# Cosmological and astrophysical probes of dark baryons

David McKeen<EMAIL_ADDRESS>TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T
2A3, Canada  Maxim Pospelov<EMAIL_ADDRESS>School of Physics and Astronomy,
University of Minnesota, Minneapolis, MN 55455, USA  William I. Fine
Theoretical Physics Institute, School of Physics and Astronomy, University of
Minnesota, Minneapolis, MN 55455, USA  Nirmal Raj<EMAIL_ADDRESS>TRIUMF, 4004
Wesbrook Mall, Vancouver, BC V6T 2A3, Canada

###### Abstract

We examine the cosmological and astrophysical signatures of a “dark baryon,” a
neutral fermion that mixes with the neutron. As the mixing is through a
higher-dimensional operator at the quark level, production of the dark baryon
at high energies is enhanced so that its abundance in the early universe may
be significant. Treating its initial abundance as a free parameter, we derive
new, powerful limits on the properties of the dark baryon. Primordial
nucleosynthesis and the cosmic microwave background provide strong constraints
due to the inter-conversion of neutrons to dark baryons through their induced
transition dipole, and due to late decays of the dark baryon. Additionally,
neutrons in a neutron star could decay slowly to dark baryons, providing a
novel source of heat that is constrained by measurements of pulsar
temperatures. Taking all the constraints into account, we identify parameter
space where the dark baryon can be a viable dark matter candidate and discuss
promising avenues for probing it.

##  I Introduction

New states with masses around a GeV that mix with the standard model (SM)
baryons have been considered recently for a number of compelling reasons, such
as the baryon asymmetry of the universe McKeen and Nelson (2016); Aitken  _et
al._ (2017); Babu  _et al._ (2013); *Allahverdi:2017edd; *Elor:2018twp;
*Nelson:2019fln; *Alonso-Alvarez:2019fym; Bringmann  _et al._ (2019), models
of dark matter McKeen and Nelson (2016); Karananas and Kassiteridis (2018);
Cline and Cornell (2018); Fornal  _et al._ (2020a), mirror matter scenarios
Berezhiani and Bento (2006); *Berezhiani:2015afa; *Berezhiani:2018eds, the
neutron lifetime anomaly Fornal and Grinstein (2018, 2020); Berezhiani
(2019b); Cline and Cornell (2018); Elahi and Mohammadi Najafabadi (2020), the
recent XENON1T excess McKeen  _et al._ (2020), $21~{}{\rm cm}$ cosmology Johns
and Koren (2020a); *Johns:2020rtp, and general baryon-number violating
phenomenology Heeck (2020); Fajfer and Susič (2020). In this paper we
investigate the cosmological and astrophysical effects of a simple, minimal
version: a single “dark baryon” $\chi$ that carries baryon number $B=1$ and
mixes with the neutron.

The phenomenology of our setup is characterized by just two parameters: the
$\chi$ mass, $m_{\chi}$, and the $n$-$\chi$ mixing angle, $\theta$. This
simple possibility has been singled out as a potential solution of the
$4\sigma$ discrepancy between “bottle” and “beam” measurements of the neutron
lifetime (although potentially introducing tension with determinations of the
nucleon axial vector coupling Czarnecki  _et al._ (2018)). Furthermore, such a
new state could have a long lifetime—either absolutely stable if
$m_{\chi}<m_{p}+m_{e}$ or cosmologically long-lived if $m_{\chi}>m_{p}+m_{e}$
and $\theta\ll 1$—and is therefore a potential dark matter (DM) candidate.

The simplicity of the “minimal” dark baryon model at low energy is somewhat
deceptive: processes in neutron stars (NS) tend to equilibrate $n$ and $\chi$
McKeen  _et al._ (2018); Baym  _et al._ (2018); *MottaNS; *EllisPattavinaNS;
*Goldman:2013qla, which together with the observations of the most massive NSs
seems to require additional repulsive interaction between $\chi$ particles
McKeen  _et al._ (2018). This may in turn require, e.g. a composite nature of
$\chi$ itself and/or the existence of a new “dark vector” force. Both these
scenarios hint at the existence of a more complicated dark sector (DS), with
some states being in the sub-GeV range. Another important consequence of the
compositeness of normal hadrons is the dimensionality of interaction that
induces the mixing angle $\theta$. The lowest dimension of the operator that
seems to mediate such a transition is six, e.g. symbolically
$\bar{\chi}(qqq)$, which in turn implies the enhancement of SM-DS inter-
conversion at high temperatures. If the cutoff scale of this operator is not
too far from the weak scale, a complete early thermalization of the SM and DS
is plausible, with subsequent chemical decoupling of the two sectors. This
motivates us to consider possible cosmological manifestations of the dark
baryon model, in addition to one that was already pointed out in the
literature, namely that $\chi$ might be a good candidate for a DM particle
McKeen and Nelson (2016); Karananas and Kassiteridis (2018); Cline and Cornell
(2018). Other cosmological effects were further studied in Refs. Karananas and
Kassiteridis (2018); Cline and Cornell (2018); Elahi and Mohammadi Najafabadi
(2020).

Here we examine key cosmological consequences of $\chi$ treating its initial
abundance as a free parameter in a wide range: $n_{\chi}\sim O(0.01-5)\times
n_{\rm baryon}$. We will see that $\chi$ can affect big bang nucleosynthesis
(BBN) of light nuclei by altering neutron-proton freeze-out, providing a
source of neutrons after the deuterium bottleneck, or by decaying and breaking
up light nuclei electromagnetically. In addition, $\chi$ that decays with a
lifetime between about $10^{7}-10^{26}~{}{\rm s}$ can disrupt the excellent
agreement between standard predictions and observations of the cosmic
microwave background (CMB) frequency spectrum and temperature anisotropies.
Further, we identify a novel late-time heating mechanism of NSs: if the decay
of stellar neutrons into dark baryons is very slow, equilibrium between the
two species is not achieved over the NS lifetime, and heat is generated by the
removal of neutrons from their Fermi sea followed by their replacement. We
exploit this phenomenon to place powerful constraints using NS temperature
measurements.

Existing limits on this model in the $\theta\gtrsim 10^{-10}$ region come from
a number of sources. The stability of the proton and ${}^{9}{\rm Be}$ require
$m_{\chi}>938.0~{}\rm MeV$ McKeen and Nelson (2016); Pfützner and Riisager
(2018). The exotic neutron decay modes $n\to\chi\gamma$ Tang  _et al._ (2018)
and $n\to\chi e^{+}e^{-}$Sun  _et al._ (2018) have also been searched for,
with null results providing limits on the parameter space. The low energy
Borexino spectrum has also been recast as a search for hydrogen decay which
can occur in this model when $m_{\chi}<m_{p}+m_{e}$ McKeen and Pospelov
(2020). As already mentioned, there is strong sensitivity to this model for
$m_{\chi}\lesssim 1.2~{}\rm GeV$ that come from the existence of heavy NSs
McKeen  _et al._ (2018); Baym  _et al._ (2018); *MottaNS; *EllisPattavinaNS;
*Goldman:2013qla. Other signatures include the diffuse $\gamma$-ray background
Berezhiani (2019b) and annihilation with nucleons in the case that the dark
matter is comprised of $\bar{\chi}$ with $B=-1$ Davoudiasl  _et al._ (2010);
*Davoudiasl:2011fj; *GaoJinAnnihil; *MarfatiaAnnihil.

This paper is laid out as follows. In Section II we introduce the dark baryon
model arising through mass mixing with the neutron, and provide expressions
for relevant rates. In Section III we outline the effects of $\chi$ on BBN
abundances and CMB measurements. We treat $\chi$ both lighter and heavier than
neutrons: the first region contains parameter space that explains the neutron
lifetime puzzle, and the second region is so far unexplored in the presence of
self-interactions obviating NS mass limits. In this section we also describe
heating of NSs via cosmologically slow decays of constituent neutrons into
$\chi$. In Section IV we derive constraints for initial abundances of $\chi$
equal to the DM abundance and 1% of the baryon abundance, which lead to
different phenomenologies. In Section V we provide discussion on future
prospects from cosmological, astrophysical, and terrestrial probes.

##  II Dark baryon model

The model consists of a charge-neutral fermion $\chi$ with $B=1$ which mixes
with the neutron,

$\mathcal{L}\supset-\delta(\bar{\chi}n+\bar{n}\chi)~{}.$ (1)

In principle, a $\bar{\chi}i\gamma_{5}n$ mixing term could exist as well. It
would modify some of the phenomenological consequences discussed below, but
not dramatically. We assume, effectively, that $\chi$ has the same internal
parity as the neutron, and that interaction (1) is parity-symmetric.

This effective hadron-level Lagrangian (1) could arise from a number of quark-
level UV completions; a simple one involves a scalar diquark with $ud$ and
$d\chi$ couplings (see, e.g. Arnold  _et al._ (2013); McKeen and Nelson
(2016)). For $\delta\ll|\Delta m|$, where $\Delta m=m_{n}-m_{\chi}$, the
$n$-$\chi$ mixing angle is $\theta=\delta/\Delta m$.

