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

# Mechanism for baryogenesis via feebly interacting massive particles

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

We present a simple mechanism which allows the simultaneous generation of the
baryon asymmetry of the Universe along with its dark matter content. To this
goal, we employ the out-of-equilibrium decays of heavy bath states into a
feebly coupled dark matter particle and Standard Model charged fermions. These
decays lead to dark matter production via the freeze-in mechanism and,
assuming that they further violate $CP$, can generate a viable matter-
antimatter asymmetry in the resonant regime. We illustrate this mechanism by
studying a particular realization of this general scenario, where the role of
the heavy bath particles is played by $SU(3)_{\text{c}}\times
SU(2)_{\operatorname{L}}$-singlet vectorlike fermions with a non-zero
hypercharge and dark matter is identified with a gauge-singlet real scalar
field. We show that in the context of this simple model the cosmological
constraints for the dark matter abundance and the baryon asymmetry are
satisfied for masses of heavy vectorlike fermion states of a few TeV,
potentially within reach of the High-Luminosity Run of the Large Hadron
Collider. Dark matter, in turn, is predicted to be rather light, with a mass
of a few keV.

##  1 Introduction

Among the numerous open questions in contemporary high-energy physics, the
origin of cosmic dark matter (DM) and that of the baryon asymmetry in the
Universe occupy a pivotal place. Not only do they constitute two of the most
striking and fundamental pieces of evidence for the existence of physics
beyond the Standard Model (BSM) of particle physics, but they do so only once
the latter is placed within a cosmological framework. Resolving either (let
alone both) of these questions most likely requires particle physics to be
viewed from a cosmological standpoint and, conversely, cosmology to be
analyzed in terms of the behavior of the fundamental constituents of matter
and their interactions. And, indeed, in doing so during the past decades model
builders have not been short of ideas concerning the nature of dark matter and
the mechanism through which matter came to dominate over antimatter in the
Universe.

Among the various dark matter candidates that have been proposed we can
mention axions Preskill:1982cy ; Abbott:1982af ; Dine:1982ah , primordial
black holes Hawking:1971ei ; Chapline:1975ojl ; Green:2020jor , a vast number
of incarnations of weakly or feebly interacting massive particles (WIMPs or
FIMPs; for reviews cf e.g. Arcadi:2017kky ; Bernal:2017kxu ), gravitationally
produced dark matter Kolb:1998ki , and asymmetric dark matter (for reviews cf
e.g. Davoudiasl_2012 ; Petraki:2013wwa ), just to name a few. Similarly, the
matter-antimatter asymmetry of the Universe has been explained in terms of
different mechanisms of baryogenesis, like baryogenesis based on grand unified
theories (GUTs) Yoshimura:1978ex , electroweak baryogenesis Kuzmin:1985mm ;
Cohen:1993nk , the Affleck-Dine mechanism Affleck:1984fy ), and leptogenesis
Fukugita:1986hr . Interestingly, during the past decade, there have also been
several attempts to actually link the two questions, most notably in the
contexts of WIMP baryogenesis McDonald_2011 ; Cui_2012 ; Cui_2013  or
asymmetric dark matter Hall:2010jx ; Unwin:2014poa ; Cui:2020dly .

Recently, the main idea behind Akhmedov-Rubakov-Smirnov (ARS) leptogenesis
Akhmedov:1998qx , namely, that of a lepton number asymmetry being generated
through $CP$-violating sterile neutrino oscillations, was exploited in
Shuve_2020 , where the authors proposed that a baryon asymmetry could,
instead, be also generated by augmenting the SM with exotic scalars and
fermions directly coupling to quarks. The fermions, which were taken to be
singlets under the SM gauge group can, moreover, play the role of viable dark
matter candidates through the freeze-in mechanism McDonald:2001vt ;
Hall:2009bx , whereas their $CP$-violating oscillations, in the presence of
electroweak sphaleron transitions, can generate the observed baryon asymmetry
of the Universe.

In this paper, we propose a mechanism which borrows ideas both from Dirac
leptogenesis Dick:1999je ; Murayama_2002 ; Cerdeno_2006 ; Gonzalez:2009  and
from this scenario of “freeze-in baryogenesis”. As we will describe in detail
in the following, we consider a heavy particle species which is charged under
(parts of) the SM gauge group and which can decay through feeble interactions
into SM fermions along with a neutral particle. The latter is our dark matter
candidate, produced upon the decays of the heavy particle through the freeze-
in mechanism, along the lines presented in Shuve_2020 . However, in our case
the decays themselves violate $CP$, in a similar manner as in leptogenesis
models. Then, in the presence of electroweak sphalerons, we will see that an
asymmetry can be generated between SM fermions and antifermions. The relevant
processes proceed through feeble couplings, preventing them from ever reaching
equilibrium and, thus, satisfying the third Sakharov condition Sakharov:1967dj
. The first condition, namely, baryon number violation, is satisfied due to
the active sphaleron processes in a way that resembles neither GUT
baryogenesis (explicit $B$ violation in decays) nor leptogenesis (explicit $L$
violation in decays), although we will also comment on the possibility of
direct $B/L$ violation as well.

The paper is structured as follows: in Section 2 we discuss the general
features of our mechanism, namely, the way through which the required dark
matter abundance and the baryon/lepton asymmetries can be generated, without
adopting any concrete microscopic model. In Section 3 we propose a simple
model as a proof-of-concept that concrete incarnations of our mechanism can,
indeed, be constructed. We compute the predicted dark matter abundance and
baryon asymmetry, quantify the effects of processes that wash out the latter
and briefly comment on the phenomenological perspectives of our model, notably
in relation to searches for long-lived particles (LLPs) at the Large Hadron
Collider (LHC). Finally, in Section 4 we summarize our main findings and
conclude. Some more technical aspects are left for the Appendix.

##  2 Dark matter and baryogenesis from freeze-in: General framework

Before presenting a concrete realization of our take on the freeze-in
baryogenesis idea, let us briefly summarize a few key notions that will be
useful for the discussion that follows: frozen-in dark matter and how the
freeze-in framework can enable us to satisfy the three Sakharov conditions
that are necessary for successful baryogenesis. A concrete implementation of
these ideas will be presented in detail in Section 3.

###  2.1 Freeze-in DM abundance

The freeze-in mechanism for dark matter production relies on two basic
premises:

  * •

The initial DM abundance is zero.

  * •

Dark matter interacts only extremely weakly (“feebly”) with the Standard Model
particles (along with any other particles that are in thermal equilibrium with
them).

Under these assumptions, and further assuming that dark matter production
takes place in a radiation-dominated Universe, dark matter never reaches
thermal equilibrium with the plasma. Instead, it is produced through the out-
of-equilibrium decays or annihilations of bath particles and all dark matter
depletion processes, the rate of which typically scales as $\left\langle\sigma
v\right\rangle\times n_{DM}^{2}$ (where $\sigma$ is the dark matter
annihilation cross section, $v$ its velocity, and $n_{DM}$ its number
density), can be ignored.

###  2.2 Baryon asymmetry abundance $Y_{B}$

In general, the decays and/or annihilations that are responsible for dark
matter production can also violate both the baryon number $B$ and
$C/CP$.111Very similar remarks can be made if the decay violates, instead, the
lepton number. This is also the option that we will adopt later in this work.
Then, as long as we stick to the freeze-in framework, these processes occur
out-of-equilibrium with the thermal plasma, thus fulfilling all three Sakharov
conditions.

Intuitively, and ignoring all washout processes, if we denote the measure of
$CP$ violation by $\epsilon_{CP}$, we would expect the generated asymmetry in
the SM fermion $Y_{\Delta f}$ to be connected to the dark matter abundance
$Y_{DM}$ through a relation of the type

$Y_{\Delta f}\,\sim\,\epsilon_{CP}\,Y_{DM}$ (1)

In reality, this limit cannot be attained given that some amount of washout is
almost inevitable, whereas concrete realizations typically require the
introduction of additional particles and decay channels. In this respect this
relation may be viewed as an upper limit to the asymmetry that can be
generated through decays that simultaneously produce dark matter.

In fact, in the following, when studying a concrete incarnation of our decay-
induced freeze-in baryogenesis idea, we will see the following.

  * •

Since we will be starting with non-self-conjugate initial states $F_{i}$
(Dirac fermions), $CPT$ conservation and unitarity impose the existence of
multiple decay channels of the $F_{i}$’s for a non-vanishing $CP$ asymmetry to
be generated. To this goal, we will exploit possible decays of the heavy
fermions into different Standard Model fermions (leptons), i.e. flavor
effects.

  * •

The freeze-in framework will necessitate extremely small values for the
couplings involved in the decay process. The predicted $CP$ violation, being
an effect that arises from the interference of tree-level and one-loop
processes is, then, even further suppressed, which will lead us to consider
resonantly enhanced configurations in self-energy-type diagrams Pilaftsis_2004
; Pilaftsis:2005rv ; Pilaftsis_1999 ; Anisimov_2006 . Therefore, at least two
heavy fermions $F_{i}$ must be added.

  * •

The baryon and/or lepton number need not be violated by the decay processes.
As proposed, e.g. in Dick:1999je , a $CP$ asymmetry can be translated to a
baryon asymmetry by the electroweak sphalerons. If, additionally, the BSM
heavy fermions are immune to the action of sphalerons, in the end a net
asymmetry will be generated in the baryon and lepton sectors.

##  3 A concrete realization

Let us now elaborate the previous considerations through a concrete, simple
model. We extend the SM by two heavy vectorlike leptons $F_{i}$ which are
singlets under $SU(3)_{\text{c}}\times SU(2)_{\operatorname{L}}$ but carry
hypercharge and a real gauge-singlet scalar $S$ which is our freeze-in DM
candidate. We moreover impose a discrete $Z_{2}$ symmetry on the Lagrangian,
under which all exotic states are taken to be odd while the SM particles are
even. Under these assumptions, the Lagrangian reads

$\mathcal{L}=\mathcal{L}_{\text{SM}}+\mathcal{L}_{S}+\mathcal{L}_{SF}$ (2)

where $\mathcal{L}_{\text{SM}}$ is the SM Lagrangian,

$\mathcal{L}_{S}=\partial_{\mu}S~{}\partial^{\mu}S-\frac{\mu_{S}^{2}}{2}S^{2}+\frac{\lambda_{S}}{4!}S^{4}+\lambda_{Sh}S^{2}\left(H^{\dagger}H\right)$
(3)

describes interactions of dark matter with itself and with the Standard Model
Higgs doublet and

$\displaystyle\mathcal{L}_{SF}\,=\,\sum_{i}\left(\bar{F}_{i}\left(i\not{D}\right)F_{i}-M_{i}\bar{F_{i}}F_{i}\right)$
$\displaystyle-\,\sum_{\alpha,i}\left(\lambda_{\alpha
i}\,S\,\bar{F}_{i}\,\operatorname{P_{R}}\,e_{\alpha}\,+\,\lambda_{\alpha
i}^{*}\,S\,\bar{e}_{\alpha}\,\operatorname{P_{L}}\,F_{i}\right)$ (4)

where $e_{\alpha}$ are the right-handed SM charged leptons of flavor
$\alpha=\\{e,\mu,\tau\\}$ and $\lambda_{\alpha i}$ denote the feeble
couplings. The heavy fermions $F_{i}$ are assumed to carry the same lepton
number with the SM leptons, so their interactions are lepton number
conserving. Note that, without loss of generality, we have neglected potential
off-diagonal couplings among the heavy fermions $F_{i}$. For simplicity in
what follows, we will also set the Higgs portal coupling, $\lambda_{Sh}$, to
zero.

The tree-level proper decay width $\Gamma_{F_{i}\rightarrow
e_{\alpha}\,S}\big{|}_{0}$ in the limit $M_{i}\gg m_{e_{\alpha}}+m_{S}$ is
given by

$\Gamma_{F_{i}\rightarrow
e_{\alpha}\,S}\big{|}_{0}\,\simeq\,\frac{\left|\lambda_{\alpha
i}\right|^{2}}{16\pi\,g_{F_{i}}}\,M_{i}$ (5)

where $g_{F_{i}}=2$ are the internal degrees of freedom of species $F_{i}$.
The equilibrium decay rate density $\gamma_{F_{i}\rightarrow e_{\alpha}\,S}$
evaluated in the decaying particle rest frame is Hall:2009bx

$\displaystyle\gamma_{F_{i}\rightarrow e_{\alpha}\,S}\,$
$\displaystyle\equiv\,\int\text{d}\Pi_{F_{i}}\,\text{d}\Pi_{e_{\alpha}}\,\text{d}\Pi_{S}\,\left(2\pi\right)^{4}\,\delta^{(4)}\left(p_{F_{i}}-p_{e_{\alpha}}-p_{S}\right)\,f^{\text{eq}}_{F_{i}}\,\left|\mathcal{M}\right|^{2}_{F_{i}\rightarrow
e_{\alpha}\,S}$
$\displaystyle=\,\frac{g_{F_{i}}}{2\pi^{2}}\,M_{i}^{2}\,\Gamma_{F_{i}\rightarrow
e_{\alpha}\,S}\,T\,K_{1}\left(\frac{M_{i}}{T}\right)$ (6)

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 1: Feynman diagrams that contribute to the generation of the baryon
asymmetry.

