# Matter-antimatter asymmetry and dark matter stability from baryon number
conservation

Mar Císcar-Monsalvatje<EMAIL_ADDRESS>Physik-Department, Technische
Universität München, James-Franck-Straße, 85748 Garching, Germany  Alejandro
Ibarra<EMAIL_ADDRESS>Physik-Department, Technische Universität München,
James-Franck-Straße, 85748 Garching, Germany  Jérôme Vandecasteele
<EMAIL_ADDRESS>Physik-Department, Technische Universität München,
James-Franck-Straße, 85748 Garching, Germany

###### Abstract

There is currently no evidence for a baryon asymmetry in our Universe.
Instead, cosmological observations have only demonstrated the existence of a
quark-antiquark asymmetry, which does not necessarily imply a baryon
asymmetric Universe, since the baryon number of the dark sector particles is
unknown. In this paper we discuss a framework where the total baryon number of
the Universe is equal to zero, and where the observed quark-antiquark
asymmetry arises from neutron portal interactions with a dark sector fermion
$N$ that carries baryon number. In order to render a baryon symmetric universe
throughout the whole cosmological history, we introduce a complex scalar
$\chi$, with opposite baryon number and with the same initial abundance as
$N$. Notably, due to the baryon number conservation, $\chi$ is absolutely
stable and could have an abundance today equal to the observed dark matter
abundance. Therefore, in this simple framework, the existence of a quark-
antiquark asymmetry is intimately related to the existence (and the stability)
of dark matter.

##  1 Introduction

The Standard Model (SM) of Particle Physics describes with outstanding
precision the results of a myriad of experiments involving particle reactions.
However, several cosmological observations suggest that the SM should be
extended. Two of the most solid evidences for New Physics beyond the SM are
the existence of dark matter in our Universe Aghanim  _et al._ (2020)

$\displaystyle\Omega_{\rm DM,0}h^{2}=0.120\pm 0.001\;,$ (1)

and the existence of a cosmic asymmetry between the number of SM matter
particles and their antiparticles, commonly expressed as the difference
between the number density of baryons and antibaryons normalized to the
entropy density Aghanim  _et al._ (2020)

$\displaystyle Y_{B,0}=\frac{n_{B}-n_{\bar{B}}}{s}\Big{|}_{0}=(8.75\pm
0.23)\times 10^{-11}\;.$ (2)

Furthermore, observations have revealed that the density in the form of dark
matter is comparable to the density of Standard Model Matter (SMM)
$\Omega_{\rm DM}/\Omega_{\rm SMM}\sim 5$.

In 1967, Sakharov presented three necessary conditions that must be
simultaneously fulfilled in order to generate a baryon asymmetry in our
Universe Sakharov (1967): (i) baryon number violation; (ii) $C$ and $CP$
violation; (iii) departure from thermal equilibrium. Many concrete models
fulfilling these three conditions have been proposed which generate a baryon
asymmetry in qualitative agreement with observations (see e.g. Yoshimura
(1978); Kuzmin  _et al._ (1985); Dodelson and Widrow (1990); Farrar and
Shaposhnikov (1993); Aitken  _et al._ (2017)). On the other hand, in these
models the dark matter is typically not accounted for and assumed to be
produced through a different mechanism, involving different particles and
interactions, and occurring at different cosmic times. Hence, the similarity
between the densities of protons and dark matter is merely coincidental. A
popular framework to explain this similarity consists in postulating that the
dark sector is asymmetric under a global dark symmetry, $U(1)_{D}$, while the
visible sector is asymmetric under a global baryon symmetry, $U(1)_{B}$. If
appropriate conditions are fulfilled in the dark sector, analogous to the
Sakharov conditions, an asymmetry between dark matter particle and
antiparticle could be generated, which is then transferred to the visible
sector Kaplan (1992); Kuzmin (1998); Barr  _et al._ (1990); Kitano and Low
(2005); Bernal  _et al._ (2016); Bringmann  _et al._ (2019); Servant and Tulin
(2013). Alternatively, both asymmetries could be generated simultaneously in
the dark and the visible sectors Oaknin and Zhitnitsky (2005); Farrar and
Zaharijas (2006); Davoudiasl  _et al._ (2010); Elor  _et al._ (2019). For
reviews on asymmetric dark matter, see Davoudiasl and Mohapatra (2012);
Petraki and Volkas (2013); Zurek (2014)).

From the observational standpoint one cannot conclude that the Universe is
baryon asymmetric, since the baryon number of the dark sector particles is
unknown. Instead, one can only asseverate that the visible sector is baryon
asymmetric, or more strictly, that the Universe contains an asymmetry between
the total number of quarks and antiquarks, given by

$\displaystyle Y_{\Delta q,0}=(2.63\pm 0.07)\times 10^{-10},$ (3)

which is obtained from multiplying Eq. (2) by 3.

In this paper we will argue that not all the Sakharov conditions are necessary
to generate a quark-antiquark asymmetry. We will assume that the baryon and
lepton numbers are exact symmetries of Nature, and we will show that a cosmic
quark-antiquark asymmetry can be generated when the three following conditions
are satisfied: (i) $C$ is violated in the dark sector, (ii) there is departure
from thermal equilibrium, (iii) there are portal interactions between the dark
sector and the quarks. These conditions are (arguably) less restrictive that
the Sakharov conditions, and seem plausible in dark sector scenarios.

To illustrate the idea, we will consider a simple framework where the dark
matter particle is a complex scalar carrying baryon number, and that the dark
sector interacts with the visible sector via a “neutron portal” Fornal and
Grinstein (2018). Imposing an initial asymmetry between the number of dark
matter particles and antiparticles, we will show that the asymmetry in the
dark sector is transmitted to the visible sector thus generating an asymmetry
between the number of quarks and antiquarks. In this way, the yield of baryons
in the visible sector and the yield of dark matter particles are naturally
comparable. Furthermore, in this simple framework the dark matter stability is
ensured by the conservation of the baryon number, and does not require
additional ad hoc symmetries. Therefore, in this scenario the existence of
dark matter in our Universe today is intimately related to the existence of a
quark-antiquark asymmetry.

The paper is organized as follows. In Section 2 we present our scenario and we
qualitatively describe its main characteristics. In section 3 we present the
Boltzmann equations for the temperature evolution of the yields of the various
particle species, and we estimate the present value of the dark matter
abundance and the quark-antiquark asymmetries in terms of the initial
conditions of our scenario. In Section 4 we discuss the constraints on the
neutron portal and the prospects of detecting signals from the dark sector,
and in Section 5 the prospects of detecting a dark matter signal. Finally, in
Section 6 we present our conclusions.

##  2 Dark sector baryons and their impact on the visible sector

We consider a hidden sector containing a complex scalar $\chi$ and a Dirac
fermion $N$, both singlets under the Standard Model gauge group, with masses
$m_{\chi}$ and $m_{N}$ respectively. We assume that these fields transform
under an exactly conserved $U(1)_{B}$ symmetry, with charges $B(\chi)=-1$ and
$B(N)=+1$. The baryon numbers of the proton and the neutron are defined as
usual as $B(p)=+1$, $B(n)=+1$ and so are the baryon numbers of the quarks and
antiquarks, $B(q)=+1/3$, $B(\bar{q})=-1/3$.

The kinetic and mass terms in the Lagrangian of the dark sector baryons read

$\displaystyle\mathcal{L}\supset\left|\partial_{\mu}\chi\right|^{2}+m_{\chi}^{2}\chi^{*}\chi+\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu\left(i\not{\partial}-m_{N}\right)N.$ (4)

Further, the Lagrangian contains interaction terms among the dark sector
particles, which we describe via dimension-5 effective operators of the form

$\displaystyle\mathcal{L}\supset\frac{1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{0}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}\chi\chi^{*}\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5muN+\frac{1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{2}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}\left(\chi\chi\mkern
1.5mu\overline{\mkern-3.5muN^{c}\mkern-1.5mu}\mkern 2.5muN+\,h.c.\right).$ (5)

where the superscript $c$ denotes charge conjugation. The first term does not
change the baryon number in the fermionic current, while the second term
changes the baryon number by two units, hence the notation for the suppression
scale of the dimension-5 operators,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{0}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{2}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
respectively.

