systems offer unparalleled
insights into both gravitational physics and particle physics, particularly in
the context of dark sectors. These systems enable precision tests of
fundamental theories, providing constraints on new physics beyond the Standard
Model. In gravitational physics, binary pulsars have been instrumental in
rigorously testing general relativity and alternative theories of gravity in
strong-field regimes Freire et al. (2012); Stairs (2003). Notably, the Hulse-
Taylor Pulsar (PSR B1913$+$16) Hulse and Taylor (1975); Weisberg and Huang
(2016) and the Double Pulsar System (PSR J0737$-$3039A/B) Lyne et al. (2004);
Kramer et al. (2021) have played key roles in validating general relativity’s
predictions Peters (1964); Peters and Mathews (1963). Furthermore, the
sensitivity of binary pulsar orbital periods to energy loss mechanisms Huang
(1963); Jeans (1924); Jeans. (1925) makes them ideal for investigating
particle interactions that extend beyond the Standard Model Goldman and
Nussinov (2010). Observations of these periods have led to constraints on
ultralight dark matter candidates Blas et al. (2017); Armaleo et al. (2020);
Kumar Poddar et al. (2019); Dror et al. (2020) and limits on dark baryon
processes involving neutron-mirror neutron mixing Goldman et al. (2019), as
well as broader constraints on various forms of quasi-equilibrium BNV Berryman
et al. (2022). The implications of these constraints, derived from binary
pulsar period observations, on quasi-equilibrium BNV and specific dark decays
of baryons are further explored in Section 4.

Pulsar Timing: A Versatile Tool for Exploring Dark Sectors The precision of
pulsar timing has become an essential tool in astrophysics, especially through
pulsar timing arrays (PTA), which are crucial in detecting stochastic
gravitational waves Agazie et al. (2023); Antoniadis et al. (2023); Reardon et
al. (2023); Xu et al. (2023). This technique also plays a pivotal role in
exploring ultralight dark matter (ULDM), as ULDM candidates with masses around
$10^{-22}$ eV are predicted to create oscillating gravitational potentials at
nanohertz frequencies Khmelnitsky and Rubakov (2014). Such oscillations lead
to periodic variations in the arrival times of radio pulses from pulsars,
overlapping with the frequency range for gravitational waves detectable by
PTAs Porayko et al. (2018). The European Pulsar Timing Array has effectively
used this aspect to significantly constrain scenarios in which dark matter
interacts only gravitationally, demonstrating that ultralight particles in the
mass range $10^{-24.0}\,{\rm eV}\lesssim m\lesssim 10^{-23.3}\,{\rm eV}$
cannot fully account for the observed local dark matter density Smarra et al.
(2023). Additionally, pulsar timing exhibits sensitivity to the effects of
heavier dark matter candidates, including primordial black holes and compact
subhalos, thereby extending its utility in constraining a wide mass range of
dark matter candidates Dror et al. (2019). In addition, pulsar timing shows
potential for probing BNV Zakeri (2023). Anomalies in pulsar spin-down rates
and braking indices could be indicative of BNV processes, possibly leading to
pulsar spin-up and the reactivation of dormant pulsars. With current timing
precision sensitive to BNV rates of $\Gamma\sim 10^{-9}\,{\rm yr}^{-1}$, PTAs
are increasingly becoming a crucial tool for future BNV detection or
constraints, particularly as their precision improves in the context of
gravitational wave detection.

###  2.3 Quasi-equilibrium dynamics of baryon loss in neutron stars

In the preceding section, we explored ways in which dark sector extensions to
the Standard Model can impact neutron stars. Building on this foundation, this
section delves further into these phenomena, with a particular focus on the
implications of specific dark decays of baryons within neutron stars.

One intriguing example involves neutron dark decays, such as $n\to\chi\gamma$.
These should occur at a rate of around $10^{-5}\,{\rm s}^{-1}$ to address the
neutron lifetime anomaly. If the dark decay products, such as $\chi$, remain
stable over short timescales, they could swiftly saturate the interior of a
neutron star shortly after the collapse of its progenitor star’s core. This
process is expected to drive dynamical effects whose outcome would be chemical
equilibrium between the decay products and the star’s baryons. Modeling the
EoS under these circumstances necessitates incorporating new degrees of
freedom associated with $\chi$, affecting the star’s pressure and energy
density. The ground state of this system would be determined by the lowest
energy state, subject to charge conservation and an extended baryon number
concept, where $B(\chi)=1$. Should $\chi$ maintain stability over long
timescales, the neutron star would eventually settle into a static state. This
scenario illustrates a static dark sector influence, altering the star’s
configuration space via its EoS. In this case, static limits such as the
maximum mass of neutron stars can be used to constrain the dark decays.

If neutron decay products such as $\chi$ further decay or engage in
annihilation leading to particles that escape the star, the situation becomes
dynamically complex. It is important to note that these depletion processes
need not be direct; they may involve a series of reactions. For instance, if
$\chi$ decays into leptons, it would explicitly violate the baryon number
conservation. This, in turn, would trigger a sequence of baryon-conserving
reactions within the neutron star. The resulting depletion of baryons
manifests itself not only through neutrino emission but also through
contributions to the thermal energy and work done on the stellar fluid. In
this case, continuous neutron decays could dynamically influence neutron star
properties over its lifetime. Ongoing processes like these could lead to
observable changes in neutron stars, including modifications in thermal
evolution, variations in pulsar spin, or alterations in the orbital periods of
binary pulsars. In this context, neutron stars offer an observational platform
for constraining and potentially revealing active dark decay processes.

