11institutetext: Laboratoire de Physique de Clermont (UMR 6533), CNRS/IN2P3,
Univ. Clermont Auvergne, 4 Av. Blaise Pascal, F-63178 Aubière Cedex,
France22institutetext: School of Physics and Astronomy, The University of
Manchester, Manchester M13 9PL, United Kingdom33institutetext: National and
Kapodistrian University of Athens, Department of Physics, Section of Nuclear
and Particle Physics, GR-157 84 Athens, Greece

# Ultraviolet freeze-in baryogenesis

Andreas Goudelis 2 , Dimitrios Karamitros 3 , Pantelis Papachristou 3 ,
Vassilis C. Spanos<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study a mechanism through which the cosmic dark matter density can be
explained simultaneously with the observed baryon asymmetry of the Universe.
At the core of our proposal lie the out-of-equilibrium scattering processes of
bath particles which are responsible for the production of feebly-interacting
dark matter. The same processes violate $CP$, which further leads to an
asymmetry between matter and antimatter being generated in the visible sector.
We focus on the possibility that these interactions are described through non-
renormalizable operators, which leads to both dark matter and the baryon
asymmetry being produced at high temperatures. The mechanism is exemplified by
studying two concrete scenarios, one involving scalar and one involving
fermion dark matter. We find that in both cases it is, indeed, possible to
achieve a common explanation for the dark matter content and the matter-
antimatter asymmetry of the Universe, provided that dark matter is in the keV
mass range.

##  1 Introduction

The nature of dark matter (DM) and the origin of the matter-antimatter
asymmetry of the Universe constitute two of the most fundamental open
questions in contemporary high-energy physics and cosmology. Despite the fact
that, to date, there is no compelling argument as to why (and how) these two
questions could be connected with each other, the idea that they might admit a
common microscopical and cosmological explanation has been consistently
entertained in several different frameworks. In the context of asymmetric DM
scenarios (for reviews cf e.g. Davoudiasl_2012 ; Petraki:2013wwa ), the
observed DM cosmic abundance is viewed as being due to an asymmetry being
created between the abundances of DM and dark anti-matter which survived after
the symmetric component annihilated away, in a manner similar to (and,
typically, connected with) the asymmetry between baryons and anti-baryons in
the visible sector. In scenarios of baryogenesis via weekly interacting
massive particles (“WIMPy baryogenesis”) McDonald_2011 ; Cui_2012 ; Cui_2013 ;
Chu:2021qwk , on the other hand, there is no asymmetry in the dark sector and
the observed baryon asymmetry is generated during the freeze-out of DM
particles, once they fall out of equilibrium with the thermal plasma.

In Hall:2010jx  it was pointed out that in DM physics there exists (at least)
one framework in which the out-of-equilibrium Sakharov condition
Sakharov:1967dj  is, in a sense, built-in from the start: the freeze-in
mechanism McDonald:2001vt ; Hall:2009bx . In this context, which was further
critically elaborated upon, e.g., in Hook:2011tk ; Unwin:2014poa , the authors
suggested that viable asymmetries could be generated (frozen-in)
simultaneously in the dark and visible sectors through out-of-equilibrium
decays or annihilations of bath particles. The framework outlined in
Hall:2010jx  involves asymmetric DM, i.e. generating asymmetries both in the
dark and in the visible sector through a common mechanism.

Instead, in Shuve_2020  another possibility was investigated: that of
freezing-in a symmetric DM abundance, along with a baryon asymmetry in the
visible sector. In its original version (cf also Berman:2022oht ) the baryon
asymmetry generation is achieved through DM oscillations in a manner similar
to the situation occurring in the Akhmedov-Rubakov-Smirnov (ARS) leptogenesis
Akhmedov:1998qx . In Goudelis:2021qla  a different approach was proposed,
according to which DM and a baryon asymmetry are simultaneously generated
through $CP$-violating decays of heavy bath particles, with $CP$ violation
arising through the interference between tree-level and one-loop diagrams in a
manner more similar to Dirac leptogenesis Dick:1999je ; Murayama:2002je ;
Cerdeno:2006ha ; Gonzalez-Garcia:2009cza . A similar idea was also recently
investigated in Chand:2022vrf .

A common feature of these proposals is that they operate at relatively low
temperatures. Indeed, the models proposed in Refs. Shuve_2020 ; Berman:2022oht
; Goudelis:2021qla ; Chand:2022vrf  are renormalizable, which leads to the
freeze-in production (both of DM and of the baryon asymmetry) being infrared
(IR)-dominated. However, it is known that in theories involving non-
renormalizable interactions parametrized by higher-dimensional Lagrangian
operators, freeze-in production occurs predominantly in the ultraviolet (UV)
regime, i.e. close to the highest considered temperature. In the context of
DM, this “ultraviolet freeze-in” scenario was first analyzed in Elahi:2014fsa
. On the side of baryogenesis, relevant constructions were presented in
Baldes:2014rda ; Baldes:2014gca  (and UV-completed in Baldes:2015lka ), in
which the authors were mainly interested in studying scenarios in which the
baryon asymmetry of the Universe is generated through scattering processes.

In this paper we study whether a viable (symmetric) DM abundance and a matter-
antimatter asymmetry can be simultaneously frozen-in in the UV through non-
renormalizable interactions. Our study will focus on two cases, those of a
scalar and a fermionic DM candidate, which will lead us to consider operators
of different dimensionality, amounting to quantitatively (albeit not
qualitatively) different results. We will see that in such a framework it is,
indeed, possible to explain both observations simultaneously as long as DM is
relatively light, while being consistent with different cosmological
constraints.

The paper is structured as follows: in Section 2 we recall some key elements
concerning the freeze-in DM production mechanism (notably its UV-dominated
version) and we discuss different constraints on the DM properties that will
be of relevance for the following. In Section 3 we study two concrete
realizations of our “ultraviolet freeze-in baryogenesis” mechanism, one in
which DM will be assumed to be a scalar and one in which it will be taken to
be a Dirac fermion. Lastly, Section 4 contains our conclusions, whereas the
Appendix that follows contains some more technical aspects of our analysis.

##  2 Ultraviolet freeze-in and constraints on the DM mass

Before studying concrete models, let us briefly recall some basic features of
frozen-in DM and discuss a set of phenomenological constraints which will be
important for the analysis that follows. Some additional issues which are more
model-dependent will be discussed later on, at the appropriate Sections.

###  2.1 Freeze-in DM production through non-renormalizable operators

The freeze-in DM production mechanism McDonald:2001vt ; Hall:2009bx  relies on
two basic features:

  * •

The initial DM abundance is negligible.

  * •

DM only interacts extremely weakly (“feebly”) with all particles in the
thermal bath.

These two assumptions in conjunction allow us to consider only DM production
processes and to ignore backreactions from the DM sector towards the visible
one, i.e. DM annihilation processes.

Moreover, as pointed out e.g. in Hall:2009bx , if the two sectors only
interact via renormalizable operators then DM production is a mostly low-
temperature (“infrared”) effect. As an example, if DM is produced through the
annihilation of light bath particles, its production peaks around a
temperature $T\sim m_{\text{DM}}/3$.

In this work we consider, instead, another possibility, in which an effective
operator $\mathcal{\hat{O}}_{\left(n\right)}$ of dimension - $(n+4)$

$\mathcal{L}\,\supset\,\frac{1}{\Lambda^{n}}\,\mathcal{\hat{O}}_{\left(n\right)}\;,$
(1)

where $n=1,2,\ldots$ and $\Lambda$ is the energy scale of the effective field
theory (EFT), is responsible for the generation of the freeze-in DM abundance
(and, as we will see, also for the baryon asymmetry of the Universe). In this
case DM is mostly produced in the ultraviolet regime, i.e. close to the
reheating temperature $T_{\rm RH}$.

Following the analysis of Hall:2009bx ; Elahi:2014fsa , and assuming that DM
is produced through the annihilation of highly relativistic particles, the
matrix element squared of the corresponding DM production process is expected
to scale as

$\sum_{\rm
idof}|\mathcal{M}|^{2}\,=A\,\left(\frac{\tilde{s}}{\Lambda^{2}}\right)^{n}\;,$
(2)

where $\sqrt{\tilde{s}}$ is energy in the center-of-mass frame and the sum
runs over all internal degrees of freedom (without averaging), $n=d-4$, and
$A$ is a model-dependent constant.

In the types of models that we will study in this paper, DM ($\chi$) and its
antiparticle ($\bar{\chi}$) are produced via pair-annihilations of particles
(say, $b_{1}$ and $b_{2}$) in thermal equilibrium with the plasma; i.e. via
processes of the type $b_{1}\,b_{2}\to\chi\,\chi$ and
$\bar{b}_{1}\,\bar{b}_{2}\to\bar{\chi}\,\bar{\chi}$. In this case, the
Boltzmann equation that describes the evolution of the number of DM particles
takes, in the limit that $\Lambda$ is larger than all the masses $m_{i}$
involved in the process, the form Hall:2009bx ; Elahi:2014fsa

$\displaystyle\dfrac{dY_{\rm
DM}}{dT}\approx-2A\,\frac{4^{n+1}n!(n+1)!}{1024\times
1.66\,\pi^{7}}\dfrac{45}{g_{\star
s}\sqrt{g_{\star\rho}}}\frac{M_{Pl}\,T^{2n-2}}{\Lambda^{2n}}\;,$ (3)

where $Y_{\rm DM}\equiv\left(n_{\chi}+n_{\bar{\chi}}\right)/s$ is the DM yield
(number density $n_{\chi}$ over the entropy density $s=2\pi^{2}g_{\star
s}T^{3}/45$), $M_{Pl}\approx 1.22\times 10^{19}~{}\operatorname{GeV}$ is the
Planck mass and $g_{\star s},\,g_{\star\rho}$ are the relativistic degrees of
freedom of the plasma with respect to the entropy and energy densities –
assumed to be constant at the temperatures of interest, respectively111The
factor $2$ is included because both $\chi$ and $\bar{\chi}$ are assumed to be
produced in equal amounts..

The form of Eq. (3) implies that DM production becomes maximal when the plasma
attains its highest temperature. Integrating, then, Eq. (3) from the reheating
temperature222Note that in some frameworks DM production may also peak at
temperatures $T>T_{\rm RH}$, i.e. during the reheating period Garcia:2017tuj .
This happens, typically, when considering effective operators of higher
dimension than the ones that we will consider in this paper. $T_{\rm RH}$ down
to some temperature $T<T_{\rm RH}$, the DM yield becomes

$\displaystyle Y_{\rm DM}(T)\approx
2A\,\frac{4^{n+1}n!(n+1)!}{2n-1}\dfrac{45}{1024\times 1.66\,\pi^{7}g_{\star
s}\sqrt{g_{\star\rho}}}\frac{M_{Pl}\,(T_{\rm
RH}^{2n-1}-T^{2n-1})}{\Lambda^{2n}}\;,$ (4)

which leads to the DM relic abundance333Here we ignore a term proportional to
the present-day temperature, since the latter is negligible compared to
$T_{\rm RH}$.