Lastly, the gauge symmetry and the baryon symmetry allow interaction terms
between the dark sector and the visible sector of the form

$\displaystyle\mathcal{L}\supset\lambda_{\chi
H}|\chi|^{2}|H|^{2}+\frac{1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{n}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}^{2}}\left(\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mud_{R}\,\overline{u_{R}^{c}}d_{R}+\,h.c\right),$ (6)

with $H$ the Standard Model Higgs doublet, and $u_{R}$ and $d_{R}$ the right-
handed up and down quarks. These two terms respectively correspond to a Higgs
portal interaction and to a neutron portal interaction Davoudiasl  _et al._
(2010); Fornal and Grinstein (2018); Fornal (2023). Portals involving heavier
generation quarks are also possible, but will not be discussed in what follows
for simplicity.

In order to generate an asymmetry in the visible sector, we assume that some
unspecified $C$-violating mechanism in the dark sector generates an excess of
$N$ over $\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu$ at
high temperatures. Since this mechanism operates exclusively in the dark
sector, the conservation of baryon number requires the generation of an excess
of $\chi$ over $\chi^{*}$, so that the total baryon number of the Universe
remains zero, as depicted in Fig. 1 (a concrete realization for that mechanism
will be presented elsewhere not ). Due to the neutron portal interaction in
Eq. (6), the scatterings $N\bar{d}\leftrightarrow ud$ and
$N\bar{u}\leftrightarrow dd$, as well as the decays $N\rightarrow udd$, inject
a net baryon number into the visible sector (which could be partially
converted into a net lepton number via sphaleron transitions, depending on the
temperature at which the asymmetry in the dark sector is generated).
Therefore, the excess of dark sector fermionic baryons over antibaryons is
leaked to the visible sector via the neutron portal, ultimately generating an
excess of quarks over antiquarks.

Note that the Higgs portal does not transmit baryon number from the dark
sector to the visible sector. Hence, we will set it to zero for simplicity,
although it could have phenomenological implications, as we will briefly
discuss in Section 5.

Notably, due to the Lorentz symmetry and the baryon number conservation,
$\chi$ and $\chi^{*}$ are absolutely stable, although they can annihilate with
one another generating a relic population of $\chi$. Therefore, in this simple
framework the existence of a quark-antiquark asymmetry in the visible sector
is intimately related to the existence of dark matter in our Universe. We
stress that the stability of the dark matter does not require any ad hoc new
symmetry, but is simply due to the conservation of the total baryon number in
the Universe111In the model presented in Eq. (4$-$6), the dark matter particle
exhibits an accidental $\mathbb{Z}_{2}$ symmetry, $\chi\rightarrow-\chi$,
which however has no influence on the dark matter stability. As it has no
importance for the phenomenology at hand, we will not address it further in
the remaining of this work..

Figure 1: Sketch of the initial abundances of the different particle species
in our model, together with the interaction terms among them. Light yellow
(blue) indicates the particles with $B-L<0$ ($B-L>0$); the total $B-L$ of the
Universe is equal to zero.

In the next section, we will describe in detail the evolution of the yields of
the different particles, and the expectations for the relic abundance of dark
matter and the quark-antiquark asymmetry.

##  3 Evolution of the particle number densities and asymmetries

The Boltzmann equations for the yields of the different particle species can
be cast as

$\displaystyle\frac{dY_{\chi}}{dx}=-\frac{\lambda}{x^{2}}\Bigg{[}$
$\displaystyle\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi\chi^{\ast}\to N\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(Y_{\chi}Y_{\chi^{\ast}}-\frac{Y_{\chi}^{\rm
eq}Y_{\chi^{\ast}}^{\rm eq}}{Y_{N}^{\rm eq}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}^{\rm eq}}Y_{N}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}\right)+2\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi\chi\to\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(Y_{\chi}^{2}-\left(\frac{Y_{\chi}^{\rm
eq}}{Y_{\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}^{\rm
eq}}\right)^{2}Y_{\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}^{2}\right)$
$\displaystyle+\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi N\to\chi^{\ast}\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Bigg{(}Y_{\chi}Y_{N}-\frac{Y_{\chi}^{\rm
eq}Y_{N}^{\rm eq}}{Y_{\chi^{\ast}}^{\rm eq}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}^{\rm
eq}}Y_{\chi^{\ast}}Y_{\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}\Bigg{)}\Bigg{]},$ (7)
$\displaystyle\frac{dY_{\chi^{\ast}}}{dx}=-\frac{\lambda}{x^{2}}\Bigg{[}$
$\displaystyle\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi\chi^{\ast}\to N\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(Y_{\chi}Y_{\chi^{\ast}}-\frac{Y_{\chi}^{\rm
eq}Y_{\chi^{\ast}}^{\rm eq}}{Y_{N}^{\rm eq}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}^{\rm eq}}Y_{N}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}\right)+2\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi^{\ast}\chi^{\ast}\to
NN}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Bigg{(}Y_{\chi^{\ast}}^{2}-\left(\frac{Y_{\chi^{\ast}}^{\rm
eq}}{Y_{N}^{\rm eq}}\right)^{2}Y_{N}^{2}\Bigg{)}$
$\displaystyle-\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi N\to\chi^{\ast}\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Bigg{(}Y_{\chi}Y_{N}-\frac{Y_{\chi}^{\rm
eq}Y_{N}^{\rm eq}}{Y_{\chi^{\ast}}^{\rm eq}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}^{\rm
eq}}Y_{\chi^{\ast}}Y_{\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}\Bigg{)}\Bigg{]},$ (8)
$\displaystyle\frac{dY_{N}}{dx}=-\frac{\lambda}{x^{2}}\Bigg{[}$
$\displaystyle-\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi\chi^{\ast}\to N\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(Y_{\chi}Y_{\chi^{\ast}}-\frac{Y_{\chi}^{\rm
eq}Y_{\chi^{\ast}}^{\rm eq}}{Y_{N}^{\rm eq}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}^{\rm eq}}Y_{N}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}\right)-2\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi^{\ast}\chi^{\ast}\to
NN}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(Y_{\chi^{\ast}}^{2}-\left(\frac{Y_{\chi^{\ast}}^{\rm
eq}}{Y_{N}^{\rm eq}}\right)^{2}Y_{N}^{2}\right)$
$\displaystyle+\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi N\to\chi^{\ast}\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(Y_{\chi}Y_{N}-\frac{Y_{\chi}^{\rm
eq}Y_{N}^{\rm eq}}{Y_{\chi^{\ast}}^{\rm eq}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}^{\rm
eq}}Y_{\chi^{\ast}}Y_{\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}\right)+\frac{1}{\mathfrak{s}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\Gamma_{N}\rangle}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{N}-Y_{N}^{\rm
eq}\Big{)}$
$\displaystyle+\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{N\bar{d}\to
ud}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{N}Y_{\bar{d}}^{\rm
eq}-Y_{u}^{\rm eq}Y_{d}^{\rm
eq}\Big{)}+\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{N\bar{u}\to
dd}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{N}Y_{\bar{u}}^{\rm
eq}-\left(Y_{d}^{\rm eq}\right)^{2}\Big{)}\Bigg{]},$ (9)
$\displaystyle\frac{dY_{\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}{dx}=-\frac{\lambda}{x^{2}}\Bigg{[}$
$\displaystyle-\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi\chi^{\ast}\to N\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(Y_{\chi}Y_{\chi^{\ast}}-\frac{Y_{\chi}^{\rm
eq}Y_{\chi^{\ast}}^{\rm eq}}{Y_{N}^{\rm eq}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}^{\rm eq}}Y_{N}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}\right)-2\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi\chi\to\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(Y_{\chi}^{2}-\left(\frac{Y_{\chi}^{\rm
eq}}{Y_{\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}^{\rm
eq}}\right)^{2}Y_{\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}^{2}\right)$
$\displaystyle-\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi N\to\chi^{\ast}\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(Y_{\chi}Y_{N}-\frac{Y_{\chi}^{\rm
eq}Y_{N}^{\rm eq}}{Y_{\chi^{\ast}}^{\rm eq}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}^{\rm
eq}}Y_{\chi^{\ast}}Y_{\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}\right)+\frac{1}{\mathfrak{s}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\Gamma_{N}\rangle}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}-Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}^{\rm eq}\Big{)}$
$\displaystyle+\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mud\to\bar{u}\bar{d}}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}Y_{d}^{\rm
eq}-Y_{\bar{u}}^{\rm eq}Y_{\bar{d}}^{\rm
eq}\Big{)}+\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5muu\to\bar{d}\bar{d}}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}Y_{u}^{\rm
eq}-\left(Y_{\bar{d}}^{\rm eq}\right)^{2}\Big{)}\Bigg{]}\;.$ (10)

