KUNS-2895

Leptonic CP asymmetry and Light flavored scalar

Yoshihiko<EMAIL_ADDRESS>Toshimasa
<EMAIL_ADDRESS>Koichi
<EMAIL_ADDRESS>
Department of Physics, Kyoto University, Kyoto 606-8502, Japan

We consider a situation where right-handed neutrinos couple to a light scalar
which is possibly a Nambu-Goldstone boson resulting from high-energy symmetry
breaking. Its coupling is typically complex-valued and flavor-dependent. In
this work, we investigate the possibility of the leptonic asymmetry generation
in the Universe from tree-level right-handed neutrino decay to flavorful light
scalar. Furthermore a new source of asymmetry generation from a single decay
process is pointed out, which is characteristic of the present setting.

###### Contents

  1. 1 Introduction
  2. 2 Lagrangian and two-body decay
  3. 3 Three-body decay for asymmetry
    1. 3.1 The lightest mode
    2. 3.2 Heavier modes
  4. 4 Lepton asymmetry with $\chi$ scalar
    1. 4.1 Boltzmann equations
    2. 4.2 Parameter space
    3. 4.3 Property of $\chi$ scalar
  5. 5 Summary
  6. A Three-body decay widths
  7. B CP asymmetry (width differences)
  8. C Boltzmann equations and asymmetry formulae
    1. C.1 Boltzmann equations
    2. C.2 Phenomenological formulae for lepton asymmetry
      1. C.2.1 Case for freeze-in from $N_{2}$
      2. C.2.2 Case for strong $N_{1}$ washout

##  1 Introduction

The Standard Model (SM) of particle physics and cosmology still have some
mysteries, e.g., the nature of dark matter, the source of matter-antimatter
asymmetry, the origin of neutrino masses, and so on. An attractive idea to
realize tiny neutrino masses is the seesaw mechanism with Majorana right-
handed (RH) neutrinos [1, 2, 3], called the type I seesaw. The additional
Majorana fermions can also be the origin of matter-antimatter asymmetry via
their decay, the scenario called leptogenesis [4].

The baryon asymmetry in the present Universe is measured by the cosmic
microwave background observation [5] as

$\displaystyle Y_{B}=\frac{n_{B}}{s}=(0.852\text{ -- }0.888)\times 10^{-10},$
(1.1)

with the number density $n_{B}$ and the entropy density $s$. In the
leptogenesis scenario, the RH neutrinos generate the lepton asymmetry, which
is then converted to the baryon asymmetry [6] via the sphaleron process [7].
In the simplest leptogenesis, the interference between the tree-level and one-
loop level RH neutrino two-body decays gives a source of the asymmetry, which
is determined by neutrino Yukawa couplings. On the other hand, Refs. [8, 9,
10, 11, 12, 13] also discuss possibilities of tree-level leptogenesis by
considering various extensions of the neutrino sector.

In this paper, we focus on a possibility of flavorful light scalar for the
lepton asymmetry generation. This is a minimal scalar extension of the type-I
seesaw model by introducing a SM singlet scalar interacting with the RH
neutrinos. This kind of scalar may be motivated by some dynamical origin of
Majorana neutrino mass scale, but in this work we do not specify it and
investigate the general form of scalar interaction to the RH neutrinos. In
this setup, we study the tree-level leptogenesis via the RH neutrino three-
body decays with this flavorful scalar field.

The rest parts are organized as follows. In Sec. 2, the model we focus on is
introduced and we derive the decay widths and the asymmetry parameters of the
two-body decay processes. In Sec. 3, the three-body decay widths and the
asymmetry parameter are evaluated and we give their approximation formulae. In
Sec. 4, we discuss the behaviors of the lepton asymmetry and its approximated
relic value by solving the Boltzmann equations, and show the allowed parameter
spaces of this model. We also give some comments on the dynamics of the
additional singlet scalar field and its realization in some phenomenological
models. Sec. 5 is devoted to the summary of this paper.