Broadening our perspective, the overall influence of baryon dark decays on
neutron stars depends on the production ($\Gamma_{P}$) and depletion
($\Gamma_{D}$) rates of dark particles, as well as the sensitivity
($\sigma_{\text{obs}}$) of neutron star observations. While the depletion
mechanisms themselves vary — manifesting as either the direct escape of dark
particles from the star or explicit BNV — the crucial factor in modeling these
effects is their rate, with one notable exception. The thermal evolution of a
neutron star does require a more detailed examination of the depletion
mechanism, since it is influenced by the specific sequence of reactions
involved in the process. These diverse scenarios necessitate distinct
computational approaches, as depicted in Figure 4.

Figure 4:  The potential pathways and resultant states stemming from dark
particle decay in neutron stars, dictated by production ($\Gamma_{\text{P}}$)
and depletion ($\Gamma_{\text{D}}$) rates. Observational sensitivity
($\sigma_{\text{obs}}$) and the star’s response rates—chemical
($\Gamma_{\text{chem}}$) and hydrodynamic ($\Gamma_{\text{hyd}}$)ones—further
influence these pathways. A departure from chemical and/or hydrostatic quasi-
equilibrium (denoted by ✗) necessitates numerical simulations for precise
stellar modeling, with the expectation that stars transition from a non-
equilibrium to a quasi-equilibrium state over time.

In cases where depletion ($\Gamma_{D}$) exceeds production ($\Gamma_{P}$), the
star is primarily comprised of Standard Model particles. Here, $\Gamma_{P}$
becomes the effective Baryon Number Violating (BNV) rate to compare against
$\sigma_{\text{obs}}$. Conversely, if production ($\Gamma_{P}$) surpasses
depletion ($\Gamma_{D}$), dark sector particles must be included in the EoS.
The depletion rate ($\Gamma_{D}$) then serves as the effective BNV rate,
because this determines the time scale over which the star changes.

In scenarios in which neutron star observations are not sensitive to the BNV
rate ($\Gamma_{\rm BNV}<\sigma_{\text{obs}}$):

  1. 1.

Production Dominance ($\Gamma_{P}>\Gamma_{D}$) leads to static effects via EoS
modifications, incorporating dark particles.

  2. 2.

Depletion Dominance ($\Gamma_{P}<\Gamma_{D}$) results in no observable
effects.

Conversely, when observations are sufficiently sensitive to possible BNV
effects ($\Gamma_{\rm BNV}>\sigma_{\text{obs}}$), and under the assumption
that in degenerate cold neutron stars the hydrodynamic response rate exceeds
the chemical rate ($\Gamma_{\rm hyd}>\Gamma_{\rm chem}$):

  1. I.

The star is in both chemical and hydrostatic quasi-equilibrium if $\Gamma_{\rm
BNV}<\Gamma_{\rm chem}$. This state warrants a quasi-equilibrium analysis.

  2. II.

The star remains only in hydrostatic quasi-equilibrium if $\Gamma_{\rm
chem}<\Gamma_{\rm BNV}<\Gamma_{\rm hyd}$. A hydrostatic equilibrium
description is valid on short timescales ($\delta t<\Gamma_{\rm BNV}^{-1}$).
However, a complete reaction chain must be considered, as chemical equilibrium
is absent in the EoS.

  3. III.

Absence of both equilibria when $\Gamma_{\rm BNV}>\Gamma_{\rm hyd}$ requires
comprehensive simulations of the star’s hydrodynamic and chemical evolution.

Notably, in the quasi-equilibrium scenario (I.), baryon number conservation on
short timescales forms a key constraint in the derivation of its EOS. The
primary distinction between production- and depletion-dominated states lies in
the inclusion or exclusion of dark sector elements and the extended definition
of baryon number.

The quasi-equilibrium framework posits that if dark-sector perturbations, such
as baryon decays, proceed at a slower rate than the star’s responses, the star
will evolve within an equilibrium configuration space defined by the EoS. In
scenarios where dark particle production dominates, this EoS incorporates the
pertinent dark degrees of freedom. Conversely, in cases where depletion
prevails, the EoS remains akin to the standard formulation, exclusive of dark
sector influences. This approach effectively balances the dynamical aspects of
dark sector interactions against the inherent stabilizing mechanisms of
neutron stars. The computational simplicity in depletion-dominated scenarios
is significant, as the EoS need not reflect intricate dark sector details,
facilitating a model-independent analysis.

Building upon this framework, the configuration space of neutron stars can be
effectively parameterized by a set of $n$ independent parameters, denoted as
$\\{\Xi_{1},\Xi_{2},\ldots,\Xi_{n}\\}$. Variations in neutron star
observables, symbolized as $\mathcal{O}$, can then be quantified through the
following expression:

$\dot{\mathcal{O}}\equiv\frac{d\mathcal{O}}{dt}=\sum_{i=1}^{n}\left(\frac{d\Xi_{i}}{dt}\right)\frac{\partial\mathcal{O}}{\partial\Xi_{i}},$
(2)

Here, each term ${\partial\mathcal{O}}/{\partial\Xi_{i}}$ represents the
sensitivity of the observable to changes in the $i$-th parameter, as governed
by the EoS. The rates of change for these parameters, ${d\Xi_{i}}/{dt}$, are
determined by imposing constraints on the system’s quasi-equilibrium dynamics.
These constraints may include theoretical principles, such as the conservation
or specific rate of violation of baryon number, alongside observational data,
like the spin-down rates of pulsars. Together, these factors provide a
comprehensive framework to model and understand the evolution of neutron star
observables within the prescribed parameter space.

Before exploring specific examples, we think it crucial to recognize that
certain neutron star characteristics, such as its surface temperature,
rotational period, or its magnetic field, may not always be central to our
framework, particularly when these factors do not significantly influence the
observable under study. Neutron stars, composed of cold, degenerate matter,
experience minimal thermal contributions to their structure. Similarly,
rotational effects are often minor for pulsars with longer periods, generally
those exceeding a few milliseconds Glendenning (1997). Therefore, dynamical
evolution within the configuration space can often be treated as independent
from changes in the magnetic field, temperature, or rotation period. However,
it is important to note that these elements should be explicitly included in
the analysis when they directly impact the observable being studied. For
example, when investigating pulsar spin-down, rotation remains a crucial
component within the formalism. Similarly, temperature may be a vital factor
in processes that are subject to Pauli blocking mechanisms.