Here $x\equiv m_{\chi}/T$,
$\mathfrak{s}=2\pi^{2}/45\,g_{\star,\mathfrak{s}}T^{3}$ is the entropy density
of the Universe at the temperature $T$, with $g_{\star,\mathfrak{s}}$ the
number of relativistic degrees of freedom. The overall numerical factor
$\lambda$ is defined as

$\lambda\equiv\frac{4\pi}{\sqrt{90}}\leavevmode\nobreak\ m_{\chi}M_{\rm
Pl}\leavevmode\nobreak\ \sqrt{g_{\star,\mathfrak{s}}}\,.$ (11)

The Boltzmann equations depend on the thermally averaged rates of different
processes. Firstly, there are reactions involving only dark sector particles:
$\chi\chi^{*}\leftrightarrow N\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu$, which is induced by
the effective operator in the first term of Eq. (5), suppressed by
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{0}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$;
$\chi\chi\leftrightarrow\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu$ and the
$C-$conjugated reaction $\chi^{*}\chi^{*}\leftrightarrow NN$, induced by the
effective operator found in the second term of Eq. (5), suppressed by
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{2}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$;
and $\chi N\leftrightarrow\chi^{*}\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu$, also induced by the
same term and suppressed by
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{2}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$.
The explicit expressions for the cross-sections of these processes read

$\displaystyle\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\sigma_{\chi\chi^{\ast}\to
N\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
$\displaystyle=\frac{\left(s-4m_{N}^{2}\right)^{3/2}}{8\pi\,s\sqrt{s-4m_{\chi}^{2}}\,\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{0}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}^{2}},$
(12)
$\displaystyle\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\sigma_{\chi\chi\to\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
$\displaystyle=\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\sigma_{\chi^{\ast}\chi^{\ast}\to
NN}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}=\frac{\left(s-4m_{N}^{2}\right)^{3/2}}{8\pi\,s\sqrt{s-4m_{\chi}^{2}}\,\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{2}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}^{2}},$
(13)
$\displaystyle\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\sigma_{\chi
N\to\chi^{\ast}\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
$\displaystyle=\frac{m_{N}^{4}-2m_{N}^{2}\left(m_{\chi}^{2}-3s\right)+\left(m_{\chi}^{2}-s\right)^{2}}{32\pi\,s^{2}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{2}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}^{2}},$
(14)

with $s$ the square of the center of mass energy. Secondly, one has reactions
between dark sector particles and visible sector particles,
$N\bar{d}\leftrightarrow ud$ and $N\bar{u}\leftrightarrow dd$, as well as the
decay $N\rightarrow udd$, all mediated by the neutron portal interaction,
suppressed by $\Lambda_{n}$. The corresponding cross-sections and decay rate
read

$\displaystyle\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\sigma_{N\bar{d}\rightarrow
ud}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
$\displaystyle=\frac{m_{N}^{2}+14s}{192\pi
s^{2}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{n}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}^{4}},$
(15)
$\displaystyle\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\sigma_{N\bar{u}\rightarrow
dd}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
$\displaystyle=\frac{2m_{N}^{2}+s}{192\pi
s^{2}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{n}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}^{4}},$
(16)
$\displaystyle\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Gamma_{N}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
$\displaystyle=\frac{1}{768\pi^{3}}\frac{m_{N}^{5}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{n}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(17)

To simplify the discussion, we will assume that the neutron portal interaction
is sufficiently strong to bring the dark sector baryons into thermal
equilibrium with the visible sector. At a temperature $T$, this condition
requires

$\displaystyle\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{n}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lesssim
5.8\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\times
10^{7}\left(\frac{T}{10^{5}\text{ GeV}}\right)^{3/4}\text{ GeV}.$ (18)

Therefore the equilibrium number density of the dark sector particle species
$i$ with mass $m_{i}$ and number of internal degrees of freedom $g_{i}$ is
given as usual by

$\displaystyle
n_{i}^{\text{eq}}=\frac{g_{i}}{2\pi^{2}}m_{i}^{2}\,TK_{2}\left(m_{i}/T\right),$
(19)

where $K_{2}(x)$ is the modified Bessel function of the $2^{\text{nd}}$ kind.

It will be convenient to work with the total yields of the different particle
species, along with their corresponding asymmetries, defined as

$\displaystyle Y_{\chi}^{\text{tot}}=Y_{\chi}+Y_{\chi^{\ast}},$ $\displaystyle
Y_{\Delta\chi}=Y_{\chi}-Y_{\chi^{\ast}},$ (20) $\displaystyle
Y_{N}^{\text{tot}}=Y_{N}+Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu},$ $\displaystyle
Y_{\Delta N}=Y_{N}-Y_{\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}.$ (21)

The Boltzmann equations describing their temperature evolution are

$\displaystyle\frac{dY_{\chi}^{\rm
tot}}{dx}=-\frac{\lambda}{x^{2}}\Bigg{[}2\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi\chi^{\ast}\to N\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{\chi}Y_{\chi^{\ast}}-PY_{N}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}\Big{)}+2\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi\chi\to\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{\chi}^{2}+Y_{\chi^{\ast}}^{2}-P\left(Y_{N}^{2}+Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}^{2}\right)\Big{)}\Bigg{]},$ (22)
$\displaystyle\frac{dY_{\Delta\chi}}{dx}=-\frac{\lambda}{x^{2}}\Bigg{[}2\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi\chi\to\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{\chi}^{2}-Y_{\chi^{\ast}}^{2}-P\left(Y_{N}^{2}-Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}^{2}\right)\Big{)}+2\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi N\to\chi^{\ast}\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{\chi}Y_{N}-Y_{\chi^{\ast}}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}\Big{)}\Bigg{]},$ (23)
$\displaystyle\frac{dY_{N}^{\rm
tot}}{dx}=-\frac{\lambda}{x^{2}}\Bigg{[}-2\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi\chi^{\ast}\to N\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{\chi}Y_{\chi^{\ast}}-PY_{N}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}\Big{)}-2\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi\chi\to\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{\chi}^{2}+Y_{\chi^{\ast}}^{2}-P\left(Y_{N}^{2}+Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}^{2}\right)\Big{)}$ (24)
$\displaystyle+\frac{1}{\mathfrak{s}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\Gamma_{N}\rangle}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{N}+Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}-Y_{N}^{\rm
eq}-Y_{\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}^{\rm
eq}\Big{)}+\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{N\bar{d}\to
ud}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{N}Y_{\bar{d}}^{\rm
eq}+Y_{\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}Y_{d}^{\rm
eq}-Y_{u}^{\rm eq}Y_{d}^{\rm eq}-Y_{\bar{u}}^{\rm eq}Y_{\bar{d}}^{\rm
eq}\Big{)}+(N\bar{u}\leftrightarrow dd)\Bigg{]},$
$\displaystyle\frac{dY_{\Delta
N}}{dx}=-\frac{\lambda}{x^{2}}\Bigg{[}2\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi\chi\to\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{\chi}^{2}-Y_{\chi^{\ast}}^{2}-P\left(Y_{N}^{2}-Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}^{2}\right)\Big{)}+2\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi N\to\chi^{\ast}\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{\chi}Y_{N}-Y_{\chi^{\ast}}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}\Big{)}$ (25)
$\displaystyle+\frac{1}{\mathfrak{s}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\Gamma_{N}\rangle}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{N}-Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}-Y_{N}^{\rm
eq}+Y_{\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}^{\rm
eq}\Big{)}+\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{N\bar{d}\to
ud}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{N}Y_{\bar{d}}^{\rm
eq}-Y_{\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}Y_{d}^{\rm
eq}-Y_{u}^{\rm eq}Y_{d}^{\rm eq}+Y_{\bar{u}}^{\rm eq}Y_{\bar{d}}^{\rm
eq}\Big{)}+(N\bar{u}\leftrightarrow dd)\Bigg{]}.$