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 $k$,
$\left|\mathcal{M}\right|^{2}$ denotes the squared matrix element summed, but
not averaged, over the internal degrees of freedom of the initial states,
$K_{1}$ is the modified Bessel function of the second kind of order one and
the distribution function of $F_{i}$ approximately follows the Maxwell-
Boltzmann distribution, $f_{F_{i}}^{\text{eq}}=e^{-E_{F_{i}}/T}$. The
corresponding thermally averaged equilibrium decay rate density
$\Braket{\gamma_{F_{i}\rightarrow e_{\alpha}\,S}}$ reads

$\Braket{\gamma_{F_{i}\rightarrow
e_{\alpha}\,S}}\,\equiv\,\frac{\gamma_{F_{i}\rightarrow
e_{\alpha}\,S}}{n_{F_{i}}^{\text{eq}}}\,=\,\frac{K_{1}\left(M_{i}/T\right)}{K_{2}\left(M_{i}/T\right)}\,\Gamma_{F_{i}\rightarrow
e_{\alpha}\,S}\,\simeq\,\left\\{\,\begin{array}[]{@{}ll@{}}\frac{M_{i}}{2T}\,\Gamma_{F_{i}\rightarrow
e_{\alpha}\,S},&T\,\gg\,M_{i}\\\ \\\ \Gamma_{F_{i}\rightarrow
e_{\alpha}\,S},&T\,\ll\,M_{i}\end{array}\right.$ (7)

where $n_{F_{i}}^{\text{eq}}$ is the equilibrium number density of $F_{i}$
assuming zero chemical potential

$n_{F_{i}}^{\text{eq}}\,\equiv\,g_{F_{i}}\int\frac{\text{d}^{3}\textbf{p}}{\left(2\pi\right)^{3}}\,f_{F_{i}}^{\text{eq}}\,=\,\frac{g_{F_{i}}}{2\pi^{2}}\,M_{i}^{2}\,T\,K_{2}\left(\frac{M_{i}}{T}\right)$
(8)

The time dilation factor $K_{1}\left(M_{i}/T\right)/K_{2}\left(M_{i}/T\right)$
implies that decays are inhibited at temperatures higher than the decaying
state mass $M_{i}$.

The dominant $2\leftrightarrow 2$ scattering processes modifying the abundance
of $S$ are those which involve the $U(1)_{Y}$ gauge boson as an external
state, i.e. $F_{i}B\leftrightarrow e_{\alpha}S$, $F_{i}S\leftrightarrow
e_{\alpha}B$ and $F_{i}\bar{e}_{\alpha}\leftrightarrow SB$, which will be
henceforth referred to as gauge scatterings. The corresponding matrix elements
depend on the product of a feeble and a gauge coupling, whereas all other
scattering processes involve higher powers of feeble couplings and are
therefore subleading. All relevant Feynman diagrams are shown in Fig. 1.

Figure 2: Tree-level decay and gauge scattering rate densities involving
$F_{1}$ and $e_{e}$ as external states, normalized to the Hubble parameter $H$
and the equilibrium number density of photons $n_{\gamma}$. The feeble
coupling of the interactions is $\lambda_{e1}\simeq 2.15\times 10^{-8}$ and
the densities have been evaluated at tree-level and in the limit $M_{1}\gg
m_{e_{\alpha}}+m_{S}$.

The equilibrium interaction rate density of a generic scattering process
$ab\rightarrow cd$ is Gondolo:1990dk

$\displaystyle\gamma_{ab\rightarrow cd}\,$
$\displaystyle\equiv\,\int\text{d}\Pi_{a}\,\text{d}\Pi_{b}\,\text{d}\Pi_{c}\,\text{d}\Pi_{d}\,\left(2\pi\right)^{4}\,\delta^{(4)}\left(p_{a}+p_{b}-p_{c}-p_{d}\right)\,f^{\text{eq}}_{a}\,f^{\text{eq}}_{b}\,\left|\mathcal{M}\right|^{2}_{ab\rightarrow
cd}$
$\displaystyle=\,\frac{T}{512\pi^{6}}\,\int_{\tilde{s}_{\text{min}}}^{\infty}\,\text{d}\tilde{s}\,\frac{1}{\sqrt{\tilde{s}}}\,\left|\textbf{p}_{\text{in}}\right|\,\left|\textbf{p}_{\text{fin}}\right|\,K_{1}\left(\frac{\sqrt{\tilde{s}}}{T}\right)\int\,\text{d}\Omega\,\left|\mathcal{M}\right|^{2}_{ab\rightarrow
cd}$ (9)

where $\textbf{p}_{\text{in}}$, $\textbf{p}_{\text{fin}}$ and
$\sqrt{\tilde{s}}$ are the initial and final momenta and the energy in the
center-of-momentum frame respectively, with
$\tilde{s}_{\text{min}}=\max\big{\\{}\left(m_{a}+m_{b}\right)^{2}\,,\,\left(m_{c}+m_{d}\right)^{2}\big{\\}}$.
In Fig. 2, we present the decay and gauge scattering rate densities involving
$F_{1}$ and the right-handed electron $e_{e}$ as external states in terms of
the dimensionless parameter $z\equiv M_{1}/T$. For the feeble coupling, we use
the value $\lambda_{e1}=2.145\times 10^{-8}$ and work at lowest order in
perturbation theory and in the limit $M_{1}\gg m_{e_{\alpha}}+m_{S}$. In order
to study their effect on the Boltzmann equations, it is convenient to
normalize them to the Hubble parameter $H$ and to the number density of
photons $n_{\gamma}=2\zeta(3)T^{3}/\pi^{2}$, where $\zeta$ is the Riemann zeta
function Kolb:1990vq . The integrals appearing in the various scattering rate
densities have been computed with the Cuba numerical library Cuba_2005 .

The s-channel resonance that appears in the scattering $F_{i}B\leftrightarrow
e_{\alpha}S$ has been regularized by the finite decay width of $F_{i}$. Also,
the IR-induced resonances due to the exchange of massless SM leptons that
appear in the u-channel of $F_{i}B\leftrightarrow e_{\alpha}S$ and in the
t-channel of $F_{i}\bar{e}_{\alpha}\leftrightarrow SB$ have both been
regulated by the thermal mass of the right-handed SM leptons
$m_{e}^{2}\left(T\right)=g_{Y}^{2}T^{2}/8$ (see Cline_1994  and references
therein). Note that, as thermal effects are irrelevant at the temperature of a
few $\operatorname{TeV}$ that is of interest to us, we include the thermal
mass only when it acts as a regulator of the IR-divergences. For a few
examples in which finite temperature effects can become important cf e.g.
Baker:2017zwx ; Dvorkin:2019zdi ; Darme:2019wpd .

###  3.1 Out-of-equilibrium decays

As we already mentioned, one of the crucial ingredients of our setup is that
all processes involving dark matter (and, for that matter, $CP$ violation)
must never attain chemical equilibrium. In order to fulfill this condition, an
upper bound on the magnitude of the freeze-in couplings can be obtained by
requiring the total thermally averaged decay rate to be smaller than the
Hubble expansion rate at the characteristic temperature $T=M_{i}$. Because of
the feeble nature of the $\lambda_{\alpha i}$ couplings, successful
baryogenesis requires a resonant enhancement of the asymmetry, which, in turn,
implies that the masses of the two heavy fermions have to be very close to
each other. Then, for $M_{1}\simeq M_{2}$ the out-of-equilibrium condition
reads

$\Big{(}\sum_{\alpha,i}\Braket{\gamma_{F_{i}\rightarrow
e_{\alpha}\,S}}\,\lesssim\,H\,\Big{)}\big{|}_{T\,=\,M_{1}}$ (10)

The Hubble parameter is given by

$H\left(T\right)\,\simeq\,\frac{1.66\,\sqrt{g_{*\rho}}}{M_{Pl}}\,T^{2}$ (11)

where $M_{Pl}\simeq 1.22\times 10^{19}\,\operatorname{GeV}$ is the Planck mass
and $g_{*\rho}\simeq 106.75$ are the effective degrees of freedom related to
the energy density. At leading order and in the limit $M_{i}\gg
m_{e_{\alpha}}+m_{S}$, the out-of-equilibrium condition (10) reads
($g_{F_{i}}=2$)

$\sum_{\alpha,i}\Gamma_{F_{i}\rightarrow
e_{\alpha}S}\,\lesssim\,2H\,\big{|}_{T\,=\,M_{1}}\;\Rightarrow\;\sum_{\alpha,i}\left|\lambda_{\alpha
i}\right|^{2}\,\lesssim\,2.83\,\times\,10^{-13}\,\left(\frac{M_{1}}{\operatorname{TeV}}\right)$
(12)

Thus, for heavy leptons at the $\operatorname{TeV}$ range, the couplings have
to be smaller than $\sim 10^{-7}$ for the decays to proceed out-of-
equilibrium.

###  3.2 Freeze-in DM abundance

The freeze-in DM abundance $Y_{S}$ that is produced from decays and
$2\rightarrow 2$ scattering processes in our model follows the Boltzmann
equation

$\displaystyle s\,\frac{\text{d}Y_{S}}{\text{d}t}\,$
$\displaystyle=\,\sum_{\alpha,i}\left\\{F_{i}\,\leftrightarrow
e_{\alpha}S\right\\}\,+\,\sum_{\alpha,i}\left\\{F_{i}B\leftrightarrow
e_{\alpha}S\right\\}\,-\,\sum_{\alpha,i}\left\\{F_{i}S\leftrightarrow
e_{\alpha}B\right\\}\,+\,\sum_{\alpha,i}\left\\{F_{i}\bar{e}_{\alpha}\leftrightarrow
SB\right\\}$
$\displaystyle+\,2\sum_{i,j}\left\\{F_{i}\bar{F}_{j}\leftrightarrow
SS\right\\}\,+\,2\sum_{\alpha,\beta}\left\\{\bar{e}_{\alpha}e_{\beta}\leftrightarrow
SS\right\\}\ .$ (13)

In writing (3.2), we have used the notations

$\displaystyle\left\\{a\,b\leftrightarrow c\,d\right\\}\,$
$\displaystyle\equiv\,\left(a\,b\leftrightarrow
c\,d\right)\,+\,\left(\bar{a}\,\bar{b}\leftrightarrow\bar{c}\,\bar{d}\right)$
(14a) $\displaystyle\left[a\,b\leftrightarrow c\,d\right]\,$
$\displaystyle\equiv\,\left(a\,b\leftrightarrow
c\,d\right)\,-\,\left(\bar{a}\,\bar{b}\leftrightarrow\bar{c}\,\bar{d}\right)$
(14b) $\displaystyle\left(a\,b\leftrightarrow c\,d\right)\,$
$\displaystyle\equiv\,\int\text{d}\Pi_{a}\text{d}\Pi_{b}\text{d}\Pi_{c}\text{d}\Pi_{d}\left(2\pi\right)^{4}\delta^{(4)}\Big{[}\left|\mathcal{M}\right|^{2}_{ab\rightarrow
cd}f_{a}f_{b}\left(1\pm f_{c}\right)\left(1\pm f_{d}\right)$
$\displaystyle\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\,\left|\mathcal{M}\right|^{2}_{cd\rightarrow
ab}f_{c}f_{d}\left(1\pm f_{a}\right)\left(1\pm f_{b}\right)\Big{]}$ (14c)

where $\delta^{\left(4\right)}$ is an abbreviation for
$\delta^{\left(4\right)}\left(p_{a}+p_{b}-p_{c}-p_{d}\right)$, $f_{k}$ is the
distribution function of species $k$ and $s$ denotes the entropy density. We
will make the following assumptions:

  * •

The initial DM abundance is zero. Combined with the feeble couplings, this
allows us to ignore the inverse decays $e_{\alpha}S\rightarrow F_{i}$, i.e.
$f_{S}\simeq 0$.

  * •

The DM production takes place during the radiation dominated era. At this
epoch, time and temperature are related by $\dot{T}\simeq-HT$, which is valid
for $\partial g_{*\rho}/\partial T\simeq 0$. Using this relation, we may
switch variables and write

$s\frac{\text{d}}{\text{d}t}Y_{S}\,=\,sHz\,\frac{\text{d}}{\text{d}z}Y_{S}$
(15)

The entropy density during the radiation dominated era is given by

$s\,=\,\frac{2\pi^{2}}{45}\,g_{*s}\,T^{3}$ (16)

where $g_{*s}$ are the effective degrees of freedom with respect to the
entropy density.