Diagonalization of the mass matrix results in a transition magnetic dipole,

$\mathcal{L}_{\rm
eff}\supset\frac{\mu_{n}}{2}\theta\bar{\chi}\sigma^{\mu\nu}nF_{\mu\nu}+{\rm
H.c.}$ (2)

In this expression, $\mu_{n}=-1.91\mu_{N}$ is the neutron magnetic dipole
moment, with $\mu_{N}=e/(2m_{n})\simeq 0.1~{}e\,{\rm fm}$ the nuclear
magneton. For $m_{\chi}<m_{n}$, the decay $n\to\chi\gamma$ occurs via (2) with
the rate

$\Gamma_{n\rightarrow\chi\gamma}=\theta^{2}\left(\frac{\mu_{n}}{\mu_{N}}\right)^{2}\frac{\alpha\omega^{3}}{m_{n}^{2}}~{},$
(3)

where $\omega=(m_{n}/2)(1-m_{\chi}^{2}/m_{n}^{2})\simeq\Delta m$ is the photon
energy in the neutron rest frame. In this paper we will use the latest world
average of the “bottle” measurement of the neutron lifetime to fix its total
width, $\tau_{n}=879.4\pm 0.6~{}\rm s$ Zyla  _et al._ (2020). In this case,
the branching fraction to $\chi\gamma$ is

${\rm Br}_{n\rightarrow\chi\gamma}\simeq 0.01\left(\frac{\theta}{5\times
10^{-10}}\right)^{2}\left(\frac{\Delta m}{\rm MeV}\right)^{3}~{}.$ (4)

For $m_{\chi}<m_{p}+m_{e}$, the conservation of baryon number ensures that
$\chi$ cannot decay. For $m_{\chi}>m_{p}+m_{e}$, $\chi$ decays through the
weak interaction like the neutron:

$\displaystyle\Gamma_{\chi\to pe^{-}\bar{\nu}}$
$\displaystyle=\frac{\theta^{2}}{\tau_{n}}\left(1-{\rm
Br}_{n\rightarrow\chi\gamma}\right)\frac{F(Q_{\chi}/m_{e})}{F(Q_{n}/m_{e})}$
(5) $\displaystyle\simeq\frac{1}{9\times 10^{22}~{}\rm
s}\left(\frac{\theta}{10^{-10}}\right)^{2}\frac{F(Q_{\chi}/m_{e})}{F(Q_{n}/m_{e})}~{}.$

Here, $Q_{\chi,n}=m_{\chi,n}-m_{p}-m_{e}$ and

$\displaystyle F(x)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{x(x+2)}}{60}(2x^{4}+8x^{3}+3x^{2}-10x-15)$ (6)
$\displaystyle+$ $\displaystyle\frac{x+1}{4}\ln(1+x+\sqrt{x^{2}+2x})~{}$

describes the available phase space. Despite not being absolutely stable,
$\chi$ can still live longer than the age of the universe $t_{U}$ for small
enough $\theta$, and can therefore potentially make up (part of) the DM even
in this part of parameter space. For $m_{\chi}>m_{n}$, the two-body mode
$n\gamma$, induced by Eq. (2), opens up. The rate for $\chi\to n\gamma$ is of
the same form as in Eq. (3) but with photon energy
$\omega=(m_{\chi}/2)(1-m_{n}^{2}/m_{\chi}^{2})$. Numerically,

$\displaystyle\Gamma_{\chi\rightarrow n\gamma}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2200~{}{\rm
s}}\bigg{(}\frac{\theta}{10^{-10}}\bigg{)}^{2}\left|\frac{\Delta m}{10~{}{\rm
MeV}}\right|^{3}~{}.$ (7)

Thus, we see that if this channel is open it dominates over the three-body
weak decay and that cosmologically interesting $\chi$ lifetimes require
smaller $\theta$. We will not discuss the regime $m_{\chi}>m_{n}+m_{\pi^{0}}$
in detail, but note that hadronic modes make up most of the $\chi$ decay width
when available and would lead to similar cosmological signatures.

##  III Signals

Given the model and rates above, we now turn to the cosmological and
astrophysical effects of a dark baryon. We describe BBN, CMB, and NS
signatures in turn.

A crucial ingredient here is the initial $\chi$ abundance,
$n_{\chi}^{0}=n_{\chi}(T_{0})$, which can depend on the details of cosmic
reheating as well as on the strength of the interactions between $\chi$ and
the SM. Take, for example, the transition dipole of Eq. (2). Through
scattering on the SM plasma at a temperature $T$, this leads to a rate of
$\chi$-number-change of roughly
$\Gamma_{\Delta\chi}\sim\theta^{2}\mu_{n}^{2}T^{3}$ which is comparable to the
expansion rate of the universe for

$T\gtrsim 100~{}{\rm MeV}\left(\frac{10^{-9}}{\theta}\right)^{2}.$ (8)

(Scattering through the strong interaction keeps $\chi$ in chemical
equilibrium down to even lower temperatures.) At temperatures above the QCD
transition, rather than neutrons one should consider processes involving
quarks and $\chi$. As mentioned above, the operator in (1) could result from
the dimension-6 operator $\bar{\chi}qqq/\Lambda^{2}$, giving $\chi$-number-
changing rates $\sim T^{5}/\Lambda^{4}$, with the cutoff scale
$\Lambda^{2}\propto(\theta\Delta m)^{-1}$. This can keep $\chi$’s in chemical
equilibrium down to GeV–PeV temperatures for $\theta\sim 10^{-10}-10^{-20}$
and $\Delta m\sim 1-100~{}{\rm MeV}$. For $T\gtrsim\Lambda$ further model-
dependence is involved since the UV theory that gives rise to the dimension-6
quark-level operator must be used.

So long as the temperature of the universe was at some point above the value
at which $\chi$ chemically decouples—which, as we see above, need not be very
large—the baryon number abundance would be shared between $\chi$ and SM baryon
species. Exact breakdown of total baryon number between $n,p$ and $\chi$ will
depend on when the decoupling happens, and the number of dark sector states.
(Besides $\chi$ there could be further metastable states with $B=1$.) If
$\chi$ is the sole “dark baryon” state, then it is reasonable to take
$n_{\chi}\sim n_{p}=n_{n}$ as an initial condition. To display our results, we
will focus on two benchmark scenarios: (i)
$n_{\chi}^{0}=5.4(n_{p}^{0}+n_{n}^{0})$ where $\chi$ could make up the
entirety of the DM if its lifetime $\tau_{\chi}\gtrsim t_{U}$111In this case
$\chi$ decays to nonrelativistic matter and the lower bound of $\sim 20\
t_{U}$ on the lifetime of DM decaying to radiation does not simply apply.
which could arise in a more detailed UV completion (see, e.g., ) and (ii)
$n_{\chi}^{0}=0.01(n_{p}^{0}+n_{n}^{0})$ which could arise if $\chi$ either
never came into chemical equilibrium with the SM or dropped out of equilibrium
at high temperatures.

These two benchmarks, for which models with differing high scale physics can
be easily incorporated, encapsulate the interesting phenomenological features
that we study.

###  III.1 BBN

The most important parts of the BBN reaction network for mass $A=1$–$4$ are
shown in the left panel of Fig. 1. The dark baryon is involved through
$n\leftrightarrow\chi$ conversions via the transition dipole of Eq. (2). Other
processes such as $pe^{-}\to\chi\nu_{e}$ and $\chi\to pe^{-}\bar{\nu}_{e}$ are
extremely slow due to weak- and $\theta$-suppressed rates and hence negligible
for BBN.

In the presence of this new interaction, we solve the coupled Boltzmann
equations that describe the evolution of the nuclide abundances seen in Fig.
1, using fits of thermally averaged weak interaction rates in Ref. Serpico
_et al._ (2004) and fits of all other thermonuclear reaction rates in Ref.
Cyburt (2004). In addition, we use the temperature evolution in the presence
of neutrino decoupling and $e^{+}e^{-}$ annihilation from Ref. Escudero Abenza
(2020). With this information we validate our BBN computation by the excellent
agreement with measured values of the abundances obtained in the $n_{\chi}\to
0$ limit. We describe the main effects for $m_{\chi}<m_{n}$ and
$m_{\chi}>m_{n}$ below.

####  III.1.1 $m_{\chi}<m_{n}$

Figure 1:  Left: The BBN reaction network. Neutron decays to dark baryons
$\chi$ could alter the branching fraction of $n$-$p$ conversion beta
processes, upsetting the timing of $n$-$p$ freeze-out. Further, $\chi$ can
overpopulate neutrons via inverse decays (for $m_{\chi}<m_{n}$) and direct
decays (for $m_{\chi}>m_{n}$), resulting in overabundances of nuclides. Right:
Helium-4 and deuterium abundances as a function of time or inverse temperature
for $\mathbf{m_{\chi}<m_{n}}$, fixing the $n$-$\chi$ mixing angle
$\theta=3\times 10^{-9}$, for $\chi$ comprising all of DM. For small
$\chi$-$n$ mass splittings inverse decays of $\chi$ greatly overproduce D,
while for large splittings the suppression of $n$-$p$ beta processes enhance D
and 4He abundances.