The yields $Y_{\chi}$, $Y_{\chi^{\ast}}$, $Y_{N}$ and $Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}$ are implicit functions
of the total yields and of the asymmetries. Further, we have introduced

$\displaystyle P\,\equiv\,\frac{Y_{\chi}^{\rm eq}Y_{\chi^{\ast}}^{\rm
eq}}{Y_{N}^{\rm eq}Y_{\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}^{\rm eq}}=\left(\frac{Y_{\chi}^{\text{eq}}}{Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}^{\text{eq}}}\right)^{2}=\Bigg{(}\frac{Y_{\chi^{\ast}}^{\text{eq}}}{Y_{N}^{\text{eq}}}\Bigg{)}^{2}\simeq\left(\frac{g_{\chi}}{g_{N}}\right)^{2}\left(\frac{m_{\chi}}{m_{N}}\right)^{4}\Bigg{(}\frac{K_{2}\left(m_{\chi}/T\right)}{K_{2}\left(m_{N}/T\right)}\Bigg{)}^{2}.$
(26)

Lastly, the asymmetry present in $N$ is transmitted to the visible sector
through the neutron portal, giving rise to an asymmetry between the total
number of quarks and antiquarks $\Delta q\equiv\sum_{i}\Delta q_{i}$. Its time
evolution is described by

$\displaystyle\frac{dY_{\Delta q}}{dx}=$
$\displaystyle\,c\,\frac{3\lambda}{x^{2}}\Bigg{[}\frac{1}{\mathfrak{s}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\Gamma_{N}\rangle}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{N}-Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}-Y_{N}^{\rm
eq}+Y_{\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}^{\rm
eq}\Big{)}$ (27)
$\displaystyle+\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{N\bar{d}\to
ud}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{N}Y_{\bar{d}}^{\rm
eq}-Y_{\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}Y_{d}^{\rm
eq}-Y_{u}^{\rm eq}Y_{d}^{\rm eq}+Y_{\bar{u}}^{\rm eq}Y_{\bar{d}}^{\rm
eq}\Big{)}+(N\bar{u}\leftrightarrow dd)\Bigg{]}.$

The constant $c$ in this expression characterises the efficiency of the
conversion of the baryon asymmetry stored in the left-handed quarks into a
lepton asymmetry stored in the left-handed leptons via sphaleron processes. If
the asymmetry in the dark sector is generated at temperatures above 130 GeV,
the point at which sphalerons drop out of thermal equilibrium D’Onofrio  _et
al._ (2014), then $c=36/111$ Harvey and Turner (1990). In turn, the asymmetry
between the total number of leptons and antileptons,
$\Delta\ell\equiv\sum_{i}\Delta\ell_{i}$ can be calculated from

$\displaystyle\frac{dY_{\Delta\ell}}{dx}=-3\,\frac{1-c}{c}\,\frac{dY_{\Delta
q}}{dx}.$ (28)

If the initial asymmetry is instead generated after sphaleron freeze-out, then
we set $c=1$ and no asymmetry is transferred to the leptons.

Let us consider a scenario where both the baryon and the lepton numbers are
exactly conserved in Nature. As initial condition, we assume that at a
temperature $T_{\rm in}\gg 130$ GeV there is no asymmetry between quarks and
antiquarks (nor between leptons and antileptons) but that there is a
$C$-violating mechanism in the dark sector that generates a primordial
asymmetry between $N$ and $\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu$. The conservation of
the baryon number requires a corresponding asymmetry between $\chi$ and
$\chi^{*}$. This initial condition is sketched in Fig. 1, where we also show
the different portals relating the various particle species in our model.
Under this plausible assumption, the dark matter relic abundance and the
quark-antiquark asymmetry are determined by the initial asymmetry in the dark
sector, and by the energy scales
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{0}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$,
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{2}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{n}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$,
which determine the strengths of the different portal interactions.

For this representative set-up, the various yields qualitatively evolve with
temperature as shown Fig. 2. At the high temperature $T_{\rm in}$ all particle
species are ultra-relativistic and their equilibrium yields are related solely
by their different number of internal degrees of freedom, see Eq. (19). For
the yields of the particles in the hidden sector, this implies
$Y_{N}^{\text{eq}}=2Y_{\chi}^{\text{eq}}$. Further, for the initial condition
sketched in Fig. 1 the yields of the asymmetries satisfy

$\displaystyle Y_{\Delta N}^{\rm in}=Y_{\Delta\chi}^{\rm in},$ (29)
$\displaystyle Y_{\Delta q}^{\rm in}=Y_{\Delta\ell}^{\rm in}=0.$ (30)

Figure 2: Representative evolution of the total yields of $\chi+\chi^{*}$ and
$N+\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu$ (dash-dotted)
and the yields of the different asymmetries between particles and
antiparticles (solid) for the initial condition depicted in Fig. 1. See the
main text for details.

Let us first discuss the evolution of the asymmetries with the temperature. At
temperatures very close to $T_{\rm in}$, the scatterings
$N\bar{u}\leftrightarrow dd$ and $N\bar{d}\leftrightarrow ud$ (with rates
depending on $\Lambda_{n}$) effectively transfer a fraction of the baryon
asymmetry from the dark sector to the visible sector, which is then
distributed between leptons and quarks by sphaleron transitions. At the same
time, the processes $\chi\chi\leftrightarrow\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu$ and $\chi
N\leftrightarrow\chi^{*}\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu$ (with rates depending on $\Lambda_{2}$) contribute to the washout of
the asymmetry within the hidden sector, thereby influencing the size of the
asymmetries in the visible sector. The effect of the washout in $\Delta N$
follows from Eq. (25). Given that at high temperatures all the yields can be
well approximated by their equilibrium values, the Boltzmann equation for
$Y_{\Delta N}$ simplifies to

$\displaystyle\frac{dY_{\Delta
N}}{dx}\simeq-\frac{3\lambda}{x^{2}}\left(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi\chi\to\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi N\to\chi^{\ast}\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\right)Y_{N}^{\text{eq}}Y_{\Delta
N},$ (31)

and similarly for $Y_{\Delta\chi}$. The analytical solution for this equations
reads

$\displaystyle Y_{\Delta N}\left(T\right)\simeq Y_{\Delta
N}^{\text{in}}\exp\Bigg{\\{}3\left[\frac{n_{N}^{\text{eq}}}{H}\left.\left(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi\chi\to\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi N\to\chi^{\ast}\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\right)\right]\right|_{T=T_{\text{in}}}\frac{T-T_{\rm
in}}{T_{\text{in}}}\Bigg{\\}}.$ (32)

The requirement that no more than 50% of the asymmetry is washed out implies
the lower limit

$\displaystyle\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{2}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gtrsim
6\times 10^{10}\left(\frac{T_{\text{in}}}{10^{5}\text{
GeV}}\right)^{1/2}\text{ GeV}.$ (33)

In order to simplify our discussion, we will assume in what follows that
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{2}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
satisfies this lower limit and that the initial asymmetry in $\Delta N$ (and
$\Delta\chi$) is very weakly washed out. A weak washout has been displayed in
Fig. 2 as well as the efficient redistribution of the asymmetry within the
visible sector.

If that is the case, the asymmetries in the different particles species at a
temperature slightly below $T_{\rm in}$ take the simple expressions

$\displaystyle Y_{\Delta N}(T)=\frac{42}{79}Y_{\Delta N}^{\rm in},$ (34)
$\displaystyle Y_{\Delta q}(T)=\frac{36}{79}Y_{\Delta N}^{\rm in},$ (35)
$\displaystyle Y_{\Delta\ell}(T)=-\frac{25}{79}Y_{\Delta N}^{\rm in}.$ (36)

These asymmetries remain constant until a temperature $T_{\rm decay}$ at which
the decays of $N$ and $\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu$ inject an additional quark-antiquark asymmetry in the visible sector
(and a lepton-antilepton asymmetry if the sphalerons are still in equilibrium
at the epoch of decays). For the range of parameters of interest (see Section
4), we find that the decays typically occur when the sphalerons are out-ot-
equilibrium, so that the remaining $\Delta N$ asymmetry is entirely converted
into a quark-antiquark asymmetry, whereas the lepton-antilepton asymmetry
stays frozen. The decrease in $Y_{\Delta N}$ at $T\sim T_{\rm in}$ and then at
$T\sim T_{\rm decay}$, along with the corresponding changes in $Y_{\Delta q}$
and $Y_{\Delta\ell}$, can be seen in Fig. 2. At the present time, the
asymmetries in the visible sector read

$\displaystyle Y_{\Delta q,0}$ $\displaystyle=3\frac{42}{79}Y_{\Delta N}^{\rm
in}+\frac{36}{79}Y_{\Delta N}^{\rm in}=\frac{162}{79}Y_{\Delta N}^{\rm in},$
(37) $\displaystyle Y_{\Delta\ell,0}$ $\displaystyle=-\frac{25}{79}Y_{\Delta
N}^{\rm in}.$ (38)

In these equations, the factor of 3 arises from the fact that $N$ generates
three quarks in its decay.