  * •

The distribution functions of the visible sector species obey the Maxwell-
Boltzmann statistics, i.e. we neglect Bose-enhancement and Pauli-blocking
factors. Hence, we can write, in general,

$\left(a\,b\leftrightarrow c\,d\right)\,=\,\gamma_{ab\rightarrow
cd}\,\frac{Y_{a}}{Y_{a}^{\text{eq}}}\,\frac{Y_{b}}{Y_{b}^{\text{eq}}}\,-\,\gamma_{cd\rightarrow
ab}\,\frac{Y_{c}}{Y_{c}^{\text{eq}}}\,\frac{Y_{d}}{Y_{d}^{\text{eq}}}$ (17)

Let us first focus on the heavy lepton decays $F_{i}\rightarrow e_{\alpha}S$
and the corresponding $CP$-conjugate processes, which provide the dominant
contribution to DM production for $z\gtrsim 1$ Hall:2009bx . Under the
aforementioned assumptions, the Boltzmann equation for the DM abundance can be
written as

$\displaystyle sHz\frac{\text{d}}{\text{d}z}Y_{S}\,$
$\displaystyle\simeq\,\sum_{\alpha,i}\left\\{F_{i}\,\leftrightarrow
e_{\alpha}S\right\\}$
$\displaystyle=\,2\,\sum_{\alpha,i}\gamma^{F_{i}}_{e_{\alpha}S}\,\frac{Y_{F_{i}+\bar{F}_{i}}}{Y^{\text{eq}}_{F_{i}+\bar{F}_{i}}}\,+\,\mathcal{O}\left(\epsilon^{2}\right)$
$\displaystyle=\,\frac{g_{F_{1}}}{\pi^{2}}\,M_{1}^{2}\,T\,K_{1}\left(\frac{M_{1}}{T}\right)\sum_{\alpha}\Gamma_{F_{1}\rightarrow
e_{\alpha}S}\,+\,\frac{g_{F_{2}}}{\pi^{2}}\,M_{2}^{2}\,T\,K_{1}\left(\frac{M_{2}}{T}\right)\sum_{\alpha}\Gamma_{F_{2}\rightarrow
e_{\alpha}S}$ (18)

where $\gamma^{F_{i}}_{e_{\alpha}S}$ is the equilibrium decay rate density at
tree-level, $\epsilon$ denotes the $CP$ asymmetry and we have used
$Y_{F_{i}+\bar{F}_{i}}\simeq Y_{F_{i}+\bar{F}_{i}}^{\text{eq}}$.222Gauge
scatterings keep $F_{i}\left(\bar{F}_{i}\right)$ close to thermal equilibrium
down to $z\sim 25$, when they eventually freeze-out. Since the baryon
asymmetry is generated prior to their freeze-out (when sphalerons become
inactive), this is a valid approximation. If we consider $M_{1}\simeq M_{2}$
(resonant case) and substitute the decay width (5), the Hubble parameter (11)
and the entropy density (16) in Eq. (3.2), then at tree-level and in the limit
$M_{i}\gg m_{e_{\alpha}}+m_{S}$ the DM abundance simplifies to

$Y_{S}\left(z\right)\,=\,\frac{45M_{Pl}}{32\times
1.66\,\pi^{5}\,g_{*s}\,\sqrt{g_{*\rho}}}\,\frac{\sum_{\alpha,i}\left|\lambda_{\alpha
i}\right|^{2}}{M_{1}}\,\int_{z_{RH}}^{z}\text{d}z^{\prime}\,{z^{\prime}}^{3}\,K_{1}\left(z^{\prime}\right)$
(19)

where $z_{RH}\equiv M_{1}/T_{RH}$ and we have considered for simplicity that
$\partial g_{*s}/\partial T\simeq 0$. In our analysis, $T_{RH}$ will be set to
$10^{12}\,\operatorname{GeV}$. At the present day $T=T_{0}\simeq 2.73\,$K, so
$T_{0}\ll M_{1}\ll T_{RH}$ and the integral contributes a factor of $3\pi/2$,
yielding

$Y_{S}\left(z_{0}\right)\,=\,\frac{135M_{Pl}}{64\times
1.66\,\pi^{4}\,g_{*s}\,\sqrt{g_{*\rho}}}\,\frac{\sum_{\alpha,i}\left|\lambda_{\alpha
i}\right|^{2}}{M_{1}}$ (20)

The experimentally observed DM abundance is

$Y_{S}\left(z_{0}\right)\,=\,\frac{\Omega_{DM}\,\rho_{c}}{m_{S}\,s_{0}}$ (21)

where $\Omega_{\text{DM}}h^{2}=0.1200\pm 0.0012$, $\rho_{c}\equiv
3H_{0}^{2}/8\pi\,G\simeq 10.537\,h^{2}\,\operatorname{GeV}m^{-3}$ is the
critical density and $s_{0}\simeq 2.9\times 10^{9}\,m^{-3}$ is the entropy
density at the present day 2020 . If we ignore dark matter production through
scattering processes, the DM mass $m_{S}$ can be related to the heavy lepton
mass $M_{1}$ and to the feeble couplings as

$m_{S}\,=\,\frac{64\times
1.66\,\pi^{4}\,g_{*s}\,\sqrt{g_{*\rho}}\,\Omega_{DM}\,\rho_{c}}{135M_{Pl}\,s_{0}}\,\frac{M_{1}}{\sum_{\alpha,i}\left|\lambda_{\alpha
i}\right|^{2}}$ (22)

Lyman-$\alpha$ forest observations can be used in order to extract a lower
bound on the DM mass, the exact value of which depends on the underlying
mechanism of DM genesis. For DM candidates that freeze-out the current bound
is $m_{DM}\gtrsim\left(1.9-5.3\right)\operatorname{keV}$ at $95\%$ C.L.
garzilli2019warm ; Palanque_Delabrouille_2020 ; Ir_i__2017 . This limit can be
mapped onto the case of freeze-in-produced DM yielding
$m_{S}\gtrsim\left(4-16\right)\operatorname{keV}$ Ballesteros_2021 ;
Boulebnane_2018 ; DEramo:2020gpr ; Decant:2021mhj . This, in turn, imposes an
upper limit on the feeble couplings

$\sum_{\alpha,i}\left|\lambda_{\alpha i}\right|^{2}\,\lesssim\,7.55\times
10^{-16}\left(\frac{M_{1}}{\operatorname{TeV}}\right)$ (23)

in order not to overclose the Universe, where we have used the more
conservative bound $m_{S}\gtrsim 4\operatorname{keV}$. Thus, we see that the
Lyman-$\alpha$ forest sets a more severe constraint than the out-of-
equilibrium condition of Eq. (10), forcing the feeble couplings to lie in the
$10^{-8}$ range and below for heavy lepton masses at the $\operatorname{TeV}$
scale.

For a more rigorous treatment one should take into account the impact of
scattering processes, which can modify the predicted DM abundance at
$z\lesssim 1$, an epoch when scatterings are dominant. Including such
scattering processes will be essential for the calculation of the baryon
asymmetry presented in the next section. We focus only on scattering processes
which involve gauge bosons as external states and ignore the subleading ones
(second row of Eq. (3.2)). In this case, the Boltzmann equation takes the form

$\displaystyle sHz\,\frac{\text{d}Y_{S}}{\text{d}z}\,$
$\displaystyle=\,\sum_{\alpha,i}\left\\{F_{i}\,\leftrightarrow
e_{\alpha}S\right\\}\,+\,\sum_{\alpha,i}\left\\{F_{i}B\leftrightarrow
e_{\alpha}S\right\\}\,-\,\sum_{\alpha,i}\left\\{F_{i}S\leftrightarrow
e_{\alpha}B\right\\}\,+\,\sum_{\alpha,i}\left\\{F_{i}\bar{e}_{\alpha}\leftrightarrow
SB\right\\}$
$\displaystyle=\,2\,\sum_{\alpha,i}\Big{(}\gamma^{F_{i}}_{e_{\alpha}S}\,+\,\gamma^{F_{i}B}_{e_{\alpha}S}\,+\,\gamma^{F_{i}\bar{e}_{\alpha}}_{SB}\,+\,\gamma^{F_{i}S}_{e_{\alpha}B}\Big{)}\,+\,\mathcal{O}\left(\epsilon^{2}\right)$
(24)

Figure 3: DM abundance generated solely by decays, as well as by decays and
scatterings for heavy leptons masses $M_{1}\simeq M_{2}=1.2\operatorname{TeV}$
and couplings $\sum_{\alpha,i}\left|\lambda_{\alpha i}\right|^{2}\simeq
6.25\times 10^{-16}$.

where we have made use of the freeze-in approximation $f_{S}\simeq 0$ and
considered $Y_{F_{i}+\bar{F}_{i}}\simeq Y_{F_{i}+\bar{F}_{i}}^{\text{eq}}$. In
Figure 3 we fix the masses of the heavy leptons $M_{1}\simeq
M_{2}=1.2\operatorname{TeV}$ and the couplings
$\sum_{\alpha,i}\left|\lambda_{\alpha i}\right|^{2}\simeq 6.25\times
10^{-16}$, and we compare the predicted DM abundance as estimated if we take
into account only decays of the heavy leptons (lower, green line) with the
result obtained through a full-blown numerical solution of Eq. (3.2) (upper,
yellow line). We observe that, as expected, the inclusion of the scattering
processes does not drastically modify the predicted amount of dark matter in
the Universe. Note that we have also cross-checked all of our results by
implementing our model in FeynRules Alloul:2013bka  and computing the
predicted freeze-in DM abundance with CalcHEP/micrOMEGAs 5 Belyaev:2012qa ;
Belanger:2018ccd . Given our findings, we conclude that the analytic estimate
of the DM mass given by Eq. (22) constitutes a reliable approximation. For the
values of the physical parameters used in Figure 3, the value of the dark
matter mass in order to reproduce the observed DM abundance in the Universe is
$m_{S}\simeq 5.81\operatorname{keV}$, which is compatible with the
Lyman-$\alpha$ forest bounds discussed previously.

###  3.3 $CP$ asymmetry

The $CP$ asymmetry generated through the decays of $F_{1,2}$ arises, at lowest
order, due to the interference of the tree-level and 1-loop self-energy
Feynman diagrams (wave-part contribution) as shown in Figures 1(a) and 1(b).
It can be defined in terms of the decay widths as

$\epsilon_{\alpha\,i}\,\equiv\,\frac{\Gamma\left(F_{i}\rightarrow
e_{\alpha}S\right)\,-\,\Gamma\left(\bar{F}_{i}\rightarrow\bar{e}_{\alpha}S\right)}{\sum_{\alpha}\Gamma\left(F_{i}\rightarrow
e_{\alpha}S\right)\,+\,\Gamma\left(\bar{F}_{i}\rightarrow\bar{e}_{\alpha}S\right)}\,=\,\frac{\Gamma\left(F_{i}\rightarrow
e_{\alpha}S\right)\,-\,\Gamma\left(\bar{F}_{i}\rightarrow\bar{e}_{\alpha}S\right)}{2\,\Gamma_{i}}$
(25)

where $\Gamma_{i}$ is the total decay width of $F_{i}$. If we 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. $\left.\mathcal{M}\right|_{k}\equiv\sum_{i=0}^{k}\,c_{i}\mathcal{A}_{i}$,
we can rewrite the $CP$ asymmetry as Davidson:2008bu

$\epsilon_{\alpha\,i}\,=\,-2\,\frac{\text{Im}\left\\{c_{0}c_{1}^{*}\right\\}}{\sum_{\alpha}\left|c_{0}\right|^{2}}\frac{\int\text{d}\Pi_{e_{\alpha}}\text{d}\Pi_{S}\left(2\pi\right)^{4}\delta^{(4)}\text{Im}\left\\{\mathcal{A}_{0}\mathcal{A}_{1}^{*}\right\\}}{\int\text{d}\Pi_{e_{\alpha}}\text{d}\Pi_{S}\left(2\pi\right)^{4}\delta^{(4)}\left|\mathcal{A}_{0}\right|^{2}}$
(26)

The imaginary part of the amplitudes is related to the discontinuity of the
corresponding graph as
$2i\text{Im}\left\\{\mathcal{A}_{0}\mathcal{A}_{1}^{*}\right\\}=\text{Disc}\\{\mathcal{A}_{0}\mathcal{A}_{1}^{*}\\}$,
which can be computed using the Cutkosky cutting rules. For a non-degenerate
$F_{i}$ spectrum, $M_{j}-M_{i}\gg\Gamma_{j}$, we obtain PackageX

$\displaystyle\epsilon_{\alpha
i}\,=\,-\frac{1}{16\pi}\,\frac{1}{1\,-\,x_{j}}\,\frac{\text{Im}\Big{\\{}\lambda_{\alpha
i}^{*}\lambda_{\alpha
j}\big{[}\lambda^{\dagger}\lambda\big{]}_{ji}\Big{\\}}}{\left[\lambda^{\dagger}\lambda\right]_{ii}}$
(27)

where $x_{j}\equiv M_{j}^{2}/M_{i}^{2}$ and we have used the standard notation
$\left[\lambda^{\dagger}\lambda\right]_{ii}\equiv\left|\lambda_{ei}\right|^{2}+\left|\lambda_{\mu
i}\right|^{2}+\left|\lambda_{\tau i}\right|^{2}$ and
$\left[\lambda^{\dagger}\lambda\right]_{ji}\equiv\lambda_{ej}^{*}\lambda_{ei}+\lambda_{\mu
j}^{*}\lambda_{\mu i}+\lambda_{\tau j}^{*}\lambda_{\tau i}$. This is half the
result obtained in standard leptogenesis (see e.g. Eq. (5.13) in
Davidson:2008bu ), because we deal with Dirac fermions where only a charged
lepton propagates in the loop, whereas Majorana neutrinos can decay to both a
charged lepton or a light neutrino accompanied by the Higgs field.

From $CPT$ invariance and unitarity, we know that the total decay width of a
state and its $CP$-conjugate are equal

$\sum_{\alpha}\Gamma\left(F_{i}\rightarrow
e_{\alpha}S\right)\,=\,\sum_{\alpha}\Gamma\left(\bar{F}_{i}\rightarrow\bar{e}_{\alpha}S\right)\,\equiv\,\Gamma_{i}$
(28)

Hence, the $CP$ asymmetry vanishes when summed over the flavors,
$\epsilon_{i}\,\equiv\,\sum_{\alpha}\epsilon_{\alpha\,i}\,=\,0$. This can be
seen by summing Eq. (27) over flavors, a case in which the argument of the
imaginary part becomes real and it vanishes identically:

$\sum_{\alpha}\text{Im}\Big{\\{}\lambda_{\alpha i}^{*}\lambda_{\alpha
j}\big{[}\lambda^{\dagger}\lambda\big{]}_{ji}\Big{\\}}\,=\,\text{Im}\left\\{\big{[}\lambda^{\dagger}\lambda\big{]}_{ij}\big{[}\lambda^{\dagger}\lambda\big{]}_{ji}\right\\}\,=\,0$
(29)

However, flavor effects can lead to a non-vanishing baryon/lepton asymmetry,
since washout processes are flavor dependent and, therefore, the lepton
asymmetries in each flavor are washed out in a different way Abada_2006 ;
Nardi_2006 ; Barbieri_2000 ; Blanchet_2007 .