$\displaystyle\Omega_{\rm DM}h^{2}\approx\dfrac{2.4\times 10^{23}}{g_{\star
s}\sqrt{g_{\star\rho}}}\ \dfrac{A\,4^{n}\,n!\,(n+1)!}{2n-1}\,\dfrac{T_{\rm
RH}^{2n-1}\ m_{\rm DM}}{\Lambda^{2n}}\;.$ (5)

From Eq. (5) we can see that we, indeed, recover the well-known result; namely
if DM is produced through the action of non-renormalizable operators its
predicted freeze-in abundance scales as a power of the reheating temperature,
with the exponent depending on the operator’s effective dimension. In Section
3 we will make extensive use of this expression, specializing it to different
types of higher-dimensional operators.

###  2.2 Constraints on the DM mass

As we will see in the following Section, requiring the simultaneous generation
of the observed DM abundance in the Universe along with a viable baryon
asymmetry will lead us to consider fairly low DM masses. Before proceeding
further with our analytical and numerical treatment let us, hence, briefly
discuss two relevant constraints on the DM mass.

####  2.2.1 Non-equilibration of the visible and dark sectors

The main premise of freeze-in is that the visible and dark sectors remain out-
of-equilibrium with each other during the cosmic expansion. In the ultraviolet
freeze-in case the condition of non-equilibration between the two sectors can
be formulated by requiring that the abundance of the DM at $T_{\text{RH}}$ be
much smaller than its equilibrium value Elahi:2014fsa , i.e.

$Y_{\text{DM}}\left(T_{\text{RH}}\right)\,\ll\,Y_{\text{DM}}^{\text{eq}}\left(T_{\text{RH}}\right)\;.$
(6)

As $T_{\text{RH}}\gg m_{\text{DM}}$ the DM can be treated as effectively
massless and its equilibrium abundance at $T_{\text{RH}}$ can be approximated
by
$Y_{\text{DM}}^{\text{eq}}(T_{\text{RH}})\approx\zeta(3)T^{3}_{\text{RH}}/(\pi^{2}s(T_{\text{RH}}))$.
Since the DM abundance effectively freezes-in close to $T_{\text{RH}}$ and,
subsequently, remains quasi-constant, it holds that
$Y_{\text{DM}}\left(T_{\text{RH}}\right)\approx
Y_{\text{DM}}\left(T_{0}\right)$, where $T_{0}\approx 2.75\text{K}$ is the
temperature at the present day. The DM relic density is, in turn, given by
$\Omega_{\text{DM}}=m_{\text{DM}}s_{0}Y_{\text{DM}}(T_{0})/\rho_{c}$, where
$\Omega_{\text{DM}}h^{2}=0.1200\pm 0.0012$, $\rho_{c}\equiv
3H_{0}^{2}/8\pi\,G\approx 10.537\,h^{2}\,\operatorname{GeV}m^{-3}$ is the
critical density and $s_{0}\approx 2.9\times 10^{9}\,m^{-3}$ is the entropy
density at the present day Planck:2018vyg . Hence, the out-of-equilibrium
constraint of Eq. (6) imposes a lower bound on the DM mass

$m_{\text{DM}}\,\gg\,\frac{\Omega_{\text{DM}}\,\rho_{c}}{Y_{\text{DM}}^{\text{eq}}\left(T_{\text{RH}}\right)\,s_{0}}\,\approx\,0.17\,\operatorname{keV}$
(7)

where we have used that in the Standard Model (SM) $g_{\star
s}\left(T_{\text{RH}}\right)=106.75$.

####  2.2.2 Lyman-$\alpha$ constraint

A more stringent constraint on the DM mass comes from Lyman-$\alpha$ forest
observations. For frozen-out DM candidates the current bound reads
$m_{\text{DM}}\gtrsim(1.9-5.3)\,\operatorname{keV}$ at $95\%$ C.L.
Garzilli:2019qki ; Palanque-Delabrouille:2019iyz ; Irsic:2017ixq . This limit
has been mapped onto different cases of freeze-in-produced DM, depending on
the type of the relevant interactions, on the dimension of the (effective)
operator involved and on the statistical distribution of the various states.
In the case of $2\leftrightarrow 2$ scatterings with two fermions annihilating
into two DM states via dimension-5 ($\mathcal{\hat{O}}_{\left(5\right)}$) and
dimension-6 ($\mathcal{\hat{O}}_{\left(6\right)}$) operators, the
Lyman-$\alpha$ lower bound on the DM mass has been shown to translate,
respectively, to Ballesteros:2020adh :

$\displaystyle\mathcal{\hat{O}}_{\left(5\right)}:$ $\displaystyle\quad
m_{\text{DM}}\,\gtrsim\,\left(4.0-15.5\right)\,\operatorname{keV}$ (8a)
$\displaystyle\mathcal{\hat{O}}_{\left(6\right)}:$ $\displaystyle\quad
m_{\text{DM}}\,\gtrsim\,\left(4.6-18.1\right)\,\operatorname{keV}\;.$ (8b)

The impact of these bounds is shown in Fig. 1, in which we depict the
$(\Lambda,T_{\rm RH})$ combinations for which it is possible to explain the
observed DM abundance in the Universe, while simultaneously satisfying the
conditions of Eqs. (8a) and (8b).

Figure 1: The Lyman-$\alpha$ constraint (shaded regions) mapped onto the
$\left(\Lambda,\,T_{\text{RH}}\right)$ plane for the
$\mathcal{\hat{O}}_{\left(5\right)}$ and $\mathcal{\hat{O}}_{\left(6\right)}$
operators. Each line corresponds to the observed DM relic density for the
lowest allowed value of the DM mass, as given by Eqs. (8a) and (8b),
respectively.

##  3 Concrete Realizations of UV freeze-in baryogenesis

Let us now examine, as a proof-of-concept, two concrete examples in which the
observed DM abundance and the matter-antimatter asymmetry of the Universe can
be simultaneously explained through the action of non-renormalizable
operators.

###  3.1 Scalar DM

We consider a simple extension of the SM by two heavy vector-like leptons
$F_{1,2}$, which are singlets under $SU(3)_{c}\times SU(2)_{\operatorname{L}}$
but carry hypercharge, and by a complex gauge-singlet scalar $\varphi$ which
will play the role of our frozen-in DM candidate. We will focus on the
following interaction Lagrangian

$\mathcal{L}_{\text{int}}\,=\,\frac{\lambda_{1}}{2\Lambda}\,\left(\bar{e}\operatorname{P_{L}}F_{1}\right)\,\varphi^{*}\varphi^{*}\,+\,\frac{\lambda_{2}}{2\Lambda}\,\left(\bar{e}\operatorname{P_{L}}F_{2}\right)\,\varphi^{*}\varphi^{*}\,+\,\frac{\kappa}{\Lambda^{2}}\,\left(\bar{e}\operatorname{P_{L}}F_{1}\right)\left(\bar{F}_{2}\operatorname{P_{R}}e\right)\,+\,\text{h.c.}$
(9)

where $e$ denotes a SM charged lepton. Such interactions can be motivated by
assuming that some heavy scalar mediator, connecting pairs of DM particles
with a SM lepton along with an exotic one, has been integrated out. DM
stability can be ensured by imposing a $\mathbb{Z}_{3}$ symmetry under which
the $F_{i}$’s and $\varphi$ are charged according to the assignments
summarized in Table 1.

The heavy fermions $F_{i}$ carry the same lepton number as the SM leptons, so
their interactions conserve the total lepton number. They are maintained in
thermal equilibrium with the thermal plasma due to their gauge interactions
(which we have omitted in Eq. (9) for the sake of brevity).

All relevant Feynman diagrams which can contribute to the generation of the
baryon asymmetry and DM are shown in Figure 2. The interactions which contain
a DM state as both initial and final states are subleading, from the freeze-in
assumption, and can be safely ignored.

Particle | Gauge | $\mathbb{Z}_{3}$
---|---|---
$\varphi$ | $(1,1)_{0}$ | $\omega$
$\varphi^{*}$ | $(1,1)_{0}$ | $\omega^{-1}$
$F_{i}$ | $(1,1)_{-1}$ | $\omega^{-1}$
$\bar{F}_{i}$ | $(1,1)_{1}$ | $\omega$
Table 1: Quantum charges of the various states, where $\omega=e^{i2\pi/3}$.

((a))

((b))

((c))

((d))

Figure 2: Feynman diagrams that contribute to the generation of the baryon
asymmetry and DM.

Let us first focus on $2\leftrightarrow 2$ processes. The equilibrium
interaction rate density of a generic scattering process $ab\rightarrow cd$ is
given by

$\displaystyle\gamma_{ab\rightarrow cd}\,$
$\displaystyle\equiv\,\frac{1}{S_{\text{in}}S_{\text{out}}}\int\text{d}\Pi_{a}\,\text{d}\Pi_{b}\,\text{d}\Pi_{c}\,\text{d}\Pi_{d}\,\left(2\pi\right)^{4}\,\delta^{(4)}\left(p_{a}+p_{b}-p_{c}-p_{d}\right)\,f^{\text{eq}}_{a}\,f^{\text{eq}}_{b}\,\left|\mathcal{M}\right|^{2}_{ab\rightarrow
cd}$
$\displaystyle=\,\frac{T}{512\pi^{6}S_{\text{in}}S_{\text{out}}}\,\int\text{d}\tilde{s}\,\frac{1}{\sqrt{\tilde{s}}}\,\left|\textbf{p}_{\text{in}}\right|\,\left|\textbf{p}_{\text{fin}}\right|\,K_{1}\left(\frac{\sqrt{\tilde{s}}}{T}\right)\int\,\text{d}\Omega\,\left|\mathcal{M}\right|^{2}_{ab\rightarrow
cd}$ (10)

where
$\text{d}\Pi_{k}=\text{d}^{3}\textbf{p}_{k}/(\left(2\pi\right)^{3}2E_{k})$ is
the elementary Lorentz-invariant phase space volume of species $k$,
$f_{k}^{\text{eq}}$ is the equilibrium distribution function of $k$ which we
assume to approximately follow the Maxwell-Boltzmann distribution with
vanishing chemical potential, $\left|\mathcal{M}\right|^{2}$ denotes the
squared matrix element summed over the internal degrees of freedom of the
particles involved, but not averaged over the internal degrees of freedom of
the initial states, $S_{\text{in}}$ and $S_{\text{out}}$ are the symmetry
factors for the incoming and outgoing states respectively, which are equal to
$N!$ in case of identical particles and $1$ otherwise, $K_{1}$ is the modified
Bessel function of the second kind of order one,
$\tilde{s}_{\text{min}}=\text{max}\big{\\{}\left(m_{a}+m_{b}\right)^{2},\left(m_{c}+m_{d}\right)^{2}\big{\\}}$,
$\tilde{s}_{\text{max}}\sim\Lambda^{2}$ and $\textbf{p}_{\text{in}}$,
$\textbf{p}_{\text{fin}}$ are the initial and final CoM momenta respectively.
An analytical approximation for the scattering rate densities at tree-level,
hereafter denoted by $\gamma^{ab}_{cd}$, can be obtained in the limit
$\Lambda\gg m_{i}$,

