# Baryonic and Leptonic GeV Dark Matter

Bartosz Fornal  Department of Physics and Astronomy, University of Utah, Salt
Lake City, UT 84112, USA  Alec Hewitt  Department of Physics and Astronomy,
University of Utah, Salt Lake City, UT 84112, USA  Yue Zhao  Department of
Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA

###### Abstract

We perform a systematic analysis of models with GeV-scale dark matter coupled
to baryons and leptons. Such theories provide a natural framework to explain
the matter-antimatter asymmetry of the universe. We find that only a few
baryonic dark matter models are free from tree-level proton decay without
explicitly imposing baryon number conservation. We enumerate those cases and
provide a brief overview of their phenomenology. We then focus on a leptonic
dark matter model for a more detailed discussion of the baryon asymmetry
generation via leptogenesis, the symmetry restoration in the dark sector and
the expected dark matter annihilation signals in indirect detection
experiments.

##  I Introduction

The evidence for dark matter in the universe is indisputable. Not only its
existence, but also its distribution and abundance are precisely determined
from various measurements, including galaxy rotation curves Rubin and Ford
(1970), cosmic microwave background de Bernardis  _et al._ (2000),
gravitational lensing Gavazzi  _et al._ (2007), etc. Nevertheless, the dark
matter mass and its interactions with Standard Model particles remain a
mystery. It is not even known if dark matter consists of elementary particles
or macroscopic objects. The mass of an elementary dark matter particle can be
anywhere between $\sim 10^{-31}\ {\rm GeV}$ (fuzzy dark matter) Press  _et
al._ (1990); Hui  _et al._ (2017) and $\sim 10^{19}\ {\rm GeV}$ (WIMPzillas)
Kolb  _et al._ (1999); Meissner and Nicolai (2019) (despite the unitarity
bound Griest and Kamionkowski (1990)), while the mass of macroscopic dark
matter objects ranges from $\sim 10^{17}\ {\rm GeV}$ (dark quark nuggets) Bai
_et al._ (2019) to $\sim 10^{59}\ {\rm GeV}$ (primordial black holes) Carr
_et al._ (2010); Bird  _et al._ (2016). In most cases, the dark matter
interactions with the known particles are small; from a theoretical
perspective such interactions can even be absent.

Interestingly, the ratio of the abundances of dark matter and ordinary matter
is on the order of five. This suggests that the two sectors may be related
and, perhaps, share a common origin. This is precisely the idea behind
theories of asymmetric dark matter Kaplan  _et al._ (2009); Zurek (2014), in
which an effective interaction between dark matter and Standard Model
particles is established. The explanation of the matter-antimatter asymmetry
of the universe in those theories relies on the asymmetries in the two sectors
being generated simultaneously and the dark matter particles being at the GeV
scale.

In this paper, we systematically analyze scenarios in which dark matter
couples to quarks and/or leptons. We study the possible effective operators
describing such interactions up to dimension eight and analyze their particle
model realizations. For a successful baryogenesis or leptogenesis, those
operators must have a nonzero baryon number contribution carried by the quarks
or a nonzero lepton number from the leptons. The effective dark matter-
Standard Model interactions require either scalar or vector mediators to be
present at the particle level of each model. For dark matter coupled to
baryons (baryonic dark matter) at least one mediator in each model is
necessarily a color triplet. For dark matter coupled to leptons (leptonic dark
matter) the mediators do not carry color.

We demonstrate that for baryonic and baryoleptonic dark matter the possible
mediators are: the scalars $(3,1)_{-4/3}$, $(3,1)_{-1/3}$, $(3,1)_{2/3}$, or
the vectors $(3,2)_{-5/6}$, $(3,2)_{1/6}$. Without imposing an additional
symmetry, all of those particles, except for the scalar $(3,1)_{2/3}$, can
trigger tree-level proton decay Arnold  _et al._ (2013); Assad  _et al._
(2018). Thus the mass of those mediators is elevated above $\sim 10^{16}\ {\rm
GeV}$ by the stringent experimental constraints on the proton lifetime,
limiting their capabilities of explaining baryogenesis and their experimental
probes.

Such large mediator masses can be avoided by fine-tuning some of the couplings
to be small or by explicitly imposing baryon and/or lepton number
conservation. However, there is no strong theoretical argument to expect
baryon or lepton number to be exact symmetries of nature. In fact, both of
them are already violated within the Standard Model itself at the non-
perturbative level by the electroweak sphalerons. Guided by the requirement of
proton stability without assuming a fine-tuning of couplings or imposing an
additional symmetry, we focus on baryonic dark matter models involving only
the scalar mediator $(3,1)_{2/3}$, as well as on leptonic dark matter models.
The latter are naturally free from tree-level proton decay, since they involve
the color singlet scalar mediators $(1,1)_{1}$ or $(1,2)_{-1/2}$.

We find that only some of the baryonic dark matter direct detection signatures
considered in the literature are realized in models that are free from proton
decay. In particular, the dark matter-nucleon annihilation, which is possible
in such models, necessarily involves a kaon in the final state. This
conclusion increases the importance of the Deep Underground Neutrino
Experiment (DUNE) Abi  _et al._ in complementing the efforts of Super-
Kamiokande Fukuda  _et al._ (2003) in the search for baryonic dark matter.
Regarding leptonic dark matter models, based on a concrete example, we analyze
their potential for generating the matter-antimatter asymmetry through
leptogenesis and highlight their unique feature of symmetry restoration in the
dark sector, achieved when one of the dark sector particles is unstable on
long time scales and decays to dark matter. Such a rebirth of symmetric dark
matter leads to the possibility of enhanced annihilation signals at present
times, previously considered in the context of heavy dark matter Cohen and
Zurek (2010); Bell  _et al._ (2011); Falkowski  _et al._ (2011) and
oscillating dark matter Buckley and Profumo (2012); Cirelli  _et al._ (2012);
Tulin  _et al._ (2012); Okada and Seto (2012); Ibe  _et al._ (2020), and
results in signatures that can be searched for by the Fermi Gamma-Ray Space
Telescope Atwood  _et al._ (2009) and the future e-ASTROGAM Tavani  _et al._
(2018).