Because of the feeble nature of the $\lambda_{\alpha i}$ couplings, a resonant
enhancement of the $CP$ asymmetry is needed in order to generate a
sufficiently large baryon asymmetry. This occurs when the mass difference
between the heavy leptons is of the order of their decay widths and is related
to the wave-part contribution to the $CP$ asymmetry (Figure 1(b)). The
resulting $CP$ asymmetry is given by

$\displaystyle\epsilon_{\alpha
i}\,=\,-\frac{1}{16\pi}\,\frac{1\,-\,x_{j}}{\left(1\,-\,x_{j}\right)^{2}\,+\,g_{j}^{2}}\,\frac{\text{Im}\Big{\\{}\lambda_{\alpha
i}^{*}\lambda_{\alpha
j}\big{[}\lambda^{\dagger}\lambda\big{]}_{ji}\Big{\\}}}{\left[\lambda^{\dagger}\lambda\right]_{ii}}$
(30)

where $g_{j}\equiv\Gamma_{j}/M_{i}$. In deriving (30), we have regularized the
divergence for small $x_{j}$ by applying the resummation procedure presented
in Pilaftsis_1999 , with the regulator given by $M_{1}\Gamma_{2}$. Note that a
more complete treatment of the resonance enhancement requires the employment
of non-equilibrium techniques, which yield, in general, different regulators.
Such an analysis has been performed _e.g._ in Dev:2017wwc , where the nearly
degenerate states are considered to be out-of-equilibrium with the bath, in
contrast to our model, where the $F_{i}$’s are kept close to equilibrium and
the required deviation occurs for the DM states. Such an analysis is beyond
the scope of this paper and is left for future work.

If we also express the coupling constants in terms of their magnitude and
phase, $\lambda_{\alpha i}\,=\,\left|\lambda_{\alpha i}\right|e^{i\phi_{\alpha
i}}$, Eq. (30) can be rewritten as

$\displaystyle\epsilon_{\alpha i}\,$
$\displaystyle=\,-\frac{1}{16\pi}\,\frac{1\,-\,x_{j}}{\left(1\,-\,x_{j}\right)^{2}\,+\,g_{j}^{2}}\,\frac{\left|\lambda_{\alpha
i}\right|\left|\lambda_{\alpha
j}\right|}{\big{[}\lambda^{\dagger}\lambda\big{]}_{ii}}\,\sum_{\beta\neq\alpha}\left|\lambda_{\beta
i}\right|\left|\lambda_{\beta j}\right|\sin\left(-\,\phi_{\alpha
i}\,+\,\phi_{\alpha j}\,-\,\phi_{\beta j}\,+\,\phi_{\beta i}\right)$
$\displaystyle\equiv\,-\frac{1}{16\pi}\,\frac{1\,-\,x_{j}}{\left(1\,-\,x_{j}\right)^{2}\,+\,g_{j}^{2}}\,\frac{\left|\lambda_{\alpha
i}\right|\left|\lambda_{\alpha
j}\right|}{\big{[}\lambda^{\dagger}\lambda\big{]}_{ii}}\,\sum_{\beta\neq\alpha}\left|\lambda_{\beta
i}\right|\left|\lambda_{\beta j}\right|\,p^{ij}_{\alpha\beta}$ (31)

where $p^{ij}_{\alpha\beta}=-p^{ji}_{\alpha\beta}=-p^{ij}_{\beta\alpha}$. The
resonance condition reads $M_{2}-M_{1}\sim\Gamma_{2}/2$ Pilaftsis_1999  and in
this case the $CP$ asymmetry can be maximally enhanced to

$\displaystyle\epsilon_{\alpha 1}^{\text{res}}\,$
$\displaystyle=\,\frac{g_{F_{1}}}{2}\,\frac{\left|\lambda_{\alpha
1}\right|\left|\lambda_{\alpha
2}\right|}{\big{[}\lambda^{\dagger}\lambda\big{]}_{11}\big{[}\lambda^{\dagger}\lambda\big{]}_{22}}\,\sum_{\beta\neq\alpha}\left|\lambda_{\beta
1}\right|\left|\lambda_{\beta 2}\right|\,p^{12}_{\alpha\beta}$ (32a)
$\displaystyle\epsilon_{\alpha 2}^{\text{res}}\,$
$\displaystyle=\,g_{F_{2}}\,\frac{\left|\lambda_{\alpha
1}\right|\left|\lambda_{\alpha
2}\right|}{\big{[}\lambda^{\dagger}\lambda\big{]}_{11}^{2}\,+\,\big{[}\lambda^{\dagger}\lambda\big{]}_{22}^{2}}\,\sum_{\beta\neq\alpha}\left|\lambda_{\beta
1}\right|\left|\lambda_{\beta 2}\right|\,p^{12}_{\alpha\beta}$ (32b)

These are the expressions that we will be using in the numerical analysis that
follows in order to compute $\epsilon_{\alpha i}$.

###  3.4 Baryon asymmetry

In the previous sections, we described how our model can satisfy two out of
the three Sakharov conditions, namely, $C/CP$ violation and out-of-equilibrium
dynamics. The last condition to be fulfilled in order to generate a baryon
asymmetry is the violation of the baryon/lepton number. In the case of
Majorana heavy states, a conserved lepton number cannot be consistently
assigned in the presence of interaction and mass terms in the Lagrangian and,
therefore, $L$ is violated. This is not the case in our model, where the heavy
states are of Dirac nature. In this case, we may rely on sphaleron departure
from equilibrium during the epoch that the lepton asymmetry is generated,
along the lines described in Gonzalez:2009 .

In the model we propose, the heavy leptons $F_{i}$ carry the same lepton
number as the SM leptons and the total lepton asymmetry is
$Y_{L}=Y_{L_{\text{SM}}}+Y_{L_{F}}$, where
$Y_{L_{\text{SM}}}\equiv\sum_{\alpha}Y_{L_{\alpha}}$ and
$Y_{L_{F}}=\sum_{i}Y_{L_{F_{i}}}$. All processes conserve the combination
$Y_{B-L}\equiv Y_{B}-Y_{L_{\text{SM}}}-Y_{L_{F}}$, i.e.
$\text{d}Y_{B-L}/\text{d}z=0$. We also assume that the Universe is initially
totally symmetric, $Y_{B-L_{\text{SM}}}|_{z_{RH}}=Y_{L_{F}}|_{z_{RH}}=0$ and,
therefore, at any $z$ it holds $Y_{B-L_{\text{SM}}}=Y_{L_{F}}$.

Since sphalerons are insensitive to the lepton asymmetry $Y_{L_{F}}$, as
$F_{i}$ are $SU(3)_{\text{c}}\times SU(2)_{\operatorname{L}}$-singlets, they
affect only the non-zero lepton asymmetry stored in the SM lepton sector
$Y_{L_{\text{SM}}}$. In particular, they convert it to a baryon asymmetry by
imposing certain relations among the chemical potentials of the various
species (see the Appendix). Once sphalerons depart from equilibrium, which
occurs at $T_{sph}=131.7\pm 2.3\operatorname{GeV}$ D_Onofrio_2014 , the baryon
and lepton numbers are separately conserved. When the heavy leptons decay
away, the total baryon asymmetry, being proportional to $Y_{B-L_{\text{SM}}}$,
vanishes. However, if sphalerons become inactive during the decay epoch of the
heavy leptons, then the baryon asymmetry freezes at $Y_{B}\propto
Y_{B-L_{\text{SM}}}|_{T_{sph}}$, which, in general, is not null.

Taking into consideration all the decay and $2\rightarrow 2$ scattering
processes, the full Boltzmann equations of the asymmetries read

$\displaystyle-sHz\frac{\text{d}Y_{\Delta F_{i}}}{\text{d}z}\,$
$\displaystyle=\,\sum_{\alpha}\left[F_{i}\,\leftrightarrow
e_{\alpha}S\right]\,+\,\sum_{\alpha}\left[F_{i}B\leftrightarrow
e_{\alpha}S\right]\,+\,\sum_{\alpha}\left[F_{i}S\leftrightarrow
e_{\alpha}B\right]\,+\,\sum_{\alpha}\left[F_{i}\bar{e}_{\alpha}\leftrightarrow
SB\right]$
$\displaystyle+\,\sum_{\alpha,\beta,j}\left[F_{i}\bar{e}_{\alpha}\leftrightarrow\bar{F}_{j}e_{\beta}\right]\,+\,\sum_{\alpha,\beta}\left[F_{i}\bar{e}_{\alpha}\leftrightarrow
F_{j}\bar{e}_{\beta}\right]\,+\,\sum_{\alpha,\beta}\left[F_{i}e_{\beta}\leftrightarrow
F_{j}e_{\alpha}\right]$
$\displaystyle+\,\sum_{\alpha,\beta}\left[F_{i}\bar{F}_{j\neq
i}\leftrightarrow
e_{\alpha}\bar{e}_{\beta}\right]\,+\,\left[F_{i}\bar{F}_{j\neq
i}\leftrightarrow
SS\right]\,+\,\sum_{\alpha,\beta,j}\left[F_{i}F_{j}\leftrightarrow
e_{\alpha}e_{\beta}\right]$ $\displaystyle+\,\left[F_{i}S\leftrightarrow
F_{j\neq i}S\right]$ (33)

$\displaystyle-sHz\frac{\text{d}Y_{\Delta_{\alpha}}}{\text{d}z}\,$
$\displaystyle=\,\sum_{i}\left[F_{i}\,\leftrightarrow
e_{\alpha}S\right]\,+\,\sum_{i}\left[F_{i}B\leftrightarrow
e_{\alpha}S\right]\,+\,\sum_{i}\left[F_{i}S\leftrightarrow
e_{\alpha}B\right]\,+\,\sum_{i}\left[F_{i}\bar{e}_{\alpha}\leftrightarrow
SB\right]$
$\displaystyle+\,\sum_{i,j,\beta}\left[F_{i}\bar{e}_{\alpha}\leftrightarrow\bar{F}_{j}e_{\beta}\right]\,+\,\sum_{i,j,\beta\neq\alpha}\left[F_{i}\bar{e}_{\alpha}\leftrightarrow
F_{j}\bar{e}_{\beta}\right]\,+\,\sum_{i,j,\beta\neq\alpha}\left[F_{i}e_{\beta}\leftrightarrow
F_{j}e_{\alpha}\right]$
$\displaystyle+\,\sum_{i,j,\beta\neq\alpha}\left[F_{i}\bar{F}_{j}\leftrightarrow
e_{\alpha}\bar{e}_{\beta}\right]\,+\,\sum_{\beta\neq\alpha}\left[\bar{e}_{\alpha}e_{\beta}\leftrightarrow
SS\right]\,+\,\sum_{i,j,\beta}\left[F_{i}F_{j}\leftrightarrow
e_{\alpha}e_{\beta}\right]$
$\displaystyle+\sum_{\beta\neq\alpha}\big{[}e_{\beta}S\leftrightarrow
e_{\alpha}S\big{]}^{\prime}$ (34)

where $Y_{\Delta F_{i}}\equiv Y_{F_{i}}-Y_{\bar{F}_{i}}$ and
$Y_{\Delta_{\alpha}}\equiv Y_{B}/3-Y_{L_{\text{SM}{\alpha}}}$. The primed term
indicates that one has to subtract the contribution due to the on-shell
propagation of $F_{i}$, usually referred to as real intermediate state
subtraction (RISS), which is already taken into account by the successive
decays $e_{\alpha}S\leftrightarrow F_{i}\leftrightarrow e_{\alpha}S$.

The various terms in the Boltzmann equations can be expressed in terms of the
$CP$ asymmetry $\epsilon_{\alpha i}$, the tree-level rate densities and the
asymmetric abundances. As is typically done, we linearize in the SM chemical
potentials Kolb:1979qa

$\frac{Y_{e_{\alpha}\left(\bar{e}_{\alpha}\right)}}{Y_{e_{\alpha}}^{\text{eq}}}\,\equiv\,e^{\pm\mu_{e_{\alpha}}/T}\,\simeq\,1\,\pm\,\frac{\mu_{e_{\alpha}}}{T}\,=\,1\,\pm\,\frac{Y_{\Delta
e_{\alpha}}}{2Y_{\gamma}}$ (35)

All non-gauge interactions are subleading in comparison to the gauge
interactions and can be safely discarded. We include only the $CP$-violating
part of the RISS term, which ensures that the source term (proportional to
$\epsilon_{\alpha i}$) takes the correct form, i.e. it vanishes when all
species are in chemical equilibrium. The Boltzmann equations can be rewritten
as333In deriving Eq. (3.2), we dropped dark matter annihilation processes
altogether. In baryo- and leptogenesis, inverse reactions can be crucial and
need to be taken into account. In order to do so in an efficient manner, in
Eqs. (36) and (37) we will approximate $f_{S}\simeq f_{S}^{\rm
eq}\frac{Y_{S}}{Y_{S}^{\rm eq}}$ – a relation which is rigorously applicable
to bath particles. We have checked that – as expected, since we have
restricted ourselves to regions of the parameter space in which the condition
of Eq. (10) is satisfied – this is, indeed, a good approximation which leads
to only a small ($\sim 2\%$) reduction of the predicted dark matter abundance
for large values of $z$.