Let us now turn to the evolution of the total yields $Y_{\chi}^{\rm tot}$ and
$Y_{N}^{\rm tot}$, Eqs. (22) and (24). Again, to simplify our discussion we
will assume that $\Lambda_{2}$ satisfies the condition in Eq. (33). Further,
we will assume
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{0}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ll\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{n}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$,
so that the rate of conversion of $N$ into quarks is slow compared to the rate
of conversion of $\chi$ into $N$. This implies that the dark matter abundance
is determined by the freeze-out of the annihilation process
$\chi\chi^{*}\rightarrow N\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu$. Under these
assumptions the Boltzmann equations Eqs. (22) and (24) simplify to

$\displaystyle\frac{dY_{\chi}^{\rm tot}}{dx}=$
$\displaystyle-\frac{\lambda}{x^{2}}\Bigg{[}2\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi\chi^{\ast}\to N\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{\chi}Y_{\chi^{\ast}}-PY_{N}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}\Big{)}\Bigg{]},$ (39)
$\displaystyle\frac{dY_{N}^{\rm tot}}{dx}=$
$\displaystyle-\frac{\lambda}{x^{2}}\Bigg{[}-2\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi\chi^{\ast}\to N\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{\chi}Y_{\chi^{\ast}}-PY_{N}Y_{\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}\Big{)}\Bigg{]},$ (40)

which amount to a secluded sector freeze-out scenario222If the effective
neutron portal is above
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{n}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\approx
4\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\times
10^{3}\text{ GeV}$, $N$ conversions into quarks drop out of equilibrium before
the time of dark matter freeze-out. Given the experimental constraints, see
Section 4, the dark sector is indeed secluded from the visible sector at the
time of dark matter freeze-out., albeit with a hidden sector temperature
identical to the one of the visible sector. The behaviour of
$Y_{\chi}^{\text{tot}}$ and $Y_{N}^{\text{tot}}$ is sketched in Fig. 2. The
total dark matter abundance at the current epoch can be calculated using
standard tools (see e.g. Gondolo and Gelmini (1991)), resulting in

$\displaystyle\Omega_{\rm DM,0}h^{2}\simeq 2.8\times 10^{8}\,Y_{\rm\chi}^{\rm
tot}(x_{\rm f.o.})\frac{m_{\chi}}{\rm GeV}.$ (41)

where the freeze-out temperature is determined by the condition
$\Gamma_{\chi\chi^{\ast}\to N\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu}(x_{\rm f.o.})=H(x_{\rm
f.o.})$, and $Y_{\chi}^{\rm tot}(T_{\rm f.o.})$ can be well approximated by
its equilibrium value at freeze-out. Besides, due to the fact that
$\chi\chi^{*}$ pairs can only annihilate into $N\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu$ pairs, one finds that
$Y_{N}^{\rm tot}(x_{\rm f.o.})\simeq
3Y_{N}^{\text{eq}}\left(T_{\text{in}}\right)$, which could additionally lead
to an epoch of early matter domination if $N$ is sufficiently long-lived
Cheung and Ishiwata (2013); Allahverdi  _et al._ (2020); Allahverdi and
Osiński (2022). Some implications of this early phase of matter domination
will be discussed in Section 4.

In the simplest scenario, all dark matter antiparticles annihilate resulting
in the final state sketched in the left diagram of Fig. 3. In this case,
$Y_{\chi}^{\rm tot}(x_{\rm f.o.})=Y_{\Delta\chi}(x_{\rm
f.o.})=Y_{\Delta\chi}^{\rm in}=Y_{\Delta N}^{\rm in}$ where in the last step
we have used Eq. (29). Therefore, in this simple scenario the dark matter
abundance and the quark-antiquark asymmetry are determined by the same
parameter, the initial asymmetry in the dark sector, $Y_{\Delta N}^{\rm in}$.
More concretely, using Eqs. (37) and (41), one finds the relation

$\displaystyle\Omega_{\rm DM,0}h^{2}\simeq 1.4\times 10^{8}\,Y_{\Delta
q,0}\frac{m_{\chi}}{\rm GeV}\;.$ (42)

One can then adjust the initial condition $Y_{\Delta\chi}^{\rm in}$ to
generate the observed quark-antiquark asymmetry, $Y_{\Delta q,0}\simeq
2.6\times 10^{-10}$, and the dark matter mass to generate the observed dark
matter abundance, $\Omega_{\rm DM,0}h^{2}\simeq 0.12$. We obtain

$\displaystyle Y_{\Delta\chi}^{\rm in}$ $\displaystyle\simeq 1.3\times
10^{-10},$ (43) $\displaystyle m_{\chi}$ $\displaystyle\simeq
3.4\,\text{GeV}.$ (44)

Figure 3: Possible final states from the initial state Fig. 1, corresponding
to a scenario where dark matter antiparticles have efficiently annihilated
(left panel), or only partially annihilated (right panel). All $N$ have
decayed into quarks. We have assumed that the initial asymmetry within the
dark sector is generated while sphalerons are still in thermal equilibrium.
Today, most leptons are in the form of neutrinos and anti-neutrinos, with a
small relative asymmetry, leaked via sphaleron processes. In both cases, the
total $B-L$ number of the Universe is conserved and equal to zero.

In the case where not all dark matter antiparticles annihilate, then
$Y_{\chi}^{\rm tot}(x_{\rm f.o.})=Y_{\Delta\chi}(x_{\rm
f.o.})+2Y_{\chi^{*}}(x_{\rm f.o.})$, and Eq. (42) must be replaced by

$\displaystyle\Omega_{\rm DM,0}h^{2}=2.8\times
10^{8}\bigg{(}2Y_{\chi^{*}}(x_{\rm f.o.})+\frac{79}{162}Y_{\Delta
q,0}\bigg{)}\frac{m_{\chi}}{\rm GeV}\,.$ (45)

This scenario is sketched in the right diagram of Fig. 3. In this case, the
initial asymmetry in the dark sector necessary to reproduce the quark-
antiquark asymmetry is still given by Eq. (37). However, since the total dark
matter yield is larger, the observed dark matter abundance is reproduced for a
smaller value of the dark matter mass,

$\displaystyle m_{\chi}\Big{|}_{Y_{\chi*}(x_{\rm f.o.})\neq
0}=\frac{Y_{\Delta\chi}(x_{\rm f.o.})}{Y_{\Delta\chi}(x_{\rm
f.o.})+2Y_{\chi^{*}}(x_{\rm f.o.})}\,m_{\chi}\Big{|}_{Y_{\chi*}(x_{\rm
f.o.})=0}.$ (46)

Here $Y_{\chi^{*}}(x)$ can be calculated from particularizing Eq. (3) to the
weak wash-out regime,

$\displaystyle\frac{dY_{\chi^{\ast}}}{dx}$
$\displaystyle=-\frac{\lambda}{x^{2}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\langle\sigma
v\rangle_{\chi\chi^{\ast}\to N\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}Y_{\chi^{\ast}}(Y_{\chi^{\ast}}+Y_{\Delta\chi}^{\rm
in})-9\,P\,Y_{\chi}^{\rm eq,in}\Big{)}.$ (47)

Since at freeze-out the dark matter antiparticles were still in thermal
equilibrium, one can approximate $Y_{\chi^{*}}(x_{\rm f.o.})=Y^{\rm
eq}_{\chi^{*}}(x_{\rm f.o.})$, and then the equilibrium distribution can be
easily obtained by setting the right hand side of Eq. (47) to zero. We obtain

$\displaystyle Y^{\rm eq}_{\chi^{*}}(x)\simeq\frac{-Y^{\rm
in}_{\Delta\chi}}{2}+\sqrt{\frac{Y_{\Delta\chi}^{\rm
in\,2}}{4}+9P(x)\left(Y_{\chi}^{\text{eq,in}}\right)^{2}}\,.$ (48)

Let us stress that these conclusions are quite insensitive to the concrete
values of the portal strengths $\Lambda_{2}$, $\Lambda_{0}$ and $\Lambda_{n}$,
provided that (i) the washout of the asymmetry is weak, (ii) $\Lambda_{n}$ is
small enough to keep the hidden sector thermalized with the visible sector and
(iii) $\Lambda_{0}\ll\Lambda_{n}$, so that $N$ is stable in the timescale of
the freeze-out. Other scenarios are also possible, by adjusting the initial
conditions in the Boltzmann equations. On the other hand, a crucial assumption
of our scenario is the existence of a neutron portal that transmits the
asymmetry in the dark sector to the visible sector. This neutron portal could
lead to experimental signatures in our scenario, which will be discussed in
detail in the next section.