We now illustrate the application of the quasi-equilibrium framework with
three distinct examples, each of which is succinctly summarized in Table 1. In
each case we note what configuration space is modified through the dynamical
change indicated as well as the constraints that act on those dynamics.

Scenario |  Configuration Space |  Constraints & Model Inputs
---|---|---
Pulsar Spin-down |  $\Xi=\\{{\cal E}_{c},\omega_{c}\\}$ |  Observed spin-down rate ($\dot{\Omega}$)
Baryon conservation ($\dot{B}=0$)
Dark Core Formation |  $\Xi=\\{{\cal E}_{c}^{V},{\cal E}_{c}^{D}\\}$ |  Baryon conversion rate ($\dot{B}_{D}=\Gamma B_{V}$)
Total baryon conservation ($\dot{B}_{D}=-\dot{B}_{V}$)
Dark Baryon Decay |  $\Xi=\\{{\cal E}_{c}\\}$ |  Total baryon loss rate ($\dot{B}=-\Gamma B$)
Table 1: Quasi-equilibrium frameworks for neutron star evolution under various
scenarios. Parameters include the central energy densities for the entire star
(${\cal E}_{c}$), visible sector (${\cal E}_{c}^{V}$), and dark sector (${\cal
E}_{c}^{D}$), as well as the central angular velocity ($\omega_{c}$). The
observed spin-down rate is indicated by $\dot{\Omega}$, while the rate of
baryon number violation or conversion in dark sector processes is denoted by
$\Gamma$.

  1. 1.

Standard Baryon-Conserving Spin-Down of a Pulsar:

     * •

Configuration Space: This is defined by $\Xi=\\{{\cal E}_{c},\omega_{c}\\}$,
where ${\cal E}_{c}$ represents the central energy density and $\omega_{c}$
denotes the central angular velocity of local inertial frames Hartle (1967);
Glendenning (1997).

     * •

Constraints: These are governed by the observed spin-down rate
($\dot{\Omega}$), while maintaining baryon number conservation ($\dot{B}=0$).

  2. 2.

Formation of a Dark Core without Depletion:

     * •

Configuration Space: This is defined by $\Xi=\\{{\cal E}_{c}^{V},{\cal
E}_{c}^{D}\\}$, which represent the central energy densities of the visible
(${V}$) and dark (${D}$) sectors at the star’s core. The formation rate is
generally assumed to be comparable to or slower than the star’s age. This rate
does not result in thermalization between the two sectors. Consequently, the
sectors are characterized by distinct energy densities and pressures,
interacting primarily through gravitational influence.

     * •

Constraints: The total baryon number conservation
($\dot{B}_{D}+\dot{B}_{V}=0$) acts as a constraint, ensuring a balanced rate
of baryon conversion between the dark and visible sectors. The conversion rate
from visible to dark baryons ($\dot{B}_{D}$) is derived from the specifics of
the particle physics governing the decay or conversion processes.

  3. 3.

Dark Baryon Decays Followed by Efficient Depletion:

     * •

Configuration Space: This space, also known as the _single parameter sequence_
(illustrated in Figure 5), is singularly defined by $\Xi=\\{{\cal E}_{c}\\}$
(if the star is not rapidly rotating). Here, ${\cal E}_{c}$ denotes the
central energy density of the neutron star. This simplified configuration
space reflects the scenario where dark baryon decays and subsequent depletion
processes dominantly influence the neutron star’s structure.

     * •

Constraints: The evolution is primarily governed by the total baryon loss rate
$\dot{B}$. This rate is instrumental in dictating the evolution of the star’s
core composition and overall structure, as indicated by the green arrow in
Figure 5, which represents the quasi-equilibrium evolution along the single
parameter sequence.

Figure 5:  Variations of neutron star properties as functions of central
energy density (${\cal E}_{c}$): mass ($M$), radius ($R$), baryon number
($B$), and moment of inertia ($I$), all normalized to typical values. Namely,
$M_{\text{typical}}=1.4\,M_{\odot}$, $R_{\text{typical}}=12\,{\rm km}$,
$B_{\text{typical}}=10^{57}$, and $I_{\text{typical}}=70\,M_{\odot}\,{\rm
km}^{2}$. The calculations employ the DS(CMF)-1 EoS Dexheimer (2021). Notable
features include the quasi-equilibrium baryon loss (indicated by the green
arrow) and the onset of hyperon appearance (denoted by cyan vertical dashed
lines) at sufficiently large central energy density.

Transitioning from the comprehensive analysis of neutron stars’ quasi-
equilibrium dynamics, we now shift our focus to specialized dark sector
models, particularly those that expand neutron dark decay scenarios to
encompass dark co-genesis. Such models propose a unified origin for the
observed baryon asymmetry and a corresponding asymmetry within the dark
sector. This approach intriguingly links the physics of neutron stars to wider
cosmological narratives.

###  2.4 Illustrative dark sector models

We have considered neutron decays in which a dark baryon $\chi$ appears, and
the manner in which we can probe this possibility in a neutron star depends on
if and how $\chi$ decays to other dark-sector particles. In this section, we
consider just one particular dark-sector interaction, but its impact on a
neutron star can vary significantly, depending on the assumed spectrum of
particles in the dark sector. Indeed with suitable choices we will see that it
can sample all the scenarios outlined in the previous section.