In this regime the rate for $n\to\chi$ is given by $\Gamma_{n\to\chi\gamma}$
in Eq. (3) while $\chi\to n$ proceeds through inverse decay,

$\Gamma_{\chi\gamma\to n}=\frac{\Gamma_{n\to\chi\gamma}}{e^{\Delta
m/T}-1}~{}.$ (9)

In this part of parameter space, $\chi$ is either stable or very long-lived on
the scale of BBN.

To describe the main effects on BBN, it is useful to briefly describe the
standard scenario. At high temperatures $n\leftrightarrow p$ reactions keep
neutrons and protons in equilibrium. When the temperature drops below
$Q_{np}\equiv m_{n}-m_{p}=1.29~{}\rm MeV$, the neutron-to-proton ratio becomes
Boltzmann-suppressed with $n_{n}/n_{p}\simeq\exp(-Q_{np}/T)$.
$n\leftrightarrow p$ reactions become inefficient when their rate,
$\Gamma_{np}\propto T^{5}/\tau_{n}$, drops below the expansion rate of the
universe, $H\propto T^{2}$. This occurs at $\overline{T}_{np}\simeq 0.7~{}\rm
MeV$ for $\tau_{n}=879.4~{}\rm s$ and the neutron-to-proton ratio freezes out
at $\exp(-Q_{np}/\overline{T}_{np})=0.17$, where, here and in what follows we
use bars to denote the values of quantities in the standard scenario. When
deuterium begins to be formed, this ratio is slightly reduced by neutron
decays so that $\overline{n_{n}/n_{p}}\simeq 0.14$. Nearly all neutrons
available at this time are processed into ${}^{4}{\rm He}$ and therefore the
resulting ${}^{4}{\rm He}$ mass fraction is
$\overline{Y}_{p}=2n_{n}/n_{p}(1+n_{n}/n_{p})\simeq 0.25$.

The presence of the $n\to\chi\gamma$ mode can modify this picture in the
following way. For fixed neutron lifetime, the $n\leftrightarrow p$ rate,
$\Gamma_{np}$, is reduced relative to the standard value by the factor ${\rm
Br}_{n\to p}=1-{\rm Br}_{n\to\chi}$ since the $n$-$p$ weak coupling must be
smaller.222We mention the impact of other determinations of the weak nucleon
coupling in Sec. IV. This means that the temperature at which
$n\leftrightarrow p$ conversion freezes out is related to the standard value
via

$T_{np}=\frac{\overline{T}_{np}}{\left(1-{\rm
Br}_{n\to\chi}\right)^{1/3}}\simeq\overline{T}_{np}\left(1+\frac{{\rm
Br}_{n\to\chi}}{3}\right)~{},$ (10)

resulting in more neutrons at freeze-out:

$\displaystyle\frac{n_{n}}{n_{p}}$
$\displaystyle\simeq\exp\left(-\frac{Q_{np}}{T_{np}}\right)\simeq\exp\left[-\frac{Q_{np}}{\overline{T}_{np}}\left(1-\frac{{\rm
Br}_{n\to\chi}}{3}\right)\right]~{},$ (11)

and the fractional increase of this ratio relative to its standard value is
therefore

$\displaystyle\frac{\delta(n_{n}/n_{p})}{\overline{n_{n}/n_{p}}}$
$\displaystyle\simeq\frac{Q_{np}{\rm Br}_{n\to\chi}}{3\overline{T}_{np}}\simeq
0.5\%\left(\frac{{\rm Br}_{n\to\chi}}{1\%}\right).$ (12)

Ignoring the small change in the fraction of neutrons that decay before
deuterium production, the fractional change in the ${}^{4}{\rm He}$ mass
fraction is simply

$\displaystyle\frac{\delta Y_{p}}{\overline{Y}_{p}}$
$\displaystyle\simeq\frac{\delta(n_{n}/n_{p})}{\overline{n_{n}/n_{p}}}\times\frac{1}{1+\overline{n_{n}/n_{p}}}$
(13) $\displaystyle\simeq 0.4\%\left(\frac{{\rm Br}_{n\to\chi}}{1\%}\right).$

Given the percent-level determination of $Y_{p}$, this limits the $n\to\chi$
branching ratio to a few percent.

For $\Delta m\lesssim 0.4~{}\rm MeV$, there is another effect: inverse decays
of neutrons through $\chi\gamma\to n$ after the deuterium bottleneck. One
should keep in mind that after the bottleneck, the density of free neutrons
plummets to low values, and even a small addition of extra neutrons could
shift the outcome of reactions for D/H. The left-over neutrons find protons
and efficiently produce deuterium that does not go on to be processed into
helium, leading to an extreme increase in the deuterium abundance. This rules
out the region of parameter space where ${\rm Br}_{n\to\chi\gamma}=1\%$ for
$\Delta m\lesssim 0.4~{}\rm MeV$.

The right panel of Fig. 1 shows the $Y_{p}$ and D/H evolution, as a function
of temperature for two representative points of the parameter space. For
$n_{\chi}^{0}/(n_{n}^{0}+n_{p}^{0})=5.4$ and mixing angle = $3\times 10^{-9}$,
a sizeable mass splitting $m_{n}-m_{\chi}=1$ MeV gives D/H close to the
standard value, but drastically overproduces $Y_{p}$. This is a direct result
of the modification of weak $n\leftrightarrow p$ conversion at the time of
neutron-proton freeze-out. Conversely, a mass splitting of 0.1 MeV predicts
$Y_{p}$ within its observational range, while giving an overproduction of D/H
by a factor $\sim 5$. This is the consequence of inverse decay,
$\chi+\gamma\to n$, and incorporation of extra neutrons into D. We conclude
that both these parameter points are excluded by the combination of $Y_{p}$
and D/H.

####  III.1.2 $m_{\chi}>m_{n}$

In this mass range, $\chi\to n\gamma$ proceeds directly while $n\to\chi$ goes
through inverse decay and is typically Boltzmann-suppressed. For
$\tau_{\chi}\lesssim 0.1~{}\rm s$, the $\chi$’s decay mostly before the onset
of BBN, and there is correspondingly no constraint. For longer lifetimes but
with $\tau_{\chi}\lesssim 10^{13}~{}{\rm s}$ such that the $\chi$’s decay
before the CMB epoch, there is potentially a mismatch in the baryon density
$\eta$ as measured in the CMB and through light element abundances. During
BBN, some baryon number is sequestered in $\chi$ which mostly plays the role
of spectator in this mass range. Each $\chi$ that later decays then produces a
baryon which contributes to the baryon density during the time relevant for
the CMB. The CMB constraint could be satisfied by dialing down the initial SM
baryon and $\chi$ abundance such that their sum is fixed to the CMB value.
However, this generally disturbs the agreement between prediction and
observation of the deuterium abundance which depends sensitively on the SM
baryon density during BBN. Since the CMB extraction of $\eta$ is at the
percent level, this means that the initial $\chi$ density must be smaller than
a few percent of the initial SM baryon density.

For a percent-level initial $n_{\chi}/n_{p}$, BBN provides nontrivial
constraints as a result of the decay $\chi\to n\gamma$. If the decays happen
at early times, this process could change $n\leftrightarrow p$ freeze-out,
altering the resulting abundances. The decays after the deuterium bottleneck
have two distinct effects: new neutrons are contributed via $\chi$ decay, and
non-thermal $\gamma$’s are created, which could split the newly formed nuclei.
Specifically, when the temperature drops below about $10~{}\rm keV$, the
electromagnetic cascade induced by the photon produced in $\chi$ decay can
lead to the photodissociation of light nuclides, changing the resulting light
element abundances in a complicated way.

The reaction network relevant for this photodissociation stage, indicating the
energy threshold for each reaction, is shown in the top panel of Fig. 2. To
obtain the nuclide abundances we use the photon spectrum $f_{\gamma}$ derived
in Refs. Forestell  _et al._ (2019); Coffey  _et al._ (2020) (which is a more
accurate description for sub-100 MeV energy injections than the “universal
spectrum” widely used). Given a production energy $\omega_{\rm
inj}=(m_{\chi}/2)(1-m_{n}^{2}/m_{\chi}^{2})$, the differential number density
of photons per unit energy is then given by

$\mathcal{N_{\gamma}}(\omega)=\frac{n^{0}_{\chi}e^{-t/\tau_{\chi}}}{\tau_{\chi\to
n\gamma}}\bigg{[}f_{\gamma}(\omega)+\frac{\delta(\omega-\omega_{\rm
inj})}{\Gamma_{\gamma}(\omega_{\rm inj})}\bigg{]}~{},$ (14)

where $\Gamma_{\gamma}$ is the photon relaxation rate taken from Ref.
Forestell  _et al._ (2019), incorporating $e^{+}e^{-}$ production, light-by-
light scattering, Compton scattering, and pair-creation on nuclei. Here
$n^{0}_{\chi}$ is the initial $\chi$ number density. We use the parametric
fits obtained in Ref. Cyburt  _et al._ (2003) for the photodissociation cross
sections.