$\displaystyle-sHz\frac{\text{d}Y_{\Delta F_{i}}}{\text{d}z}\,$
$\displaystyle=\,\sum_{\alpha}\Big{(}y_{F_{i}}\,-\,\frac{Y_{S}}{Y_{S}^{\text{eq}}}\,y_{e_{\alpha}}\Big{)}\,\Big{(}\gamma^{F_{i}}_{e_{\alpha}S}\,+\,\gamma^{F_{i}B}_{e_{\alpha}S}\Big{)}\,+\,\sum_{\alpha}\Big{(}\frac{Y_{S}}{Y_{S}^{\text{eq}}}\,y_{F_{i}}\,-\,y_{e_{\alpha}}\Big{)}\,\gamma^{F_{i}S}_{e_{\alpha}B}$
$\displaystyle+\,\sum_{\alpha}\Big{(}y_{F_{i}}\,-\,y_{e_{\alpha}}\Big{)}\,\gamma^{F_{i}\bar{e}_{\alpha}}_{SB}$
(36) $\displaystyle-sHz\frac{\text{d}Y_{\Delta_{\alpha}}}{\text{d}z}\,$
$\displaystyle=\,2\,\sum_{i}\epsilon_{\alpha
i}\,\Big{[}\,\Big{(}1\,-\,\frac{Y_{S}}{Y_{S}^{\text{eq}}}\Big{)}\,\sum_{\rho}\Big{(}\gamma^{F_{i}}_{e_{\rho}S}\,+\,\gamma^{F_{i}B}_{e_{\rho}S}\,+\,\gamma^{F_{i}\bar{e}_{\rho}}_{SB}\,-\,\gamma^{F_{i}S}_{e_{\rho}B}\Big{)}\Big{]}$
$\displaystyle+\,\sum_{i}\Big{[}y_{F_{i}}\,-\,\frac{Y_{S}}{Y_{S}^{\text{eq}}}\,\sum_{\beta}\left(\text{B}^{F_{i}}_{e_{\beta}S}\,y_{e_{\beta}}\right)\Big{]}\,\gamma^{F_{i}}_{e_{\alpha}S}\,+\,\sum_{i}\Big{(}y_{F_{i}}\,-\,\frac{Y_{S}}{Y^{\text{eq}}_{S}}\,y_{e_{\alpha}}\Big{)}\,\gamma^{F_{i}B}_{e_{\alpha}S}$
$\displaystyle+\,\sum_{i}\Big{(}\frac{Y_{S}}{Y_{S}^{\text{eq}}}\,y_{F_{i}}\,-\,y_{e_{\alpha}}\Big{)}\,\gamma^{F_{i}S}_{e_{\alpha}B}\,+\,\sum_{i}\Big{(}y_{F_{i}}\,-\,y_{e_{\alpha}}\Big{)}\,\gamma^{F_{i}\bar{e}_{\alpha}}_{SB}$
(37)

where $y_{F_{i}}\equiv Y_{\Delta F_{i}}/Y_{F_{i}}^{\text{eq}}$,
$y_{e_{\alpha}}\equiv Y_{\Delta e_{\alpha}}/Y_{\gamma}$,
$\text{B}^{F_{i}}_{e_{\beta}S}$ denotes the branching ratio of the decay
$F_{i}\rightarrow e_{\beta}S$ and we have used $Y_{F_{i}+\bar{F}_{i}}\simeq
Y_{F_{i}+\bar{F}_{i}}^{\text{eq}}$. In deriving the equations above, we have
taken into account that the $CP$ asymmetry in scattering processes stemming
from self-energy diagrams (Figure 1(b)) is always equal to the $CP$ asymmetry
from decays Nardi_2007 . As described in Davidson:2008bu  one must also
include the contributions from the RISS $2\rightarrow 3$ scatterings involving
gauge bosons, in order to obtain the correct form for the scattering source
terms. These processes can, however, be neglected as subleading with regards
to their impact on washout. Note that the source term in the Boltzmann
equation of the asymmetry in $F_{i}$ vanishes as a consequence of $CPT$ and
unitarity, as explained in Section 3.3. Lastly, these equations are not
totally independent, as they are related through the conservation of
$Y_{B-L}$, resulting in $\sum_{i}Y_{\Delta
F_{i}}=\sum_{\alpha}Y_{\Delta\alpha}$.

One can express the asymmetries in each SM flavor $y_{e_{\alpha}}$ in terms of
$Y_{\Delta\alpha}$, solving a system of algebraic equations which relate the
chemical potentials and abundances in equilibrium of the various species (see
the Appendix). For the temperatures of interest we find

$\begin{pmatrix}y_{e_{e}}\\\ y_{e_{\mu}}\\\
y_{e_{\tau}}\end{pmatrix}\,=\,-\frac{1}{711}\,\left[\begin{array}[]{ccc}230&-7&-7\\\
-7&230&-7\\\ -7&-7&230\end{array}\right]\,\begin{pmatrix}Y_{\Delta_{e}}\\\
Y_{\Delta_{\mu}}\\\ Y_{\Delta_{\tau}}\end{pmatrix}\frac{1}{Y_{\gamma}}$ (38)

In the same way, we have calculated the amount of the SM lepton asymmetry
converted to baryon asymmetry by the sphaleron transitions to be

$Y_{B}\,=\,\frac{22}{79}\,\sum_{\alpha}Y_{\Delta\alpha}$ (39)

Figure 4: The generated DM abundance $Y_{S}$, the asymmetries
$Y_{\Delta\alpha}$, $Y_{\Delta F_{1}}$ and the baryon asymmetry $Y_{B}$ as a
function of the dimensionless parameter $z=M_{1}/T$, for the heavy leptons
masses and feeble couplings given in Eq. (40). The dashed line illustrates the
evolution that $Y_{B}$ would follow in the absence of sphaleron decoupling,
while the vertical line denotes the temperature at which sphalerons depart
from equilibrium.

The system of the five coupled Boltzmann equations of the asymmetries (36) and
(37) has been solved numerically, using the DM abundance generated by decays
and scattering processes of Eq. (3.2). We confirm that the proposed model can
indeed generate the observed baryon asymmetry $Y_{\text{B}}=8.71\pm 0.06\times
10^{-11}$ 2020  in the resonant regime. As an illustration, we present in
Figure 4 an explicit example, using the following values of the parameters:

$\begin{split}M_{1}\,&=\,1.2\operatorname{TeV},\\\
\left|\lambda_{e1}\right|\,&=\,2.145\times 10^{-8},\\\ \left|\lambda_{\mu
1}\right|\,&=\,\left|\lambda_{\tau 1}\right|\,=\,9\times 10^{-9},\\\
p^{12}_{e\mu}\,&=\,p^{12}_{e\tau}\,=\,p^{12}_{\mu\tau}\,=\,1\end{split}\qquad\begin{split}M_{2}\,&=\,M_{1}\,+\,\Gamma_{2}/2\\\
\left|\lambda_{e2}\right|\,&=\,\left|\lambda_{\tau 2}\right|\,=\,9\times
10^{-10}\\\ \left|\lambda_{\mu 2}\right|\,&=\,8\times 10^{-10}\\\
&\end{split}$ (40)

The corresponding resonant $CP$ asymmetries $\epsilon_{\alpha i}$ turn out to
be: $\epsilon_{e1}\simeq 2.1\times 10^{-1},\,\epsilon_{\mu 1}\simeq-5.7\times
10^{-2},\,\epsilon_{\tau 1}\simeq-1.53\times 10^{-1}$ and $\epsilon_{e2}\simeq
1.53\times 10^{-3},\,\epsilon_{\mu 2}\simeq-4.2\times 10^{-4},\,\epsilon_{\tau
2}\simeq-1.11\times 10^{-3}$. We have explicitly verified that $Y_{B-L}$ is
conserved for all values of $z$ or, equivalently, that the relation
$\sum_{i}Y_{\Delta F_{i}}=\sum_{\alpha}Y_{\Delta\alpha}$ holds. Similarly, the
baryon asymmetry vanishes in the absence of flavor effects, i.e. when
$|\lambda_{e1}|=|\lambda_{\mu 1}|=|\lambda_{\tau 1}|$ and
$|\lambda_{e2}|=|\lambda_{\mu 2}|=|\lambda_{\tau 2}|$.

Note that, contrary to the standard leptogenesis scenarios, the SM flavor
asymmetries $Y_{\Delta\alpha}$ are constantly generated, since the
$CP$-violating processes always occur out-of-equilibrium. They eventually
attain their final value as soon as the DM state freezes-in at $z\sim 3-5$. On
the other hand, as the heavy leptons decay, their asymmetries $Y_{\Delta
F_{i}}$ decrease and eventually vanish at high $z$. The asymmetry in $F_{2}$
is many orders of magnitude smaller than the one in $F_{1}$, due to its
smaller couplings $\lambda_{\alpha 2}\ll\lambda_{\alpha 1}$, and is not shown
in Figure 4. Once the $F_{i}$’s have decayed away, conservation of $Y_{B-L}$
implies that also $Y_{B-L_{SM}}\rightarrow 0$. Thus, the model predicts an
equal amount of baryon $Y_{B}$ and SM lepton $Y_{L_{\text{SM}}}$ asymmetries
left in the Universe Gonzalez:2009 .

Figure 5: Combinations of the feeble couplings and the heavy lepton mass which
can generate the observed baryon asymmetry, for the set of parameters depicted
in Table 1. The dashed lines depict representative DM masses
$m_{S}=\\{2,4,6,16\\}\operatorname{keV}$ for which the observed DM abundance
of the Universe can be reproduced. Masses below $4\operatorname{keV}$ are
excluded from current Lyman-$\alpha$ forest observations (gray shaded area).

Our numerical analysis reveals that at least one of the feeble couplings must
have a magnitude larger than $\sim 10^{-8}$ in order to account for the
observed baryon asymmetry. Our main results are summarized in Figure 5, where
we show the combinations of the feeble couplings and the heavy lepton mass for
which a viable baryon asymmetry can be generated, for the two illustrative
scenarios depicted in Table 1.

| $\left|\lambda_{\mu 1}\right|$ | $\left|\lambda_{\tau 1}\right|$ | $\left|\lambda_{e2}\right|$ | $\left|\lambda_{\mu 2}\right|$ | $\left|\lambda_{\tau 2}\right|$
---|---|---|---|---|---
Scenario 1 | $10^{-9}$ | $2\times 10^{-9}$ | $9\times 10^{-10}$ | $8\times 10^{-10}$ | $9\times 10^{-10}$
Scenario 2 | $9\times 10^{-9}$ | $9\times 10^{-9}$ | $9\times 10^{-10}$ | $8\times 10^{-10}$ | $9\times 10^{-10}$
Table 1: Values of the parameters used for the two scenarios shown in Figure
5.

In both cases, the $CP$ asymmetry is resonantly enhanced, i.e. the mass of
$F_{2}$ is $M_{2}=M_{1}+\Gamma_{2}/2$ and we consider
$p^{12}_{e\mu}=p^{12}_{e\tau}=p^{12}_{\mu\tau}=1$ for the phases of the feeble
couplings. The $\lambda_{e1}$ coupling is being treated as a free parameter
and lies in the $10^{-8}$ ballpark.

The dark matter relic density constraint can be satisfied throughout the
parameter ranges depicted in the figure, for an appropriate choice of the mass
$m_{S}$. However, in the shaded region, the required dark matter mass turns
out to be in conflict with the Lyman-$\alpha$ bound of Eq. (23).

We observe that the first scenario (upper, blue line) is excluded from
Lyman-$\alpha$ constraints for all values of the heavy lepton mass $M_{1}$. On
the contrary, the second scenario (lower, orange line) is able to account for
both the baryon asymmetry and the DM abundance for a wide range of $M_{1}$,
$550\operatorname{GeV}\lesssim M_{1}\lesssim 2800\operatorname{GeV}$. Note
that, interestingly, we find that for any combination of parameter values the
relic density constraint is satisfied for DM masses that do not exceed $\sim
6\operatorname{keV}$. This implies that if Lyman-$\alpha$ constraints become
stronger in the future, they may be able to fully exclude the model’s viable
parameter space.

###  3.5 Phenomenological aspects

In the previous paragraphs, we saw that our simple model, described by the
Lagrangian of Eq. (2), can provide a common explanation for the observed dark
matter content and baryon asymmetry of the Universe. As we showed, this can be
achieved by assuming highly degenerate heavy fermions $F_{i}$ with masses
$M_{i}\sim 1$ TeV and Yukawa-like couplings of the order of $\lambda_{\alpha
1}\sim 10^{-8}-10^{-7}$ and $\lambda_{\alpha 2}\sim 10^{-10}$ for $F_{1}$ and
$F_{2}$, respectively.