##  II The models

In this section, we enumerate the possible particle physics models for the
effective operators describing the interactions of dark matter with just the
quarks (baryonic dark matter), both quarks and leptons (baryoleptonic dark
matter), and with leptons only (leptonic dark matter), and briefly discuss
their phenomenology. We adopt the four-component Dirac spinor notation for the
fermion fields, indicating with a subscript $L$ or $R$ their left- or right-
handed chirality. The conjugate fields are denoted by a bar symbol. We focus
on those operators which carry a nonzero baryon number contribution from
quarks or a nonzero lepton number contribution from leptons, since only those
theories provide promising asymmetric dark matter frameworks. In particular,
we do not consider operators of the type $q\bar{q}\chi\bar{\chi}$ or
$l\bar{l}\chi\bar{\chi}$, which arise, e.g., in certain theories with gauge
bosons coupled to baryon or lepton number Duerr  _et al._ (2013); Schwaller
_et al._ (2013). In the following analysis, $\chi$ and $\widetilde{\chi}$ are
Dirac fermions, whereas $\phi$ and $\Phi$ are complex scalars. Both $\chi$ and
$\phi$ are Standard Model singlets.

Baryonic dark matter

The simplest effective operator describing the interaction of dark matter with
quarks is the dimension six $qqq\chi$, where $q$ stands for $Q_{L}$, $d_{R}$
or $u_{R}$. The possible gauge-invariant realizations are
$u_{R}d_{R}d_{R}\chi$ and $Q_{L}Q_{L}d_{R}\chi$. There are three models one
can write down for those operators, labeled as Models 1A–1C in Table 1. They
involve the mediators: scalar $(3,1)_{2/3}$, scalar $(3,1)_{-1/3}$, and vector
$(3,2)_{1/6}$, respectively. Among those particles, only the scalar
$(3,1)_{2/3}$ in Model 1A does not give rise to tree-level proton decay, since
the quantum numbers do not allow $\Phi$ to couple to a quark and a lepton
Arnold  _et al._ (2013); Assad  _et al._ (2018). This is in contrast to the
other two cases, where the scalar $(3,1)_{-1/3}$ can have the couplings
$\Phi\,u_{R}d_{R}$ and $\Phi^{*}u_{R}e_{R}$, whereas the vector $(3,2)_{1/6}$
can couple via $X_{\mu}Q_{L}\gamma^{\mu}d_{R}$ and
$X^{\dagger}_{\mu}L_{L}\gamma^{\mu}u_{R}$, both leading to $p\to
e^{+}\pi^{0}$. For this reason, we focus on Model 1A below. An example of a
diagram generating the operator $qqq\chi$ is shown in Fig. 1.

At dimension seven, the possible effective interactions are $qqq\chi\phi$ and
$qqqH\chi$, where $H$ is the Higgs field. In this study, we do not consider
operators involving the Higgs field and only focus on the ones in the first
category. There are six particle models for the operator $qqq\chi\phi$,
denoted as Models 2A–2F in Table 1. They involve the same scalar and vector
mediators as introduced for the operator $qqq\chi$. In addition, an
intermediate fermion $\widetilde{\chi}$ is required (see, Fig. 2). We discuss
in more detail Model 2A, since, similarly to Model 1A, it does not suffer from
tree-level proton decay.

At dimension eight, the operators are $qqq\chi\phi^{2}$, $qqqH\chi\phi$ and
$qqqH^{2}\chi$. Again, we only consider the ones in the first category. Models
for the operator $qqq\chi\phi^{2}$ can be constructed by introducing an
additional scalar singlet field $\phi_{e}$, replacing $\phi$ with $\phi_{e}$
in Models 2A–2F, and adding the interaction $\phi_{e}^{*}\phi^{2}$. The
particle $\phi$ is then automatically stable without imposing specific
relations between the masses. The particle $\chi$, on the other hand, may be
unstable if sufficiently heavy.

Baryonic dark matter
---
Model | Interactions | Mediators
${\bm{qqq\chi}}$
1A | $\Phi\,d_{R}d_{R}$ , $\Phi^{*}\\!u_{R}\chi$ | $\Phi=(3,1)_{\frac{2}{3}}$
1B | $\big{(}\Phi\,u_{R}d_{R}$ or $\Phi\,Q_{L}Q_{L}\big{)}$, | $\Phi=(3,1)_{-\frac{1}{3}}$
| $\Phi^{*}\\!d_{R}\chi$ |
1C | $\\!\\!\\!X_{\mu}Q_{L}\gamma^{\mu}d_{R}$ , $X^{\dagger}_{\mu}Q_{L}\gamma^{\mu}\chi$ | $X_{\mu}=(3,2)_{\frac{1}{6}}$
${\bm{qqq\chi\phi}}$
2A | $\\!\\!\\!\Phi\,d_{R}d_{R}$ , $\Phi^{*}\\!u_{R}\widetilde{\chi}$ , $\overline{\widetilde{\chi}}\chi\phi$ | $\Phi=(3,1)_{\frac{2}{3}}$ | $\widetilde{\chi}=(1,1)_{0}$
2B | $\\!\\!\\!\Phi\,d_{R}d_{R}$ , $\Phi^{*}\widetilde{\chi}\chi$ , $\overline{\widetilde{\chi}}u_{R}\phi$ |  | $\widetilde{\chi}=(3,1)_{\frac{2}{3}}$
2C | $\big{(}\Phi\,u_{R}d_{R}$ or $\Phi\,Q_{L}Q_{L}$), | $\\!\Phi=(3,1)_{-\frac{1}{3}}\\!\\!\\!$ | $\widetilde{\chi}=(1,1)_{0}$
| $\Phi^{*}\\!d_{R}\widetilde{\chi}$ , $\overline{\widetilde{\chi}}\chi\phi\ \ $ |  |
2D | $\big{(}\Phi\,u_{R}d_{R}$ or $\Phi\,Q_{L}Q_{L}$), |  | $\\!\\!\widetilde{\chi}=(3,1)_{-\frac{1}{3}}\\!\\!\\!\\!$
|  $\Phi^{*}\widetilde{\chi}\chi$ , $\overline{\widetilde{\chi}}d_{R}\phi$ |  |
2E | $\\!\\!\\!X_{\mu}Q_{L}\gamma^{\mu}d_{R}$ , $\overline{\widetilde{\chi}}\chi\phi$ , | $\\!\\!X_{\mu}=(3,2)_{\frac{1}{6}}\\!\\!\\!\\!$ | $\widetilde{\chi}=(1,1)_{0}$
| $X^{\dagger}_{\mu}Q_{L}\gamma^{\mu}\widetilde{\chi}\\!\\!\\!$ |  |
2F | $\\!\\!\\!X_{\mu}Q_{L}\gamma^{\mu}d_{R}$ , $X^{\dagger}\widetilde{\chi}\chi$ , |  | $\widetilde{\chi}=(3,2)_{\frac{1}{6}}$
| $\overline{\widetilde{\chi}}Q_{L}\phi$ |  |
Table 1: Effective operators describing the interaction of baryonic dark
matter with quarks and their model realizations. Figure 1: Model 1A
realization of the operator $qqq\chi$.