##  2 Lagrangian and two-body decay

In this work, we denote the RH neutrinos as $N_{i}$ in the mass diagonal
basis. They are the Majorana fermions with the Majorana masses $m_{N_{i}}$. An
SM gauge singlet scalar $\chi$ is introduced and assumed to have the coupling
to the RH neutrinos. The lagrangian we consider is

$\displaystyle\mathcal{L}=\mathcal{L}_{\mathrm{SM}}+\frac{1}{2}\overline{N_{i}}(i\not{\partial}-m_{N_{i}})N_{i}-y^{\nu}_{ij}\tilde{H}^{\dagger}\overline{N_{i}}P_{L}L_{j}+\mathrm{h.c.}-\frac{1}{2}\chi\overline{N_{i}}\bigl{(}\xi_{ij}P_{R}+\xi^{*}_{ij}P_{L}\bigr{)}N_{j},$
(2.1)

where $L_{j}$ mean the SM lepton doublets and $H$ the Higgs one, and $P_{L/R}$
is the chirality projection operator. The coupling constants $\xi_{ij}$ are
$i,j$ symmetric and generally complex-valued. They can be decomposed into the
scalar (real part) and pseudo-scalar (imaginary part) couplings to $N_{i}$. As
discussed in Ref. [14], with the seesaw relation, the neutrino Yukawa coupling
$y^{\nu}$ can be generally parameterized as

$\displaystyle
y^{\nu}=\frac{\sqrt{2}i}{v}\sqrt{M_{N}}R\sqrt{m_{\nu}}U^{\dagger}_{\text{MNS}},$
(2.2)

with the Majorana mass matrix
$M_{N}=\mathop{\mathrm{diag}}(m_{N_{1}},m_{N_{2}},m_{N_{3}})$ and the neutrino
mass matrix
$m_{\nu}=\mathop{\mathrm{diag}}(m_{\nu_{1}},m_{\nu_{2}},m_{\nu_{3}})$, and the
neutrino flavor mixing matrix $U_{\text{MNS}}$. $v\approx 246~{}\mathrm{GeV}$
is the electroweak scale. For the neutrino masses and mixing angles, we use
the experimentally observed values summarized in Ref. [15]. $R$ is an
arbitrary complex orthogonal matrix describing the degrees of freedom of
neutrino Yukawa couplings which cannot be reached with the seesaw relation. We
parameterize $R$ as

$\displaystyle R=\begin{pmatrix}1&0&0\\\ 0&\cos\omega_{23}&\sin\omega_{23}\\\
0&-\sin\omega_{23}&\cos\omega_{23}\end{pmatrix}\begin{pmatrix}\cos\omega_{13}&0&\sin\omega_{13}\\\
0&1&0\\\
-\sin\omega_{13}&0&\cos\omega_{13}\end{pmatrix}\begin{pmatrix}\cos\omega_{12}&\sin\omega_{12}&0\\\
-\sin\omega_{12}&\cos\omega_{12}&0\\\ 0&0&1\end{pmatrix},$ (2.3)

introducing the complex angles $\omega_{ij}$ (called the Casas-Ibarra (CI)
parameters in the following).

Through the neutrino Yukawa coupling, the RH neutrinos interact with the SM
thermal bath. For the two-body decay of $N_{i}$ to the SM particles, the
partial widths at tree level are given by

$\displaystyle\Gamma_{N_{i}\to
L_{j}H}=\Gamma_{N_{i}\to\bar{L}_{j}\bar{H}}=\frac{y^{\nu}_{ij}y^{\nu\dagger}_{ji}}{16\pi}m_{N_{i}},$
(2.4)

where $\bar{L}$, $\bar{H}$ means the corresponding anti-particles. We denote
the total width of the two-body decay via neutrino Yukawa coupling as

$\displaystyle\tilde{\Gamma}_{i}=\sum_{j}\bigl{(}\Gamma_{N_{i}\to
L_{j}H}+\Gamma_{N_{i}\to\bar{L}_{j}\bar{H}}\bigr{)}.$ (2.5)

When $m_{N_{2}}>m_{N_{1}}+m_{\chi}$, a heavier mode $N_{2}$ can also decay to
$N_{1}$ and $\chi$ via the off-diagonal coupling $\xi_{12}$ and its width is
evaluated as

$\displaystyle\Gamma_{N_{2}\to N_{1}\chi}=$
$\displaystyle\;\frac{m_{N_{2}}}{16\pi}\frac{\sqrt{\lambda(m_{N_{2}}^{2},m_{N_{1}}^{2},m_{\chi}^{2})}}{m_{N_{2}}^{2}}\biggl{\\{}\Bigl{[}\Bigl{(}1-\frac{m_{N_{1}}}{m_{N_{2}}}\Bigr{)}^{2}-\frac{m_{\chi}^{2}}{m_{N_{2}}^{2}}\Bigr{]}(\mathop{\mathrm{Im}}\xi_{12})^{2}$
$\displaystyle\quad+\Bigl{[}\Bigl{(}1+\frac{m_{N_{1}}}{m_{N_{2}}}\Bigr{)}^{2}-\frac{m_{\chi}^{2}}{m_{N_{2}}^{2}}\Bigr{]}(\mathop{\mathrm{Re}}\xi_{12})^{2}\biggr{\\}},$
(2.6)

with the kinematic function $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2yz-2yx$.