$\displaystyle\gamma^{F_{i}\bar{e}}_{\varphi\varphi}\,$
$\displaystyle\approx\,\frac{\left|\lambda_{i}\right|^{2}T}{1024\,\pi^{5}\,\Lambda^{2}}\int_{0}^{\infty}\text{d}\tilde{s}\,\,\tilde{s}^{3/2}\,K_{1}\left(\frac{\sqrt{\tilde{s}}}{T}\right)\,=\,\frac{\left|\lambda_{i}\right|^{2}}{32\,\pi^{5}}\frac{T^{6}}{\Lambda^{2}}$
(11a) $\displaystyle\gamma^{F_{i}\bar{e}}_{F_{j}\bar{e}}\,$
$\displaystyle\approx\,\frac{\left|\kappa\right|^{2}T}{512\,\pi^{5}\,\Lambda^{4}}\int_{0}^{\infty}\text{d}\tilde{s}\,\,\tilde{s}^{5/2}\,K_{1}\left(\frac{\sqrt{\tilde{s}}}{T}\right)\,=\,\frac{3\left|\kappa\right|^{2}}{2\,\pi^{5}}\frac{T^{8}}{\Lambda^{4}}\;.$
(11b)

In what follows we will consider $F_{i}$ masses close to the TeV range and
values of $\Lambda\gtrsim 10^{11}\operatorname{GeV}$ (see Figure 1). The
$F_{i}$ decay width can, in turn, be approximated as

$\Gamma_{F_{i}\rightarrow
e\varphi\varphi}\,\approx\,\dfrac{\left|\lambda_{i}\right|^{2}}{3072\
\pi^{3}}\frac{M_{i}^{3}}{\Lambda^{2}},\qquad\Gamma_{F_{2}\rightarrow
F_{1}e\bar{e}}\,\approx\,\dfrac{\left|\kappa\right|^{2}}{6144\
\pi^{3}}\frac{M_{2}^{5}}{\Lambda^{4}}\,,$ (12)

where the latter expression holds assuming $M_{1}\ll M_{2}$. Then, for the
parameter ranges that will be of interest to us ($\\{\Lambda,T_{\rm RH}\\}\gg
M_{i}$), the $1\leftrightarrow 3$ decay processes $F_{i}\leftrightarrow
e\varphi\varphi$ and $F_{2}\leftrightarrow F_{1}e\bar{e}$ are extremely
suppressed and can be safely ignored. Instead, the dominant processes for the
generation of the DM abundance and the asymmetries are the $2\leftrightarrow
2$ scatterings $F_{i}\bar{e}\leftrightarrow\varphi\varphi$ and
$F_{i}\bar{e}\leftrightarrow F_{j}\bar{e}$ which are, moreover, mostly
efficient at high temperatures (cf Eqs. (11)).

The freeze-in DM abundance in our model can be read-off directly from Eq. (4),
substituting $n=1$ and $A=|\lambda_{1}|^{2}+|\lambda_{2}|^{2}$. That is,

$Y_{\text{DM}}\left(T\right)\,=\,\frac{45\big{(}\left|\lambda_{1}\right|^{2}+\left|\lambda_{2}\right|^{2}\big{)}M_{Pl}}{16\times
1.66\pi^{7}g_{\star
s}\sqrt{g_{\star\rho}}}\frac{T_{\text{RH}}-T}{\Lambda^{2}}$ (13)

where $g_{\star s}=g_{\star\rho}=113.75$, due to the addition of the two Dirac
fermions $F_{i}$ besides the usual SM degrees of freedom. The DM relic density
$\Omega_{\text{DM}}h^{2}$ can, in turn, be cast into the simple form

$\Omega_{\rm DM}h^{2}\approx
0.12\times\Big{(}|\lambda_{1}|^{2}+|\lambda_{2}|^{2}\Big{)}\Big{(}\dfrac{1.61\times
10^{16}~{}\operatorname{GeV}}{\Lambda}\Big{)}^{2}\Big{(}\dfrac{T_{\rm
RH}}{2\times
10^{15}~{}\operatorname{GeV}}\Big{)}\Big{(}\dfrac{m_{\varphi}}{10~{}\operatorname{keV}}\Big{)}\;.$
(14)

####  3.1.1 CP asymmetries

The $CP$ asymmetry is generated, at leading order, due to the interference of
the tree-level and 1-loop diagrams presented in Figure 2. As we mentioned, we
focus on the dominant $2\leftrightarrow 2$ scattering processes depicted in
Figures 2(a) and 2(c).

In order to derive an expression for the relevant $CP$ asymmetry let us first
consider a generic $2\leftrightarrow 2$ process, denoted by $ab\leftrightarrow
cd$. The $CP$ violation can be parametrized as

$\epsilon\,=\,\frac{\gamma_{ab\rightarrow
cd}\,-\,\gamma_{\bar{a}\bar{b}\rightarrow\bar{c}\bar{d}}}{\gamma_{ab\rightarrow
cd}\,+\,\gamma_{\bar{a}\bar{b}\rightarrow\bar{c}\bar{d}}}\,\approx\,\frac{\int\text{d}\tilde{s}\,\frac{1}{\sqrt{\tilde{s}}}\,\left|\textbf{p}_{\text{in}}\right|\,\left|\textbf{p}_{\text{fin}}\right|\,K_{1}\left(\frac{\sqrt{\tilde{s}}}{T}\right)\,\left[\left|\mathcal{M}\right|^{2}_{ab\rightarrow
cd}\,-\,\left|\mathcal{M}\right|^{2}_{\bar{a}\bar{b}\rightarrow\bar{c}\bar{d}}\right]}{2\int\text{d}\tilde{s}\,\frac{1}{\sqrt{\tilde{s}}}\,\left|\textbf{p}_{\text{in}}\right|\,\left|\textbf{p}_{\text{fin}}\right|\,K_{1}\left(\frac{\sqrt{\tilde{s}}}{T}\right)\,\left|\mathcal{M}\right|^{2}_{ab\rightarrow
cd}\big{|}_{0}}$ (15)

where $\bar{i}$ denotes the $CP$-conjugate state of $i$ and we have used the
expression for the scattering rate density given in Eq. (3.1). It is
convenient to separate the matrix element $\left.\mathcal{M}\right|_{k}$, with
$k$ being the loop order $\left(k=0,1,\ldots\right)$, into a coupling constant
part $c_{k}$ (denoting a collection of coupling constants) and an amplitude
part $\mathcal{A}_{k}$, i.e.
$\left.\mathcal{M}\right|_{k}\equiv\sum_{i=0}^{k}\,c_{i}\mathcal{A}_{i}$. To
leading order the $CP$ asymmetry can be rewritten as Davidson:2008bu

$\epsilon\,=\,-2\,\frac{\text{Im}\big{\\{}c_{0}^{*}c_{1}\big{\\}}}{\left|c_{0}\right|^{2}}\frac{\int\text{d}\tilde{s}\;\frac{1}{\sqrt{\tilde{s}}}\,\left|\textbf{p}_{\text{in}}\right|\,\left|\textbf{p}_{\text{fin}}\right|\,K_{1}\left(\frac{\sqrt{\tilde{s}}}{T}\right)\,\text{Im}\big{\\{}\mathcal{A}_{0}^{*}\mathcal{A}_{1}\big{\\}}}{\int\text{d}\tilde{s}\,\frac{1}{\sqrt{\tilde{s}}}\,\left|\textbf{p}_{\text{in}}\right|\,\left|\textbf{p}_{\text{fin}}\right|\,K_{1}\left(\frac{\sqrt{\tilde{s}}}{T}\right)\,\left|\mathcal{A}_{0}\right|^{2}}$
(16)

where
$\text{Im}\big{\\{}\mathcal{A}_{0}^{*}\mathcal{A}_{1}\big{\\}}=\frac{1}{2i}\mathcal{A}_{0}^{*}\sum_{\text{cuts}}\mathcal{A}_{1}$.
In the model under consideration there are three $CP$ asymmetries generated by
scattering processes, which can be defined as

$\displaystyle\epsilon_{i}\,$
$\displaystyle\equiv\frac{\gamma_{F_{i}\bar{e}\rightarrow\varphi\varphi}\,-\,\gamma_{\bar{F}_{i}e\rightarrow\varphi^{*}\varphi^{*}}}{\gamma_{F_{i}\bar{e}\rightarrow\varphi\varphi}\,+\,\gamma_{\bar{F}_{i}e\rightarrow\varphi^{*}\varphi^{*}}}$
(17a) $\displaystyle\epsilon_{3}\,$
$\displaystyle\equiv\frac{\gamma_{F_{1}\bar{e}\rightarrow
F_{2}\bar{e}}\,-\,\gamma_{\bar{F}_{1}e\rightarrow\bar{F}_{2}e}}{\gamma_{F_{1}\bar{e}\rightarrow
F_{2}\bar{e}}\,+\,\gamma_{\bar{F}_{1}e\rightarrow\bar{F}_{2}e}}$ (17b)

where $i=1,2$. Let us note that whereas the processes corresponding to the
$\epsilon_{i}$ quantities generate correlated $CP$ asymmetries in the two
sectors (i.e. in the SM and the exotic fermions), the reactions of the type
$F_{1}\bar{e}\leftrightarrow F_{2}\bar{e}$ do not directly affect the
asymmetry generated in the SM leptons.

The imaginary part of the amplitudes entering the $\epsilon_{i}$ asymmetries
can be calculated from the optical theorem to be (see Appendix A)

$\text{Im}\big{\\{}\mathcal{A}_{0}^{*}\mathcal{A}_{1}\big{\\}}_{F_{i}\bar{e}\rightarrow\varphi\varphi}\,=\,\frac{\big{(}\tilde{s}-M_{i}^{2}-m_{e}^{2}\big{)}\big{(}\tilde{s}-M_{j}^{2}-m_{e}^{2}\big{)}\,\sqrt{\lambda\big{(}\tilde{s},M_{j}^{2},m_{e}^{2}\big{)}}}{16\pi\tilde{s}}$
(18)

where $\lambda$ is the Källén (or triangle) function. This expression has been
crossed-checked using Package-X Patel:2015tea , as well as by implementing the
Lagrangian of Eq. (9) in FeynRules Alloul:2013bka  and then performing the
computation through Feynarts/FormCalc Hahn:2000kx ; Hahn:1998yk .

An analytical estimation for the $\epsilon_{1}$ and $\epsilon_{2}$ $CP$
asymmetries can be obtained in the limit $\Lambda\gg m_{i}$, yielding

$\displaystyle\epsilon_{1}\,$
$\displaystyle\approx\,-\frac{3}{\pi}\frac{\left|\kappa\right|\left|\lambda_{2}\right|}{\left|\lambda_{1}\right|}\sin\left(-\Delta\phi\right)\frac{T^{2}}{\Lambda^{2}}$
(19a) $\displaystyle\epsilon_{2}\,$
$\displaystyle\approx\,-\frac{3}{\pi}\frac{\left|\kappa\right|\left|\lambda_{1}\right|}{\left|\lambda_{2}\right|}\sin\left(\Delta\phi\right)\frac{T^{2}}{\Lambda^{2}}$
(19b)

where we have expressed the coupling constants in terms of their magnitude and
phase, i.e. $\lambda_{i}=|\lambda_{i}|e^{i\phi_{i}}$ and
$\kappa=|\kappa|e^{i\phi_{3}}$, and we have defined
$\Delta\phi=\phi_{1}-\phi_{3}-\phi_{2}$. In the subsequent analysis, we will
set $\Delta\phi=\pi/2$ in order to maximize the contribution of the phases to
the $CP$ violation.