Model 1A ($\bm{qqq\chi}$)

The Lagrangian for Model 1A is given by

$\displaystyle-\mathcal{L}_{1}$ $\displaystyle\supset$
$\displaystyle\lambda_{q}^{ab}\epsilon^{ijk}\Phi_{i}d_{Rj}^{a}d_{Rk}^{b}+\lambda_{\chi}^{a}\Phi^{*i}{\chi}u_{Ri}^{a}\
+\ {\rm h.c.}\ ,\ \ \ \ \ $ (1)

where $i,j,k$ are color indices and $a,b$ are flavor indices. Due to the
antisymmetric nature of the $\epsilon$ tensor, the coupling $\lambda_{q}^{ab}$
must be antisymmetric in flavor.

Although there is no tree-level proton decay to a final state consisting of
only Standard Model particles, the model still suffers from proton decay if
$m_{\chi}<m_{p}-m_{e}=937.761\ {\rm MeV}$, since the proton can then undergo
the dark decay $p\to\bar{\chi}\,e^{+}\nu_{e}$. The mass range $937.761\ {\rm
MeV}<m_{\chi}<937.993\ {\rm MeV}$ is problematic as well – although the proton
remains stable, the ${}^{9}{\rm Be}$ nucleus, known to be stable, can then
undergo the nuclear dark decay ${}^{9}{\rm Be}\to\chi+2\,\alpha$ Pfutzner and
Riisager (2018).

The mass range $937.993\ {\rm MeV}<m_{\chi}<938.783\ {\rm MeV}$ is of
particular interest, since then the proton, all stable nuclei and the dark
matter $\chi$ remain stable, but the neutron can undergo the dark decay
$n\to\bar{\chi}\,\gamma$. This decay channel was proposed in Fornal and
Grinstein (2018) as a possible solution to the neutron lifetime discrepancy.
It was shown that Model 1B allows the neutron to have a dark decay branching
fraction ${\rm Br}(n\to\bar{\chi}\,\gamma)=1\%$, which corresponds to
$m_{\Phi}\sim\mathcal{O}(100\ {\rm TeV})$ for order one couplings; see Fornal
and Grinstein (2020) for a detailed review of this proposal along with its
experimental signatures. It was recently argued that within the framework of
Model 1A, the neutron dark decay branching fraction is constrained to be ${\rm
Br}(n\to\bar{\chi}\,\gamma)<10^{-6}$ Fajfer and Susic .

In Models 1A–1C, the dark matter $\chi$ can annihilate with the neutron,
leading to signatures such as $\chi\,n\to\gamma+{\rm meson(s)}$ at Super-
Kamiokande and the future DUNE Keung  _et al._ (2019). We will discuss this in
more detail in the context of Model 2A below. Finally, in the case
$m_{\chi}>m_{p}+m_{e}=938.783\ {\rm MeV}$ the dark particle $\chi$ is
unstable, since the decay channel $\chi\to\bar{p}\,e^{+}\nu_{e}$ opens up
kinematically. Although in this scenario $\chi$ is not the dark matter, it can
still be produced in experiments and lead to detectable signatures in $B$
factories, e.g., missing energy signals from dark decays of heavy baryons and
mesons Heeck .

Model 2A ($\bm{qqq\chi\phi}$)

The Lagrangian for Model 2A is given by a simple extension of the interactions
in Eq. (1),

$\displaystyle-\mathcal{L}_{2}\ $ $\displaystyle\supset$ $\displaystyle\
\lambda_{q}^{ab}\epsilon^{ijk}\Phi_{i}d_{Rj}^{a}d_{Rk}^{b}+\lambda_{\widetilde{\chi}}^{a}\Phi^{*i}{\widetilde{\chi}}u_{Ri}^{a}$
(2) $\displaystyle+\ \lambda_{\phi}\overline{\widetilde{\chi}}\chi\phi\,+{\rm
h.c.}\ .$

Once again, the coupling $\lambda_{q}^{ab}$ is antisymmetric in flavor. The
stability of the proton and all stable nuclei is guaranteed by the condition
$m_{\chi}+m_{\phi}>937.993\ {\rm MeV}$. In the special case $937.993\ {\rm
MeV}<m_{\chi}+m_{\phi}<m_{n}=939.565\ {\rm MeV}$, the neutron can undergo the
decay $n\to\bar{\chi}\,\phi^{*}$. This is the second neutron dark decay
channel proposed in Fornal and Grinstein (2018) to solve the neutron lifetime
discrepancy, and leads to unique signals in nuclear decays Pfutzner and
Riisager (2018). In contrast to Model 1A, a large mass of $\chi$ or $\phi$
does not necessarily lead to dark matter decay;provided that
$|m_{\chi}-m_{\phi}|<m_{p}+m_{e}=938.783\ {\rm MeV}$, both $\chi$ and $\phi$
remain stable.