As we will see in the following, the decay of the lightest $N_{1}$ is
dominated by the two-body decay, but for a heavier mode the resonant
contribution of three-body decay is comparable to that of $N_{2}\to
N_{1}\chi$. Then the total decay widths $\Gamma_{N_{i}}$ of the lightest and
heavier RH neutrinos are generally written by

$\displaystyle\Gamma_{N_{1}}\approx\tilde{\Gamma}_{1},\qquad\Gamma_{N_{2}}=\tilde{\Gamma}_{2}+\Gamma_{N_{2}\to
N_{1}\chi}+\Gamma_{2},$ (2.7)

where $\Gamma_{i}$ are the three-body decay widths of $N_{i}$ and will be
described in the next section.

The CP asymmetry in the ordinary leptogenesis comes from the interference of
the RH neutrino’s two-body decays (the tree-level and the right two diagrams
in Fig. 1), where the phases of neutrino Yukawa couplings play an important
role. The asymmetry parameters $\tilde{\epsilon}^{(LH)}_{i}$ associated with
this interference is evaluated as (for a review [16]),

$\displaystyle\tilde{\epsilon}^{(LH)}_{i}=\frac{\Gamma_{N_{i}\to
LH}-\Gamma_{N_{i}\to\bar{L}\bar{H}}}{\Gamma_{N_{i}\to
LH}+\Gamma_{N_{i}\to\bar{L}\bar{H}}}=\frac{1}{8\pi(y^{\nu}y^{\nu{\dagger}})_{ii}}\sum_{j}\mathop{\mathrm{Im}}\bigl{[}(y^{\nu}y^{\nu{\dagger}})_{ji}\bigr{]}^{2}\mathrm{g}(m_{N_{j}}^{2}/m_{N_{i}}^{2}),$
(2.8)

where $\mathrm{g}(x)$ in the SM is given by
$\mathrm{g}(x)=\sqrt{x}\bigl{[}\frac{2-x}{1-x}-(1+x)\ln\big{(}\frac{1+x}{x}\bigr{)}\bigr{]}$.
It is noticed that, when $\omega_{ij}$ are real or pure imaginary,
$\tilde{\epsilon}^{(LH)}_{i}$ vanish and the two-body decay via the neutrino
Yukawa loop does not generate the CP asymmetry.

In addition to the SM particle loops, lighter RH neutrinos and the scalar
field $\chi$ give an additional asymmetry in the two-body decay of heavier RH
neutrinos. The left diagram in Fig. 1 shows the relevant one-loop contribution
to the $N_{i}$ decay.

Figure 1: The one-loop two-body decays of a RH neutrino $N_{i}$ to the SM
fields. (Left) from lighter RH neutrinos and the scalar $\chi$. (Middle and
Right) from the SM loops.

The asymmetry reads from the difference between the decay widths to particles
and anti-particles,

$\Delta\Gamma_{N\chi}=\bigl{|}\mathcal{M}_{0}+\mathcal{M}_{1}\bigr{|}^{2}-\bigl{|}\mathcal{M}_{0}^{C}+\mathcal{M}_{1}^{C}\bigr{|}^{2},$
(2.9)

where $\mathcal{M}_{0}^{(C)}$ denotes the amplitude of the tree-level decay to
the (anti-)particles and $\mathcal{M}_{1}^{(C)}$ that of the one-loop diagram
including $\chi$, respectively. This is evaluated as

$\displaystyle\Delta\Gamma_{N\chi}=\frac{m_{N_{i}}^{4}}{4\pi^{2}}\sum_{k,l}\frac{1}{m_{N_{l}}^{2}-m_{N_{i}}^{2}}\int_{0}^{1}dx\,\mathop{\mathrm{Im}}\bigl{(}\ln\Delta^{2}_{N_{k}\chi}\bigr{)}\mathop{\mathrm{Im}}\bigl{[}f_{li}(x)(y^{\nu}y^{\nu\dagger})_{il}\bigr{]},$
(2.10)

where

$\displaystyle\Delta^{2}_{N_{k}\chi}=$
$\displaystyle\;x(x-1)m_{N_{i}}^{2}+(1-x)m_{N_{k}}^{2}+xm_{\chi}^{2},$ (2.11)
$\displaystyle f_{li}(x)=$
$\displaystyle\;\biggl{(}\xi_{lk}^{*}\xi_{ki}+\xi_{lk}\xi^{*}_{ki}\frac{m_{N_{l}}}{m_{N_{i}}}\biggr{)}x+\xi_{lk}^{*}\xi_{ki}^{*}\frac{m_{N_{k}}}{m_{N_{i}}}+\xi_{lk}\xi_{ki}\frac{m_{N_{k}}m_{N_{l}}}{m_{N_{i}}^{2}}.$
(2.12)