Although we certainly expect $\chi$ to decay were it to play a role in
mesogenesis, it need not — perhaps $\chi$ is stable and a possible dark matter
candidate in and of itself. This would require, of course, that the $\chi$ is
light enough that it cannot decay to light anti-baryons, and we presume it
does not decay just to mesons and/or leptons alone, so that true baryon-
number-violating ($|\Delta B|=1$) processes do not appear in this context.
Thus as in Elor et al. (2019); Alonso-Álvarez et al. (2021) we introduce two
new particles, a scalar baryon $\phi_{B}$ with $B=1$ and a Majorana fermion
$\xi$. It is important that neither $\phi_{B}$ nor $\xi$ decay, nor that they
transform into each other. However, $\chi$ may decay via Elor et al. (2019)

${\cal L}_{\rm dark}\supset y_{d}{\bar{\chi}}\phi_{B}\xi+\rm h.c.\,,$ (3)

where either $\phi_{B}$ or $\xi$ or both could be dark-matter candidates. This
interaction can destabilize the proton or produce $\chi$ decay or $\chi\chi$
annihilation, with examples as shown in Figures 6a,6b. For different choices
of the particles’ masses, different processes appear. For example,

  * •

If $M_{\phi_{B}},M_{\xi}>M_{\chi}$, then $\chi$ does not decay.

  * •

If $M_{\phi_{B}}+M_{n}>M_{\xi}+M_{\pi^{0}}$, then induced nucleon decay
($\phi_{B}^{*}+n\to\xi+\pi^{0}$) can appear as in Figure 6a. Similarly if
$M_{\xi}+M_{n}>M_{\phi_{B}}+M_{\pi^{0}}$, then $\xi+n\to\phi_{B}+\pi^{0}$ can
appear.

  * •

If $M_{\chi}>M_{\phi_{B}}$ but $M_{\xi}>M_{\chi}$, then
$\chi\chi\to\phi_{B}\phi_{B}$ can occur, as shown in Figure 6b, though
$\chi\to\phi_{B}\xi$ cannot. Similarly, if $M_{\chi}>M_{\xi}$ but
$M_{\phi_{B}}>M_{\chi}$, then $\chi\chi\to\xi\xi$ can occur instead.

  * •

If $M_{\chi}>M_{\phi_{B}}+M_{\chi}$, then $\chi\to\phi_{B}\xi$ can occur.

If $\chi$ does not decay, the neutron dark decay rate is constrained, or
alternatively, $\chi$ self-interactions must be present, as previously noted
in Section 1, to align with observed neutron star masses. This scenario
corresponds to the production dominance route depicted in Figure 4, leading to
static effects such as the modification of the maximum theoretical neutron
star mass.

However, with a suitable choice of dark-sector masses, subsequent to neutron
decay to a final state with a $\chi$ particle, either $\phi_{B}$ and/or $\xi$
can appear. These outcomes can act to different effect within a neutron star.
Either sort of particle could fall to the core of the star, and modify its
evolution — and possibly act to destabilizing effect McDermott et al. (2012).
Alternatively, if $\phi_{B}$ and $\xi$ are light enough they can escape the
star. Thus the scenarios described in the previous section can all be sampled
through the dark sector interaction of Eq. (3). In our own investigations
Berryman et al. (2022, 2023), we have used a quasi-static approach to
determine the evolution of the star in the presence of slow apparent BNV to
dark final states. The consistency of this analysis, that new-physics dynamics
do not impact the evolution of the star, requires that we have an effective
mechanism to remove energy from the star upon decay to a dark final state.
Accordingly, our analysis is predicated on the assumptions inherent in the
depletion dominance pathway of Figure 4, which culminates in a quasi-
equilibrium evolution of the star. As a concrete example of such, we have
supposed that $M_{\xi}\gg M_{\phi_{B}},M_{\chi}$, realizing $\chi$ removal
through $\chi\chi\to\phi_{B}\phi_{B}$ decay, with $\phi_{B}$ light enough to
escape the star.

Additional physical constraints appear if we work within the context of a dark
co-genesis model. For example, if $\xi\to\phi_{B}\chi$ could occur, and it
would under the conditions we have just outlined for quasi-static evolution of
the star, then if $\chi$ is heavy enough, $\chi\to{p}\pi^{-}$ and
$\chi\to{n}\pi^{0}$ can also appear. In a broader context, such processes
would act to dilute a cosmic baryon asymmetry. Thus either the flavor
structure of the pertinent quark-$\chi$ couplings must be sufficiently small,
or the mass of $\chi$ is further constrained. Following the latter path, we
note that successful baryogenesis within the dark sector scenario of Eq. (3)
can be realized if $m_{\chi}\leq 1.07784\,\rm GeV$. This and nuclear stability
considerations gives us the following window on $\chi$ Berryman et al. (2023):

$0.937993\,{\rm GeV}<m_{\chi}<1.07784\,{\rm GeV}\,,$ (4)

though in what follows we consider neutron-star constraints on $\chi$-baryon
mixing over a larger mass window, in anticipation of other model-building
solutions. We now turn to the evaluation of these decays within the dense
medium of a neutron star.

(a)  (b)

Figure 6: (a) Induced nucleon decay. (b) $\chi\chi$ annihilation, from
Berryman et al. (2023).

##  3 Neutron stars and particle decays in dense matter

The consideration of dense matter phenomena is crucial for investigating
baryon interactions in neutron stars, where intense gravitational forces
compress baryonic matter to densities exceeding nuclear saturation. Such
extreme conditions push the system into the strongly repulsive domain of
nuclear interactions, resulting in an elevation of the ground state energies
of baryons. This increase in energy levels potentially allows for decay
processes that are otherwise kinematically forbidden in both vacuum and within
nuclei Berryman et al. (2023). Consequently, neutron stars present themselves
as unique astrophysical laboratories, offering the possibility to observe dark
baryon decays that bypass the constraints usually encountered in earthbound
experiments such as Super-Kamiokande or KamLAND Takhistov et al. (2015); Araki
et al. (2006). To gain a more comprehensive and quantitative insight into this
phenomenon, we delve into a particular formalism of dense matter. This
exploration aims to uncover the unique role of heavy neutron stars as cosmic
laboratories, offering new probes of dark sectors.