Upon including an additional heavy particle $\widetilde{\chi}^{\prime}$ with
the coupling $\overline{\widetilde{\chi}^{\prime}}\chi\phi$, non-trivial $CP$
phases can lead to different decay probabilities for $\widetilde{\chi}\to udd$
versus $\overline{\widetilde{\chi}}\to\bar{u}\bar{d}\bar{d}$ through the
interference between tree-level and loop-level decay channels. Model 2A is
then capable of explaining the matter-antimatter asymmetry of the universe
through the hylogenesis mechanism Davoudiasl  _et al._ (2010) if
$m_{\chi}+m_{\phi}\sim 5\ {\rm GeV}$.

This model also predicts striking signatures in direct detection experiments:
$\chi\,N\to\phi^{*}+{\rm meson(s)}$, where $N$ is a proton or a neutron. Such
signals were studied in Davoudiasl  _et al._ (2011); Demidov and Gorbunov
(2016); Jin and Gao (2018); Keung  _et al._ (2019) and include:
$\chi\,n\to\phi^{*}\,\pi^{0}$, $\chi\,p\to\phi^{*}\,\pi^{+}$,
$\chi\,n\to\phi^{*}\,K^{0}$ and $\chi\,p\to\phi^{*}\,K^{+}$. Interestingly,
not all of them are present in Model 2A. Given the antisymmetric structure of
$\lambda_{q}^{ab}$ in Eq. (2), the scalar mediator does not couple to two down
quarks, which implies that signatures involving solely pions in the final
state are not possible in Model 2A.

In conclusion, we find that there is a specific prediction of baryonic dark
matter models without tree-level proton decay: dark matter-nucleon
annihilation leads to at least one kaon in the final state. This is an
especially relevant observation in light of DUNE’s expected exquisite
sensitivity to kaons. In certain regions of parameter space, DUNE will be able
to probe $\Phi$ masses up to $m_{\Phi}\sim\mathcal{O}(100\ {\rm TeV})$ Keung
_et al._ (2019).

Baryoleptonic dark matter

The simplest class of baryoleptonic dark matter models arises from dimension
seven effective operators $qqql\phi$ and $qqq\bar{l}\phi$, where $l$ is a
Standard Model lepton representation. Minimal particle physics realizations of
$qqql\phi$ involve one of the scalar mediators $(3,1)_{-1/3}$, $(3,1)_{-4/3}$,
or one of the vectors $(3,2)_{-5/6}$, $(3,2)_{1/6}$, whereas particle models
for $qqq\bar{l}\phi$ require the scalar mediator $(3,1)_{-1/3}$ or the vector
$(3,2)_{1/6}$. In models with only a single mediator, it has to couple to the
bilinears $qq$ and $ql$ (or $q\bar{l}$), resulting in tree-level proton decay
which cannot be forbidden by any symmetry.

A possible way to overcome this issue is to introduce two mediators, one
coupled only to $qq$, and the other coupled only to $ql$ (or $q\bar{l}$) Heeck
, or include an additional heavy fermion Huang and Zhao (2014). However, we
will not consider those scenarios here, since they require imposing
baryon/lepton number conservation. A simple extension of this class of models,
described by the effective operator $qqql\phi^{2}$, i.e., replacing $\phi$ by
$\phi_{e}$ and adding the interaction term $\phi_{e}^{*}\phi^{2}$ to stabilize
$\phi$, provides a working mechanism for baryogenesis as in Model 2A and also
exhibits nonstandard nucleon destruction signatures Huang and Zhao (2014). A
more general analysis of the baryon asymmetry generation in those types of
models was performed in Bernal  _et al._ (2016).

Figure 2: Model 2A realization of the operator $qqq\chi\phi$.

Leptonic dark matter

In order to explain the matter-antimatter asymmetry through leptogenesis,
operators carrying a nonzero lepton number contribution from the leptons are
needed. Those operators can generate an asymmetry in the lepton sector, which
is then transferred to the baryon sector by electroweak sphalerons Klinkhamer
and Manton (1984). The dimension four and five operators $l\bar{l}\phi$ and
$l\bar{l}\phi^{2}$ are not of interest to us since $\phi$ does not carry
lepton number. Also, as in the baryonic dark matter case, we will not consider
operators involving the Higgs, e.g., the dimension four operator $HL_{L}\chi$.

The first operator of interest is the dimension six $ll\bar{l}\chi$, where $l$
can be either $L_{L}$ or $e_{R}$. The only gauge-invariant realization is
$L_{L}L_{L}\mkern 1.5mu\overline{\mkern-1.5mue_{R}\mkern-1.5mu}\mkern
1.5mu\chi$. Note that the operator $lll\chi$ is not invariant under
hypercharge. There are two particle models for the operator $ll\bar{l}\chi$,
denoted as Models 3A and 3B in Table 2. The corresponding mediators are the
scalars $(1,1)_{1}$ and $(1,2)_{-1/2}$, respectively. They are colorless, thus
they do not mediate proton decay. Because of the similarity between the two
models, we write down the Lagrangian only for Model 3A. Its realization of the
operator $ll\bar{l}\chi$ is shown in Fig. 3.