In the same way we can calculate the imaginary part of the Feynman amplitudes
for the case of the $\epsilon_{3}$ asymmetry to be

$\text{Im}\big{\\{}\mathcal{A}_{0}^{*}\mathcal{A}_{1}\big{\\}}_{F_{1}\bar{e}\rightarrow
F_{2}\bar{e}}\,=\,\frac{\big{(}\tilde{s}-M_{1}^{2}-m_{e}^{2}\big{)}\big{(}\tilde{s}-M_{2}^{2}-m_{e}^{2}\big{)}\,\sqrt{\lambda\big{(}\tilde{s},m_{\varphi}^{2},m_{\varphi}^{2}\big{)}}}{16\pi\tilde{s}}\;.$
(20)

In the limit $\Lambda\gg m_{i}$ we can, again, derive an analytical
approximation for $\epsilon_{3}$ as

$\epsilon_{3}\,\approx\,-\frac{1}{16\pi}\frac{\left|\lambda_{1}\right|\left|\lambda_{2}\right|}{\left|\kappa\right|}\sin\left(\Delta\phi\right)\;.$
(21)

Contrary to $\epsilon_{i}$ case the $\epsilon_{3}$ asymmetry is constant
during the cosmic expansion, which can be understood by dimensional arguments
noting that the combination of the couplings that enters $\epsilon_{3}$
eliminates the EFT energy scale $\Lambda$.

Figure 3: Temperature evolution of the $CP$ asymmetries assuming
$\Lambda=1.28\times 10^{16}\operatorname{GeV}$, $T_{\text{RH}}=2\times
10^{15}\operatorname{GeV}$, $M_{1}=1\operatorname{TeV}$,
$M_{2}=5\operatorname{TeV}$, $|\lambda_{1}|=0.5$, $|\lambda_{2}|=|\kappa|=1$
and $\Delta\phi=\pi/2$. The coloured solid lines correspond to the full
numerical results, while the dashed lines to the analytical approximations as
described in the text. The inset focuses on the evolution of the $\epsilon$’s
at low temperatures and the horizontal black solid line corresponds to the
$CP$ asymmetry generated by the decay $F_{2}\rightarrow e\varphi\varphi$.

In Figure 3 we illustrate the $CP$ asymmetries predicted by our model as a
function of the dimensionless parameter $x\equiv T_{\text{RH}}/T$, for a
benchmark set of parameter values: $\Lambda=1.28\times
10^{16}\operatorname{GeV}$, $T_{\text{RH}}=2\times 10^{15}\operatorname{GeV}$,
$M_{1}=1\operatorname{TeV}$, $M_{2}=5\operatorname{TeV}$, $|\lambda_{1}|=0.5$,
$|\lambda_{2}|=|\kappa|=1$ and $\Delta\phi=\pi/2$, which, as we will see
shortly, lead to the observed baryon asymmetry of the Universe. The solid
lines correspond to the full numerical results obtained through Eqs. (16),
(18) and (20), whereas the dashed lines to the analytical approximations of
Eqs. (19) and (21). In the temperature range which is relevant for our study
($T<T_{\rm RH}$), we observe that the numerical results are in very good
agreement with the analytical ones. Besides, as the temperature drops below
the mass $M_{2}$ of the heavier fermion, i.e. below $x_{M}\equiv
T_{\text{RH}}/M_{2}$, the $\epsilon_{1}$ asymmetry becomes kinematically
suppressed, as there is not enough energy for the $F_{2}$ fermion to propagate
on-shell in the loop. On the other hand, $\epsilon_{2}$ takes values close to
the $CP$ asymmetry generated by the heavier fermion decay $F_{2}\rightarrow
e\phi\phi$, which can be approximated as
$\epsilon_{D}\approx\frac{\left|\lambda_{2}\right|^{2}}{32\pi}\frac{M_{2}^{2}}{\Lambda^{2}}$.

Before setting off to compute the baryon asymmetry of the Universe predicted
by our model let us, finally, note that unitarity of the $S$-matrix together
with invariance under $CPT$ impose specific relations between the various
asymmetries $\epsilon_{1,2,3}$ and the corresponding equilibrium reaction rate
densities. For a generic process $a_{i}\leftrightarrow b_{j}$, these two
conditions imply that Kolb:1979qa

$\sum_{b_{j},\bar{b}_{j}}\gamma_{a_{i}\rightarrow
b_{j}}\,=\,\sum_{b_{j},\bar{b}_{j}}\gamma_{b_{j}\rightarrow
a_{i}}\,=\,\sum_{b_{j},\bar{b}_{j}}\gamma_{\bar{a}_{i}\rightarrow\bar{b}_{j}}\,=\,\sum_{b_{j},\bar{b}_{j}}\gamma_{\bar{b}_{j}\rightarrow\bar{a}_{i}}$
(22)

where $a_{i}$ and $b_{j}$ denote a set of states with $i=1,\dots,N$ and
$j=1,\dots,M$ and the bar denotes the $CP$-conjugate states. Hence, $CPT$ and
unitarity ensure that if thermal equilibrium is attained, then, no asymmetry
can be produced. In our model, these requirements translate into the following
conditions, which we will apply throughout our subsequent analysis

$\displaystyle\epsilon_{1}\gamma^{F_{1}\bar{e}}_{\varphi\varphi}\,+\,\epsilon_{3}\gamma^{F_{1}\bar{e}}_{F_{2}\bar{e}}\,$
$\displaystyle=\,0$ (23a)
$\displaystyle\epsilon_{2}\gamma^{F_{2}\bar{e}}_{\varphi\varphi}\,-\,\epsilon_{3}\gamma^{F_{1}\bar{e}}_{F_{2}\bar{e}}\,$
$\displaystyle=\,0$ (23b)
$\displaystyle\epsilon_{1}\gamma^{F_{1}\bar{e}}_{\varphi\varphi}\,+\,\epsilon_{2}\gamma^{F_{2}\bar{e}}_{\varphi\varphi}\,$
$\displaystyle=\,0\;.$ (23c)

Moreover, the reaction $F_{1}\bar{F}_{2}\leftrightarrow e\bar{e}$, as well as
all gauge interactions, are $CP$-conserving which is the reason why, with some
amount of hindsight, we have neglected them in our discussion.

####  3.1.2 Baryon asymmetry

Having discussed how $C/CP$ violation and out-of-equilibrium dynamics arise in
the model described by the Lagrangian of Eq. (9), let us now turn to baryon
(or lepton) number violation.

First, note that in our model the heavy fermions $F_{i}$ are of Dirac nature
and carry the same lepton number as the SM leptons. Then, once sector-wise
asymmetries are generated in the SM and exotic fermion sectors, the total
lepton asymmetry is given by $Y_{L}=Y_{L_{\text{SM}}}+Y_{L_{F}}$, where
$Y_{L_{F}}\equiv\sum_{i}Y_{L_{F_{i}}}$. On the other hand, all processes
(including non-perturbative sphaleron transitions) conserve the combination
$Y_{B-L}\equiv Y_{B}-Y_{L_{\text{SM}}}-Y_{L_{F}}$, i.e.
$\text{d}Y_{B-L}/\text{d}T=0$. If we also assume that the Universe is
initially symmetric, $Y_{B-L_{\text{SM}}}=Y_{L_{F}}=0$, then at any $T$ it
holds that $Y_{B-L_{\text{SM}}}=Y_{L_{F}}$, i.e. equal and opposite lepton
asymmetries are generated in the two sectors and the net lepton/baryon
asymmetry can only be zero.

The situation changes once sphaleron transitions are taken into account.
Whereas sphalerons are insensitive to the heavy lepton asymmetry $Y_{L_{F}}$,
as the $F_{i}$ are $SU(3)_{c}\times SU(2)_{\operatorname{L}}$ singlets, they
only affect the non-zero lepton asymmetry $Y_{L_{\text{SM}}}$ stored in the SM
sector. In particular, they convert it into a baryon asymmetry by imposing
certain relations among the chemical potentials of the various species. Once
sphalerons depart from equilibrium, which occurs at $T_{sph}=131.7\pm
2.3\operatorname{GeV}$ DOnofrio:2014rug , the baryon and lepton numbers are
separately conserved. In principle, when the relic heavy leptons eventually
decay away the net baryon asymmetry, being proportional to
$Y_{B-L_{\text{SM}}}$, would vanish. However, if the sphalerons are inactive
during the decay epoch of $F_{i}$ (which, as we will see in Section 3.1.3, is
the case throughout our viable parameter space), then the baryon asymmetry
remains frozen at a value $Y_{B}\propto Y_{B-L_{\text{SM}}}|_{T_{sph}}$,
which, in general, is non-zero.

Taking into consideration only the $2\leftrightarrow 2$ scatterings and
ignoring the subleading $1\leftrightarrow 3$ decay processes, the Boltzmann
equations of the asymmetries read

$\displaystyle-sHT\frac{\text{d}Y_{\Delta F_{i}}}{\text{d}T}\,$
$\displaystyle=\,-\big{[}F_{i}\bar{e}\leftrightarrow\varphi\varphi\big{]}\,-\,\big{[}F_{i}\bar{e}\leftrightarrow
F_{j}\bar{e}\big{]}\,+(-1)^{i}\,\big{[}F_{i}\bar{F}_{j}\leftrightarrow
e\bar{e}\big{]}$ (24a) $\displaystyle-
sHT\frac{\text{d}Y_{B-L_{\text{SM}}}}{\text{d}T}\,$
$\displaystyle=\,-\sum_{i}\big{[}F_{i}\bar{e}\leftrightarrow\varphi\varphi\big{]}$
(24b)

where $Y_{\Delta F_{i}}\equiv Y_{F_{i}}-Y_{\bar{F}_{i}}$ and we have used the
notations:

$\displaystyle\left[a\,b\leftrightarrow c\,d\right]\,$
$\displaystyle\equiv\,\left(a\,b\leftrightarrow
c\,d\right)\,-\,\left(\bar{a}\,\bar{b}\leftrightarrow\bar{c}\,\bar{d}\right)$
(25a) $\displaystyle\left(a\,b\leftrightarrow c\,d\right)\,$
$\displaystyle\equiv\,\int\text{d}\Pi_{a}\,\text{d}\Pi_{b}\,\text{d}\Pi_{c}\,\text{d}\Pi_{d}\,\left(2\pi\right)^{4}\delta^{(4)}\Big{[}\left|\mathcal{M}\right|^{2}_{ab\rightarrow
cd}f_{a}f_{b}\left(1\pm f_{c}\right)\left(1\pm f_{d}\right)$
$\displaystyle\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\,\left|\mathcal{M}\right|^{2}_{cd\rightarrow
ab}f_{c}f_{d}\left(1\pm f_{a}\right)\left(1\pm f_{b}\right)\Big{]}$ (25b)

with $\delta^{\left(4\right)}$ being an abbreviation for
$\delta^{\left(4\right)}\left(p_{a}+p_{b}-p_{c}-p_{d}\right)$. We further
approximate the distribution functions of the visible sector species by
Maxwell-Boltzmann statistics, i.e. we neglect Bose-enhancement and Pauli-
blocking factors. Hence, we can write

$\left(a\,b\leftrightarrow c\,d\right)\,=\,\gamma_{ab\rightarrow
cd}\,\frac{Y_{a}}{Y_{a}^{\text{eq}}}\,\frac{Y_{b}}{Y_{b}^{\text{eq}}}\,-\,\gamma_{cd\rightarrow
ab}\,\frac{Y_{c}}{Y_{c}^{\text{eq}}}\,\frac{Y_{d}}{Y_{d}^{\text{eq}}}\;.$ (26)

Let us also point out that throughout the subsequent analysis we will ignore
thermal corrections to the masses of the particles involved in the various
processes. Since these corrections should affect in a similar manner the
masses of the $F_{i}$’s and the SM leptons, we expect them not to
qualitatively alter our results.