Our model is intended to serve mainly as a proof-of-concept for the freeze-in
baryogenesis mechanism that we propose. In this spirit, a full-blown analysis
of its phenomenological predictions goes well beyond the scope of the present
work, especially since alternative constructions assuming, e.g., different
quantum numbers for the various particles involved, can lead to wildly
different phenomenological signatures. Still, despite its simplicity, this
model does exhibit an interesting phenomenology, which we will briefly comment
upon. Note that a simpler variant of this model involving a single vectorlike
heavy fermion $F$ was already studied, e.g., in Belanger:2018sti ;
Brooijmans:2020yij , mainly from the viewpoint of its predicted dark matter
abundance as well as its signatures in new physics searches at the LHC.

First, the heavy fermions can mediate lepton flavor-violating decays of the
type $\ell_{\beta}\rightarrow\ell_{\alpha}\gamma$. In particular, given the
constraint $Br(\mu\rightarrow e\gamma)<4.2\times 10^{-13}$ MEG:2016leq  and
assuming that (as was the case throughout the analysis presented in the
previous sections) $\lambda_{e1}\sim\lambda_{\mu 1}$, the analysis presented
in Brooijmans:2020yij  restricts the product of the Yukawa-like couplings of
$F_{1}$ to the first two generation leptons as

$\frac{\sqrt{\lambda_{e1}\lambda_{\mu 1}}}{M_{1}}\lesssim 3.6\times 10^{-5}\
{\rm GeV}^{-1}$ (41)

which is easily satisfied in the cosmologically relevant part of our parameter
space. In deriving this bound, we have neglected the interference between
Feynman diagrams involving different species of heavy fermions running in the
loop, since the couplings $\lambda_{\alpha 2}\ll\lambda_{\alpha 1}$. Besides,
constraints stemming from measurements of the muon lifetime turn out to be
subleading Brooijmans:2020yij .

On the side of collider searches, let us first point out the fact that, given
the feeble nature of the $\lambda_{\alpha i}$ couplings, the mean proper decay
length of $F_{1}$ is of the order of a few centimeters, whereas that of
$F_{2}$ is of the order of several meters. Given these lifetime ranges, both
$F_{1}$ and $F_{2}$ can be looked for in searches for long-lived particles at
the LHC. In particular, as shown in Belanger:2018sti ; Brooijmans:2020yij ,
searches for displaced leptons accompanied by missing energy can target the
production and decay of $F_{1}$ and, for the values of $\lambda_{\alpha 1}$
that we assume, already exclude $F_{1}$ masses up to $\sim 400-500$ GeV.
$F_{2}$, on the other hand, survives long enough to escape the detector and
can be probed by searches for heavy stable charged particles, with masses
smaller than $\sim 500-600$ GeV being already excluded. The corresponding
searches at the High-Luminosity Run of the LHC will probe $F_{1,2}$ masses
reaching up to $\sim 800$ GeV and $\sim 1.5$ TeV, respectively
Belanger:2018sti .

In summary, if our model is to simultaneously explain the dark matter
abundance and the baryon asymmetry of the Universe while remaining consistent
with Lyman-$\alpha$ bounds, then it predicts interesting signatures for LLP
searches at the LHC. In conjunction with the fact that, as we showed,
explaining the matter-antimatter asymmetry of the Universe tends to favor
$F_{1,2}$ masses lying in the range of a few TeV, we can hope that a
substantial part of the cosmologically favored part of the parameter space
will be probed by the LHC within the next few years.

##  4 Conclusions and Outlook

In this paper, we discussed a mechanism for the simultaneous generation of the
dark matter density and the baryon asymmetry of the Universe. To this goal, we
relied on the out-of-equilibrium decays of heavy bath particles into a (feebly
coupled) dark matter state along with Standard Model charged fermions, which
leads to dark matter production via the freeze-in mechanism. If, moreover,
$CP$ is violated by these same decay processes, a viable matter-antimatter
asymmetry can also be generated either directly (if the decays also lead to
baryon/lepton number violation) or through the interplay of the generated $CP$
asymmetry with the SM electroweak sphalerons.

As a proof-of-concept, we employed a simple model in which the role of the
heavy bath particles was played by two $SU(3)_{c}\times
SU(2)_{\operatorname{L}}$-singlet vectorlike fermions with a non-zero
hypercharge, and dark matter was identified with a gauge singlet real scalar
field. We showed that, indeed, such a simple construction can lead to both
dark matter production and successful baryogenesis and we briefly discussed
the phenomenological prospects of such a construction, particularly in
relation to searches for long-lived particles at the LHC.

Our proposal draws inspiration from the idea presented in Shuve_2020 : First,
the dark matter content and the baryon asymmetry of the Universe are generated
simultaneously through the freeze-in mechanism. Second, at least within the
framework of our concrete incarnation of this proposal, in both cases the
electroweak sphalerons are used in order to convert a $CP$ asymmetry into a
baryonic one. In our case, however, instead of considering oscillations – as
in ARS leptogenesis – we rather relied on decays in order to generate this
initial $CP$ asymmetry – a process which is somewhat reminiscent of GUT
baryogenesis. Moreover, we presented a general thermodynamical treatment that
can also be applied in cases in which baryon/lepton number is violated already
at the decay level.

There are several topics which we did not expand upon in this paper, and which
would merit much more detailed investigation. First, more elaborate models can
certainly be developed in this context of freeze-in baryogenesis, potentially
establishing connections with theoretically well-motivated UV completions of
the Standard Model. As an example, we could mention the fact that we assumed
rather ad hoc values for the couplings of the heavy bath particles to dark
matter and to the SM fermions. More realistic flavor structures can be
envisaged in quite a straightforward manner, whereas nothing forbids dark
matter itself to be asymmetric, along the lines described in Hall:2010jx .
Second, albeit related to the previous point, our discussion of more
phenomenological aspects has been fairly elementary. Concrete, well-motivated
constructions can have important implications for flavor physics (e.g. flavor-
violating decays of or contributions to electric/magnetic moments of SM
fermions), collider physics (e.g. new resonances and/or final states in LHC
searches, potentially involving long-lived particles Alimena:2019zri ), or
cosmological measurements (e.g. direct detection experiments or cosmic
microwave background observations). Such considerations, which are typically
fairly model dependent, are left for future work.

###### Acknowledgements.

The authors acknowledge fruitful exchanges with Geneviève Bélanger, Marcos A.
G. Garcia, Kalliopi Petraki and Alexander Pukhov. We are particularly grateful
toward Sacha Davidson for enlightening discussions and comments on an earlier
version of this manuscript and toward Dimitrios Karamitros for useful comments
on our findings. 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). This research work 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).

##  Appendix A Spectator processes

During the baryon and/or lepton number violation era, there are various
processes which can modify the number densities of the particles species. They
are called ”spectator processes” because they affect the baryon/lepton number
indirectly by distributing the asymmetry among all SM fermions and Higgs
fields. These processes include the gauge interactions, the Yukawa
interactions and the electroweak and QCD non-perturbative sphaleron
transitions. If these processes are in thermal equilibrium with the cosmic
plasma, then their effect is to impose certain relations among the chemical
potentials of the various particle species. Recall that the asymmetry in the
particle and antiparticle equilibrium number densities of species $k$, denoted
as $n_{\Delta k}\equiv n_{k}-n_{\bar{k}}$, is given in the ultra-relativistic
limit $m_{k}\ll T$ and $\mu_{k}\ll T$ by

$\displaystyle n_{\Delta
k}\,\equiv\,n_{k}-n_{\bar{k}}\,=\,\left\\{\,\begin{array}[]{@{}ll@{}}g_{k}\mu_{k}\frac{T^{2}}{3},&k:\text{boson}\\\
\\\ g_{k}\mu_{k}\frac{T^{2}}{6},&k:\text{fermion}\end{array}\right.$ (45)

where $\mu_{k}$ is the chemical potential and $g_{k}$ are the independent
degrees of freedom of species $k$. The total baryon $n_{B}$ and lepton $n_{L}$
asymmetries can be written as

$n_{B}\,=\,\sum_{k}\,B_{k}\,n_{\Delta k},\qquad
n_{L}\,=\,\sum_{k}\,L_{k}\,n_{\Delta k}$ (46)

where $B_{k}$ and $\,L_{k}$ are the baryon and lepton numbers of species $k$,
respectively.

In our case, the lepton asymmetry is mainly generated during $T_{EW}\leq T\ll
10^{8}\,\operatorname{GeV}$ when all spectator processes are active, including
the sphaleron processes. These violate the $n_{B+L}\equiv n_{B}+n_{L}$
asymmetry but conserve the orthogonal combination $n_{B-L}\equiv n_{B}-n_{L}$.
A rough estimation is that sphalerons will erase the $n_{B+L}$ asymmetry,
resulting in

$\displaystyle n_{B}\,=\,\frac{n_{B+L}}{2}\,+\,\frac{n_{B-L}}{2}\,$
$\displaystyle\overset{\text{sphalerons}}{\longrightarrow}\,n_{B}\,=\,\frac{n_{B-L}}{2}$
(47a) $\displaystyle n_{L}\,=\,\frac{n_{B+L}}{2}\,-\,\frac{n_{B-L}}{2}\,$
$\displaystyle\overset{\text{sphalerons}}{\longrightarrow}\,n_{L}\,=\,-n_{B}\,=\,-\frac{n_{B-L}}{2}$
(47b)

A more careful treatment shows that, in spite of the fact that sphalerons
violate $n_{B+L}$, thermodynamic equilibrium requires a non-zero value, i.e.
$n_{B+L}\neq 0$. To quantify the asymmetry distribution, one can always relate
the $n_{B-L}$ number density asymmetry with $n_{B}$ and $n_{L}$ as follows:

$n_{B-L}\,\equiv\,n_{B}\,-\,n_{L}\,=\,c\,n_{B-L}\,-\,\left(c-1\right)\,n_{B-L}\,\Rightarrow\,\left\\{\,\begin{array}[]{@{}ll@{}}n_{B}\,=\,c\,n_{B-L}\\\
\\\ n_{L}\,=\,\left(c-1\right)\,n_{B-L}\end{array}\right.$ (48)

where $c\lesssim 1$ is the constant of proportionality. To evaluate it, we
assign chemical potentials to the particle spectrum of the model. This
consists of all the SM species with three families and one Higgs doublet,
together with the two BSM vectorlike fermions $F_{i}$.

####  A.0.1 SM

Let us first consider only the SM spectrum and assign the following non-
vanishing chemical potentials Harvey:1990qw ; Buchmuller:2005eh

$\displaystyle\mu_{q_{\alpha}}\qquad$ $\displaystyle\left(\text{LH quark
doublets}\right)$ (49a) $\displaystyle\mu_{u_{\alpha}}\qquad$
$\displaystyle\left(\text{RH up-quark singlets}\right)$ (49b)
$\displaystyle\mu_{d_{\alpha}}\qquad$ $\displaystyle\left(\text{RH down-quark
singlets}\right)$ (49c) $\displaystyle\mu_{{\ell}_{\alpha}}\qquad$
$\displaystyle\left(\text{LH lepton doublets}\right)$ (49d)
$\displaystyle\mu_{e_{\alpha}}\qquad$ $\displaystyle\left(\text{RH charged
lepton singlets}\right)$ (49e) $\displaystyle\mu_{H}\qquad$
$\displaystyle\left(\text{Higgs
doublet,}\,H=\left(h^{+},h^{0}\right)^{T}\right)$ (49f)

where $\alpha=\\{1,2,3\\}$ is a SM flavor index.444Note that in this Appendix
we are introducing a slightly different notation with respect to the rest of
the paper, denoting the SM right-handed leptons by $\\{e_{1},e_{2},e_{3}\\}$
instead of $\\{e_{e},e_{\mu},e_{\tau}\\}$. The corresponding antiparticle
states have opposite chemical potentials, e.g. $\mu_{\tilde{H}}=-\mu_{H}$,
where $\tilde{H}=\left(h^{0*},-h^{-}\right)^{T}$. Note that the electrically
chargeless gauge bosons have vanishing chemical potentials, as they carry no
conserved quantum number, while the chemical potential of the $W^{\pm}$ gauge
bosons vanishes, because the third - component of the weak isospin is zero at
$T>T_{EW}$ Harvey:1990qw . Hence, the number density asymmetries of the
various components can be written, according to Eq. (45), as

$\displaystyle n_{\Delta q_{\alpha}}\,$
$\displaystyle\equiv\,n_{q_{\alpha}}\,-\,n_{\bar{q}_{\alpha}}\,=\,6\,\mu_{q_{\alpha}}\,\frac{T^{2}}{6}$
(50a) $\displaystyle n_{\Delta u_{\alpha}}\,$
$\displaystyle\equiv\,n_{u_{\alpha}}\,-\,n_{\bar{u}_{\alpha}}\,=\,3\,\mu_{u_{\alpha}}\,\frac{T^{2}}{6}$
(50b) $\displaystyle n_{\Delta d_{\alpha}}\,$
$\displaystyle\equiv\,n_{d_{\alpha}}\,-\,n_{\bar{d}_{\alpha}}\,=\,3\,\mu_{d_{\alpha}}\,\frac{T^{2}}{6}$
(50c) $\displaystyle n_{\Delta{\ell}_{\alpha}}\,$
$\displaystyle\equiv\,n_{{\ell}_{\alpha}}\,-\,n_{\bar{\ell}_{\alpha}}\,=\,2\,\mu_{{\ell}_{\alpha}}\,\frac{T^{2}}{6}$
(50d) $\displaystyle n_{\Delta e_{\alpha}}\,$
$\displaystyle\equiv\,n_{e_{\alpha}}\,-\,n_{\bar{e}_{\alpha}}\,=\,\mu_{e_{\alpha}}\,\frac{T^{2}}{6}$
(50e) $\displaystyle n_{\Delta H}\,$
$\displaystyle\equiv\,n_{H}\,-\,n_{\tilde{H}}\,=\,2\,\mu_{H}\,\frac{T^{2}}{3}\,=\,4\,\mu_{H}\,\frac{T^{2}}{6}$
(50f)