In order to produce non-trivial asymmetry, the conditions $i\neq k$ for the
interference and $\Delta^{2}_{N_{k}\chi}<0$ are needed. The latter condition
is satisfied if the internal loop particles $N_{k}$ and $\chi$ are lighter
than the parent particle $N_{i}$, and then only the case of $i=2,~{}k=l=1$
gives a non-vanishing asymmetry parameter. As a result, we obtain the
$N$-$\chi$ loop contribution to asymmetry parameter

$\displaystyle\tilde{\epsilon}^{(N\chi)}_{2}=\frac{-m_{N_{2}}}{64\pi^{2}\tilde{\Gamma}_{2}}\int_{x_{-}}^{x_{+}}\\!dx\,\mathop{\mathrm{Im}}\bigl{[}f_{12}(x)(y^{\nu}y^{\nu\dagger})_{21}\bigl{]}\;\approx\,\frac{1}{16\pi(y^{\nu}y^{\nu\dagger})_{22}}\mathop{\mathrm{Im}}\bigl{[}\xi_{11}\xi^{*}_{12}(y^{\nu}y^{\nu\dagger})_{12}\bigr{]},$
(2.13)

where $x_{\pm}$ are the solutions for $\Delta^{2}_{N_{1}\chi}=0$. In the
latter approximation, we have assumed the hierarchy $m_{N_{1}}\ll m_{N_{2}}$.
The total CP asymmetry from two-body decay of the lightest and heavier RH
neutrinos are given by

$\tilde{\epsilon}_{1}\,=\,\tilde{\epsilon}^{(LH)}_{1},\qquad\tilde{\epsilon}_{2}\,=\,\tilde{\epsilon}^{(LH)}_{2}+\tilde{\epsilon}^{(N\chi)}_{2}.$
(2.14)

With general complex phases of neutrino Yukawa couplings,
$\tilde{\epsilon}^{(LH)}_{i}$ are not necessarily vanishing simultaneously. In
the following analysis, we simply assume that the CI parameters are real or
pure imaginary to examine the CP asymmetry effect of the flavorful scalar
$\chi$.

##  3 Three-body decay for asymmetry

Figure 2: The three-body decay of a RH neutrino $N_{i}$ at tree level
including the scalar $\chi$.

When there exist a scalar $\chi$ and its (pseudo-) scalar coupling to the RH
neutrinos $N$, non-trivial CP asymmetry can be generated at the three-body
decay $N\to LH\chi$ (Fig. 2). Its amplitude is

$\displaystyle\mathcal{M}_{N_{i}\to
L_{j}H\chi}=-\sum_{k}\overline{u_{L_{j}}}(k_{1})y^{\nu\dagger}_{jk}P_{R}\,\frac{1}{\not{q}-\bar{m}_{N_{k}}}(\xi_{ki}P_{R}+\xi^{*}_{ki}P_{L})u_{N_{i}}(p),$
(3.1)

where $\bar{m}_{N_{k}}=m_{N_{k}}-i\Gamma_{N_{k}}/2$. The initial and final
state momenta $p$, $k_{i}$ are shown in the figure ($q=k_{1}+k_{2}$). The
renormalized mass parameters are
$m_{N_{i}}^{r\,2}=m_{N_{i}}^{2}-\Gamma_{N_{i}}^{2}/4$ for $\Gamma_{N_{i}}$
being the width of the resonance $N_{i}$, but we quantitatively drop any
difference of $\mathcal{O}((\Gamma/m_{N})^{2})$ throughout this paper. Summing
up the final charged-lepton flavor and gauge charge, the three-body decay
width becomes