In the bottom panels of Fig. 2 we show the evolution of 4He and D abundances,
fixing $m_{\chi}-m_{n}$ = 10 and 50 MeV and varying $\theta$ to scan over
$\tau_{\chi}$. The initial abundance of $\chi$ is set to 1% the sum of the
initial proton and neutron abundances. For both masses values the final
$Y_{p}$ is well within observed uncertainties.

For $m_{\chi}-m_{n}=10$ MeV, the post-bottleneck stage of BBN at a few$\times
100-10^{4}\,{\rm s}$ is very sensitive to $\chi$ decays to neutrons, which can
easily overproduce D/H. The effect can be appreciable, even if the majority of
$\chi\to n\gamma$ decays happen much later. As for the non-thermal photon
injection, even though $\omega_{\rm inj}\simeq 10~{}\rm MeV$ is large enough
to destroy D, which has a photodissociation threshold of 2.2 MeV, this does
not occur efficiently due to an insufficient number of photons produced by the
small $\chi$ abundance. Photodissociation of 4He does not occur either since
its threshold is about 20 MeV. Fig. 2 shows that D/H can be sensitive to the
combination of lifetimes and abundance of dark baryons at the level of
$\tau_{\chi}\sim n_{\chi}^{0}/(n_{n}^{0}+n_{p}^{0})\times 10^{9}$ s.

For $m_{\chi}-m_{n}=50$ MeV, $\omega_{\rm inj}$ is above the 4He
photodissociation threshold, and indeed we see a slightly decrease in $Y_{p}$
when $\chi$ decays occur. The photodissociation of 4He creates D and 3He. As
the 4He abundance is $\mathcal{O}(10^{4})$ greater than those of D and 3He,
breaking up even a small fraction of 4He results in considerable excesses of
the latter nuclides. Thus in the region $m_{\chi}-m_{n}\gtrsim$ 20 MeV, our
constraints are driven by the excess abundances of D and 3He that result from
$\chi$ decays.

For both benchmark masses we observe that larger $\theta$ results in larger
D/H due to either larger rates of neutron injection at early times or 4He
photodissociation at $t\simeq\tau_{\chi}$.

Figure 2: Top: The reaction network for photodissociation of nuclides by the
decay $\chi\to n\gamma$ for $\mathbf{m_{\chi}>m_{n}}$, relevant for
$t>10^{4}~{}$s. The threshold for each photodissociation reaction is labelled.
Bottom: Nuclide abundances as a function of time/inverse temperature for
benchmark points corresponding to negligible photodissociation of D [left] and
appreciable photodissociation of 4He leading to D (and 3He) excesses [right].
An initial $\chi$ abundance of 1% is assumed, a choice that is safe from CMB
measurements of the total baryon density $\Omega_{b}$. See Sec. III.1 for
further details.

###  III.2 CMB

If dark baryons decay during or after the recombination epoch at around
$10^{13}~{}\rm s$, they can alter CMB observables. In both the
$m_{p}+m_{e}<m_{\chi}<m_{n}$ and $m_{\chi}>m_{n}$ regimes, the final state in
$\chi$ decay contains an energetic electromagnetically interacting particle,
respectively either $e^{-}$ or $\gamma$, that can alter the ionization history
of the universe. This has long been recognized as a sensitive probe of
particles that decay during or after recombination Scott, D.  _et al._ (1991);
*Dodelson:1991bz; *Adams:1998nr; *Chen:2003gz; *Kasuya:2003sm;
*Pierpaoli:2003rz. Current precision on the CMB temperature anisotropies can
access states such as $\chi$ with DM-level energy density for $10^{12}~{}{\rm
s}\lesssim\tau_{\chi}\lesssim 10^{26}~{}{\rm s}$.

In addition to distortions of the temperature anisotropies, long-lived
particles that source photons before recombination can alter the energy
spectrum of CMB photons from that of a simple black body. These photons have
different effects on the spectrum depending on the time at which they are
injected. For particles decaying to photons after photon-number-changing
double Compton scattering becomes inefficient at $\tau_{\rm dC}\simeq
6.1\times 10^{6}~{}{\rm s}$, $\mu$ distortions result. Particle decays after
Compton scattering turns off at $\tau_{\rm C}\simeq 8.8\times 10^{9}~{}{\rm
s}$ lead to so-called $y=\delta\rho_{\gamma}/\rho_{\gamma}$ distortions. These
are constrained by the COBE satellite measurement of the CMB spectrum and
could be improved by future satellite missions; see, e.g., Ref. Chluba (2013).

Figure 3:  Constraints on the $n$-$\chi$ mixing angle $\theta$ as a function
of the dark baryon mass $m_{\chi}$ from various cosmological and astrophysical
probes for initial abundance
$n_{\chi}^{0}=5.4\left(n_{p}^{0}+n_{n}^{0}\right)$, i.e. for $\chi$ making up
all dark matter if it is cosmologically long-lived. Note the difference in
scale above and below $m_{n}$. The gray, dash-dotted curves show values of
$\theta$ where $\tau_{\chi}=t_{U}$, the age of the universe; above (below)
this curve the $\chi$ lifetime is shorter (longer). The initial neutron and
proton abundances are chosen so that the baryon density at the CMB epoch
agrees with observation; this depends on the $\chi$ lifetime, cf. Eq. (17).
Asterisks denote limits derived in this paper, which arise from (1) the
requirement that the ${}^{9}{\rm Be}$ lifetime be longer than $t_{U}$, (2) not
overheating the coldest neutron star measured, (3) the helium-4 mass fraction,
$Y_{p}$, and the deuterium-to-hydrogen ratio in the region $m_{\chi}<m_{n}$,
(4) the mismatch between the CMB determination of the baryon-to-photon ratio
and the deuterium abundance for $m_{\chi}>m_{n}$ given this choice of
$n_{\chi}^{0}$, $n_{p}^{0}$, and $n_{n}^{0}$, and (5) CMB observations of
reionization history from $\chi\to pe^{-}\bar{\nu}$ and $\chi\to n\gamma$
decays. Existing limits from the UCNA search for the decay $n\to\chi\gamma$
and a recast of Borexino data constraining the radiative decay of atomic
hydrogen are also shown. The gray, dashed curve labelled “1%” shows where
${\rm Br}_{n\to\chi\gamma}=1\%$, explaining the neutron lifetime anomaly. See
Sec. IV for further details.  Figure 4: As in Fig. 3, but with a reduced
initial $\chi$ abundance $n_{\chi}^{0}=0.01\left(n_{p}^{0}+n_{n}^{0}\right)$.
In this case, for $m_{\chi}>m_{n}$ we display limits from primordial
abundances of $\rm D$, ${}^{3}{\rm He}$, and ${}^{4}{\rm He}$, which can come
from either photodissociation of ${}^{4}{\rm He}$ or enhanced production
through $\chi\to n\gamma$. In addition to limits from CMB temperature
anisotropy limits, we show those from the COBE measurement of the CMB
blackbody frequency spectrum; the hatched region could be probed with an
improved measurement of this spectrum by the PIXIE satellite. See Sec. IV for
further details.

###  III.3 Neutron stars

New sensitivity to the parameter space of dark baryons, irrespective of their
cosmological abundance, can be derived from processes in neutron stars. In
particular, if $n\to\chi$ conversion is occurring on cosmological time scales,
it would lead to a new energy generation mechanism, and could raise the
temperatures of NSs. This way, the region $10^{-21}\lesssim\theta\lesssim
10^{-15}$ can be probed by measurements of NS temperatures, and in particular
the coldest NS observed, PSR J2144–3933 Guillot  _et al._ (2019). For slow
enough decays, the mode $n\to\chi\gamma$ would not populate $\chi$’s within
the lifetime of the star, and thus the $\chi$ Fermi sea in the stellar core is
largely unfilled, allowing this decay mode to be ongoing. These decays would
deposit enormous amounts of energy in the star as holes (or vacancies) left
behind in the neutron Fermi levels are refilled by the de-excitation of
higher-energy neutrons333Since this mechanism relies principally on removal of
neutrons from the Fermi sea, it also applies to models with other exotic
neutron decay modes such as $n\to\chi$ \+ dark photon.. This energy deposition
raises the stellar luminosity by an amount

$L_{n\to\chi\gamma}=N_{n}\widetilde{\Gamma}_{n\to\chi\gamma}\Delta E~{},$ (15)

where $\Delta E\approx$ 100 MeV is the average energy deposited per decay
obtained from the average neutron Fermi momentum of $\mathcal{O}(100)$ MeV,
$N_{n}=1.5\times 10^{57}$ is the number of neutrons in PSR J2144–3933 (whose
mass is estimated as 1.4 $M_{\odot}$), and
$\widetilde{\Gamma}_{n\to\chi\gamma}$ is the decay rate in the dense NS medium
accounting for the self-energy $\Sigma$, which gives rise to the effective
mass and coupling

$m_{n}\rightarrow m_{n}+\Sigma,\ \theta\rightarrow\theta\frac{\Delta
m}{\sqrt{\Sigma^{2}+\Delta m^{2}}}~{}.$ (16)

For our order-of-magnitude estimate, we take an average $\Sigma$ of 10 MeV
Glendenning (1997). In the next section we describe constraints arising from
this phenomenon.