Leptonic dark matter
---
Model | Interactions | Mediators
${\bm{ll\bar{l}\chi}}$
3A | $\Phi L_{L}L_{L}$ , $\Phi^{*}\mkern 1.5mu\overline{\mkern-1.5mue_{R}\mkern-1.5mu}\mkern 1.5mu\chi$ | $\Phi=(1,1)_{1}$
3B | $\Phi L_{L}\mkern 1.5mu\overline{\mkern-1.5mue_{R}\mkern-1.5mu}\mkern 1.5mu$ , $\Phi^{*}\\!{L_{L}}\chi$ | $\Phi=(1,2)_{-\frac{1}{2}}$
${\bm{ll\bar{l}\chi\phi}}$
4A | $\Phi L_{L}L_{L}$ , $\Phi^{*}\mkern 1.5mu\overline{\mkern-1.5mue_{R}\mkern-1.5mu}\mkern 1.5mu\widetilde{\chi}$ , $\overline{\widetilde{\chi}}\chi\phi$ | $\Phi=(1,1)_{1}$ | $\widetilde{\chi}=(1,1)_{0}$
4B | $\Phi L_{L}L_{L}$ , $\Phi^{*}\widetilde{\chi}\chi$ , $\overline{\widetilde{\chi}}\mkern 1.5mu\overline{\mkern-1.5mue_{R}\mkern-1.5mu}\mkern 1.5mu\phi$ |  | $\widetilde{\chi}=(1,1)_{1}$
4C | $\Phi L_{L}\mkern 1.5mu\overline{\mkern-1.5mue_{R}\mkern-1.5mu}\mkern 1.5mu$ , $\Phi^{*}L_{L}\widetilde{\chi}$ , $\overline{\widetilde{\chi}}\chi\phi$ | $\\!\Phi=(1,2)_{-\frac{1}{2}}\\!\\!\\!$ | $\widetilde{\chi}=(1,1)_{0}$
4D | $\Phi L_{L}\mkern 1.5mu\overline{\mkern-1.5mue_{R}\mkern-1.5mu}\mkern 1.5mu$ , $\Phi^{*}\widetilde{\chi}\chi$ , $\overline{\widetilde{\chi}}L_{L}\phi$ |  |  $\widetilde{\chi}=(1,2)_{-\frac{1}{2}}$
Table 2: Effective operators describing the interaction of leptonic dark
matter with leptons and their model realizations. Figure 3: Model 3A
realization of the operator $ll\bar{l}\chi$.

At dimension seven, the effective operator of interest is $ll\bar{l}\chi\phi$,
again with only a single gauge-invariant realization $L_{L}L_{L}\mkern
1.5mu\overline{\mkern-1.5mue_{R}\mkern-1.5mu}\mkern 1.5mu\chi\phi$. The
corresponding particle models are labeled as Models 4A–4D in Table 2. An
intermediate particle $\widetilde{\chi}$ is required. We focus on Model 4A
below for a quantitative discussion of its properties. The realization of the
operator $ll\bar{l}\chi\phi$ within this model is presented in Fig. 4.

Regarding other options, not involving the Higgs field, the dimension seven
operator $ll\bar{l}\bar{l}\phi$ does not carry nonzero lepton number, thus it
cannot fit into the framework of asymmetric dark matter. At dimension eight,
the only operator carrying lepton charge is $ll\bar{l}\chi\phi^{2}$. As in the
baryonic dark matter case, models for this operator are obtained from Models
4A–4D by substituting $\phi$ with $\phi_{e}$, and introducing the interaction
$\phi_{e}^{*}\phi^{2}$. This makes $\phi$ automatically stable, but offers no
other advantages compared to the $ll\bar{l}\chi\phi$ case.

Model 3A ($\bm{ll\bar{l}\chi}$)

The Lagrangian for Model 3A is

$\displaystyle-\mathcal{L}_{3}\ $ $\displaystyle\supset$ $\displaystyle\
\lambda_{l}^{ab}\Phi(L_{L}^{a}\epsilon
L_{L}^{b})+\lambda_{\chi}^{a}\,\Phi^{*}\\!{\chi}\mkern
1.5mu\overline{\mkern-1.5mue_{R}\mkern-1.5mu}\mkern 1.5mu^{a}+{\rm h.c.}\ ,\ \
\ \ \ $ (3)

where again $a,b$ are flavor indices and the parenthesis denotes the
contraction of ${\rm SU}(2)_{L}$ indices. Because of the antisymmetric
structure of the $LL$ bilinear, the coupling $\lambda_{l}^{ab}$ must be
antisymmetric in flavor.

This model is not phenomenologically attractive because it is hard to ensure
the stability of the dark matter particle. Especially, without imposing any
lepton flavor symmetry, $\chi$ can undergo the decay $\chi\to\nu\gamma$
through a loop diagram. Thus, we do not consider this class of models further.

Model 4A ($\bm{ll\bar{l}\chi\phi}$)

A phenomenologically viable model which contains a dark matter candidate and
can successfully explain the matter-antimatter of the universe is obtained by
introducing an additional scalar particle $\phi$. The resulting Lagrangian is
a minimal extension of $\mathcal{L}_{3}$,

$\displaystyle-\mathcal{L}_{4}\ $ $\displaystyle\supset$ $\displaystyle\
\lambda_{l}^{ab}\Phi(L_{L}^{a}\epsilon
L_{L}^{b})+\lambda_{\widetilde{\chi}}^{a}\,\Phi^{*}{\widetilde{\chi}}\mkern
1.5mu\overline{\mkern-1.5mue_{R}\mkern-1.5mu}\mkern
1.5mu^{a}+\lambda_{\phi}\overline{\widetilde{\chi}}\chi\phi+{\rm h.c.}\ .\ \ \
\ \ $ (4)

The same symmetry arguments as for Model 3A apply, i.e., $\lambda_{l}^{ab}$ is
antisymmetric in flavor. In addition, we imposed a $Z_{2}$ parity in the dark
sector, under which $\phi$ and $\chi$ are odd, so that the lighter of them
remains stable and can constitute the dark matter. Model 4A is the subject of
the subsequent section.

Figure 4: Model 4A realization of the operator $ll\bar{l}\chi\phi$.

##  III Phenomenology of leptonic dark matter

The crucial property of leptonic dark matter models is that they do not suffer
from proton decay. Below we discuss the baryon asymmetry generation in Model
4A via leptogenesis, the symmetry restoration in the dark sector and the
signatures expected in indirect detection experiments. Our conclusions apply
to Models 4B–4D as well.

Leptogenesis

The generation of lepton asymmetry within the framework of Model 4A is similar
to the hylogenesis mechanism Davoudiasl  _et al._ (2010). The process starts
immediately after inflation. Once the inflaton field $\Psi$ falls into the
potential minimum, it starts oscillating and leads to the reheating of the
universe. We assume that during reheating the particles $\widetilde{\chi}$ and
$\overline{\widetilde{\chi}}$ are produced in equal amounts. As the
temperature drops, the particles $\widetilde{\chi}$ decay via the channels
$\widetilde{\chi}\to\chi\,\phi$ and $\widetilde{\chi}\to e_{R}\mkern
1.5mu\overline{\mkern-1.5mue_{L}\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5mu\nu_{L}\mkern-1.5mu}\mkern 1.5mu$, where $\mkern
1.5mu\overline{\mkern-1.5mue_{L}\mkern-1.5mu}\mkern 1.5mu$ and $\mkern
1.5mu\overline{\mkern-1.5mu\nu_{L}\mkern-1.5mu}\mkern 1.5mu$ have different
flavors. The antiparticles $\overline{\widetilde{\chi}}$ decay through
conjugate processes.