##  4 Constraints on the neutron portal

The particle $N$ can have implications in our visible sector through the
neutron portal Eq. (6), e.g. through the decay of $N$ into quarks, the
production of $N$ in proton-proton collisions, or the generation of a mass
mixing term with the neutron below the QCD confinement scale. The various
constraints are summarized in Fig. 4, in the parameter space defined by the
mass of $N$ ($m_{N}$) and the energy scale of the neutron portal
($\Lambda_{n}$). The region allowed by all the constraints, shown in white, is
bounded and could in principle be probed in its totality. Let us describe in
detail the multiple constraints from the parameter space.

Figure 4: Constraints on the neutron portal energy scale ($\Lambda_{n}$) and
mass of $N$ ($m_{N}$) from cosmology, proton stability, and collider
experiments, along with contours of production cross-section and decay length
at the LHC with $\sqrt{s}=14$ TeV. The allowed region is shown in white.

The standard Big Bang Nucleosynthesis (BBN) scenario is extremely successful
in describing the evolution of the Universe after $\sim 1$ s. In particular,
observations indicate that the quark-antiquark asymmetry at the time of BBN
does not differ significantly from the quark-antiquark asymmetry at the time
of recombination. In order to preserve the standard BBN scenario, we will
require that the yield of $N$ is largely depleted at $\sim 1$ s, so that their
decays have practically no impact at later times. Using the fact that the
width of $N$ is given by

$\displaystyle\Gamma^{-1}_{N\rightarrow udd}\approx 1.6\text{
s}\left(\frac{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{n}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}{10^{5}\text{
GeV}}\right)^{4}\left(\frac{\text{GeV}}{m_{N}}\right)^{5},$ (49)

and requiring conservatively $\Gamma^{-1}_{N\rightarrow udd}\lesssim 0.1$ s,
we exclude the region $\Lambda_{n}\gtrsim 10^{5}\,{\rm GeV}(m_{N}/{\rm
GeV})^{5/4}$, indicated in Fig. 4 as a hatched orange region.

The neutron portal also leads to the production of $N$ in proton-proton
collisions through the partonic processes $ud\rightarrow N\bar{d}$ and
$dd\rightarrow N\bar{u}$. We estimate the non-resonant production cross-
section at $\sqrt{s}=14$ TeV to be

$\displaystyle\sigma_{pp\rightarrow N+\text{jet}}\approx 2\,{\rm
fb}\left(\frac{f_{\rm PDF}}{10^{-2}}\right)\left(\frac{10^{4}\,{\rm
GeV}}{\Lambda_{n}}\right)^{4}\;,$ (50)

where we have estimated the effect of the partonic distributions in the
protons in the parameter to be $f_{\rm PDF}\approx 10^{-2}$ Clark  _et al._
(2017). Depending on the lifetime of $N$, the signal at colliders could be in
the form of missing $p_{T}$ (if stable within the detector’s volume), in the
form of a displaced vertex (if the decay length is macroscopic), or in the
form of dijets (if the decay length is microscopic). We show in Fig. 4 the
line for which the decay length lies outside of the ATLAS or CMS detectors,
$ct_{\rm lab}=100$ m, where we have taken a Lorentz factor
$\gamma=\sqrt{\hat{s}}/(2m_{N})$ with $\sqrt{\hat{s}}\sim 2$ TeV for the
partonic center of mass energy. We also indicate in the plot the values of
$\Lambda_{n}$ corresponding to a production cross-section
$\sigma_{pp\rightarrow N+\text{jet}}=1$ fb, 10 fb, 100 fb and 1 pb, and in
green the ballpark area of values that can be probed at the LHC with an
integrated luminosity of ${\cal L}=100$ fb-1, which corresponds to effective
interactions with strength $\Lambda_{n}\lesssim 6$ TeV. We note that for small
values of $\Lambda_{n}$ the effective field theory breaks down at LHC
energies, and instead a dedicated search for the new particles mediating the
neutron portal should be performed. A detailed collider analysis is however
beyond the scope of this paper. We also note that the collider constraints
become weaker if the portal between the dark sector and the visible sector
involves sea quarks, e.g. $\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mus_{R}\overline{c_{R}^{c}}s_{R}$ instead of $\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mud_{R}\overline{u_{R}^{c}}d_{R}$; this variant of the portal could be
probed in the decays of heavy mesons and baryons Heeck (2021).

The mass of $N$ is bounded from below from the requirement that the proton
must be the lightest fermion carrying baryon number. Otherwise the proton
could decay, e.g. $p\rightarrow N\pi^{+}$. This requirement translates into
the lower limit $m_{N}\geq 938$ MeV (this limit on the mass could be avoided
if the width is suppressed by a large $\Lambda_{n}$, however this region is in
tension with BBN McKeen  _et al._ (2021)). Lastly, the mass of $N$ is bounded
from above from the requirement that the dark matter is not overproduced. More
specifically, the annihilation process $\chi\chi^{*}\rightarrow N\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu$ must be efficient
enough to deplete most of the dark matter density. Naively, this requires
$m_{N}<m_{\chi}$, however, as argued in Griest and Seckel (1991); D’Agnolo and
Ruderman (2015), the annihilation can also occur in a “forbidden” channel, due
to the existence of sufficiently energetic dark matter particles in the tail
of the Maxwell-Boltzmann distribution. The thermally averaged annihilation
cross section in a forbidden channel is approximately given by D’Agnolo and
Ruderman (2015)

$\displaystyle\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\left\langle\sigma
v\right\rangle}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\chi\chi^{\ast}\rightarrow
N\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}\approx
8\pi
f(\Delta)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\left\langle\sigma
v\right\rangle}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{N\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu\rightarrow\chi\chi^{\ast}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}e^{-2\Delta
x},$ (51)

where $\Delta=\left(m_{N}-m_{\chi}\right)/m_{\chi}$ is the relative mass
splitting and $f(\Delta)$ is a function of $\Delta$, which approximates to
$f(\Delta)\approx 1+\Delta$ for $\Delta\gg 1$. In Eq. (51) the Boltzmann
suppression of the dark matter annihilation cross section is explicit. A more
rigorous upper limit on $m_{N}$ is derived by ensuring that dark matter
overproduction is avoided for the largest possible value of the annihilation
cross-section, Eq. (51). From the s-wave unitarity requirement
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\left\langle\sigma
v\right\rangle}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{N\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu\rightarrow\chi\chi^{\ast}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}\leq
4\pi/m_{N}^{2}(x_{\text{f.o.}}/\pi)^{1/2}$, we obtain $m_{N}\lesssim 4.8$ GeV.
For the milder requirement that the effective field theory remains valid,
which corresponds to
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{0}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sim
m_{\chi}$, we obtain $m_{N}\lesssim 4.3$ GeV. This limit is shown in Fig. 4 in
red, where the relaxation of constraint at large $\Lambda_{n}$ values is
attributed to the freeze-out of dark matter taking place during a period of
matter domination. In this case, the value of the total annihilation cross
section required to obtain the correct dark matter relic abundance is smaller
than in the standard WIMP paradigm Hamdan and Unwin (2018).