##  IV Constraints

Using all the signal estimates above, we display in Figs. 3 and 4 the current
constraints and future sensitivities of our setup in the space of $n$-$\chi$
mixing angle $\theta$ versus dark baryon mass $m_{\chi}$. The initial $\chi$
abundance is set to $n_{\chi}^{0}=5.4(n_{p}+n_{n})$ in Fig. 3 and
$n_{\chi}^{0}=0.01(n_{p}+n_{n})$ in Fig. 4. If $\chi$ is cosmologically
stable, this corresponds to $\chi$ comprising all of the dark matter or just a
percent-level fraction, respectively. For a given ratio of initial densities
of $\chi$ to SM baryons, we fix the SM density so that the baryon density at
recombination agrees with that measured by the Planck collaboration using CMB
data, $\eta_{\rm CMB}=\left(6.13\pm 0.04\right)\times 10^{-10}$ Aghanim  _et
al._ (2020). Because $\chi$ decays produce an SM baryon, this means the
initial SM baryon density we use depends on $\tau_{\chi}$. Namely, we set

$\frac{n_{p}^{0}+n_{n}^{0}}{n_{\gamma}}=\left\\{\begin{array}[]{ll}\eta_{\rm
Planck},&\tau_{\chi}>10^{13}~{}{\rm s}\\\ \eta_{\rm
Planck}\left(1+\frac{n_{\chi}^{0}}{n_{p}^{0}+n_{n}^{0}}\right)^{-1},&\tau_{\chi}<10^{13}~{}{\rm
s}\end{array}\right..$ (17)

For our BBN limits we use the measured values Fields  _et al._ (2020)

$\displaystyle Y_{p}$ $\displaystyle=$ $\displaystyle 0.245\pm 0.004~{},$
$\displaystyle{\rm D/H}$ $\displaystyle=$ $\displaystyle(2.55\pm 0.03)\times
10^{-5}~{},$ (18) ${}^{3}{\rm He/H}$ $\displaystyle=$ $\displaystyle(1.0\pm
0.5)\times 10^{-5}~{},$

and quote limits at the 2$\sigma$ level.

We use the results of Refs. Slatyer (2013); Cline and Scott (2013) for CMB
temperature anisotropy limits. For $\tau_{\chi}$ longer than $t_{\rm
rec}\simeq 10^{13}$ s this translates to

$f_{\chi}/\tau_{\chi}\lesssim 10^{-25}~{}{\rm s}^{-1}~{},$ (19)

where $f_{\chi}$ is the ratio of the $\chi$ energy density to that measured in
DM. For shorter lifetimes the bound weakens rapidly, disappearing for
$\tau_{\chi}\lesssim 10^{7}~{}\rm s$. The above constraint is roughly constant
for energy injections $1~{}{\rm keV}\lesssim\omega_{\rm inj}\lesssim
100~{}{\rm MeV}$ and comes from studies of $e^{+}e^{-}$ or $\gamma$ injection
via DM decay. For $m_{\chi}>m_{n}$, $\chi$ decay produces a photon and this
limit applies directly. For $m_{p}+m_{e}<m_{\chi}<m_{n}$, the energetic,
interacting particle in the final state is a single electron; however, we
adopt the above limit as an order-of-magnitude estimate. Note that for energy
depositions of 200–250 MeV, Milky Way gamma-ray background measurements give
lifetime limits quite similar to the CMB limit above Cohen  _et al._ (2017),
and would apply to larger $m_{\chi}$.

To obtain exclusion limits on $\theta$ from NS heating we use Eq. (15) and
demand

$L_{\rm n\to\chi\gamma}\leq 4\pi\sigma_{\rm SB}R^{2}_{\star}T^{4}_{\star}~{},$
(20)

where $R_{\star}=11$ km and $T_{\star}=4.3\times 10^{4}$ K are respectively
the radius estimate and upper bound on blackbody temperature of PSR
J2144–3933. The NS heating constraints are shown by a horizontal grey band on
Fig. 3 and 4 that covers the $\theta\sim 10^{-20}-10^{-16}$ range for
$m_{n}>m_{\chi}$. The ceiling on $\theta$ is obtained by simply demanding that
the decays are longer than the age of the star, estimated to be $3\times
10^{8}$ yr. This ceiling could be improved by luminosity measurements of
younger NSs; similarly, the floor on $\theta$ could be improved by future
measurements of old NSs’ temperatures down to $\mathcal{O}(10^{2}-10^{3})$ K
in infrared telescopes Baryakhtar  _et al._ (2017); *Raj:2017wrv;
*Garani:2018kkd; *Acevedo:2019agu; *Bell:2018pkk; *Camargo:2019wou;
*Hamaguchi:2019oev; *Bell:2019pyc; *Garani:2019fpa; *Joglekar:2019vzy;
*Keung:2020teb; *Joglekar:2020liw; *Bell:2020jou; *Garani:2020wge; for
example, observing a 1000 K NS would imply a $\chi$ lifetime bound
$(42000/1000)^{4}\sim 10^{6}$ times stronger, i.e. a $\theta$ bound $\sim
10^{3}$ times stronger. In forthcoming work McKeen  _et al._ we undertake
these tasks, as well as a more careful treatment of density effects, equation-
of-state (EoS) uncertainties, and other effects. We would like to emphasize
that unlike the case with accumulating/annihilating DM Baryakhtar  _et al._
(2017); *Raj:2017wrv; *Garani:2018kkd; *Acevedo:2019agu; *Bell:2018pkk;
*Camargo:2019wou; *Hamaguchi:2019oev; *Bell:2019pyc; *Garani:2019fpa;
*Joglekar:2019vzy; *Keung:2020teb; *Joglekar:2020liw; *Bell:2020jou;
*Garani:2020wge, where actual sensitivity to DM is predicated on future
observational progress, dark baryons are already constrained by existing NS
temperature measurements, and regardless of whether they form DM.

We also display the constraint from the stability of ${}^{9}{\rm Be}$ due to
neutron decays inside weakly bound 9Be. As pointed out in Ref. Pfützner and
Riisager (2018), if $m_{\chi}<m_{{}^{9}{\rm Be}}-2m_{\alpha}=938.0~{}\rm MeV$,
with $\alpha$ the ${}^{4}{\rm He}$ nucleus, the decay ${}^{9}{\rm
Be}\to\chi\alpha\alpha$ can proceed via

${}^{9}{\rm Be}\to 2\,^{4}{\rm He}+\chi;~{}~{}^{9}{\rm Be}\to 2\,^{4}{\rm
He}+\chi+\gamma$ (21)

reactions. The energy release in both cases is given by

$Q_{{}^{9}{\rm Be}}=m_{{}^{9}{\rm Be}}-2m_{\alpha}-m_{\chi}=\Delta
m-1.574\,{\rm MeV}.$ (22)

Both reaction rates are proportional to the cube of the momentum of lightest
emitted particle, which translates to a different scaling with $Q_{{}^{9}{\rm
Be}}$: 3/2-power for the first rate, and 1/3 for the second rate. This makes
the first rate vastly larger, which we estimate using a simple ${}^{8}{\rm
Be}$-$n$ cluster model for ${}^{9}{\rm Be}$ where the neutron has binding
energy $E_{b}$. Taking the perturbation that mediates the transition as the
mixing angle times the potential energy of the ${}^{8}{\rm Be}$-$n$ system, we
reduce the resulting transition rate,

$\Gamma_{{}^{9}{\rm
Be}}\simeq\frac{\theta^{2}m_{\chi}p_{\chi}(E_{\chi}+|E_{b}|)^{2}}{4\pi^{2}}\times|\langle
n|\chi\rangle|^{2}d\Omega_{\vec{p_{\chi}}},$ (23)

to a simple projection of bound $|n\rangle$ state onto the final state wave
function $|\chi\rangle$. The energy of the final state dark baryon $E_{\chi}$
is approximated by $Q_{{}^{9}{\rm Be}}$. Taking the neutron wave function in
the $p_{3/2}$ wave, and ensuring that its spatial extent is similar to the
${}^{9}{\rm Be}$ nuclear radius, in the limit of small energy release we
estimate the decay rate as

$\Gamma_{{}^{9}{\rm Be}}\sim 50\,{\rm
keV}\times\theta^{2}\times\left(\frac{Q_{{}^{9}{\rm Be}}}{1\,{\rm
MeV}}\right)^{3/2}~{}.$ (24)

Consequently our prediction for the lifetime of ${}^{9}{\rm Be}$ is

$\tau_{{}^{9}{\rm Be}}\sim 4\times 10^{10}~{}{\rm
yr}\left(\frac{10^{-19}}{\theta}\right)^{2}\left(\frac{1\,\rm
MeV}{Q_{{}^{9}{\rm Be}}}\right)^{3/2}~{},$ (25)

to be compared with the age of extremely metal-poor (thus very old) stars in
which the beryllium-9 atoms are observed Rich and Boesgaard (2009). Very
conservatively requiring $\tau_{{}^{9}{\rm Be}}<3\times 10^{9}\,{\rm yr}$,
which would give $\sim O(100)$ suppression of beryllium abundance in the
oldest stars, we arrive at the stability limits shown in Fig. 3 and 4. Notice
the significant overlap of NS and beryllium lifetime constraints, which
provides an upper limit on the $\Delta m$ parameter for a very wide range of
$\theta$. Also, fortuitously, the combined NS and ${}^{9}{\rm Be}$ lifetime
exclusion region does not depend on an order-of-magnitude variation in the
limiting value for $\tau_{{}^{9}{\rm Be}}$ and/or in the nuclear decay rate
(24). We note in passing that superior limits on beryllium lifetime can be
derived in the laboratory by exploiting the ionization/heat created by
outgoing $\alpha$-particles.