In order to generate $CP$ violation in the model, we introduce the particle
$\widetilde{\chi}^{\prime}$ with the following interaction terms,

$\displaystyle-\mathcal{L}^{\prime}_{4}\ \supset\
\lambda_{\widetilde{\chi}^{\prime}}^{a}\,\Phi^{*}{\widetilde{\chi}^{\prime}}\mkern
1.5mu\overline{\mkern-1.5mue_{R}\mkern-1.5mu}\mkern
1.5mu^{a}+\lambda^{\prime}_{\phi}\,\overline{\widetilde{\chi}^{\prime}}\chi\phi+{\rm
h.c.}\ ,$ (5)

and we take $m_{\widetilde{\chi}^{\prime}}\gg m_{\widetilde{\chi}}$. The
asymmetry between the decays of ${\widetilde{\chi}}$ and
$\overline{\widetilde{\chi}}$ arises from the interference between the tree-
level and one-loop diagram shown in Fig. 5. If the leading decay channel is
$\widetilde{\chi}\to\chi\,\phi$, the generated lepton asymmetry is 111In this
calculation, we only consider the decay channel with one particular lepton
flavor choice. The rescaling can be easily done to include more flavor
channels.

$\displaystyle\Delta_{L}\ $ $\displaystyle=$ $\displaystyle\
\frac{\Gamma(\widetilde{\chi}\to e_{R}\mkern
1.5mu\overline{\mkern-1.5mue_{L}\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5mu\nu_{L}\mkern-1.5mu}\mkern
1.5mu)-\Gamma(\overline{\widetilde{\chi}}\to\mkern
1.5mu\overline{\mkern-1.5mue_{R}\mkern-1.5mu}\mkern
1.5mu{e}_{L}\nu_{L})}{2\,\Gamma(\widetilde{\chi}\to\chi\,\phi)}$ (6)
$\displaystyle\approx$ $\displaystyle\ \frac{|\lambda_{l}|^{2}\,{\rm
Im}\big{(}\lambda_{\widetilde{\chi}}^{*}\lambda_{\widetilde{\chi}^{\prime}}\lambda_{\phi}\lambda_{\phi}^{\prime*}\big{)}}{{1536}\,\pi^{3}|\lambda_{\phi}|^{2}}\frac{m_{\widetilde{\chi}}^{5}}{m_{\Phi}^{4}\,m_{\widetilde{\chi}^{\prime}}}\
.\ \ \ $

In order to avoid the washout of the asymmetry by $\chi\,\phi\to e_{R}\mkern
1.5mu\overline{\mkern-1.5mue_{L}\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5mu\nu_{L}\mkern-1.5mu}\mkern 1.5mu$ scattering, the
reheating temperature, for couplings ${\mathcal{O}(1)}$, needs to satisfy
Davoudiasl  _et al._ (2010)

$\displaystyle T_{R}\lesssim{(11\ {\rm
TeV})}\bigg{[}{\frac{m_{\Phi}^{4}m_{\widetilde{\chi}}^{2}}{(1\ {\rm
PeV})^{6}}}\bigg{]}^{\\!\frac{1}{5}}\\!.$ (7)

Figure 5: Diagrams contributing to the decay $\widetilde{\chi}\to e_{R}\mkern
1.5mu\overline{\mkern-1.5mue_{L}\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5mu\nu_{L}\mkern-1.5mu}\mkern 1.5mu$.

Provided that $T_{R}$ is above the electroweak symmetry breaking scale, i.e.,
$T_{R}\gtrsim 200\ {\rm GeV}$, a portion of the created lepton asymmetry will
be converted into a baryon asymmetry by the electroweak sphalerons. Therefore,
as long as the masses of $\Phi$ and $\widetilde{\chi}$ fulfill the condition

$\displaystyle(m_{\Phi}^{4}m_{\widetilde{\chi}}^{2})^{\frac{1}{6}}\gtrsim{40\
{\rm TeV}}\ ,$ (8)

the sphalerons efficiently transfer the asymmetry to the baryonic sector. We
find that the final baryon asymmetry is

$\displaystyle\Delta_{B}\approx\frac{28}{79}\,\Delta_{L}\ .$ (9)

The exact relation between the baryon-to-photon ratio $\eta_{B}$ and the
baryon asymmetry depends on the model of reheating; up to $\mathcal{O}(1)$
factors it is given by Asaka  _et al._ (1999); Giudice  _et al._ (1999);
Buchmuller  _et al._ (2005)

$\displaystyle\eta_{B}\approx\frac{\Delta_{B}T_{R}}{M_{\Psi}}\ ,$ (10)

where $M_{\Psi}$ is the inflaton mass. The observed value of the baryon-to-
photon ratio in the universe of $\eta_{B}\approx 6\times 10^{-10}$ Aghanim
_et al._ (2020) is obtained, e.g., for $\mathcal{O}(1)$ couplings,
$m_{\Phi}=m_{\widetilde{\chi}}=1\ {\rm PeV}$,
$m_{\widetilde{\chi}^{\prime}}=5\ {\rm PeV}$, $M_{\Psi}=20\ {\rm PeV}$ and
$T_{R}=8\ {\rm TeV}$.

The sum of $\chi$ and $\phi$ masses is set by the observed ratio of the dark
matter and baryon abundances. Since the generated dark matter asymmetry
$\Delta_{\rm DM}=\Delta_{L}$, one arrives at

$\displaystyle m_{\chi}+m_{\phi}=m_{p}\frac{\Omega_{\rm
DM}}{\Omega_{B}}\,\bigg{|}\frac{\Delta_{B}}{\Delta_{\rm DM}}\bigg{|}\approx
1.8\ {\rm GeV}\ .$ (11)

Figure 6: Annihilation of the dark matter symmetric component.