##  5 Constraints from Dark Matter searches

###  5.1 Dark matter signals via the Higgs portal

As our dark matter candidate is a complex scalar, interactions between $\chi$
and the SM via the Higgs portal are not forbidden, see the first term in Eq.
(6). In turn, this could lead to potential signals for a dark matter
discovery.

In the present scenario, the predicted dark matter mass of $3.4\text{ GeV}$
allows for the Higgs to decay invisibly into a dark matter particle-
antiparticle pair, $h\rightarrow\chi\chi^{\ast}$. This search is the most
sensitive probe of the portal coupling at colliders and excludes values of
$\lambda_{\chi H}$ above $10^{-2}$ Arcadi  _et al._ (2020). The asymmetric
nature of dark matter forbids any signals in indirect searches via the Higgs
portal. Additionally, its low mass makes nuclear recoil direct detection
experiments attain comparable sensitivity to the portal coupling when compared
to the collider constraints Arcadi  _et al._ (2020).

One can also consider the following constraint on the portal coupling
$\lambda_{\chi H}$. After electroweak symmetry breaking, the square of the
mass of the dark matter candidate will receive an extra contribution of the
form $\lambda_{\chi H}v^{2}$, with $v\approx 246\text{ GeV}$ being the vacuum
expectation value of the Higgs field in the broken electroweak phase. In the
simplest scenario, assuming dark matter to be fully asymmetric, we find that
the portal coupling should be less than

$\displaystyle\lambda_{\chi H}<2\times 10^{-4},$ (52)

as to prevent the Higgs portal contribution to the bare mass to lead to dark
matter overabundance, see Eq. (44).

Given the above constraints on $\lambda_{\chi H}$, it can be verified that
interactions between $\chi$ and the visible sector do not change the dynamics
of dark matter freeze-out. Within our scenario, the possibility to detect
$\chi$ via the Higgs portal is open.

###  5.2 Dark matter signals via the neutron portal

Within the present scenario, we expect that our dark matter candidate $\chi$
will dominantly interact with neutrons through scattering. The two possible
processes are depicted in Fig. 5. At energies below the QCD confinement scale,
the interactions between $N$ ($\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu$) and (anti-)neutrons
are mediated by a mass mixing $\delta m$ Fornal and Grinstein (2020), which is
depicted in Fig. 5 as a crossed circle. This mass mixing can be expressed in
terms of the neutron portal effective scale as

$\displaystyle\delta m=1.4\times 10^{-8}\left(\frac{10^{3}\text{
GeV}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{n}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}\right)^{2}\text{
GeV}.$ (53)

As mentioned in the case of dark matter scatterings via the Higgs portal,
direct detection experiments searching for nuclear recoils quickly lose their
sensitivity in the regime of light dark matter masses, $m_{\text{DM}}\lesssim
10\text{ GeV}$. Given the predicted dark matter mass of $3.4\text{ GeV}$, we
find that the strongest current limits on the DM-nucleon scattering cross
section Agnes  _et al._ (2018); Aprile  _et al._ (2022) do not place any
constraints in our scenario.

Recently, neutron stars have been put forward as a new laboratory for
detecting dark matter. They effectively act as calorimeters, and the energy
injected via annihilation of captured dark matter Kouvaris (2008) and/or
kinetic heating by dark matter Baryakhtar  _et al._ (2017) could provide a
signal in the form of anomalous surface temperature. The neutron star
sensitivity to dark matter scatterings is comparable to Earth-based
experiments Kouvaris and Tinyakov (2010) and they can accommodate for much
larger mass-splitting $(\lesssim 300\text{ MeV})$ in dark matter inelastic
scattering processes Bell  _et al._ (2018). A priori, neutron stars could be
the most sensitive probe of dark matter scatterings within our realisation. A
dark matter mass in the range of a few GeV is not significantly hindered by
Pauli-blocking effects Garani  _et al._ (2019), which become important when
$m_{\chi}<m_{n}$, where $m_{n}$ is the neutron mass. In this scenario, the
dominant capture process could be the inelastic scattering $\chi
n\rightarrow\chi N$, suppressed only by one insertion of $N-n$ mass mixing.
This holds for values of $m_{N}$ reasonably close to the neutron mass.
However, due to medium effects, the effective neutron mass inside the compact
star is significantly reduced compared to its vacuum value Anzuini  _et al._
(2021) and the dark matter kinetic energy is insufficient to overcome the
effective mass difference between $N$ and $n$. Within our scenario, neutron
stars are therefore not expected to yield any constraints on the parameters of
the framework.

Our scenario allows for exotic scatterings in which dark matter transmutes a
neutron into an antineutron333Previous works have considered this effect in
the context where $N$ is the dark matter candidate Jin and Gao (2018); Fornal
_et al._ (2021). Other scenarios in which dark matter and baryogenesis are
linked, can also generate similar exotic decays of baryons Berger and Elor
(2023)., $\chi n\rightarrow\chi^{\ast}\mkern
1.5mu\overline{\mkern-3.5mun\mkern-1.5mu}\mkern 2.5mu$, see Fig. 5 b). It is
possible to constrain dark matter-induced neutron transmutation with an
existing search for neutron-antineutron oscillation in Super-Kamiokande Abe
_et al._ (2021). However, in the present realization, due to the large
suppression coming from the values of
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{n}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
and
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{2}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$,
the potential dark matter signal sits well below existing and future limits.

The prospects for indirect detection of dark matter are strongly influenced by
whether the symmetric component has been depleted at freeze-out. In the
asymmetric case the only annihilation channel available is
$\chi\chi\rightarrow\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern
2.5mu\mkern 1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu$. This
process is highly suppressed in the present realization due to the large
effective scale
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Lambda_{2}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$,
and no constraints on the model parameters are expected. Nevertheless, it is
noteworthy that an asymmetric complex scalar dark matter candidate could have
an open direct annihilation channel today. In the symmetric scenario, it is
expected that strong $\chi\chi^{\ast}\rightarrow N\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu$ annihilations will
occur, as $\chi$ is a light and thermal dark matter candidate. However, due to
the conservation of baryon number, $N$ must eventually decay into neutrons,
whose masses would already account for a significant portion of the energy
budget in the final state of the annihilation. The decay of $N$ into semi-
relativistic neutrons would emit a soft cascade of various hadrons and/or a
gamma-ray of energy $E_{\gamma}=m_{N}-m_{n}$, which could provide a window
into both the nature of dark matter and its link to the neutron portal.

$\chi$$\chi$$\frac{1}{\Lambda_{0}}$$\delta m$$\delta m$${\rm Xe}$${\rm
Xe}$$n$$N$$n$$N$a)$\chi$$\chi^{\ast}$$\frac{1}{\Lambda_{2}}$$\delta m$$\delta
m$$n$$\mkern 1.5mu\overline{\mkern-3.5mun\mkern-1.5mu}\mkern 2.5mu$$N$$\mkern
1.5mu\overline{\mkern-3.5muN\mkern-1.5mu}\mkern 2.5mu$b)

Figure 5: a) Scattering process for direct detection of dark matter. b) Dark
matter scattering process which converts a neutron into antineutron. Time
flows from left to right in both diagrams.

##  6 Conclusions

We have presented a scenario that accommodates both dark matter and a quark-
antiquark asymmetry. We have postulated the existence of a spin-0 and a
spin-1/2 particle in the dark sector, both singlets under the Standard Model
gauge group, and carrying baryon number. As initial conditions, we have
assumed that at very high temperatures the Universe has zero baryon and lepton
numbers, but that the dark sector contains an asymmetry between particles and
antiparticles. We have argued that the asymmetry in the spin 1/2 particle can
be transmitted to the visible sector through (baryon conserving) neutron
portal interactions, thus resulting into a quark-antiquark asymmetry (and
possibly a lepton-antilepton asymmetry via sphalerons). On the other hand, the
spin-0 particle is stable due to the baryon number conservation and
constitutes a dark matter candidate. In this framework, the $B-L$ number of
the visible sector is exactly compensated by an opposite asymmetry in the dark
sector, thus linking the observed quark-antiquark asymmetry to the existence
of dark matter.