We also show the regions where $\chi$ (against decay to $pe^{-}\bar{\nu}_{e}$)
and H (against decay to $\chi\nu_{e}$) are stable. For reference we have shown
the contour along which the branching fraction of $n\rightarrow\chi\gamma$ is
1%, as required to explain the neutron lifetime anomaly.

In the $m_{\chi}<m_{n}$ regime for both initial $\chi$ densities displayed,
the BBN constraints largely follow a contour of constant $n\to\chi\gamma$
rate, through its effect on $n\leftrightarrow p$ freeze out (for fixed
$\tau_{n}$). As mentioned previously, for mass splittings less than a few
hundred keV, production of neutrons after the deuterium bottleneck through
inverse decay is important and strengthens the limit. In this regime, the
limits from CMB anisotropies on EM energy injections during and after
recombination are quite important for splittings less than several hundred
keV.

For comparison we also show limits in this mass range that come from a search
for $n\to\chi\gamma$ using ultracold neutrons at LANL Tang  _et al._ (2018)
and a recast McKeen and Pospelov (2020) of Borexino data Agostini  _et al._
(2015) to constrain the radiative decay of atomic hydrogen to
$\chi\bar{\nu}_{e}\gamma$. We also show in both figures the value of the
mixing angle that would give rise to ${\rm Br}_{n\to\chi\gamma}=1\%$, which
would explain the neutron lifetime anomaly. While terrestrial data rules out
large parts of this parameter space for mass splittings above roughly
$700~{}\rm keV$, cosmological data from the CMB and BBN probe larger
$m_{\chi}$, even for a small initial $\chi$ abundance, where the small $\Delta
m$ leaves little energy for visible particles produced in association with
$\chi$ in neutron decay.

For $m_{\chi}>m_{n}$, $\chi\to n\gamma$ opens up and, as a result, CMB
anisotropy limits apply to small mixing angles, for
$10^{-22}\lesssim\theta\lesssim 10^{-16}$. For larger mixing angles in this
mass regime, $\chi$’s decay before the CMB epoch and therefore contribute to
the baryon density measured in the CMB. As seen in Eq. (17), this means we
have to set $(n_{\chi}^{0}+n_{p}^{0}+n_{n}^{0})/n_{\gamma}^{0}=\eta_{\rm
CMB}$. For large $n_{\chi}^{0}/(n_{p}^{0}+n_{n}^{0})$, if $\chi$ lives long
enough such that it decays after BBN, this means that the SM baryon density
during BBN is much smaller than in the standard case and disturbs the good
agreement with observational data, particularly the primordial deuterium
abundance. This is seen to be the case in Fig. 4 where
$n_{\chi}^{0}/(n_{p}^{0}+n_{n}^{0})=5.4$.

In the case of a smaller initial $\chi$ abundance, the $m_{\chi}>m_{n}$ regime
can have more interesting consequences. CMB temperature anisotropies still
constrain small mixing angles where $\tau_{\chi}\gtrsim 10^{12}~{}\rm s$.
Shorter lifetimes can impact the apparent blackbody spectrum of CMB photons.
We show constraints from the COBE satellite as well as the region that could
be probed by the proposed PIXIE experiment, which mostly come from the so-
called $y$ distortions that occur when $\chi$ decays after Compton scattering
freezes out at $\tau_{C}$ Chluba (2020); Forestell  _et al._ (2019). In
addition, there are important BBN limits. For relatively long lifetimes up to
$10^{13}~{}\rm s$, late $\chi\to n\gamma$ decays photo-dissociate light
elements, and the resulting limits are driven largely by not disturbing the
agreement with expectation of the measured D/H ratio; these limits tend to
follow contours of constant $\tau_{\chi}$, up to effects due to thresholds.
For shorter lifetimes, down to $\tau_{\chi}\gtrsim 4s$, the injection of
neutrons at late times can change the synthesis of light elements; the maximum
$\theta$ probed here is determined by the ${}^{4}{\rm He}$ abundance.

We now remark on some limits that are not displayed here. Measurements of the
diffuse gamma-ray background constrain $\chi$ decays, albeit more weakly than
CMB limits we have shown on the reionization history of the universe
Berezhiani (2019b). Additionally, the impact on the CMB from just $\chi$
decay, ignoring its decay products, is much less important because $\chi$
decays to nonrelativistic matter and electromagnetically interacting
particles.

Finally, Ref. Czarnecki  _et al._ (2018) derived an upper limit on the
neutron’s branching fraction to exotic modes of 0.27% at 95% C.L. This limit
was obtained by using the most up-to-date radiative corrections as well as
measurements of the nucleon axial coupling $g_{A}$ from nuclear $\beta$-decay
asymmetries, comparing the predicted neutron $\beta$-decay lifetime to the
world average of the “bottle” measurements. This limit would appear very close
to the ${\rm Br}_{n\to\chi\gamma}=1\%$ contour in Fig. 3 or 4. As emphasized
in Ref. Berezhiani (2019b), any solution to the neutron lifetime anomaly using
exotic neutron decay would then imply a tension with recent measurements of
$g_{A}$, which would be further highlighted if evidence were found for the
dark baryon in this part of parameter space.

##  V Discussion

Our results in the $m_{\chi}<m_{n}$ region for $\Omega_{\chi}=\Omega_{\rm DM}$
in Fig. 3 show that only a narrow 100 keV window between $m_{\chi}\simeq$
938.8$-$938.9 MeV survives that could explain the neutron lifetime anomaly,
motivating laboratory searches to increase their sensitivity to cover this
window and settle the question. This window is only 300 keV-wide for
$n_{\chi}^{0}=0.01(n_{p}+n_{n})$ in Fig. 4, implying that experimental
improvements will probe this scenario even for percent-level DM populations of
dark baryons. We remark here that if an additional decay mode to a dark photon
$n\to\chi A_{\rm D}$ were to exist, as present when large self-interactions of
$\chi$ mediated by $A_{\rm D}$ are required to evade NS EoS constraints Cline
and Cornell (2018), then a 1% total exotic branching fraction would correspond
to smaller $\theta$ than shown in our plots, widening the surviving window.

Analogously, for $n_{\chi}^{0}=0.01(n_{p}+n_{n})$ only a small window is open
in the range $0<m_{\chi}-m_{n}<20$ MeV, which will be probed by the PIXIE
experiment in the CMB frequency spectrum. Other parts of parameter space may
be probed in the future via exotic decays of promising nuclei such as 11Be
Pfützner and Riisager (2018), nuclear capture of $\chi$ at dark matter and
neutrino experiments Fornal  _et al._ (2020b), and measurements of neutron
star luminosities at $\mathcal{O}(10^{2}-10^{3})$K temperatures Baryakhtar
_et al._ (2017); *Raj:2017wrv; *Garani:2018kkd; *Acevedo:2019agu;
*Bell:2018pkk; *Camargo:2019wou; *Hamaguchi:2019oev; *Bell:2019pyc;
*Garani:2019fpa; *Joglekar:2019vzy; *Keung:2020teb; *Joglekar:2020liw;
*Bell:2020jou; *Garani:2020wge. In non-minimal setups, the “hydrogen portal”
could be relevant: future DM experiments could detect the decays of H in the
Earth’s hydrosphere and the Sun McKeen  _et al._ (2020).