The baryon and lepton number density asymmetries for each flavor
$\left(n_{B_{\alpha}},\,n_{L_{\alpha}}\right)$, as well as the corresponding
total ones $\left(n_{B},\,n_{L}\right)$, can be written in terms of the
chemical potentials in Eq. (49) as follows:

$\displaystyle n_{B_{\alpha}}\,$
$\displaystyle\equiv\,\sum_{k}\,B_{k_{\alpha}}\,n_{\Delta
k_{\alpha}}\,=\,\frac{1}{3}\left(n_{\Delta q_{\alpha}}\,+\,n_{\Delta
u_{\alpha}}\,+\,n_{\Delta
d_{\alpha}}\right)\,=\,\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\right)\,\frac{T^{2}}{6}$
(51a) $\displaystyle n_{B}\,$
$\displaystyle\equiv\,\sum_{\alpha=1}^{3}\,n_{B_{\alpha}}\,=\,\sum_{\alpha=1}^{3}\,\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\right)\,\frac{T^{2}}{6}$
(51b) $\displaystyle n_{L_{\alpha}}\,$
$\displaystyle\equiv\,\sum_{k}\,L_{k_{\alpha}}\,n_{\Delta
k_{\alpha}}\,=\,n_{\Delta\ell_{\alpha}}\,+\,n_{\Delta
e_{\alpha}}\,=\,\left(2\mu_{\ell_{\alpha}}\,+\,\mu_{e_{\alpha}}\right)\,\frac{T^{2}}{6}$
(51c) $\displaystyle n_{L}\,$
$\displaystyle\equiv\,\sum_{\alpha=1}^{3}\,n_{L_{\alpha}}\,=\,\sum_{\alpha=1}^{3}\left(2\mu_{\ell_{\alpha}}\,+\,\mu_{e_{\alpha}}\right)\,\frac{T^{2}}{6}$
(51d)

These 16 SM chemical potentials are not totally independent; they are related
through the processes that attain chemical equilibrium during the era of the
asymmetry generation. At $T\ll 10^{8}$ these are

  * •

$SU(2)_{L}$ sphaleron transitions which induce an effective 12-fermion
effective operator
$\mathcal{O}_{(B-L)_{\text{SM}}}\,=\,\prod_{i=1}^{N_{f}}\left(q_{i}q_{i}q_{i}\ell_{i}\right)$,
which implies

$\sum_{\alpha=1}^{3}\left(3\mu_{q_{\alpha}}\,+\,\mu_{\ell_{\alpha}}\right)\,=\,0$
(52)

  * •

$SU(3)_{c}$ sphaleron transitions which give rise to

$\sum_{\alpha=1}^{3}\left(2\mu_{q_{\alpha}}\,-\,\mu_{u_{\alpha}}\,-\,\mu_{d_{\alpha}}\right)\,=\,0$
(53)

  * •

Yukawa interactions which imply

$\displaystyle\mu_{q_{\alpha}}\,-\,\mu_{H}\,-\,\mu_{d_{\alpha}}\,=\,0,\qquad$
$\displaystyle\left(\,q_{\alpha}\,\tilde{H}\,\bar{d}_{\alpha}\,+\,\text{h.c.}\,\right)$
(54a)
$\displaystyle\mu_{q_{\alpha}}\,+\,\mu_{H}\,-\,\mu_{u_{\alpha}}\,=\,0,\qquad$
$\displaystyle\left(\,q_{\alpha}\,H\,\bar{u}_{\alpha}\,+\,\text{h.c.}\,\right)$
(54b)
$\displaystyle\mu_{{\ell}_{\alpha}}\,-\,\mu_{H}\,-\,\mu_{e_{\alpha}}\,=\,0,\qquad$
$\displaystyle\left(\,{\ell}_{\alpha}\,\tilde{H}\,\bar{e}_{\alpha}\,+\,\text{h.c.}\,\right)$
(54c)

Out of the 11 constraints of Eqs. (52) - (54), only ten of them are linearly
independent, as the QCD sphaelron-induced relation (53) can be obtained by
adding Eqs. (54a) and (54b). An additional independent constraint can be
obtained from the hypercharge $\left(\Upsilon=Q-t_{3}\right)$ neutrality
condition, which implies

  * •

Hypercharge constraint

$\displaystyle n_{\Upsilon}\,$
$\displaystyle\equiv\,\sum_{k}\,\Upsilon_{k}\,n_{\Delta k}\,=\,0$
$\displaystyle\Rightarrow\,\sum_{\alpha=1}^{3}\Big{(}\frac{1}{6}n_{\Delta
q_{\alpha}}\,+\,\frac{2}{3}n_{\Delta u_{\alpha}}\,-\,\frac{1}{3}n_{\Delta
d_{\alpha}}\,-\,\frac{1}{2}\,n_{\Delta\ell_{\alpha}}\,-\,n_{\Delta
e_{\alpha}}\Big{)}\,+\,\frac{1}{2}n_{\Delta H}\,=\,0$
$\displaystyle\Rightarrow\,\sum_{\alpha=1}^{3}\left(\mu_{q_{\alpha}}\,+\,2\mu_{u_{\alpha}}\,-\,\mu_{d_{\alpha}}\,-\,\mu_{\ell_{\alpha}}\,-\,\mu_{e_{\alpha}}\right)\,+\,2\mu_{H}\,=\,0$
(55)

where $\Upsilon_{k}$ is the hypercharge of species $k$. Two more constraints
are imposed by the equality of the baryon flavor asymmetries Davidson:2008bu .

  * •

Baryon flavor asymmetry equality

$\displaystyle n_{B_{1}}\,=\,n_{B_{2}}\,=\,n_{B_{3}}$
$\displaystyle\Rightarrow\,$ $\displaystyle
2\,\mu_{q_{3}}\,+\,\mu_{u_{3}}\,+\,\mu_{d_{3}}\,=\,2\,\mu_{q_{2}}\,+\,\mu_{u_{2}}\,+\,\mu_{d_{2}}\,=\,2\,\mu_{q_{1}}\,+\,\mu_{u_{1}}\,+\,\mu_{d_{1}}$
(56)

Hence, there are in total 13 independent constraints (Eqs. (52), (54), (• ‣
A.0.1) and (• ‣ A.0.1)) and, therefore, one can express the 16 SM chemical
potentials (49) in terms of three chemical potentials, that we chose to be
$\\{\mu_{\ell_{\alpha}}\\}$. Solving the system of equations, one obtains

$\displaystyle\mu_{q_{\alpha}}\,$
$\displaystyle=\,-\frac{1}{9}\left(\mu_{\ell_{1}}\,+\,\mu_{\ell_{2}}\,+\,\mu_{\ell_{3}}\right)$
(57a) $\displaystyle\mu_{u_{\alpha}}\,$
$\displaystyle=\,\frac{5}{63}\left(\mu_{\ell_{1}}\,+\,\mu_{\ell_{2}}\,+\,\mu_{\ell_{3}}\right)$
(57b) $\displaystyle\mu_{d_{\alpha}}\,$
$\displaystyle=\,-\frac{19}{63}\left(\mu_{\ell_{1}}\,+\,\mu_{\ell_{2}}\,+\,\mu_{\ell_{3}}\right)$
(57c) $\displaystyle\mu_{e_{1}}\,$
$\displaystyle=\,\frac{1}{21}\left(17\mu_{\ell_{1}}\,-\,4\mu_{\ell_{2}}\,-\,4\mu_{\ell_{3}}\right)$
(57d) $\displaystyle\mu_{e_{2}}\,$
$\displaystyle=\,\frac{1}{21}\left(-4\mu_{\ell_{1}}\,+\,17\mu_{\ell_{2}}\,-\,4\mu_{\ell_{3}}\right)$
(57e) $\displaystyle\mu_{e_{3}}\,$
$\displaystyle=\,\frac{1}{21}\left(-4\mu_{\ell_{1}}\,-\,4\mu_{\ell_{2}}\,+\,17\mu_{\ell_{3}}\right)$
(57f) $\displaystyle\mu_{H}\,$
$\displaystyle=\,\frac{4}{21}\left(\mu_{\ell_{1}}\,+\,\mu_{\ell_{2}}\,+\,\mu_{\ell_{3}}\right)$
(57g)

Now the baryon and lepton number density asymmetries can be written in terms
of the chemical potentials $\mu_{\ell_{\alpha}}$ as

$\displaystyle n_{B_{\alpha}}\,$
$\displaystyle=\,\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\right)\,\frac{T^{2}}{6}\,=\,-\frac{4}{9}\left(\mu_{\ell_{1}}\,+\,\mu_{\ell_{2}}\,+\,\mu_{\ell_{3}}\right)\,\frac{T^{2}}{6}$
(58a) $\displaystyle n_{B}\,$
$\displaystyle=\,3\,n_{B_{\alpha}}\,=\,-\frac{4}{3}\left(\mu_{\ell_{1}}\,+\,\mu_{\ell_{2}}\,+\,\mu_{\ell_{3}}\right)\,\frac{T^{2}}{6}$
(58b) $\displaystyle n_{L_{\alpha}}\,$
$\displaystyle=\,\left(2\mu_{\ell_{\alpha}}\,+\,\mu_{e_{\alpha}}\right)\,\frac{T^{2}}{6}$
(58c) $\displaystyle n_{L}\,$
$\displaystyle=\,\sum_{\alpha=1}^{3}\left(2\mu_{\ell_{\alpha}}\,+\,\mu_{e_{\alpha}}\right)\,\frac{T^{2}}{6}\,=\,\frac{17}{7}\left(\mu_{\ell_{1}}\,+\,\mu_{\ell_{2}}\,+\,\mu_{\ell_{3}}\right)\,\frac{T^{2}}{6}$
(58d)

The flavor asymmetry combination $n_{\Delta_{\alpha}}\equiv
n_{B_{\alpha}}-n_{L_{\alpha}}=n_{B}/3-n_{L_{\alpha}}$ can be written as

$\displaystyle n_{\Delta_{\alpha}}\,$
$\displaystyle=\,\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\,-\,2\mu_{\ell_{\alpha}}\,-\,\mu_{e_{\alpha}}\right)\,\frac{T^{2}}{6}$
$\displaystyle\Rightarrow\,\begin{pmatrix}n_{\Delta_{1}}\\\ n_{\Delta_{2}}\\\
n_{\Delta_{3}}\end{pmatrix}\,=\,-\frac{1}{63}\,\left[\begin{array}[]{ccc}205&16&16\\\
16&205&16\\\ 16&16&205\end{array}\right]\,\begin{pmatrix}\mu_{\ell_{1}}\\\
\mu_{\ell_{2}}\\\ \mu_{\ell_{3}}\end{pmatrix}\,\frac{T^{2}}{6}$

Inverting the expression above, one finds

$\begin{pmatrix}\mu_{\ell_{1}}\\\ \mu_{\ell_{2}}\\\
\mu_{\ell_{3}}\end{pmatrix}\,=\,-\frac{1}{711}\,\left[\begin{array}[]{ccc}221&-16&-16\\\
-16&221&-16\\\ -16&-16&221\end{array}\right]\,\begin{pmatrix}n_{\Delta_{1}}\\\
n_{\Delta_{2}}\\\ n_{\Delta_{3}}\end{pmatrix}\,\frac{6}{T^{2}}$ (59)

Using Eq. (50), we can express the chemical potentials $\mu_{\ell_{\alpha}}$
in terms of the flavor asymmetries $n_{\Delta\ell_{\alpha}}$ and relate them
to $n_{\Delta\alpha}$ as 555Note that our result agrees with reference
Nardi_2006 , while the authors of reference Gonzalez:2009  define
$n_{\Delta\ell_{\alpha}}$ per gauge degree of freedom and, therefore, their
result is smaller by a factor of 2.