Removing the symmetric component

To ensure the annihilation of the symmetric component of dark matter, it is
sufficient to introduce one new particle, e.g., a vector $V$, lighter than
$\chi$ and $\phi$, with the interaction terms

$\displaystyle-\mathcal{L}^{\prime\prime}_{4}\ $ $\displaystyle\supset$
$\displaystyle\ g_{\chi}V_{\mu}\,\mkern
1.5mu\overline{\mkern-1.5mu\chi\mkern-1.5mu}\mkern
1.5mu\gamma^{\mu}\chi+g_{\phi}V_{\mu}\,\phi\,\partial^{\mu}\phi^{*}$ (12)
$\displaystyle+\ g_{V}V_{\mu}\,\mkern
1.5mu\overline{\mkern-1.5mue_{R}\mkern-1.5mu}\mkern
1.5mu^{a}\gamma^{\mu}e_{R}^{a}+{\rm h.c.}\ .$

This allows for the annihilation channels $\mkern
1.5mu\overline{\mkern-1.5mu\chi\mkern-1.5mu}\mkern 1.5mu\chi\to VV^{\dagger}$
and $\phi\,\phi^{*}\to VV^{\dagger}$, along with the subsequent decays $V\to
l^{+}l^{-}$. Diagrams corresponding to $\mkern
1.5mu\overline{\mkern-1.5mu\chi\mkern-1.5mu}\mkern 1.5mu\chi\to VV^{\dagger}$
are shown in Fig. 6. The annihilation cross section $\sigma_{\mkern
1.5mu\overline{\mkern-1.5mu\chi\mkern-1.5mu}\mkern 1.5mu\chi}$ is related to
the symmetric component of the $\chi$ relic density as

$\displaystyle\Omega_{\chi}h^{2}\simeq\left(\frac{9\times 10^{-11}}{{\rm
GeV^{2}}}\right)\frac{1}{\langle\sigma_{\mkern
1.5mu\overline{\mkern-1.5mu\chi\mkern-1.5mu}\mkern
1.5mu\chi}v\rangle\sqrt{g_{*}}}\frac{m_{\chi}}{T_{f}}\ ,$ (13)

where $g_{*}$ is the number of relativistic degrees of freedom and $T_{f}$ is
the freeze-out temperature Kolb and Turner . Given the observed value of the
dark matter relic density $\Omega_{\rm DM}h^{2}=0.12$ Tanabashi  _et al._
(2018), we find that for $m_{V}\ll m_{\chi}$ an efficient annihilation of the
symmetric component of $\chi$ is achieved when

$\displaystyle g_{\chi}\gtrsim 0.01\ .$ (14)

Similar arguments apply to the annihilation $\phi\,\phi^{*}\to VV^{\dagger}$.
The decays of $V$, such as $V\to e^{+}e^{-}$, depending on its coupling and
mass, can be a slow process. Thus a small coupling between $V$ and leptons is
allowed, and can easily be consistent with all experimental constraints Jeong
_et al._ (2016).

Another possibility is to introduce a light scalar $\phi^{\prime}$, rather
than the vector boson $V$, to remove the symmetric components of $\chi$ and
$\phi$. The annihilation $\mkern
1.5mu\overline{\mkern-1.5mu\chi\mkern-1.5mu}\mkern
1.5mu\chi\to\phi^{\prime}\phi^{\prime*}$ is a $p$-wave process, thus it is
suppressed by the relative velocity. Such a velocity suppression does not
significantly affect the annihilation of the symmetric components, since both
$\chi$ and $\phi$ are still semi-relativistic during their freeze-out.
However, it has interesting implications for the experimental constraints and
signatures, which will be discussed below.

Symmetry restoration

Interestingly, within Model 4A the asymmetry in the dark sector is not
preserved during the evolution of the universe. If $m_{\phi}>m_{\chi}$, the
interactions in Eq. (4) render the particle $\phi$ unstable, resulting in the
two-body decay $\phi\to\mkern
1.5mu\overline{\mkern-1.5mu\chi\mkern-1.5mu}\mkern 1.5mu\,\mkern
1.5mu\overline{\mkern-1.5mu\nu\mkern-1.5mu}\mkern 1.5mu$ and the four-body
decay $\phi\to\mkern 1.5mu\overline{\mkern-1.5mu\chi\mkern-1.5mu}\mkern
1.5mu\,\mkern 1.5mu\overline{\mkern-1.5mu\nu\mkern-1.5mu}\mkern
1.5mu\,e^{+}e^{-}$. The two-body channel is dominant and proceeds through the
diagram shown in Fig. 7. The resulting decay rate is

$\displaystyle\Gamma(\phi\to\mkern
1.5mu\overline{\mkern-1.5mu\chi\mkern-1.5mu}\mkern 1.5mu\,\bar{\nu})$
$\displaystyle\sim$
$\displaystyle\frac{|\lambda_{l}\,\lambda_{\widetilde{\chi}}\lambda_{\phi}|^{2}}{4096\,\pi^{5}}\,\frac{m_{\tau}^{2}\,m_{\phi}}{m_{\widetilde{\chi}}^{2}}\bigg{(}1-\frac{m_{\chi}^{2}}{m_{\phi}^{2}}\bigg{)}^{\\!2}\
.\ \ \ $ (15)