Our setup in the $m_{\chi}>m_{n}$ regime could potentially address the lithium
problem, the long-standing discrepancy in the 7Li abundance between standard
BBN predictions and low-metallicity stars’ Spite plateau measurements Fields
_et al._ (2020):

${}^{7}{\rm Li}/{\rm H}=\begin{cases}4.7\pm 0.7\times 10^{-10},\ \ \ {\rm
standard\ BBN}~{},\\\ (1.6\pm 0.3)\times 10^{-10},\ {\rm Spite\
plateau}~{}.\end{cases}$ (26)

These values account for the fact that 7Be created during BBN is later
converted to 7Li via electron capture. As reviewed in Ref. Pospelov and
Pradler (2010a), an injection of $\gtrsim 10^{-5}$ neutrons per baryon from
$\chi\to n\gamma$ just after 7Be synthesis via ${}^{3}{\rm He}+^{4}{\rm
He}\to^{7}{\rm Be}+\gamma$ (i.e. $3\times 10^{2}~{}{\rm s}<t<10^{3}~{}{\rm s}$
or 60 keV$>T>$ 30 keV) would burn 7Be via the reaction ${}^{7}{\rm
Be}+n\to^{7}{\rm Li}+p$ and bring the 7Be+7Li abundance down to the observed
values. For our $n^{0}_{\chi}=0.01(n^{0}_{p}+n^{0}_{n})$ scenario (that is not
ruled out by CMB $\eta_{b}$ measurements), we obtain $n_{n}/n_{p}>10^{-5}$
injection within $t<10^{3}$ s if the $\chi$ lifetime $\tau_{\chi}\lesssim
10^{6}$ s. Unfortunately such short lifetimes are deeply excluded by excess
D/H in our setup, as in many other solutions that utilize “extra neutrons” Coc
_et al._ (2014). For example, in Fig. 4, $\tau_{\chi}=10^{6}$ s corresponds to
a contour just above the PIXIE reach, which is in the excluded region. One
could naively imagine solving the lithium problem while preserving D/H by
photodissociating excess deuterium down to observed levels, achieved by
increasing $n^{0}_{\chi}/(n^{0}_{p}+n^{0}_{n})$ and consequently increasing
$\gamma$ injection via $\chi\to n\gamma$. (Other solutions to lithium problem
with unstable particles, where neutron-injection-induced increase in D/H is
followed by its reduction via the electromagnetic injection, are known
Pospelov and Pradler (2010b).) But as we had discussed in detail, increasing
$n^{0}_{\chi}/(n^{0}_{p}+n^{0}_{n})$ above the percent level for the dark
baryon model at $\tau_{\chi}<t_{\rm rec}$ would run afoul of the agreement
between the CMB $\Omega_{b}$ measurement with the observed D/H ratio. Thus we
conclude that the minimal dark baryon model cannot reduce primordial
${}^{7}{\rm Li}$ abundance.

We finally note that a relatively large portion of parameter space exists that
might be compatible with the dark baryon $\chi$ being the entirety of DM; see
Fig. 3. Since it is one of the relatively economical/natural models, this
topic would warrant future investigations McKeen  _et al._ . In particular,
this entire area induces a relatively fast $n\leftrightarrow\chi$
equilibration inside neutron stars. And while the NS mass-radius relation in
equilibrium can be brought to within observed ranges using a repulsive dark
force for $\chi$, it is far from certain how quickly this equilibrium is
achieved, and if there are any other outstanding observational consequences of
$n\leftrightarrow\chi$ equilibration processes that could disfavor this part
of the parameter space.

## Acknowledgments

We are indebted to David Morrissey for providing us with photon spectra from
electromagnetic energy injection, and for fruitful conversations. The work of
D. M. and N. R. is supported by the Natural Sciences and Engineering Research
Council of Canada. Triumf receives federal funding via a contribution
agreement with the National Research Council Canada. M.P. is supported in part
by U.S. Department of Energy Grant No. desc0011842.