In this vein, a hadronic relativistic mean-field (RMF) theory offers a robust
framework for addressing dense matter interactions in such extreme
environments Walecka (1974); Serot and Walecka (1986, 1997); Dexheimer and
Schramm (2008). Within RMF theory, baryon interactions are mediated through
various Lorentz scalar and vector mesons. The Euler-Lagrange equations that
govern the dynamics of these meson fields include baryon currents as source
terms, taking the form of $\overline{\psi}\psi$ for scalar mesons and
$\overline{\psi}\gamma^{\mu}\psi$ for vector mesons. As the baryon density
increases, these currents intensify, leading to a significant vacuum
expectation value (VEV) for the meson fields. Within this framework, the
mesonic degrees of freedom are thus effectively represented by their classical
VEVs, a simplification that sees scalar meson VEVs contributing to the baryon
masses and vector mesons to their four-momentum. This theoretical construct
provides the foundation for understanding baryon behavior in the dense,
locally uniform medium of neutron stars.

In such a medium, the wave-function for a baryon with a canonical four-
momentum $p^{\mu}$ is expressed as

$\psi(x)=e^{-ip\cdot x}u(p^{*},\lambda),$ (5)

where $p^{*\mu}\equiv
p^{\mu}-\Sigma^{\mu}=\left\\{E^{*}(p^{*}),\vec{p}-\vec{\Sigma}\right\\}$
represents the kinetic four-momentum. The vector self-energy ($\Sigma^{\mu}$),
generated by the vector meson VEVs, is characterized by $\vec{\Sigma}=0$ in
the nuclear matter frame. The time-component of $p^{*\mu}$, is given by
$E^{*}(p^{*})\equiv\sqrt{{m^{*}}^{2}+|\vec{p}^{*}|^{2}}$, with $m^{*}$
signifying the baryon’s effective mass. The baryon spinor $u(p^{*},\lambda)$,
satisfying the Dirac equation

$\left(\not{p}^{*}-m^{*}\right)u(p^{*},\lambda)=0,$ (6)

is solved in the Dirac-Pauli representation as

$u(p^{*},\lambda)=\sqrt{E^{*}(p^{*})+m^{*}}\begin{pmatrix}1\\\
\frac{\vec{\sigma}\cdot\vec{p}^{\,*}}{E^{*}(p^{*})+m^{*}}\end{pmatrix}\eta_{\lambda},$
(7)

where $\vec{\sigma}$ are the Pauli matrices, and $\eta_{\lambda}$ is the Pauli
spinor with $\eta_{\uparrow}=(1,0)^{T}$ and $\eta_{\downarrow}=(0,1)^{T}$. The
baryons’ energy spectrum ($p^{0}$) is delineated by

$E(p)=\sqrt{{m^{*}}^{2}+|\vec{p}-\vec{\Sigma}|^{2}}+\Sigma^{0},$ (8)

where we note that in the RMF approximation, $\Sigma^{\mu}$ and $m^{*}$ are
density-dependent but independent of $p^{\mu}$. The CM energy of baryons,
denoted as $E_{\rm cm}(p)$, is a function of their effective mass $m^{*}$ and
momentum $\vec{p}$ in the nuclear matter frame. It is expressed by the
equation:

$\displaystyle E_{\rm
cm}(p)=\sqrt{\left[E(p)\right]^{2}-\left|\vec{p}\right|^{2}}=\sqrt{\left(\Sigma^{0}\right)^{2}+\left(m^{*}\right)^{2}+2\Sigma^{0}\sqrt{\left(m^{*}\right)^{2}+\left|\vec{p}\right|^{2}}}.$
(9)

This energy reaches its minimum when the momentum $|\vec{p}|$ is zero and
reaches its maximum at the Fermi momentum. The numerical values of $m^{*}$,
$\Sigma^{0}$, and $E_{\rm cm}$ for baryons within neutron stars under the
DS(CMF)-1 EoS Dexheimer (2021), which comes from a more sophisticated
descendent Dexheimer and Schramm (2008) of the RMF model we have noted, are
depicted as functions of density in Figure 7. (We anticipate further
evolutionary development of the RMF approach through a determination of its
parameters from a description of pure neutron matter within chiral effective
theory Alford et al. (2022).) As evidenced in Figure 7, a notable reduction in
effective baryon masses ($m^{*}$) at higher densities is observed, which is
indicative of chiral symmetry restoration in these high-density regimes in
this model. Furthermore, the dynamics between $m^{*}$ and $\Sigma^{0}$ (in the
nuclear matter frame) are such that while $m^{*}$ decreases, $\Sigma^{0}$
increases, cumulatively leading to an increase in the total energy of baryons
as delineated by Eq. (8) at higher densities. The core (maximum) density of
four pulsars are denoted by vertical dot-dashed lines for comparison. This
sets the stage for understanding how each neutron star, with its unique
density profile, represents a distinct astrophysical laboratory.

Figure 8 further demonstrates how each neutron star offers a unique probe
intrinsically linked to its specific density profile. The CM energy spectrum
of baryons ($\Lambda$, neutron ($n$), and proton ($p$)) in neutron stars PSR
J1614$-$2230, PSR J0348+0432, and PSR J0737$-$3039A is observed to surpass
their vacuum rest mass within the stars’ cores, as shown in the figure.
Notably, heavier neutron stars such as PSR J1614$-$2230 and PSR J0348+0432
exhibit a higher energy density reach, underscoring the relationship between a
star’s mass and its potential for exploring baryon dark decays. This
characteristic of neutron stars, particularly the more massive ones,
highlights their capability to facilitate baryonic decays into states
kinematically forbidden in vacuum conditions. The unique energy reaches
offered by these pulsars offer invaluable insights into dark matter models
beyond the scope of terrestrial experiments.In the ensuing section, we
demonstrate how observations of binary pulsars, in conjunction with the
considerations of dense matter outlined herein, pave the way for novel
constraints on dark decays of baryons.