The various terms in the Boltzmann equation can be expressed in terms of the
$CP$ asymmetries, the tree-level scattering rate densities and the asymmetric
abundances. Due to the small value of the observed baryon asymmetry, it is
typical to linearize in the SM chemical potentials Kolb:1979qa

$\frac{Y_{e(\bar{e})}}{Y_{e}^{\text{eq}}}\,\equiv\,e^{\pm\mu_{e}/T}\,\approx\,1\,\pm\,\frac{\mu_{e}}{T}\,=\,1\,\pm\,\frac{Y_{\Delta
e}}{2Y_{\gamma}}$ (27)

Using (27) as well as the $CPT$ and unitarity conditions of Eqs. (23), the
Boltzmann equations can be rewritten as

$\displaystyle-sHT\frac{\text{d}Y_{\Delta
F_{i}}}{\text{d}T}\,=\,2\,\epsilon_{i}\gamma^{F_{i}\bar{e}}_{\varphi\varphi}\left[\frac{Y_{F_{j}+\bar{F}_{j}}}{Y_{F_{j}+\bar{F}_{j}}^{\text{eq}}}\,-\,\left(\frac{Y_{\varphi}}{Y_{\varphi}^{\text{eq}}}\right)^{2}\right]-\gamma^{F_{i}\bar{e}}_{\varphi\varphi}\left(y_{F_{i}}-y_{e}\right)+$
$\displaystyle+(-1)^{i}\,\gamma^{F_{1}\bar{e}}_{F_{2}\bar{e}}\left(y_{F_{1}}-\frac{Y_{F_{1}+\bar{F}_{1}}}{Y_{F_{1}+\bar{F}_{1}}^{\text{eq}}}\,y_{e}-y_{F_{2}}+\frac{Y_{F_{2}+\bar{F}_{2}}}{Y_{F_{2}+\bar{F}_{2}}^{\text{eq}}}\,y_{e}\right)\,+(-1)^{i}\,\gamma^{F_{1}\bar{F}_{2}}_{e\bar{e}}\left(\frac{Y_{F_{1}}}{Y_{F_{1}}^{\text{eq}}}\frac{Y_{\bar{F}_{2}}}{Y_{F_{2}}^{\text{eq}}}\,-\,\frac{Y_{\bar{F}_{1}}}{Y_{F_{1}}^{\text{eq}}}\frac{Y_{F_{2}}}{Y_{F_{2}}^{\text{eq}}}\right)$
(28a) $\displaystyle-
sHT\frac{\text{d}Y_{B-L_{\text{SM}}}}{\text{d}T}\,=\,2\,\epsilon_{1}\gamma^{F_{1}\bar{e}}_{\varphi\varphi}\left(\frac{Y_{F_{2}+\bar{F}_{2}}}{Y_{F_{2}+\bar{F}_{2}}^{\text{eq}}}-\frac{Y_{F_{1}+\bar{F}_{1}}}{Y_{F_{1}+\bar{F}_{1}}^{\text{eq}}}\right)\,-\,\gamma^{F_{1}\bar{e}}_{\varphi\varphi}\left(y_{F_{1}}-y_{e}\right)\,-\,\gamma^{F_{2}\bar{e}}_{\varphi\varphi}\left(y_{F_{2}}-y_{e}\right)$
(28b)

where $y_{F_{i}}\equiv Y_{\Delta F_{i}}/Y_{F_{i}}^{\text{eq}}$ and
$y_{e}\equiv Y_{\Delta e}/Y_{\gamma}$. Note that we have used the $CPT$ and
unitarity conditions in order to show explicitly that the source term of each
asymmetry vanishes when all species follow their equilibrium abundance. In
other words, no asymmetries are generated unless there is a departure from
equilibrium, in accordance with the third Sakharov condition.

Under the freeze-in assumption $Y_{\varphi}/Y_{\varphi}^{\text{eq}}\approx 0$
and, if we also consider the heavy leptons $F_{i}$ to be kept close to their
equilibrium abundances due to their gauge interactions, the Boltzmann
equations of the asymmetries simplify to

Figure 4: The generated asymmetries $Y_{\Delta F_{i}},Y_{B-L_{\text{SM}}}$ and
the DM abundance $Y_{\text{DM}}$, in terms of the dimensionless parameter
$x\equiv T_{\text{RH}}/T$. Observe that the asymmetric yields freeze-in very
close to the reheating temperature.

$\displaystyle-sHT\frac{\text{d}Y_{\Delta F_{i}}}{\text{d}T}\,$
$\displaystyle=\,2\epsilon_{i}\gamma^{F_{i}\bar{e}}_{\varphi\varphi}\,-\,\gamma^{F_{i}\bar{e}}_{\varphi\varphi}\left(y_{F_{i}}-y_{e}\right)\,+(-1)^{i}\,\gamma^{F_{1}\bar{e}}_{F_{2}\bar{e}}\left(y_{F_{1}}-y_{F_{2}}\right)$
(29a) $\displaystyle-sHT\frac{\text{d}Y_{B-L_{\text{SM}}}}{\text{d}T}\,$
$\displaystyle=\,-\gamma^{F_{1}\bar{e}}_{\varphi\varphi}\left(y_{F_{1}}-y_{e}\right)\,-\,\gamma^{F_{2}\bar{e}}_{\varphi\varphi}\left(y_{F_{2}}-y_{e}\right)\;.$
(29b)

In passing, let us point out that the first term of Eq. (28b) vanishes, since
$F_{1,2}$ are kept in thermal equilibrium due to their gauge interactions. In
contrast, the heavy leptons asymmetries have a non-vanishing $CP$-source term
and conservation of $Y_{B-L}$ implies that $\sum_{i}Y_{\Delta
F_{i}}=Y_{B-L_{\text{SM}}}$.

Due to the fact that the interactions responsible for the creation of all
asymmetries are non-renormalizable, the asymmetries freeze-in close to the
reheating temperature – which in our case will turn out to be higher than
$10^{13}\operatorname{GeV}$. In this regime all relevant spectator processes
which could redistribute the asymmetries initially created in the right-handed
SM leptons are inactive Nardi:2005hs , which implies that $Y_{\Delta
e}=-Y_{B-L_{\text{SM}}}$. We will solve the Boltzmann Eqs. (29) making use of
this observation. At temperatures higher than $10^{12}\operatorname{GeV}$ the
electroweak sphalerons are out-of-equilibrium and therefore no baryon
asymmetry can be generated. However, as the temperature drops below this
value, the sphalerons become active and the baryon asymmetry becomes
proportional to $Y_{B-L_{\text{SM}}}$. The proportionality constant depends on
all of the spectator processes (e.g. Yukawa interactions, sphaleron
transitions) which are active and relate the chemical potentials of the
participating species PhysRevD.42.3344 ; Nardi:2005hs , including the beyond
SM heavy fermions. At $T\ll 10^{8}\operatorname{GeV}$, when all spectator
processes are in equilibrium, the baryon asymmetry is related to
$Y_{B-L_{\text{SM}}}$ as Goudelis:2021qla

$Y_{B}\,=\,\frac{22}{79}Y_{B-L_{\text{SM}}}\,.$ (30)

The system of the three Boltzmann equations, Eqs. (29), has been solved
numerically using both the numerical and the analytical expressions for the
various rates and $CP$ asymmetries, with the results being in good agreement.
As an illustration, we present an explicit example in Figure 4, where the
asymmetric yields and the DM abundance (the latter being included for
completeness) have been calculated using the same set of parameters as in Fig.
3. These parameters reproduce the observed baryon asymmetry of the Universe,
while in order to obtain the observed DM relic density the mass of the DM
state, as can be seen from Eq. (14), has to be set to $m_{\varphi}\approx
5.1\operatorname{keV}$.

One might note that, strictly speaking, once spectator processes become
active, the relation $Y_{\Delta e}=-Y_{B-L_{\text{SM}}}$ no longer holds and
rather translates to a proportionality relation, with the proportionality
constant depending on the temperature under consideration. For example below
$T\sim 10^{8}\operatorname{GeV}$ the two are related through $Y_{\Delta
e}=-(28/79)Y_{B-L_{\text{SM}}}$. We have numerically verified that, at least
for the values of the reheating temperature that we will consider, replacing
$Y_{\Delta e}$ with $-Y_{B-L_{\text{SM}}}$ in our Boltzmann equations
constitutes a good approximation within an accuracy better than $1\%$. This is
due to the fact that, as can be seen from Fig. 4, the $B-L_{\text{SM}}$
asymmetry is generated extremely close to the reheating temperature (which, in
our example, is much higher than $10^{13}$ GeV). We will, hence, adopt this
approximation throughout the rest of our numerical treatment.

####  3.1.3 Results

Our results for the predictions of the model described by the Lagrangian of
Eq. (9) are summarized in Figure 5. In the left hand-side panel, we
numerically solve the relevant Boltzmann equations, Eqs. (29), while adopting
the analytical approximations for the $CP$ asymmetries and the scattering rate
densities that we presented previously. We show the combinations of the
reheating temperature and the EFT energy scale for which the observed baryon
asymmetry can be obtained, fixing the relevant couplings at the values:
$|\lambda_{1}|=0.5$, $|\lambda_{2}|=|\kappa|=1$ and $\Delta\phi=\pi/2$.

((a))

((b))

Figure 5: (a) Combinations of the reheating temperature $T_{\text{RH}}$ and
the energy scale $\Lambda$, which can generate the observed baryon asymmetry.
The coupling constants have been set to $|\lambda_{1}|=0.5$,
$|\lambda_{2}|=|\kappa|=1$ and $\Delta\phi=\pi/2$. The dashed lines depict
representative DM masses $m_{\varphi}=\\{4,15.5\\}\operatorname{keV}$ for
which the observed DM relic density can be reproduced. Masses below
$4\operatorname{keV}$ are excluded from current Lyman-$\alpha$ forest
observations (yellow-shaded area). (b) The values of $\Lambda$ and $T_{\rm
RH}$ that result in the observed DM relic and baryon asymmetry for
$|\kappa|=|\lambda_{2}|=1$ and $\Delta\phi=\pi/2$, while $|\lambda_{1}|$ is
chosen so that $\Omega_{\text{DM}}h^{2}=0.12$. The darker-shaded region
corresponds to $m_{\varphi}>15.5~{}\operatorname{keV}$, and it is extended to
the lighter-shaded one for $m_{\varphi}>4~{}\operatorname{keV}$. The blue
lines show the upper bound on $T_{\rm RH}$ for which the observed DM abundance
in the Universe can be reproduced for each value of $\Lambda$, for the two DM
mass limits.