Under reasonable assumptions, we expect the dark matter mass to be $\sim 3.4$
GeV if it is fully asymmetric, or potentially lighter if there remains a
population of dark matter antiparticles after freeze-out. The scenario also
predicts the mass of the exotic spin-1/2 particle to be comparable to that of
the dark matter. Such a particle could be produced at the LHC or in flavor
physics experiments through the neutron portal, generically leaving the
detector before decaying. This particle would then produce a signal of missing
energy, and an apparent violation of baryon number, due to the imbalance in
the baryon number of the visible sector particles involved in the reaction.

Finally, we have briefly discussed the prospects for observing dark matter
signals in our scenario. We find that the most promising avenue to detect
signals lies in the Higgs portal, either through the invisible Higgs decay
width or in direct detection experiments, akin to the singlet scalar dark
matter model.

## Acknowledgments

This work is supported by the Collaborative Research Center SFB1258 and by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy - EXC-2094 - 390783311. We thank Motoko
Fujiwara, Florian Jörg, Robert McGehee and David Rousso for useful
discussions.

## References

  * Aghanim  _et al._ (2020) N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO] .
  * Sakharov (1967) A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5, 32 (1967).
  * Yoshimura (1978) M. Yoshimura, Phys. Rev. Lett. 41, 281 (1978), [Erratum: Phys.Rev.Lett. 42, 746 (1979)].
  * Kuzmin  _et al._ (1985) V. A. Kuzmin, V. A. Rubakov, and M. E. Shaposhnikov, Phys. Lett. B 155, 36 (1985).
  * Dodelson and Widrow (1990) S. Dodelson and L. M. Widrow, Phys. Rev. D 42, 326 (1990).
  * Farrar and Shaposhnikov (1993) G. R. Farrar and M. E. Shaposhnikov, Phys. Rev. Lett. 70, 2833 (1993), [Erratum: Phys.Rev.Lett. 71, 210 (1993)], arXiv:hep-ph/9305274 .
  * 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] .
  * Kaplan (1992) D. B. Kaplan, Phys. Rev. Lett. 68, 741 (1992).
  * Kuzmin (1998) V. A. Kuzmin, Phys. Part. Nucl. 29, 257 (1998), arXiv:hep-ph/9701269 .
  * Barr  _et al._ (1990) S. M. Barr, R. S. Chivukula, and E. Farhi, Phys. Lett. B 241, 387 (1990).
  * Kitano and Low (2005) R. Kitano and I. Low, Phys. Rev. D 71, 023510 (2005), arXiv:hep-ph/0411133 .
  * Bernal  _et al._ (2016) N. Bernal, C. S. Fong, and N. Fonseca, JCAP 09, 005, arXiv:1605.07188 [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] .
  * Servant and Tulin (2013) G. Servant and S. Tulin, Phys. Rev. Lett. 111, 151601 (2013), arXiv:1304.3464 [hep-ph] .
  * Oaknin and Zhitnitsky (2005) D. H. Oaknin and A. Zhitnitsky, Phys. Rev. D 71, 023519 (2005).
  * Farrar and Zaharijas (2006) G. R. Farrar and G. Zaharijas, Phys. Rev. Lett. 96, 041302 (2006).
  * 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] .
  * Elor  _et al._ (2019) G. Elor, M. Escudero, and A. Nelson, Phys. Rev. D 99, 035031 (2019), arXiv:1810.00880 [hep-ph] .
  * Davoudiasl and Mohapatra (2012) H. Davoudiasl and R. N. Mohapatra, New J. Phys. 14, 095011 (2012), arXiv:1203.1247 [hep-ph] .
  * Petraki and Volkas (2013) K. Petraki and R. R. Volkas, Int. J. Mod. Phys. A 28, 1330028 (2013), arXiv:1305.4939 [hep-ph] .
  * Zurek (2014) K. M. Zurek, Phys. Rept. 537, 91 (2014), arXiv:1308.0338 [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 (2023) B. Fornal, (2023), arXiv:2306.11349 [hep-ph] .
  * (24) In preparation.
  * D’Onofrio  _et al._ (2014) M. D’Onofrio, K. Rummukainen, and A. Tranberg, Phys. Rev. Lett. 113, 141602 (2014), arXiv:1404.3565 [hep-ph] .
  * Harvey and Turner (1990) J. A. Harvey and M. S. Turner, Phys. Rev. D 42, 3344 (1990).
  * Gondolo and Gelmini (1991) P. Gondolo and G. Gelmini, Nucl. Phys. B 360, 145 (1991).
  * Cheung and Ishiwata (2013) C. Cheung and K. Ishiwata, Phys. Rev. D 88, 017901 (2013), arXiv:1304.0468 [hep-ph] .
  * Allahverdi  _et al._ (2020) R. Allahverdi _et al._ 10.21105/astro.2006.16182 (2020), arXiv:2006.16182 [astro-ph.CO] .
  * Allahverdi and Osiński (2022) R. Allahverdi and J. K. Osiński, Phys. Rev. D 105, 023502 (2022), arXiv:2108.13136 [hep-ph] .
  * Clark  _et al._ (2017) D. B. Clark, E. Godat, and F. I. Olness, Comput. Phys. Commun. 216, 126 (2017), arXiv:1605.08012 [hep-ph] .
  * Heeck (2021) J. Heeck, Phys. Lett. B 813, 136043 (2021), arXiv:2009.01256 [hep-ph] .
  * McKeen  _et al._ (2021) D. McKeen, M. Pospelov, and N. Raj, Phys. Rev. D 103, 115002 (2021), arXiv:2012.09865 [hep-ph] .
  * Griest and Seckel (1991) K. Griest and D. Seckel, Phys. Rev. D 43, 3191 (1991).
  * D’Agnolo and Ruderman (2015) R. T. D’Agnolo and J. T. Ruderman, Phys. Rev. Lett. 115, 061301 (2015), arXiv:1505.07107 [hep-ph] .
  * Hamdan and Unwin (2018) S. Hamdan and J. Unwin, Mod. Phys. Lett. A 33, 1850181 (2018), arXiv:1710.03758 [hep-ph] .
  * Arcadi  _et al._ (2020) G. Arcadi, A. Djouadi, and M. Raidal, Phys. Rept. 842, 1 (2020), arXiv:1903.03616 [hep-ph] .
  * Fornal and Grinstein (2020) B. Fornal and B. Grinstein, Mod. Phys. Lett. A 35, 2030019 (2020), arXiv:2007.13931 [hep-ph] .
  * Agnes  _et al._ (2018) P. Agnes _et al._ (DarkSide), Phys. Rev. Lett. 121, 081307 (2018), arXiv:1802.06994 [astro-ph.HE] .
  * Aprile  _et al._ (2022) E. Aprile _et al._ (XENON), Phys. Rev. Lett. 129, 161805 (2022), arXiv:2207.11330 [hep-ex] .
  * Kouvaris (2008) C. Kouvaris, Phys. Rev. D 77, 023006 (2008), arXiv:0708.2362 [astro-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] .
  * Kouvaris and Tinyakov (2010) C. Kouvaris and P. Tinyakov, Phys. Rev. D 82, 063531 (2010), arXiv:1004.0586 [astro-ph.GA] .
  * Bell  _et al._ (2018) N. F. Bell, G. Busoni, and S. Robles, JCAP 09, 018, arXiv:1807.02840 [hep-ph] .
  * Garani  _et al._ (2019) R. Garani, Y. Genolini, and T. Hambye, JCAP 05, 035, arXiv:1812.08773 [hep-ph] .
  * Anzuini  _et al._ (2021) F. Anzuini, N. F. Bell, G. Busoni, T. F. Motta, S. Robles, A. W. Thomas, and M. Virgato, JCAP 11 (11), 056, arXiv:2108.02525 [hep-ph] .
  * Jin and Gao (2018) M. Jin and Y. Gao, Phys. Rev. D 98, 075026 (2018), arXiv:1808.10644 [hep-ph] .
  * Fornal  _et al._ (2021) B. Fornal, A. Hewitt, and Y. Zhao, Phys. Lett. B 815, 136151 (2021), arXiv:2011.09014 [hep-ph] .
  * Berger and Elor (2023) J. Berger and G. Elor, (2023), arXiv:2301.04165 [hep-ph] .
  * Abe  _et al._ (2021) K. Abe _et al._ (Super-Kamiokande), Phys. Rev. D 103, 012008 (2021), arXiv:2012.02607 [hep-ex] .