$\begin{pmatrix}n_{\Delta\ell_{1}}\\\ n_{\Delta\ell_{2}}\\\
n_{\Delta\ell_{3}}\end{pmatrix}\,=\,-\frac{2}{711}\,\left[\begin{array}[]{ccc}221&-16&-16\\\
-16&221&-16\\\ -16&-16&221\end{array}\right]\,\begin{pmatrix}n_{\Delta_{1}}\\\
n_{\Delta_{2}}\\\ n_{\Delta_{3}}\end{pmatrix}$ (60)

The total $B-L$ asymmetry is

$n_{B-L}\,\equiv\,n_{B}\,-\,n_{L}\,=\,-\frac{79}{21}\left(\mu_{\ell_{1}}\,+\,\mu_{\ell_{2}}\,+\,\mu_{\ell_{3}}\right)\,\frac{T^{2}}{6}$
(61)

Hence, in the SM and for $T_{sph}\ll T\ll 10^{8}\,\operatorname{GeV}$, the
total baryon $n_{B}$ and the $n_{L}$ asymmetries are related to $n_{B-L}$ by

$\displaystyle
n_{B}\,=\,\frac{28}{79}\,n_{B-L}\,=\,\frac{28}{79}\,\sum_{\alpha=1}^{3}n_{\Delta\alpha}$
(62) $\displaystyle
n_{L}\,=\,-\frac{51}{79}\,n_{B-L}\,=\,-\frac{51}{79}\,\sum_{\alpha=1}^{3}n_{\Delta\alpha}$
(63)

####  A.0.2 Our model

We extend the SM to include also two heavy vectorlike leptons $F_{1}$ and
$F_{2}$ with chemical potentials $\mu_{F_{1}}$ and $\mu_{F_{2}}$,
respectively, and asymmetry

$n_{\Delta F_{i}}\,=\,\mu_{F_{i}}\,\frac{T^{2}}{6}$ (64)

Since the BSM fermions are gauged under $U(1)_{\Upsilon}$, with
$\Upsilon_{F_{i}}=\Upsilon_{e}=-1$, the hypercharge constraint (• ‣ A.0.1) is
modified to

$\displaystyle n_{\Upsilon}\,$
$\displaystyle\equiv\,\sum_{k}\,\Upsilon_{k}\,n_{\Delta k}\,=\,0$
$\displaystyle\Rightarrow\,\sum_{\alpha=1}^{3}\Big{(}\frac{1}{6}n_{\Delta
q_{\alpha}}\,+\,\frac{2}{3}n_{\Delta u_{\alpha}}\,-\,\frac{1}{3}n_{\Delta
d_{\alpha}}\,-\,\frac{1}{2}\,n_{\Delta\ell_{\alpha}}\,-\,n_{\Delta
e_{\alpha}}\Big{)}\,+\,\frac{1}{2}n_{\Delta H}\,-\,\sum_{i=1}^{2}\,n_{\Delta
F_{i}}\,=\,0$
$\displaystyle\Rightarrow\,\sum_{\alpha=1}^{3}\left(\mu_{q_{\alpha}}\,+\,2\mu_{u_{\alpha}}\,-\,\mu_{d_{\alpha}}\,-\,\mu_{\ell_{\alpha}}\,-\,\mu_{e_{\alpha}}\right)\,+\,2\mu_{H}\,-\,\left(\mu_{F_{1}}\,+\,\mu_{F_{2}}\right)\,=\,0$
(65)

The theory conserves the total $n_{B-L}\equiv
n_{B}-n_{L}=n_{B}-n_{L_{SM}}-n_{L_{F}}$ asymmetry, where

$\displaystyle n_{B_{\alpha}}\,$
$\displaystyle\equiv\,\sum_{k}\,B_{k_{\alpha}}\,n_{\Delta
k_{\alpha}}\,=\,\frac{1}{3}\left(n_{\Delta q_{\alpha}}\,+\,n_{\Delta
u_{\alpha}}\,+\,n_{\Delta
d_{\alpha}}\right)\,=\,\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\right)\,\frac{T^{2}}{6}$
(66a) $\displaystyle n_{B}\,$
$\displaystyle\equiv\,\sum_{\alpha=1}^{3}\,n_{B_{\alpha}}\,=\,\sum_{\alpha=1}^{3}\,\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\right)\,\frac{T^{2}}{6}\,=\,3\,\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\right)\,\frac{T^{2}}{6}$
(66b) $\displaystyle n_{L_{\alpha}}\,$
$\displaystyle\equiv\,\sum_{k:\,SM}\,L_{k_{\alpha}}\,n_{\Delta
k_{\alpha}}\,=\,n_{\Delta\ell_{\alpha}}\,+\,n_{\Delta
e_{\alpha}}\,=\,\left(2\mu_{\ell_{\alpha}}\,+\,\mu_{e_{\alpha}}\right)\,\frac{T^{2}}{6}$
(66c) $\displaystyle n_{L_{SM}}\,$
$\displaystyle\equiv\,\sum_{\alpha=1}^{3}\,n_{L_{\alpha}}\,=\,\sum_{\alpha=1}^{3}\left(2\mu_{\ell_{\alpha}}\,+\,\mu_{e_{\alpha}}\right)\,\frac{T^{2}}{6}$
(66d) $\displaystyle n_{L_{F}}\,$
$\displaystyle\equiv\,\sum_{i=1}^{2}\,n_{\Delta
F_{i}}\,=\,\sum_{i=1}^{2}\,\mu_{F_{i}}\,\frac{T^{2}}{6}$ (66e) $\displaystyle
n_{L}\,$
$\displaystyle\equiv\,n_{L_{SM}}\,+\,n_{L_{F}}\,=\,\left(\sum_{\alpha=1}^{3}\left(2\mu_{\ell_{\alpha}}\,+\,\mu_{e_{\alpha}}\right)\,+\,\sum_{i=1}^{2}\,\mu_{F_{i}}\right)\,\frac{T^{2}}{6}$
(66f)

If we also assume that the early Universe is totally symmetric,
$n_{B-L_{SM}}|_{0}=0$ and $n_{F}|_{0}=0$, then we obtain an additional
constraint to $\mu_{F_{1}}+\mu_{F_{2}}$, that is

$\displaystyle
n_{B}\,-\,n_{L_{SM}}\,-\,n_{L_{F}}\,=\,\sum_{\alpha=1}^{3}\,n_{\Delta\alpha}\,-\,\sum_{i=1}^{2}\,n_{\Delta
F_{i}}\,=\,0$ $\displaystyle\Rightarrow$
$\displaystyle\sum_{\alpha=1}^{3}\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\,-\,2\mu_{\ell_{\alpha}}\,-\,\mu_{e_{\alpha}}\right)\,-\,\left(\mu_{F_{1}}\,+\,\mu_{F_{2}}\right)\,=\,0$
(67)

Replacing this in Eq. (A.0.2), we obtain the modified hypercharge constraint
expressed in terms of the SM chemical potentials.

  * •

Modified hypercharge constraint

$\sum_{\alpha=1}^{3}\left(-\,\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,-\,2\mu_{d_{\alpha}}\,+\,\mu_{\ell_{\alpha}}\right)\,+\,2\mu_{H}\,=\,0$
(68)

Now the 16 SM chemical potentials are constrained under the $SU(2)$ sphaleron
(52) and Yukawa (54) processes (10 constraints), as well as under the baryon
flavor asymmetry equality (• ‣ A.0.1) and the modified hypercharge (68)
conditions (3 constraints). Solving the system of equations, we may express
all chemical potentials in terms of $\mu_{e_{\alpha}}$

$\displaystyle\mu_{q_{\alpha}}\,$
$\displaystyle=\,-\frac{11}{144}\left(\mu_{e_{1}}\,+\,\mu_{e_{2}}\,+\,\mu_{e_{3}}\right)$
(69a) $\displaystyle\mu_{u_{\alpha}}\,$
$\displaystyle=\,-\frac{13}{72}\left(\mu_{e_{1}}\,+\,\mu_{e_{2}}\,+\,\mu_{e_{3}}\right)$
(69b) $\displaystyle\mu_{d_{\alpha}}\,$
$\displaystyle=\,\frac{1}{36}\left(\mu_{e_{1}}\,+\,\mu_{e_{2}}\,+\,\mu_{e_{3}}\right)$
(69c) $\displaystyle\mu_{\ell_{1}}\,$
$\displaystyle=\,\frac{1}{48}\left(43\mu_{e_{1}}-5\mu_{e_{2}}-5\mu_{e_{3}}\right)$
(69d) $\displaystyle\mu_{\ell_{2}}\,$
$\displaystyle=\,\frac{1}{48}\left(-5\mu_{e_{1}}+43\mu_{e_{2}}-5\mu_{e_{3}}\right)$
(69e) $\displaystyle\mu_{\ell_{3}}\,$
$\displaystyle=\,\frac{1}{48}\left(-5\mu_{e_{1}}-5\mu_{e_{2}}+43\mu_{e_{3}}\right)$
(69f) $\displaystyle\mu_{H}\,$
$\displaystyle=\,-\frac{5}{48}\left(\mu_{e_{1}}\,+\,\mu_{e_{2}}\,+\,\mu_{e_{3}}\right)$
(69g)

Now the baryon and lepton number density asymmetries can be written in terms
of the chemical potentials $\mu_{e_{\alpha}}$ as

$\displaystyle n_{B_{\alpha}}\,$
$\displaystyle=\,\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\right)\,\frac{T^{2}}{6}\,=\,-\frac{11}{36}\left(\mu_{e_{1}}\,+\,\mu_{e_{2}}\,+\,\mu_{e_{3}}\right)\,\frac{T^{2}}{6}$
(70a) $\displaystyle n_{B}\,$
$\displaystyle=\,3\,n_{B_{\alpha}}\,=\,-\frac{11}{12}\left(\mu_{e_{1}}\,+\,\mu_{e_{2}}\,+\,\mu_{e_{3}}\right)\,\frac{T^{2}}{6}$
(70b) $\displaystyle n_{L_{\alpha}}\,$
$\displaystyle=\,\left(2\mu_{\ell_{\alpha}}\,+\,\mu_{e_{\alpha}}\right)\,\frac{T^{2}}{6}$
(70c) $\displaystyle n_{L_{SM}}\,$
$\displaystyle=\,\sum_{\alpha=1}^{3}\left(2\mu_{\ell_{\alpha}}\,+\,\mu_{e_{\alpha}}\right)\,\frac{T^{2}}{6}\,=\,\frac{19}{8}\left(\mu_{e_{1}}\,+\,\mu_{e_{2}}\,+\,\mu_{e_{3}}\right)\,\frac{T^{2}}{6}$
(70d) $\displaystyle n_{L_{F}}\,$
$\displaystyle=\,\left(\mu_{F_{1}}\,+\,\mu_{F_{2}}\right)\,\frac{T^{2}}{6}\,=\,\sum_{\alpha=1}^{3}\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\,-\,2\mu_{\ell_{\alpha}}\,-\,\mu_{e_{\alpha}}\right)\,\frac{T^{2}}{6}$
$\displaystyle=\,-\frac{79}{24}\left(\mu_{e_{1}}\,+\,\mu_{e_{2}}\,+\,\mu_{e_{3}}\right)\,\frac{T^{2}}{6}\,=\,n_{B}\,-\,n_{L_{SM}}$
(70e) $\displaystyle n_{L}\,$
$\displaystyle\equiv\,n_{L_{SM}}\,+\,n_{L_{F}}\,=\,-\frac{11}{12}\left(\mu_{e_{1}}\,+\,\mu_{e_{2}}\,+\,\mu_{e_{3}}\right)\,\frac{T^{2}}{6}\,=\,n_{B}$
(70f)

The flavor asymmetry combination $n_{\Delta_{\alpha}}\equiv
n_{B_{\alpha}}-n_{L_{\alpha}}=n_{B}/3-n_{L_{\alpha}}$ can be written as

$\displaystyle n_{\Delta_{\alpha}}\,$
$\displaystyle=\,\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\,-\,2\mu_{\ell_{\alpha}}\,-\,\mu_{e_{\alpha}}\right)\,\frac{T^{2}}{6}$
$\displaystyle\Rightarrow\,\begin{pmatrix}n_{\Delta_{1}}\\\ n_{\Delta_{2}}\\\
n_{\Delta_{3}}\end{pmatrix}\,=\,-\frac{1}{72}\,\left[\begin{array}[]{ccc}223&7&7\\\
7&223&7\\\ 7&7&223\end{array}\right]\,\begin{pmatrix}\mu_{e_{1}}\\\
\mu_{e_{2}}\\\ \mu_{e_{3}}\end{pmatrix}\,\frac{T^{2}}{6}$

Inverting the expression above, one finds

$\begin{pmatrix}\mu_{e_{1}}\\\ \mu_{e_{2}}\\\
\mu_{e_{3}}\end{pmatrix}\,=\,-\frac{1}{711}\,\left[\begin{array}[]{ccc}230&-7&-7\\\
-7&230&-7\\\ -7&-7&230\end{array}\right]\,\begin{pmatrix}n_{\Delta_{1}}\\\
n_{\Delta_{2}}\\\ n_{\Delta_{3}}\end{pmatrix}\,\frac{6}{T^{2}}$ (71)

Using Eq. (50), we can express the chemical potentials $\mu_{e_{\alpha}}$ in
terms of the flavor asymmetries $n_{\Delta e_{\alpha}}$ and relate them to
$n_{\Delta\alpha}$ as

$\begin{pmatrix}n_{\Delta e_{1}}\\\ n_{\Delta e_{2}}\\\ n_{\Delta
e_{3}}\end{pmatrix}\,=\,-\frac{1}{711}\,\left[\begin{array}[]{ccc}230&-7&-7\\\
-7&230&-7\\\ -7&-7&230\end{array}\right]\,\begin{pmatrix}n_{\Delta_{1}}\\\
n_{\Delta_{2}}\\\ n_{\Delta_{3}}\end{pmatrix}$ (72)

Hence, in this model and for $T_{sph}\ll T\ll 10^{8}\,\operatorname{GeV}$, the
total baryon $n_{B}$ and the SM lepton $n_{L_{\text{SM}}}$ asymmetries are
related to $n_{B-L_{\text{SM}}}$ by

$\displaystyle
n_{B}\,=\,\frac{22}{79}\,n_{B-L_{SM}}\,=\,\frac{22}{79}\,\sum_{\alpha=1}^{3}n_{\Delta\alpha}$
(73) $\displaystyle
n_{L_{SM}}\,=\,-\frac{57}{79}\,n_{B-L_{SM}}\,=\,-\frac{57}{79}\,\sum_{\alpha=1}^{3}n_{\Delta\alpha}$
(74)

This result agrees with that obtained in reference Shuve_2020 .

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