In order to extend our scan of the parameter space in an efficient manner, we
first note that the third Boltzmann equation admits a formal solution as

$Y_{B-L_{\text{SM}}}=C\,e^{\frac{4}{5}\,(|\lambda_{1}|^{2}+|\lambda_{2}|^{2})\,C\,T}\int_{T_{\rm
RH}}^{T}dT^{\prime}\,e^{-\frac{4}{5}\,(|\lambda_{1}|^{2}+|\lambda_{2}|^{2})\,C\,T^{\prime}}\left(|\lambda_{1}|^{2}Y_{\Delta
F_{1}}+|\lambda_{2}|^{2}Y_{\Delta F_{2}}\right)\;,$ (31)

where

$C=\sqrt{\frac{3}{2}}\frac{45}{128\
\pi^{15/2}}\frac{M_{Pl}}{\Lambda^{2}}\frac{1}{Y_{F_{1,2}}^{\rm eq}g_{\star
s}\sqrt{g_{\star\rho}}}\Big{|}_{T=T_{\rm RH}}\;.$ (32)

Moreover, an estimate of $Y_{\Delta F_{i}}$ can be obtained by keeping only
the first terms in the RHS of Eqs. (29a). The reason for this is that while
the sector-wise asymmetries are produced close to the reheating temperature
(cf the previous discussion and Fig. 4), the second and third terms become
active at later times when the yields become sizeable; however, by this time
the interaction rates have already become considerably smaller, since they
drop quickly with the temperature. Therefore,

$Y_{\Delta
F_{i}}\approx-2\,\int_{T_{\text{RH}}}^{T}dT^{\prime}\,\epsilon_{i}\,\dfrac{\gamma^{F_{i}\bar{e}}_{\varphi\varphi}}{HsT^{\prime}}\;.$
(33)

By substituting Eq. (33) into Eq. (31), expanding for $\Lambda\gg T_{\rm RH}$
and keeping only the leading term, we obtain

$\displaystyle\frac{22}{79}Y_{B-L_{\text{SM}}}\approx-\,8.71\times
10^{-11}\,|\kappa||\lambda_{1}||\lambda_{2}|\left(|\lambda_{1}|^{2}-|\lambda_{2}|^{2}\right)\sin\left(\Delta\phi\right)\times$
(34) $\displaystyle\times\left(\dfrac{T_{\rm RH}}{3.3\times
10^{16}~{}\operatorname{GeV}}\right)^{4}\left(\dfrac{10^{17}~{}\operatorname{GeV}}{\Lambda}\right)^{6}$
.

As expected from the $\epsilon_{i}$ CP asymmetries in Eqs. (19), the asymmetry
in the SM lepton vanishes if the couplings $\lambda_{1,2}$ are equal, since
the production of lepton asymmetry from both $F_{1,2}$ becomes equal and
opposite.

By requiring the DM relic abundance and the baryon asymmetry to match their
corresponding observed values, $\Omega_{\text{DM}}h^{2}\approx 0.12$ and
$Y_{B}\approx 8.71\times 10^{-11}$, and using Eqs. (14) and (34), we find that
the cut-off and the reheating temperature have to satisfy

$\displaystyle\Lambda\approx 2\times 10^{16}~{}\operatorname{GeV}\
\dfrac{\left(|\lambda_{1}|^{2}+|\lambda_{2}|^{2}\right)^{2}}{\sqrt{\left||\kappa||\lambda_{1}||\lambda_{2}|^{2}(|\lambda_{1}|^{2}-|\lambda_{2}|^{2})\sin\left(\Delta\phi\right)\right|}}\left(\dfrac{m_{\varphi}}{10~{}\operatorname{keV}}\right)^{2}$
(35a) $\displaystyle T_{\rm RH}\approx 3\times 10^{15}~{}\operatorname{GeV}\
\dfrac{\left(|\lambda_{1}|^{2}+|\lambda_{2}|^{2}\right)^{3}}{\left||\kappa||\lambda_{1}||\lambda_{2}|^{2}(|\lambda_{1}|^{2}-|\lambda_{2}|^{2})\sin\left(\Delta\phi\right)\right|}\left(\dfrac{m_{\varphi}}{10~{}\operatorname{keV}}\right)^{3}\;.$
(35b)

Scanning over the parameter space, both numerically and analytically using
Eqs. (35), we obtain the allowed values of $\Lambda$ and $T_{\rm RH}$
delineated in Figure 5(b) (shaded regions).

A few comments are in order. First, as we discussed in Section 3.1.2, in order
for the generated baryon asymmetry to survive until the present-day era, the
decay of the relic $F_{i}$’s has to occur after sphaleron decoupling. By
employing Eq. (12) we find this to indeed be the case within the entire viable
parameter space depicted in Figure 5. In particular, we find that if the mass
of the heavy fermions is smaller than a few TeV, they tend to decay during the
big bang nucleosynthesis (BBN) era or even later. At the same time, the
(temporary, i.e. prior to their eventual decay) relic abundance of the
$F_{i}$’s is comparable to that of thermal WIMPs, since they interact
electromagnetically. Based on the analysis performed, e.g., in
Kawasaki:2017bqm , particles of such abundances decaying into leptons should
possess a lifetime shorter than $\sim 100$ seconds. In our case this
constraint can, however, be easily satisfied either by simply requiring the
masses of the $F_{i}$’s to be larger than $\sim 5$ TeV or by introducing an
additional (faster) decay channel. Lastly, as we observe in Fig. 5(b), the
combination of different cosmological and astrophysical constraints lead us to
consider rather large values of the reheating temperature. The current cosmic
microwave background constraint on the scalar-to-tensor ratio $(r<0.056)$
Planck:2018jri , combined with the requirement that the inflaton decays during
the inflationary period, imposes an upper bound on the reheating temperature
which is roughly of order $10^{16}\operatorname{GeV}$. In this sense, the
region of the parameter space where $T_{\text{RH}}$ approaches close to
$10^{17}\operatorname{GeV}$ may be considered to be only marginally
cosmologically acceptable.

Note that throughout the parameter space depicted in both panels of Figure 5,
the DM abundance constraint can be satisfied by appropriately choosing the DM
mass.

###  3.2 Fermion DM

Let us, now, turn to the case of a Dirac fermion DM candidate $\chi$, which is
pair-created through a dimension-6 operator. The interaction Lagrangian that
we consider is

$\mathcal{L}\;\supset\;\frac{\lambda_{1}}{2\Lambda^{2}}\,\left(\bar{e}\operatorname{P_{L}}F_{1}\right)\,\left(\bar{\chi}\operatorname{P_{R}}\chi^{c}\right)\,+\,\frac{\lambda_{2}}{2\Lambda^{2}}\,\left(\bar{e}\operatorname{P_{L}}F_{2}\right)\,\left(\bar{\chi}\operatorname{P_{R}}\chi^{c}\right)\,+\,\frac{\kappa}{\Lambda^{2}}\,\left(\bar{e}\operatorname{P_{L}}F_{1}\right)\left(\bar{F}_{2}\operatorname{P_{R}}e\right)\,+\,\text{h.c.}$
(36)

Such a Lagrangian can, like in the case of the model described in Section 3.1,
be understood as stemming from a theory in which a scalar mediator has been
integrated out. DM stability can, again, be ensured by imposing a
$\mathbb{Z}_{3}$ symmetry under which the $F_{i}$’s and $\chi$ are charged
according to the assignments summarized in Table 2.

Particle | Gauge | $\mathbb{Z}_{3}$
---|---|---
$\chi$ | $(1,1)_{0}$ | $\omega$
$\bar{\chi}$ | $(1,1)_{0}$ | $\omega^{-1}$
$F_{i}$ | $(1,1)_{-1}$ | $\omega^{-1}$
$\bar{F}_{i}$ | $(1,1)_{1}$ | $\omega$
Table 2: Quantum charges of the various states, where $\omega=e^{i2\pi/3}$.

Much like before, $1\leftrightarrow 3$ decay processes $F_{i}\leftrightarrow
e\chi\chi$ and $F_{2}\leftrightarrow F_{1}e\bar{e}$ are expected to be highly
suppressed for $M_{i}\ll T_{\rm RH}$ and $\Lambda\gtrsim
10^{11}\operatorname{GeV}$ (see Figure 1), with respect to $2\leftrightarrow
2$ scattering ones. The corresponding decay widths can be approximated to
scale as

$\Gamma_{F_{i}\rightarrow
e\chi\chi}\,\approx\,\dfrac{\left|\lambda_{i}\right|^{2}}{12288\
\pi^{3}}\frac{M_{i}^{5}}{\Lambda^{4}},\qquad\Gamma_{F_{2}\rightarrow
F_{1}e\bar{e}}\,\approx\,\dfrac{\left|\kappa\right|^{2}}{6144\
\pi^{3}}\frac{M_{2}^{5}}{\Lambda^{4}}\;,$ (37)

where we have assumed that $M_{1}\ll M_{2}$.

The dominant processes for the generation of the DM abundance (and the $CP$
asymmetries) are the $2\leftrightarrow 2$ scatterings
$F_{i}\bar{e}\leftrightarrow\chi\chi$ and $F_{i}\bar{e}\leftrightarrow
F_{j}\bar{e}$. An analytical approximation for the corresponding scattering
rate densities can be obtained in the limit $\Lambda\gg m_{i}$, yielding

$\gamma^{F_{i}\bar{e}}_{\chi\chi}\,\approx\,\frac{\left|\lambda_{i}\right|^{2}T}{1024\,\pi^{5}\,\Lambda^{4}}\int_{0}^{\infty}\text{d}\tilde{s}\,\,\tilde{s}^{5/2}\,K_{1}\left(\frac{\sqrt{\tilde{s}}}{T}\right)\,=\,\frac{3\left|\lambda_{i}\right|^{2}}{4\,\pi^{5}}\frac{T^{8}}{\Lambda^{4}}$
(38)

whereas $\gamma^{F_{i}\bar{e}}_{F_{j}\bar{e}}$ is given by Eq. (11b).

The freeze-in DM abundance in this model can, again, be read-off from Eq. (4)
setting $n=2$ and $A=|\lambda_{1}|^{2}+|\lambda_{2}|^{2}$, i.e.

$Y_{\text{DM}}\left(T\right)\,=\,\frac{45\big{(}\left|\lambda_{1}\right|^{2}+\left|\lambda_{2}\right|^{2}\big{)}M_{Pl}}{2\times
1.66\pi^{7}g_{\star
s}\sqrt{g_{\star\rho}}}\frac{T_{\text{RH}}^{3}-T^{3}}{\Lambda^{4}}\;.$ (39)

The DM relic density $\Omega_{\text{DM}}h^{2}$ can, in turn, be cast into the
simple form

$\Omega_{\rm DM}h^{2}\approx
0.12\times\Big{(}|\lambda_{1}|^{2}+|\lambda_{2}|^{2}\Big{)}\Big{(}\dfrac{2.3\times
10^{15}~{}\operatorname{GeV}}{\Lambda}\Big{)}^{4}\Big{(}\dfrac{T_{\rm
RH}}{3\times
10^{14}~{}\operatorname{GeV}}\Big{)}^{3}\Big{(}\dfrac{m_{\chi}}{10~{}\operatorname{keV}}\Big{)}\;.$
(40)

####  3.2.1 CP asymmetries

The $CP$ asymmetries generated by the $2\leftrightarrow 2$ scattering
processes can be defined, similarly to the previous scenario that we examined,
as

$\displaystyle\epsilon_{i}\,$
$\displaystyle\equiv\frac{\gamma_{F_{i}\bar{e}\rightarrow\chi\chi}\,-\,\gamma_{\bar{F}_{i}e\rightarrow\bar{\chi}\bar{\chi}}}{\gamma_{F_{i}\bar{e}\rightarrow\chi\chi}\,+\,\gamma_{\bar{F}_{i}e\rightarrow\bar{\chi}\bar{\chi}}}$
(41a) $\displaystyle\epsilon_{3}\,$
$\displaystyle\equiv\frac{\gamma_{F_{1}\bar{e}\rightarrow
F_{2}\bar{e}}\,-\,\gamma_{\bar{F}_{1}e\rightarrow\bar{F}_{2}e}}{\gamma_{F_{1}\bar{e}\rightarrow
F_{2}\bar{e}}\,+\,\gamma_{\bar{F}_{1}e\rightarrow\bar{F}_{2}e}}$ (41b)

where $i=1,2$.