Figure 7:  Variations in baryonic properties under the DS(CMF)-1 EoS, depicted
as functions of density. (a) Effective masses of various baryons. (b) Vector
self-energies of the baryons in the nuclear matter frame. (c) Range of center-
of-mass (CM) frame energies for the baryons. Vertical lines in all panels
represent the central number densities ($n_{c}$) of the pulsars examined in
Section 4, while horizontal lines correspond to the vacuum masses of the
respective baryons.  Figure 8:  Energy spectra of baryons ($\Lambda$, neutron
($n$), and proton ($p$)) in their center-of-mass (CM) frames within specific
neutron stars (a) PSR J0348+0432, (b) PSR J1614$-$2230, and (c) PSR
J0737$-$3039A, plotted as functions of stellar radius. The plot highlights how
the CM energy in high-density regions exceeds the baryons’ vacuum rest mass,
hinting at the possibility of decay into states that are kinematically
forbidden in vacuum. Detailed analysis of these phenomena is provided in the
main text.

##  4 Baryon-loss constraints from energy-loss limits in pulsar binaries

Binary pulsar orbital periods, sensitive to energy loss mechanisms, present an
effective observational tool for probing quasi-equilibrium BNV processes. In
scenarios with efficient depletion, such as Example 3 outlined in Section 2.3,
quasi-equilibrium BNV at a rate $\dot{B}$ contributes to an energy loss
$\dot{M}^{\rm eff}$ from neutron stars as Berryman et al. (2022):

$\dot{M}^{\rm eff}=\left(\partial_{{\cal
E}_{c}}M+\left(\frac{\Omega^{2}}{2}\right)\partial_{{\cal
E}_{c}}I\right)\left(\frac{\dot{B}}{\partial_{{\cal E}_{c}}B}\right),$ (10)

where $I$ denotes the moment of inertia and $\Omega$ the observed spin
frequency. This energy loss affects the binary pulsar orbital period Huang
(1963); Jeans (1924); Jeans. (1925):

$\dot{P}_{b}^{\dot{E}}=-2\left(\frac{\dot{M}_{1}^{\rm eff}+\dot{M}_{2}^{\rm
eff}}{M_{1}+M_{2}}\right)\,P_{b},$ (11)

in which $1$ and $2$ represent binary components. The observed orbital period
decay rate, $\dot{P}_{b}^{\rm obs}$, combines gravitational radiation
($\dot{P}_{b}^{\rm GR}$), intrinsic energy-loss ($\dot{P}_{b}^{\dot{E}}$), and
extrinsic factors ($\dot{P}_{b}^{\rm ext}$):

$\dot{P}_{b}^{\rm obs}=\dot{P}_{b}^{\rm
GR}+\dot{P}_{b}^{\dot{E}}+\dot{P}_{b}^{\rm ext}.$ (12)

The precision in these terms determines the sensitivity of BNV limits, making
binary pulsars with accurate timing ideal for stringent constraints. Moreover,
as outlined in Section 3, each pulsar, especially the more massive ones,
offers a unique opportunity for BNV study. Their larger mass extends the
density reach, enabling constraints on higher mass ranges of dark particles
like $\chi$ in processes such as $n\to\chi\gamma$. Furthermore, selecting
binary systems with heavier pulsars is essential for constraining dark decays
of hyperons. In EoSs that incorporate hyperon degrees of freedom, hyperons
start populating the medium only in extremely dense environments Ambartsumyan
and Saakyan (1960), specifically in the cores of heavier neutron stars (as
illustrated in Figure 8).

In our pursuit for comprehensive coverage of the model parameter space, a
strategic combination of binary pulsars—both heavy and precise—is essential.
Ideal candidates are systems with accurately determined masses, orbital
periods, and well-quantified contributions to each term in Eq. (12). It is
critical to select binary systems free from mass transfer between their
components to avoid complex alterations in the observed orbital period decay
rate, as mass transfer can significantly influence $\dot{P}_{b}^{\text{obs}}$
Rawley et al. (1986). Excluding the heavy candidate PSR J0952–0607, a “black
widow” pulsar with substantial mass uncertainty $M_{p}=(2.35\pm
0.17)\,M_{\odot}$ and significant rotational effects on its structure Romani
et al. (2022), our attention turns to the following candidates:

  * •

PSR J0740+6620: $M_{p}=(2.08\pm 0.07)\,M_{\odot}$ Fonseca et al. (2021),

  * •

PSR J0348$+$0432: $M_{p}=(2.01\pm 0.04)\,M_{\odot}$ Antoniadis et al. (2013),

  * •

PSR J1614–2230: $M_{p}=(1.908\pm 0.016)\,M_{\odot}$ Arzoumanian et al. (2018).

We ultimately select PSR J0348$+$0432 and PSR J1614–2230, which offer the
advantages of smaller errors in $M_{p}$ and $\dot{P}_{b}^{\text{obs}}$,
ensuring more precise constraints. Complementing this selection, we also
consider pulsar systems renowned for their high-precision measurements. Among
these, the double pulsar J0737$-$3039A/B Kramer et al. (2021) and the Hulse-
Taylor pulsar system Hulse and Taylor (1975) stand out. Since both are
characterized by similar mass profiles, we focus on the double pulsar
J0737$-$3039A/B, chosen for its superior precision.