The observed dark matter relic density imposes a constraint on this rate. More
explicitly, the decays $\phi\to\mkern
1.5mu\overline{\mkern-1.5mu\chi\mkern-1.5mu}\mkern 1.5mu\,\mkern
1.5mu\overline{\mkern-1.5mu\nu\mkern-1.5mu}\mkern 1.5mu$ restore the symmetry
between $\chi$ and $\bar{\chi}$ in the relic abundance. Such a restoration
must happen sufficiently late so that $\chi$ and $\mkern
1.5mu\overline{\mkern-1.5mu\chi\mkern-1.5mu}\mkern 1.5mu$ do not efficiently
annihilate with each other to cause $\mathcal{O}(1)$ change to the dark matter
relic abundance. We find that this requirement is met if the decay happens at
temperatures $T\lesssim 50$ MeV or, equivalently, after $t\simeq O(10^{-4})$
s. This can be achieved if the mass splitting between $\chi$ and $\phi$ is
small. For example, taking $\mathcal{O}(1)$ couplings,
$m_{\widetilde{\chi}}\sim 1\ {\rm PeV}$ and $(m_{\phi}-m_{\chi})/m_{\phi}\sim
10^{-3}$, the lifetime of $\phi$ is $\tau_{\phi}\sim$ $0.1\ {\rm s}$. Since
the decays of $\phi$’s only produce slow-moving $\chi$’s and low-energy
neutrinos due to the small mass splitting, the cosmological constraints, e.g.,
from Big Bang Nucleosynthesis, are easily evaded Kanzaki  _et al._ (2007);
Acharya and Khatri .

Figure 7: Diagram for the decay $\phi\to\mkern
1.5mu\overline{\mkern-1.5mu\chi\mkern-1.5mu}\mkern 1.5mu\,\mkern
1.5mu\overline{\mkern-1.5mu\nu\mkern-1.5mu}\mkern 1.5mu$.

The restored symmetry in the dark sector revives $\mkern
1.5mu\overline{\mkern-1.5mu\chi\mkern-1.5mu}\mkern 1.5mu\chi$ annihilation at
present times in regions with large dark matter concentration, e.g., in the
Galactic Center, which does not happen in standard asymmetric dark matter
models. A similar scenario of symmetry restoration in the dark sector was
proposed in the context of heavy dark matter Cohen and Zurek (2010); Bell  _et
al._ (2011); Falkowski  _et al._ (2011) and oscillating dark matter Buckley
and Profumo (2012); Cirelli  _et al._ (2012); Tulin  _et al._ (2012); Okada
and Seto (2012); Ibe  _et al._ (2020). In addition, late decays of the heavier
dark matter component, $\tau_{\phi}\sim 10^{9}\ {\rm years}$, along with a
small mass splitting between the components, are consistent with observation
Peter  _et al._ (2010); Bell  _et al._ (2010). Such a late decay is also
proposed as a possible solution to the missing satellites problem Abdelqader
and Melia (2008) and the core-cusp problem Cline  _et al._ .

Indirect detection signatures

The symmetry restoration in the dark sector induces nontrivial dark matter
indirect detection signals. For example, if $V$ or $\phi^{\prime}$ couples to
leptons and if its mass is larger than $2m_{\mu}$, the dark matter
annihilation leads to final states involving four leptons, i.e.,
$e^{+}e^{-}e^{+}e^{-}$, $e^{+}e^{-}\mu^{+}\mu^{-}$ or
$\mu^{+}\mu^{-}\mu^{+}\mu^{-}$. Those leptons can further undergo inverse
Compton scattering and bremsstrahlung, producing photons that can be measured
in indirect detection experiments.

The gamma ray constraints from the Fermi Gamma-Ray Space Telescope for a GeV-
scale dark matter annihilating to four leptons were studied in Abazajian  _et
al._ (2010). Depending on the species of the final state leptons, as well as
on the mass of the light mediator, the dark matter annihilation rate is
constrained to be $\langle\sigma_{\mkern
1.5mu\overline{\mkern-1.5mu\chi\mkern-1.5mu}\mkern 1.5mu\chi}v\rangle\lesssim
10^{-25}\,-\,10^{-24}\ {\rm cm^{3}}/{\rm s}$.

A more stringent constraint arises from the measurements of the Cosmic
Microwave Background (CMB) provided by the Planck satellite Aghanim  _et al._
(2020). More explicitly, $\langle\sigma_{\mkern
1.5mu\overline{\mkern-1.5mu\chi\mkern-1.5mu}\mkern 1.5mu\chi}v\rangle$ is
required to be smaller than $10^{-27}\ {\rm cm^{3}}/{\rm s}$ Slatyer (2016).
However, such a constraint is evaded if the light mediator is a scalar, i.e.,
$\phi^{\prime}$, since the dark matter annihilation cross section is largely
reduced due to the velocity suppression at late times. In addition, if $\phi$
or the light mediator ($V$ or $\phi^{\prime}$) is very long-lived due to
either a small mass splitting or tiny couplings, the energy deposition to the
Standard Model thermal bath only happens after the recombination, and the CMB
constraint is also avoided.

Lastly, dark matter annihilation to a four-lepton final state will be probed
independently by the future e-ASTROGAM experiment in the region
$\langle\sigma_{\mkern 1.5mu\overline{\mkern-1.5mu\chi\mkern-1.5mu}\mkern
1.5mu\chi}v\rangle\gtrsim 4\times 10^{-27}\ {\rm cm^{3}}/{\rm s}$ Bartels  _et
al._ (2017).

##  IV Summary

The nature of dark matter, origin of the matter-antimatter asymmetry of the
universe and proton stability are certainly among the greatest open questions
in modern particle physics. Theories in which dark matter couples to quarks
and leptons introduce a natural framework for solving the first two of those
puzzles. However, many of such models suffer from tree-level proton decay
unless a conservation of baryon and/or lepton number is imposed by hand.

In this paper, we chose proton stability as the primary criterion for a
model’s naturalness. We demonstrated that this reduces the number of viable
baryonic dark matter theories to just a few models involving the scalar
mediator $(3,1)_{2/3}$. This has an interesting impact on the potential
experimental signatures – we found that for baryonic models naturally free
from proton decay, the final state of dark matter-nucleon annihilation
necessarily involves a kaon.

We also considered a model of leptonic dark matter, which explains the matter-
antimatter asymmetry of the universe through leptogenesis and predicts
symmetry restoration in the dark sector. In this theory the dark matter
annihilation may be enhanced at late times, providing signals that can be
searched for in future indirect detection experiments.

### Acknowledgments

The authors are supported in part by the U.S. Department of Energy under Award
No. ${\rm DE}$-${\rm SC0009959}$.

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