The imaginary part of the amplitudes can be calculated from the optical
theorem to be (see Appendix A)

$\text{Im}\big{\\{}\mathcal{A}_{0}^{*}\mathcal{A}_{1}\big{\\}}_{F_{i}\bar{e}\rightarrow\chi\chi}\,=\,\frac{\big{(}\tilde{s}-M_{i}^{2}-m_{e}^{2}\big{)}\big{(}\tilde{s}-M_{j}^{2}-m_{e}^{2}\big{)}\left(\tilde{s}-2m_{\chi}^{2}\right)\,\sqrt{\lambda\big{(}\tilde{s},M_{j}^{2},m_{e}^{2}\big{)}}}{16\pi\tilde{s}}\;.$
(42)

Once again, this expression has been crossed-checked both using Package-X and
through the FeynRules-FeynArts-FormCalc chain.

Figure 6: The $CP$ asymmetries generated by the scattering processes for
$\Lambda=2.85\times 10^{15}\operatorname{GeV}$, $T_{\text{RH}}=4\times
10^{14}\operatorname{GeV}$, $M_{1}=1\operatorname{TeV}$,
$M_{2}=5\operatorname{TeV}$, $|\lambda_{1}|=0.5$, $|\lambda_{2}|=|\kappa|=1$
and $\Delta\phi=\pi/2$. The coloured solid lines correspond to the numerical
results, the dashed lines to the analytical approximations, while the
horizontal black solid line corresponds to the $CP$ asymmetry from the heavier
fermion decay $F_{2}\rightarrow e\chi\chi$, i.e.
$\epsilon_{D}\sim\frac{\left|\lambda_{2}\right|^{2}}{32\pi}\frac{M_{2}^{2}}{\Lambda^{2}}$.

An analytical approximation for the $CP$ asymmetries $\epsilon_{1}$ and
$\epsilon_{2}$ can be obtained in the limit $\Lambda\gg m_{i}$, yielding

$\displaystyle\epsilon_{1}\,$
$\displaystyle\approx\,-\frac{6}{\pi}\frac{\left|\kappa\right|\left|\lambda_{2}\right|}{\left|\lambda_{1}\right|}\sin\left(-\Delta\phi\right)\frac{T^{2}}{\Lambda^{2}}$
(43a) $\displaystyle\epsilon_{2}\,$
$\displaystyle\approx\,-\frac{6}{\pi}\frac{\left|\kappa\right|\left|\lambda_{1}\right|}{\left|\lambda_{2}\right|}\sin\left(\Delta\phi\right)\frac{T^{2}}{\Lambda^{2}}\;.$
(43b)

In the case of the $\epsilon_{3}$ asymmetry the imaginary part of the Feynman
amplitudes is

$\text{Im}\big{\\{}\mathcal{A}_{0}^{*}\mathcal{A}_{1}\big{\\}}_{F_{1}\bar{e}\rightarrow
F_{2}\bar{e}}\,=\,\frac{\big{(}\tilde{s}-M_{1}^{2}-m_{e}^{2}\big{)}\big{(}\tilde{s}-M_{2}^{2}-m_{e}^{2}\big{)}\left(\tilde{s}-2m_{\chi}^{2}\right)\,\sqrt{\lambda\big{(}\tilde{s},m_{\chi}^{2},m_{\chi}^{2}\big{)}}}{16\pi\tilde{s}}$
(44)

and can be analytically approximated by

$\epsilon_{3}\,\approx\,-\frac{3}{\pi}\frac{\left|\lambda_{1}\right|\left|\lambda_{2}\right|}{\left|\kappa\right|}\sin\left(\Delta\phi\right)\frac{T^{2}}{\Lambda^{2}}\;.$
(45)

In Figure 6 we illustrate the values of the $CP$ asymmetries as a function of
$x\equiv T_{\rm RH}/T$, for the representative set of parameter values:
$\Lambda=2.85\times 10^{15}\operatorname{GeV}$, $T_{\text{RH}}=4\times
10^{14}\operatorname{GeV}$, $M_{1}=1\operatorname{TeV}$,
$M_{2}=5\operatorname{TeV}$, $|\lambda_{1}|=0.5$, $|\lambda_{2}|=|\kappa|=1$
and $\Delta\phi=\pi/2$. Again, the numerical and analytical results are
identical at the temperatures of interest.

Finally, invariance under $CPT$ and $S$-matrix unitarity result to the same
conditions as before, Eq. (23), with $\varphi$ replaced by the Dirac DM state
$\chi$.

####  3.2.2 Baryon asymmetry

The Boltzmann equations describing the temperature evolution of the
asymmetries in the $F_{i}$’s and in the SM can be written in a completely
analogous manner as in the scalar DM case. We repeat them here for
convenience:

$\displaystyle-sHT\frac{\text{d}Y_{\Delta F_{i}}}{\text{d}T}\,$
$\displaystyle=\,2\epsilon_{i}\gamma^{F_{i}\bar{e}}_{\chi\chi}\,-\,\gamma^{F_{i}\bar{e}}_{\chi\chi}\left(y_{F_{i}}-y_{e}\right)\,+(-1)^{i}\,\gamma^{F_{1}\bar{e}}_{F_{2}\bar{e}}\left(y_{F_{1}}-y_{F_{2}}\right)$
(46a) $\displaystyle-sHT\frac{\text{d}Y_{B-L_{\text{SM}}}}{\text{d}T}\,$
$\displaystyle=\,-\gamma^{F_{1}\bar{e}}_{\chi\chi}\left(y_{F_{1}}-y_{e}\right)\,-\,\gamma^{F_{2}\bar{e}}_{\chi\chi}\left(y_{F_{2}}-y_{e}\right)\,.$
(46b)

Figure 7: The generated asymmetries $Y_{\Delta F_{i}},Y_{B-L_{\text{SM}}}$ and
the DM abundance $Y_{\text{DM}}$, in terms of the dimensionless parameter
$x\equiv T_{\text{RH}}/T$.

In Figure 7 we present an explicit example of the temperature evolution of the
various yields, in which we have assumed the same set of parameters as those
used in Figure 6. In order to obtain the observed DM relic density, the mass
of the DM state, as can be seen from Eq. (40), is $m_{\chi}\approx
8\operatorname{keV}$.

####  3.2.3 Results

Our numerical results are summarized in Figure 8 following a methodology
similar to the one described in Section 3.1.3. In the left-hand side panel, we
show the combinations of the reheating temperature and the EFT energy scale
for which the observed baryon asymmetry can be obtained, keeping the couplings
fixed at the values: $|\lambda_{1}|=0.5$, $|\lambda_{2}|=|\kappa|=1$ and
$\Delta\phi=\pi/2$. Here, the Boltzmann equations Eqs. (46) have been solved
numerically, using the analytical approximations for the different rate
densities and $CP$ asymmetries that we extracted in Sections 3.2 and 3.2.1.

In the right hand-side panel, we again present a more extended scan of the
available parameter space. We can derive an approximate solution for
$Y_{B-L_{\text{SM}}}$ by keeping only the first terms of Eqs. (46a). The
leading-order approximation in this case reads

$\displaystyle\frac{22}{79}Y_{B-L_{\text{SM}}}\approx-\,8.71\times
10^{-11}\,|\kappa||\lambda_{1}||\lambda_{2}|\left(|\lambda_{1}|^{2}-|\lambda_{2}|^{2}\right)\sin\left(\Delta\phi\right)\times$
(47) $\displaystyle\times\left(\dfrac{T_{\rm RH}}{1.45\times
10^{15}~{}\operatorname{GeV}}\right)^{8}\left(\dfrac{10^{16}~{}\operatorname{GeV}}{\Lambda}\right)^{10}$
.

By requiring the DM abundance and the baryon asymmetry to correspond to their
respective observed values, from Eqs. (40) and (47) we find that the cut-off
$\Lambda$ and the reheating temperature $T_{\rm RH}$ must be related
approximately through

$\displaystyle\Lambda\approx 7.8\times 10^{14}~{}\operatorname{GeV}\
\dfrac{\left(|\lambda_{1}|^{2}+|\lambda_{2}|^{2}\right)^{4}}{\left(\left||\kappa||\lambda_{1}||\lambda_{2}|^{2}(|\lambda_{1}|^{2}-|\lambda_{2}|^{2})\sin\left(\Delta\phi\right)\right|\right)^{3/2}}\left(\dfrac{m_{\chi}}{10~{}\operatorname{keV}}\right)^{4}$
(48a) $\displaystyle T_{\rm RH}\approx 7\times 10^{13}~{}\operatorname{GeV}\
\dfrac{\left(|\lambda_{1}|^{2}+|\lambda_{2}|^{2}\right)^{5}}{\left||\kappa||\lambda_{1}||\lambda_{2}|(|\lambda_{1}|^{2}-|\lambda_{2}|^{2})\sin\left(\Delta\phi\right)\right|^{2}}\left(\dfrac{m_{\chi}}{10~{}\operatorname{keV}}\right)^{5}\;.$
(48b)

Our results (which we have verified to agree within a $10\%$ accuracy with a
fully numerical resolution of the Boltzmann equations) are shown in Fig. 8(b).

((a))

((b))

Figure 8: (a) Combinations of the reheating temperature $T_{\text{RH}}$ and
the energy scale $\Lambda$, which can generate the observed baryon asymmetry.
The coupling constants have been set to $|\lambda_{1}|=0.5$,
$|\lambda_{2}|=|\kappa|=1$ and $\Delta\phi=\pi/2$. The dashed lines depict
representative DM masses $m_{\chi}=\\{4.6,18.1\\}\operatorname{keV}$ for which
the observed DM relic density can be reproduced. Masses below
$4.6\operatorname{keV}$ are excluded from current Lyman-$\alpha$ forest
observations (yellow-shaded area). (b) The values of $\Lambda$ and $T_{\rm
RH}$ that result in the observed DM relic and baryon asymmetry for
$|\kappa|=|\lambda_{1}|=1$ and $\Delta\phi=\pi/2$, while $|\lambda_{2}|$ is
chosen so that $\Omega_{\text{DM}}h^{2}=0.12$. The darker-shaded region
corresponds to corresponds to $m_{\chi}>18.1~{}\operatorname{keV}$, and it is
extended to the lighter-shaded one for $m_{\chi}>4.6~{}\operatorname{keV}$.
The blue lines show the upper bound on $T_{\rm RH}$ for which the observed DM
abundance in the Universe can be reproduced for each value of $\Lambda$, for
the two DM mass limits.