The constraints derived from our selected pulsar systems are applied to a
specific model involving baryon decays ${\cal B}\to\chi\gamma$, dictated by an
effective mixing parameter $\varepsilon_{{\cal B}\chi}$. The in-medium
Lagrangian governing these decays is represented as:

$\mathcal{L}=\overline{\psi}_{\cal B}\left(i\not{\partial}-\not{\Sigma}_{\cal
B}-m_{\cal B}^{*}\right)\psi_{\cal
B}+\overline{\psi}_{\chi}\left(i\not{\partial}-m_{\chi}\right)\psi_{\chi}-\varepsilon_{{\cal
B}\chi}\left(\overline{\psi}_{\cal
B}\psi_{\chi}+\overline{\psi}_{\chi}\psi_{\cal B}\right)\,.$ (13)

To construct a combined limit ($\varepsilon^{\rm comb}_{{\cal B}\chi}$) from
individual pulsar constraints ($\varepsilon^{\,i}_{{\cal B}\chi}$), we utilize
the following relation:

$\varepsilon^{\rm\,comb}_{{\cal
B}\chi}=\Big{[}\sum_{i}\big{(}\varepsilon^{\,i}_{{\cal
B}\chi}\big{)}^{-4}\Big{]}^{-1/4}.$ (14)

The resulting limits, as derived in Ref. Berryman et al. (2023), are depicted
for individual systems and their combined limits under the assumption of the
DS(CMF)-1 EoS Dexheimer (2021), from the hadronic EoS of Dexheimer and Schramm
(2008) with a crust Gulminelli and Raduta (2015) from the CompOSE database
Typel et al. (2022) in Figure 9.

Figure 9:  Two-$\sigma$ exclusion constraints on the mixing parameters for
baryon-dark baryon interactions, derived under the depletion dominance
scenario Berryman et al. (2023) as illustrated in Figure 4 and based on the
DS(CMF)-1 EoS Berryman et al. (2023). The vertical purple band indicates the
range for the dark baryon mass ($m_{\chi}$), essential for nuclear stability
and successful baryogenesis as defined in Eq. (4). (a) Neutron ($n$)-dark
baryon ($\chi$) interaction constraints as functions of $m_{\chi}$. The
colored lines represent the exclusion limits from PSR J0348+0432 (green), PSR
J1614–2230 (blue), and the double pulsar system PSR J0737–3039A/B (orange),
with the combined limits depicted by a dotted black curve. The vertical dashed
lines mark the vacuum rest mass of the neutron. (b) Similar constraints for
the $\Lambda$-$\chi$ interaction, using the same color scheme and
presentation. Here, the vertical dashed lines represent the vacuum rest mass
of the $\Lambda$ hyperon.

As depicted in Figure 9, pulsars provide constraints on models with dark
baryon $\chi$ masses extending several hundred MeV above the mass of the
decaying baryon. Specifically, the heaviest pulsar, PSR J0348$+$0432,
demonstrates the broadest energy reach, while the most precisely measured
system, the double pulsar J0737$-$3039A/B, yields the most stringent limits
for lower $\chi$ mass ranges. By varying the choice of EoS we can explore the
robustness of our assessments. We have explored this for the entire family of
DS(CMF) EoS, DS(CMF)-1-8 Dexheimer (2021, 2021) in Ref. Berryman et al.
(2023). Interestingly, the combined $\varepsilon_{n\chi}$ constraint changes
negligibly below $\sim 1\,\rm GeV$, effectively covering our window of
interest, Eq. (4), whereas variations in $\varepsilon_{\Lambda\chi}$ of a
factor of a few appear over the entire mass range studied — and only
DS(CMF)-1,3,7 support the appearance of strange baryons for the parameters of
our candidate pulsars. We note that the $\epsilon_{n\chi}$ parameter at
$2\sigma$ is bounded to be no larger than $2\times 10^{-17}$ for the entire
window in $\chi$ mass we have supposed in a dark co-genesis scenario.

We consider the implications of these results in the next section.

##  5 Perspectives on the neutron lifetime anomaly and dark co-genesis

We now determine the outcomes from our neutron-star analysis in regard to the
in-vacuum limits on neutron dark decay. We can translate the constraints on
$\varepsilon_{{\cal B}\chi}$ we report in Figure 9 to ones on the in-vacuum
decay rate of ${\cal B}\to\chi\gamma$, using

$\displaystyle\Gamma\left({\cal B}\to\chi\gamma\right)=\frac{g_{{\cal
B}}^{2}\,e^{2}\,\varepsilon_{{\cal B}\chi}^{2}}{128\pi}\frac{\left(m_{{\cal
B}}+m_{\chi}\right)^{2}}{m_{{\cal B}}^{5}}\left(m_{{\cal
B}}^{2}-m_{\chi}^{2}\right).$ (15)

$g_{\cal B}$ signifies the baryon magnetic moment (Landé) $g$-factor, where
$g_{n}\simeq 3.826$ for neutrons and $g_{\Lambda}\simeq-1.226$ for $\Lambda$
baryons Workman et al. (2022). The resulting constraints on the vacuum
branching ratios are illustrated in Figure 10.

In Figure 11 we focus on the particular range of masses pertinent to the
neutron lifetime anomaly, reporting both $\epsilon_{n\chi}$ and ${\rm
B}(n\to\chi\gamma)$. It is apparent that terrestrial searches for baryon-
number-violating processes in underground detectors give the most stringent
limits on the possibility of $n\to\chi\gamma$, yet the windows on $\chi$ mass
open in such searches do not overlap with the possible mass window for the
neutron lifetime anomaly — the nuclear stability constraint we have noted
yields the lowest possible $\chi$ mass, reported in Eq. (4), and this
precludes that possibility. Nevertheless, we include the KamLAND Araki et al.
(2006) and SuperKamiokande Takhistov et al. (2015) limits, as well as the
limit on ${\rm B}(\Lambda\to\chi\gamma)$ from BESIII Ablikim et al. (2022) and
that from the SN1987A study of Alonso-Álvarez et al. (2021) for context.
Specifically, the KamLAND constraints are relevant to our study of the decay
$n\to\chi\gamma$; in these scenarios, the prompt photon would not meet the
correlation cuts, rendering it invisible. We observe that our neutron-star
limit is about $10^{18}$ times more severe than the accelerator limit and is
similarly less sensitive to BNV than the underground experiments.
Additionally, we incorporate constraints arising from the influence of
$\Lambda$ decay on the neutrino signal duration observed from SN1987A Alonso-
Álvarez et al. (2021). These constraints are derived from a supernova
simulation involving an $18.6\,M_{\odot}$ progenitor leading to a
$1.553\,M_{\odot}$ remnant Gar . Theoretically, these constraints could extend
to lower $\chi$ masses as well. Notably, our derived limits significantly
surpass the bounds established by the supernova data Alonso-Álvarez et al.
(2021).