## References

  * McKeen and Nelson (2016) D. McKeen and A. E. Nelson, Phys. Rev. D 94, 076002 (2016), arXiv:1512.05359 [hep-ph] .
  * Aitken  _et al._ (2017) K. Aitken, D. McKeen, T. Neder, and A. E. Nelson, Phys. Rev. D 96, 075009 (2017), arXiv:1708.01259 [hep-ph] .
  * Babu  _et al._ (2013) K. Babu, P. Bhupal Dev, E. C. Fortes, and R. Mohapatra, Phys. Rev. D 87, 115019 (2013), arXiv:1303.6918 [hep-ph] .
  * Allahverdi  _et al._ (2018) R. Allahverdi, P. S. B. Dev, and B. Dutta, Phys. Lett. B 779, 262 (2018), arXiv:1712.02713 [hep-ph] .
  * Elor  _et al._ (2019) G. Elor, M. Escudero, and A. Nelson, Phys. Rev. D 99, 035031 (2019), arXiv:1810.00880 [hep-ph] .
  * Nelson and Xiao (2019) A. E. Nelson and H. Xiao, Phys. Rev. D 100, 075002 (2019), arXiv:1901.08141 [hep-ph] .
  * Alonso-Álvarez  _et al._ (2020) G. Alonso-Álvarez, G. Elor, A. E. Nelson, and H. Xiao, JHEP 03, 046 (2020), arXiv:1907.10612 [hep-ph] .
  * Bringmann  _et al._ (2019) T. Bringmann, J. M. Cline, and J. M. Cornell, Phys. Rev. D 99, 035024 (2019), arXiv:1810.08215 [hep-ph] .
  * Karananas and Kassiteridis (2018) G. K. Karananas and A. Kassiteridis, JCAP 09, 036 (2018), arXiv:1805.03656 [hep-ph] .
  * Cline and Cornell (2018) J. M. Cline and J. M. Cornell, JHEP 07, 081 (2018), arXiv:1803.04961 [hep-ph] .
  * Fornal  _et al._ (2020a) B. Fornal, A. Hewitt, and Y. Zhao, (2020a), arXiv:2011.09014 [hep-ph] .
  * Berezhiani and Bento (2006) Z. Berezhiani and L. Bento, Phys. Rev. Lett. 96, 081801 (2006), arXiv:hep-ph/0507031 .
  * Berezhiani (2016) Z. Berezhiani, Eur. Phys. J. C 76, 705 (2016), arXiv:1507.05478 [hep-ph] .
  * Berezhiani (2019a) Z. Berezhiani, Eur. Phys. J. C 79, 484 (2019a), arXiv:1807.07906 [hep-ph] .
  * Fornal and Grinstein (2018) B. Fornal and B. Grinstein, Phys. Rev. Lett. 120, 191801 (2018), [Erratum: Phys.Rev.Lett. 124, 219901 (2020)], arXiv:1801.01124 [hep-ph] .
  * Fornal and Grinstein (2020) B. Fornal and B. Grinstein, Mod. Phys. Lett. A 35, 2030019 (2020), arXiv:2007.13931 [hep-ph] .
  * Berezhiani (2019b) Z. Berezhiani, LHEP 2, 118 (2019b), arXiv:1812.11089 [hep-ph] .
  * Elahi and Mohammadi Najafabadi (2020) F. Elahi and M. Mohammadi Najafabadi, Phys. Rev. D 102, 035011 (2020), arXiv:2005.00714 [hep-ph] .
  * McKeen  _et al._ (2020) D. McKeen, M. Pospelov, and N. Raj, (2020), arXiv:2006.15140 [hep-ph] .
  * Johns and Koren (2020a) L. Johns and S. Koren, (2020a), arXiv:2012.06584 [hep-ph] .
  * Johns and Koren (2020b) L. Johns and S. Koren, (2020b), arXiv:2012.06591 [hep-ph] .
  * Heeck (2020) J. Heeck, (2020), arXiv:2009.01256 [hep-ph] .
  * Fajfer and Susič (2020) S. Fajfer and D. Susič, (2020), arXiv:2010.08367 [hep-ph] .
  * Czarnecki  _et al._ (2018) A. Czarnecki, W. J. Marciano, and A. Sirlin, Physical Review Letters 120 (2018), 10.1103/physrevlett.120.202002.
  * McKeen  _et al._ (2018) D. McKeen, A. E. Nelson, S. Reddy, and D. Zhou, Phys. Rev. Lett. 121, 061802 (2018), arXiv:1802.08244 [hep-ph] .
  * Baym  _et al._ (2018) G. Baym, D. Beck, P. Geltenbort, and J. Shelton, Phys. Rev. Lett. 121, 061801 (2018), arXiv:1802.08282 [hep-ph] .
  * Motta  _et al._ (2018) T. Motta, P. Guichon, and A. Thomas, J. Phys. G 45, 05LT01 (2018), arXiv:1802.08427 [nucl-th] .
  * Ellis  _et al._ (2018) J. Ellis, G. Hütsi, K. Kannike, L. Marzola, M. Raidal, and V. Vaskonen, Phys. Rev. D 97, 123007 (2018), arXiv:1804.01418 [astro-ph.CO] .
  * Goldman  _et al._ (2013) I. Goldman, R. Mohapatra, S. Nussinov, D. Rosenbaum, and V. Teplitz, Phys. Lett. B 725, 200 (2013), arXiv:1305.6908 [astro-ph.CO] .
  * Pfützner and Riisager (2018) M. Pfützner and K. Riisager, Phys. Rev. C 97, 042501 (2018), arXiv:1803.01334 [nucl-ex] .
  * Tang  _et al._ (2018) Z. Tang, M. Blatnik, L. J. Broussard, J. H. Choi, S. M. Clayton, C. Cude-Woods, S. Currie, D. E. Fellers, E. M. Fries, P. Geltenbort, F. Gonzalez, K. P. Hickerson, T. M. Ito, C.-Y. Liu, S. W. T. MacDonald, M. Makela, C. L. Morris, C. M. O’Shaughnessy, R. W. Pattie, B. Plaster, D. J. Salvat, A. Saunders, Z. Wang, A. R. Young, and B. A. Zeck, Phys. Rev. Lett. 121, 022505 (2018).
  * Sun  _et al._ (2018) X. Sun _et al._ (UCNA), Phys. Rev. C 97, 052501 (2018), arXiv:1803.10890 [nucl-ex] .
  * McKeen and Pospelov (2020) D. McKeen and M. Pospelov, “How long does the hydrogen atom live?” (2020), arXiv:2003.02270 [hep-ph] .
  * Davoudiasl  _et al._ (2010) H. Davoudiasl, D. E. Morrissey, K. Sigurdson, and S. Tulin, Phys. Rev. Lett. 105, 211304 (2010), arXiv:1008.2399 [hep-ph] .
  * Davoudiasl  _et al._ (2011) H. Davoudiasl, D. E. Morrissey, K. Sigurdson, and S. Tulin, Phys. Rev. D 84, 096008 (2011), arXiv:1106.4320 [hep-ph] .
  * Jin and Gao (2018) M. Jin and Y. Gao, Phys. Rev. D 98, 075026 (2018), arXiv:1808.10644 [hep-ph] .
  * Keung  _et al._ (2019) W.-Y. Keung, D. Marfatia, and P.-Y. Tseng, JHEP 09, 053 (2019), arXiv:1905.03401 [hep-ph] .
  * Arnold  _et al._ (2013) J. M. Arnold, B. Fornal, and M. B. Wise, Phys. Rev. D 87, 075004 (2013), arXiv:1212.4556 [hep-ph] .
  * Zyla  _et al._ (2020) P. Zyla _et al._ (Particle Data Group), PTEP 2020, 083C01 (2020).
  * Serpico  _et al._ (2004) P. D. Serpico, S. Esposito, F. Iocco, G. Mangano, G. Miele, and O. Pisanti, JCAP 12, 010 (2004), arXiv:astro-ph/0408076 .
  * Cyburt (2004) R. H. Cyburt, Phys. Rev. D 70, 023505 (2004), arXiv:astro-ph/0401091 .
  * Escudero Abenza (2020) M. Escudero Abenza, JCAP 05, 048 (2020), arXiv:2001.04466 [hep-ph] .
  * Forestell  _et al._ (2019) L. Forestell, D. E. Morrissey, and G. White, JHEP 01, 074 (2019), arXiv:1809.01179 [hep-ph] .
  * Coffey  _et al._ (2020) J. Coffey, L. Forestell, D. E. Morrissey, and G. White, JHEP 07, 179 (2020), arXiv:2003.02273 [hep-ph] .
  * Cyburt  _et al._ (2003) R. H. Cyburt, J. R. Ellis, B. D. Fields, and K. A. Olive, Phys. Rev. D 67, 103521 (2003), arXiv:astro-ph/0211258 .
  * Scott, D.  _et al._ (1991) Scott, D., Rees, M. J., and Sciama, D. W., Astron. & Astrophys. 250, 295 (1991).
  * Dodelson and Jubas (1992) S. Dodelson and J. M. Jubas, Phys. Rev. D 45, 1076 (1992).
  * Adams  _et al._ (1998) J. A. Adams, S. Sarkar, and D. Sciama, Mon. Not. Roy. Astron. Soc. 301, 210 (1998), arXiv:astro-ph/9805108 .
  * Chen and Kamionkowski (2004) X.-L. Chen and M. Kamionkowski, Phys. Rev. D 70, 043502 (2004), arXiv:astro-ph/0310473 .
  * Kasuya  _et al._ (2004) S. Kasuya, M. Kawasaki, and N. Sugiyama, Phys. Rev. D 69, 023512 (2004), arXiv:astro-ph/0309434 .
  * Pierpaoli (2004) E. Pierpaoli, Phys. Rev. Lett. 92, 031301 (2004), arXiv:astro-ph/0310375 .
  * Chluba (2013) J. Chluba, Mon. Not. Roy. Astron. Soc. 436, 2232 (2013), arXiv:1304.6121 [astro-ph.CO] .
  * Guillot  _et al._ (2019) S. Guillot, G. Pavlov, C. Reyes, A. Reisenegger, L. Rodriguez, B. Rangelov, and O. Kargaltsev, Astrophys. J. 874, 175 (2019), arXiv:1901.07998 [astro-ph.HE] .
  * Glendenning (1997) N. Glendenning,  _Compact stars: Nuclear physics, particle physics, and general relativity_ (1997).
  * Aghanim  _et al._ (2020) N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A6 (2020), arXiv:1807.06209 [astro-ph.CO] .
  * Fields  _et al._ (2020) B. D. Fields, K. A. Olive, T.-H. Yeh, and C. Young, Journal of Cosmology and Astroparticle Physics 2020, 010–010 (2020).
  * Slatyer (2013) T. R. Slatyer, Physical Review D 87 (2013), 10.1103/physrevd.87.123513.
  * Cline and Scott (2013) J. M. Cline and P. Scott, JCAP 03, 044 (2013), [Erratum: JCAP 05, E01 (2013)], arXiv:1301.5908 [astro-ph.CO] .
  * Cohen  _et al._ (2017) T. Cohen, K. Murase, N. L. Rodd, B. R. Safdi, and Y. Soreq, Phys. Rev. Lett. 119, 021102 (2017), arXiv:1612.05638 [hep-ph] .
  * Baryakhtar  _et al._ (2017) M. Baryakhtar, J. Bramante, S. W. Li, T. Linden, and N. Raj, Phys. Rev. Lett. 119, 131801 (2017), arXiv:1704.01577 [hep-ph] .
  * Raj  _et al._ (2018) N. Raj, P. Tanedo, and H.-B. Yu, Phys. Rev. D 97, 043006 (2018), arXiv:1707.09442 [hep-ph] .
  * Garani  _et al._ (2019) R. Garani, Y. Genolini, and T. Hambye, JCAP 1905, 035 (2019), arXiv:1812.08773 [hep-ph] .
  * Acevedo  _et al._ (2020) J. F. Acevedo, J. Bramante, R. K. Leane, and N. Raj, JCAP 03, 038 (2020), arXiv:1911.06334 [hep-ph] .
  * Bell  _et al._ (2018) N. F. Bell, G. Busoni, and S. Robles, JCAP 1809, 018 (2018), arXiv:1807.02840 [hep-ph] .
  * Camargo  _et al._ (2019) D. A. Camargo, F. S. Queiroz, and R. Sturani, JCAP 1909, 051 (2019), arXiv:1901.05474 [hep-ph] .
  * Hamaguchi  _et al._ (2019) K. Hamaguchi, N. Nagata, and K. Yanagi, Phys. Lett. B795, 484 (2019), arXiv:1905.02991 [hep-ph] .
  * Bell  _et al._ (2019) N. F. Bell, G. Busoni, and S. Robles, JCAP 1906, 054 (2019), arXiv:1904.09803 [hep-ph] .
  * Garani and Heeck (2019) R. Garani and J. Heeck, Phys. Rev. D100, 035039 (2019), arXiv:1906.10145 [hep-ph] .
  * Joglekar  _et al._ (2019) A. Joglekar, N. Raj, P. Tanedo, and H.-B. Yu, (2019), arXiv:1911.13293 [hep-ph] .
  * Keung  _et al._ (2020) W.-Y. Keung, D. Marfatia, and P.-Y. Tseng, JHEP 07, 181 (2020), arXiv:2001.09140 [hep-ph] .
  * Joglekar  _et al._ (2020) A. Joglekar, N. Raj, P. Tanedo, and H.-B. Yu, (2020), arXiv:2004.09539 [hep-ph] .
  * Bell  _et al._ (2020) N. F. Bell, G. Busoni, S. Robles, and M. Virgato, (2020), arXiv:2004.14888 [hep-ph] .
  * Garani  _et al._ (2020) R. Garani, A. Gupta, and N. Raj, (2020), arXiv:2009.10728 [hep-ph] .
  * (74) D. McKeen, M. Pospelov, and N. Raj, In preparation .
  * Rich and Boesgaard (2009) J. A. Rich and A. M. Boesgaard, Astrophys. J. 701, 1519 (2009), arXiv:0906.3296 [astro-ph.GA] .
  * Agostini  _et al._ (2015) M. Agostini _et al._ (Borexino), Phys. Rev. Lett. 115, 231802 (2015), arXiv:1509.01223 [hep-ex] .
  * Chluba (2020) J. Chluba, Proc. Int. Sch. Phys. Fermi 200, 265 (2020), arXiv:1806.02915 [astro-ph.CO] .
  * Fornal  _et al._ (2020b) B. Fornal, B. Grinstein, and Y. Zhao, Phys. Lett. B 811, 135869 (2020b), arXiv:2005.04240 [hep-ph] .
  * Pospelov and Pradler (2010a) M. Pospelov and J. Pradler, Ann. Rev. Nucl. Part. Sci. 60, 539 (2010a), arXiv:1011.1054 [hep-ph] .
  * Coc  _et al._ (2014) A. Coc, M. Pospelov, J.-P. Uzan, and E. Vangioni, Phys. Rev. D 90, 085018 (2014), arXiv:1405.1718 [hep-ph] .
  * Pospelov and Pradler (2010b) M. Pospelov and J. Pradler, Phys. Rev. D 82, 103514 (2010b), arXiv:1006.4172 [hep-ph] .