We observe that in this case, it is possible to satisfy the baryon asymmetry
and DM abundance bounds for a fairly wider range of $\Lambda$ and $T_{\rm RH}$
with respect to the scalar DM case that we examined previously. Besides,
similar remarks as the ones we made in Section 3.1.3 apply in this case as
well: we indeed find that the heavy fermions decay long after sphaleron
decoupling which, in turn, ensures that the baryon asymmetry survives until
the present day. In order to prevent the $F_{i}$’s from decaying too late and,
hence, interfering with primordial nucleosynthesis, they must either be heavy
(in this case, heavier than several thousands of TeV) or, alternatively, an
additional decay channel can be introduced which enables the $F_{i}$’s to
decay before BBN. All in all, this setup appears to offer more possibilities
than the scalar DM one, since it allows for greater freedom in the choice of
the EFT scale and the reheating temperature.

##  4 Conclusions and Outlook

In this paper we presented a mechanism in order to explain simultaneously the
observed DM abundance and matter-antimatter asymmetry of the Universe. A
symmetric DM density is created through the freeze-in mechanism, relying on
highly suppressed $2\leftrightarrow 2$ scattering processes described by non-
renormalizable operators. At the same time, these scattering processes also
violate $CP$ which amounts, in synergy with the action of electroweak
sphalerons, to an asymmetry between SM baryons and anti-baryons. The fact that
the relevant interactions are non-renormalizable leads to both abundances
being generated at high temperatures, conversely to other scenarios of freeze-
in baryogenesis which have been considered in the literature Shuve_2020 ;
Goudelis:2021qla ; Chand:2022vrf .

As a proof-of-concept we studied two simple scenarios of scalar and fermion
DM, in which the interactions between the dark and visible particles are
described by dimension-5 and dimension-6 operators, respectively, involving
the DM particles themselves, SM fermions and exotic vector-like fermions. An
additional dimension-6 operator between the SM and vector-like fermions,
allowed by all symmetries in the theory, leads to interference between the
leading-order and next-to-leading-order Feynman diagrams, giving rise to $CP$
violation. Our numerical analysis showed that in both cases it is, indeed,
possible to simultaneously freeze-in the necessary DM density along with a
viable matter-antimatter asymmetry, as long as DM is relatively light (but
within existing Lyman-$\alpha$ forest bounds), the interaction scale $\Lambda$
is larger than $O(10^{16})$ ($O(10^{14})$) GeV and the reheating temperature
$T_{\rm RH}$ is higher than $O(10^{15})$ ($O(10^{13})$) GeV in the case of
scalar (fermion) DM. The masses of the vector-like fermions can be quite low,
down to a few TeV, although in the case of fermion DM this would require the
existence of an additional decay channel so that they decay before the era of
BBN.

There are several ways through which our analysis could be extended. First of
all, the simple scenarios that we presented were intended to serve mostly as
proofs-of-concept concerning the fact that successful freeze-in baryogenesis
can be realized in the UV. Clearly, much more elaborate models can be
developed, based on well-motivated extensions of the SM. In a similar vein, it
would be interesting to examine whether the parameter space can be extended to
lower cutoff and/or reheating temperature values as well as to accommodate
heavier DM. Indeed, preference for relatively light DM appears to be a common
(albeit to different extents, depending on the precise mechanism that is
responsible for $CP$ violation) in the freeze-in baryogenesis scenarios that
have appeared in the literature so far Shuve_2020 ; Goudelis:2021qla . Is this
a generic feature, or not? We hope to address these questions in future work.

###### Acknowledgements.

The authors acknowledge useful discussions with Ioannis Dalianis. DK
acknowledges support by the Lancaster-Manchester-Sheffield Consortium for
Fundamental Physics under STFC Grant No. ST/T001038/1. The work of VCS was
supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.)
under the “First Call for H.F.R.I. Research Projects to support Faculty
members and Researchers and the procurement of high-cost research equipment
grant” (Project Number: 824). This research is co-financed by Greece and the
European Union (European Social Fund- ESF) through the Operational Programme
“Human Resources Development, Education and Lifelong Learning” in the context
of the project “Strengthening Human Resources Research Potential via Doctorate
Research - 2nd Cycle” (MIS-5000432), implemented by the State Scholarships
Foundation (IKY).

##  Appendix A Appendix A

In our model $CP$-violating effects arise, at lowest order, due to the
interference between tree-level and 1-loop diagrams. In addition, the
intermediate states in the loop must propagate on-shell so that the
corresponding Feynman amplitudes obtain an imaginary part Kolb:1979qa . A
direct way to calculate them is by using the optical theorem. In this appendix
we present more details on the calculations of the imaginary parts of the
$\epsilon_{i}(i=1,2)$ and $\epsilon_{3}$ $CP$ asymmetries that appear in the
models under consideration.

  * •

$F_{i}\bar{e}\rightarrow F_{j}\bar{e}\rightarrow\varphi\varphi$

$\displaystyle
2i\text{Im}\big{\\{}\mathcal{A}_{0}^{*}\mathcal{A}_{1}\big{\\}}$
$\displaystyle=i\mathcal{A}_{0}^{*}\left(F_{i}\bar{e}\rightarrow\varphi\varphi\right)\int\text{d}\Pi_{F_{j}}\text{d}\Pi_{\bar{e}}\left(2\pi\right)^{4}\delta^{(4)}\mathcal{A}_{0}\left(F_{i}\bar{e}\rightarrow
F_{j}\bar{e}\right)\mathcal{A}_{0}\left(F_{j}\bar{e}\rightarrow\varphi\varphi\right)$
$\displaystyle=4i\int\text{d}\Pi_{F_{j}}\,\text{d}\Pi_{\bar{e}}\,\left(2\pi\right)^{4}\delta^{(4)}\left(p_{F_{i}}\cdot
p_{\bar{e}}\right)\left(p_{F_{j}}\cdot p_{\bar{e}}\right)$
$\displaystyle=\frac{i}{8\pi\tilde{s}}\big{(}\tilde{s}-M_{i}^{2}-m_{e}^{2}\big{)}\big{(}\tilde{s}-M_{j}^{2}-m_{e}^{2}\big{)}\,\sqrt{\lambda\big{(}\tilde{s},M_{j}^{2},m_{e}^{2}\big{)}}$
(49)

  * •

$F_{i}\bar{e}\rightarrow\varphi\varphi\rightarrow F_{j}\bar{e}$

$\displaystyle
2i\text{Im}\big{\\{}\mathcal{A}_{0}^{*}\mathcal{A}_{1}\big{\\}}$
$\displaystyle=i\mathcal{A}_{0}^{*}\left(F_{i}\bar{e}\rightarrow
F_{j}\bar{e}\right)\int\text{d}\Pi_{\varphi}\,\text{d}\Pi_{\varphi}\,\left(2\pi\right)^{4}\delta^{(4)}\mathcal{A}_{0}\left(F_{i}\bar{e}\rightarrow\varphi\varphi\right)\mathcal{A}_{0}\left(\varphi\varphi\rightarrow
F_{j}\bar{e}\right)$
$\displaystyle=4i\int\text{d}\Pi_{\varphi}\,\text{d}\Pi_{\varphi}\,\left(2\pi\right)^{4}\delta^{(4)}\left(p_{F_{i}}\cdot
p_{\bar{e}}\right)\left(p_{F_{j}}\cdot p_{\bar{e}}\right)$
$\displaystyle=\frac{i}{8\pi\tilde{s}}\big{(}\tilde{s}-M_{i}^{2}-m_{e}^{2}\big{)}\big{(}\tilde{s}-M_{j}^{2}-m_{e}^{2}\big{)}\,\sqrt{\lambda\big{(}\tilde{s},m_{\varphi}^{2},m_{\varphi}^{2}\big{)}}$
(50)

  * •

$F_{i}\bar{e}\rightarrow F_{j}\bar{e}\rightarrow\chi\chi$

$\displaystyle
2i\text{Im}\big{\\{}\mathcal{A}_{0}^{*}\mathcal{A}_{1}\big{\\}}$
$\displaystyle=i\mathcal{A}_{0}^{*}\left(F_{i}\bar{e}\rightarrow\chi\chi\right)\int\text{d}\Pi_{F_{j}}\,\text{d}\Pi_{\bar{e}}\,\left(2\pi\right)^{4}\delta^{(4)}\mathcal{A}_{0}\left(F_{i}\bar{e}\rightarrow
F_{j}\bar{e}\right)\mathcal{A}_{0}\left(F_{j}\bar{e}\rightarrow\chi\chi\right)$
$\displaystyle=8i\int\text{d}\Pi_{F_{j}}\,\text{d}\Pi_{\bar{e}}\,\left(2\pi\right)^{4}\delta^{(4)}\left(p_{F_{i}}\cdot
p_{\bar{e}}\right)\left(p_{\chi_{1}}\cdot
p_{\chi_{2}}\right)\left(p_{F_{j}}\cdot p_{\bar{e}}\right)$
$\displaystyle=\frac{i}{8\pi\tilde{s}}\big{(}\tilde{s}-M_{i}^{2}-m_{e}^{2}\big{)}\big{(}\tilde{s}-M_{j}^{2}-m_{e}^{2}\big{)}\left(\tilde{s}-2m_{\chi}^{2}\right)\,\sqrt{\lambda\big{(}\tilde{s},M_{j}^{2},m_{e}^{2}\big{)}}$
(51)

  * •

$F_{i}\bar{e}\rightarrow\chi\chi\rightarrow F_{j}\bar{e}$

$\displaystyle
2i\text{Im}\big{\\{}\mathcal{A}_{0}^{*}\mathcal{A}_{1}\big{\\}}$
$\displaystyle=i\mathcal{A}_{0}^{*}\left(F_{i}\bar{e}\rightarrow
F_{j}\bar{e}\right)\int\text{d}\Pi_{\chi}\,\text{d}\Pi_{\chi}\,\left(2\pi\right)^{4}\delta^{(4)}\mathcal{A}_{0}\left(F_{i}\bar{e}\rightarrow\chi\chi\right)\mathcal{A}_{0}\left(\chi\chi\rightarrow
F_{j}\bar{e}\right)$
$\displaystyle=8i\int\text{d}\Pi_{\chi}\,\text{d}\Pi_{\chi}\,\left(2\pi\right)^{4}\delta^{(4)}\left(p_{F_{i}}\cdot
p_{\bar{e}}\right)\left(p_{F_{j}}\cdot
p_{\bar{e}}\right)\left(p_{\chi_{1}}\cdot p_{\chi_{2}}\right)$
$\displaystyle=\frac{i}{8\pi\tilde{s}}\big{(}\tilde{s}-M_{i}^{2}-m_{e}^{2}\big{)}\big{(}\tilde{s}-M_{j}^{2}-m_{e}^{2}\big{)}\left(\tilde{s}-2m_{\chi}^{2}\right)\,\sqrt{\lambda\big{(}\tilde{s},m_{\chi}^{2},m_{\chi}^{2}\big{)}}$
(52)

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