The outcome of our analysis appears to exclude a new-physics interpretation of
the neutron lifetime anomaly, but the context in which we perform this
analysis is crucial: different choices within the dark sector could lead to
alternate outcomes, as explored in Section 2.3. In our case, we have chosen a
dark sector content and interaction as per Eq. (3), and accompanying
discussion, so that the SM dynamics of the neutron star determines the
evolution of the star under a change in its baryon number — with the injected
energy from the exotic neutron decay, particularly from $\chi$ production,
removed through $\chi\chi\to\phi_{B}\phi_{B}$ decay, with $\phi_{B}$
subsequently removed from the star. Our approach aligns with the depletion
dominance route depicted in Figure 4, resulting in a quasi-static evolution of
the neutron star. However, in alternative scenarios, establishing limits would
depend on varying pathways of dark decay, as outlined in the same figure.

As for dark-cogenesis models, to which our neutron dark decay studies connect,
such as $B$-mesogenesis Elor et al. (2019), we constrain the flavor-structure
of the quark-$\chi$ couplings through the limits we report Berryman et al.
(2023) — and we show our limits in our favored mass region by the shaded band
in Figure 9.

Figure 10:  Two-$\sigma$ exclusion limits on the vacuum branching fraction for
${\cal B}\to\chi\gamma$ processes are depicted in panels (a) for neutrons and
(b) for $\Lambda$ baryons. The green region represents limits from
SuperKamiokande Takhistov et al. (2015). We highlight that the KamLAND
constraint (red) Araki et al. (2006) is pertinent to our study of
$n\to\chi\gamma$ decay, as the prompt photon would not pass the correlation
cuts and would remain undetected. Panel (b) also presents limits from neutron
star studies for the DS(CMF)-7, DS(CMF)-3, and DS(CMF)-1 EoS in order of
increasing severity, along with constraints from the BES-III experiment (blue)
Ablikim et al. (2022), noting that the $\gamma$ in this case would also be
undetected, and the SN1987A limits (gold) Alonso-Álvarez et al. (2021). Figure
11:  Exclusion limits at 2$\sigma$ on (a) $\varepsilon_{n\chi}$ and (b) the
vacuum branching fraction for $n\to\chi\gamma$, from our quasi-static neutron-
star analysis in the particular $\chi$ mass region pertinent to an explanation
of the neutron lifetime anomaly. Additional constraints and expected limits
(dashed) as per McKeen and Pospelov (2020) have been included. This figure has
been taken from Berryman et al. (2023).

##  6 Summary

There is an increasing recognition that the solution to the dark-matter
problem may involve an entire sector of dark particles, with dark dynamics.
The presence of such dynamics can be probed in terrestrial experiments
sensitive to the determination of missing momentum Berlin et al. (2019) in
hadron decays, e.g., and in this article we have considered this broader
possibility in the context of neutron stars. Particularly, we have noted the
consequences from limits on the observation of energy loss in neutron stars,
particularly those that emerge from the timing of the period of binary
pulsars, on baryon dark decays, particularly of neutrons and $\Lambda$
baryons.

Neutron-star observables have multiple points of contact with the problem of
the neutron lifetime anomaly, and to the broader possibility of simultaneous
generation of dark matter and the BAU. We have noted models that would give
rise to $n\to\chi\gamma$, or to all dark decays, as described in Fornal (2023)
and Husain et al. (2023) in this volume, are limited by the need to form
neutron stars of sufficient mass. An unwanted softening of the neutron EoS
through such neutron-decay effects can be mitigated through the possibility of
$\chi$ self-interactions, or by reducing the neutron dark decay rate. We have
studied the possibility of using dark-sector interactions to eradicate $\chi$
particles produced through dark decays through $\chi\chi$ annihilation to
light, dark-sector particles that can escape from the star. Here we study the
limit in which the dark decays proceed slowly with respect to the usual Urca
processes that operate in the neutron star, enforcing chemical and thermal
equilibrium. Thus in this quasi-static framework we are able to study the
consequences of the observed energy-loss constraints from pulsar timing
studies on dark decays in the neutron stars. The constraints we have found
thus far are severe, but broader possibilities remain — and beckon. Certainly,
there are models of the neutron lifetime anomaly that are not constrained by
our concerns. Here we note the model that yields $n\to\chi\chi\chi$ Strumia
(2022). That model does not soften the EoS unduely, nor does it give rise to
substantial energy loss effects that could be constrained through our pulsar
timing studies. Thus the possibility of a new-physics solution to the neutron
lifetime anomaly remains, with the future neutron decay correlation and
lifetime studies offering the possibility of setting the controversy to rest,
though effects in neutron stars, also isolated ones, through observed heating
or spin effects may finally serve to identify the effects of dark sectors in a
positive way.

All authors contributed equally to this work.

This research was supported by the U.S. Department of Energy Office of Nuclear
Physics under contract DE-FG02-96ER40989.

No new data was generated in the production of this article.

###### Acknowledgements.

We thank Jeffrey M. Berryman for his past collaboration on many of the topics
in this overview. The authors declare no conflict of interest. References

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