$\displaystyle\Gamma_{N_{i}\to LH\chi}=$
$\displaystyle\sum_{k,\,l}\frac{(y^{\nu}y^{\nu\dagger})_{lk}}{256\pi^{3}m_{N_{i}}^{3}}\int_{0}^{(m_{N_{i}}-m_{\chi})^{2}}\\!\\!\\!dm_{12}^{2}\int_{[m_{23}^{2}]_{-}}^{[m_{23}^{2}]_{+}}dm_{23}^{2}\,\frac{1}{(m_{12}^{2}-\bar{m}_{N_{l}}^{*2})(m_{12}^{2}-\bar{m}_{N_{k}}^{2})}$
$\displaystyle\;\times\Bigl{[}\xi_{il}\xi^{*}_{ki}m_{12}^{2}(m_{12}^{2}+m_{23}^{2}-m_{\chi}^{2})+(\xi_{il}\xi_{ki}\bar{m}_{N_{k}}+\xi^{*}_{il}\xi^{*}_{ki}\bar{m}_{N_{l}}^{*})m_{N_{i}}m_{12}^{2}$
$\displaystyle\qquad+\xi^{*}_{il}\xi_{ki}\bar{m}_{N_{l}}^{*}\bar{m}_{N_{k}}(m_{N_{i}}^{2}-m_{23}^{2})\Bigr{]}.$
(3.2)

The invariant mass parameters are defined by $m_{12}^{2}=(k_{1}+k_{2})^{2}$,
$m_{23}^{2}=(k_{2}+k_{3})^{2}$, and the maximum and minimum values of
$m_{23}^{2}$ with a fixed $s$ are given by [17, 15]

$\displaystyle[m_{23}^{2}]_{\pm}=\frac{1}{2}\left[m_{N_{i}}^{2}+m_{\chi}^{2}-m_{12}^{2}\pm\sqrt{\lambda(m_{N_{i}}^{2},m_{\chi}^{2},m_{12}^{2})}\,\right].$
(3.3)

The three-body decay to the anti-particles
$\Gamma_{N_{i}\to\bar{L}\bar{H}\chi}$ is evaluated in the same manner. The
total decay width and the CP asymmetry parameter for the three-body decay are
defined by

$\displaystyle\Gamma_{i}=\Gamma_{N_{i}\to
LH\chi}+\Gamma_{N_{i}\to\bar{L}\bar{H}\chi},\qquad\epsilon_{i}=\frac{\Gamma_{N_{i}\to
LH\chi}-\Gamma_{N_{i}\to\bar{L}\bar{H}\chi}}{\Gamma_{N_{i}\to
LH\chi}+\Gamma_{N_{i}\to\bar{L}\bar{H}\chi}}.$ (3.4)

In the following, we will discuss dominant contributions to the widths and
asymmetry parameters and their approximate forms.

###  3.1 The lightest mode

Among the intermediate states $N_{k}$, the lightest one $N_{1}$ generally
gives the dominant contribution to the $N_{1}$ three-body decay, i.e., the
$i=k=l=1$ part in (3.2). For a typical hierarchy of mass parameters,
$m_{\chi}\ll m_{N_{1}}\ll m_{N_{2}}$, the $N_{1}$ decay width reduces to

$\displaystyle\Gamma_{1}^{(11)}\approx\frac{m_{N_{1}}}{64\pi^{3}}(y^{\nu}y^{\nu\dagger})_{11}\big{[}|\xi_{11}|^{2}+\mathop{\mathrm{Re}}(\xi^{2}_{11})\big{]}\ln\Bigl{(}\frac{m_{N_{1}}}{m_{\chi}}\Bigr{)},$
(3.5)

which is mainly determined by the diagonal coupling $\xi_{11}$. Compared with
the two-body decay $N_{1}\to LH$, the width is multiplied by a factor
$\sim(\xi_{11})^{2}/4\pi\,\ln(m_{N_{1}}/m_{\chi})$, which can be of order one.
If $\xi_{11}$ is negligibly small, the width is instead given by the $k=l=2$
part,

$\displaystyle\Gamma_{1}^{(22)}\approx\frac{m_{N_{1}}^{3}}{384\pi^{3}m_{N_{2}}^{2}}(y^{\nu}y^{\nu\dagger})_{22}|\xi_{12}|^{2}.$
(3.6)

The CP asymmetry, the difference between the decay widths to particles and
anti-particles, is usually described by the quantum interference of processes
with different intermediate states. For the current three-body decay, such
interference occurs from the diagrams with different intermediate neutrinos
$N_{k,l}$ $(k\neq l)$, that is, the cross terms in the amplitude squared. It
is important that, in addition to this usual asymmetry, the present model
induces CP asymmetry from a single diagram with one intermediate state. For
example, we find the $N_{1}$ three-body decay with virtual $N_{1}$ generates a
nonzero CP asymmetry parameter

$\displaystyle\epsilon_{1}^{(11)}\approx\frac{\mathop{\mathrm{Im}}(\xi_{11}^{2})}{|\xi_{11}|^{2}+\mathop{\mathrm{Re}}(\xi^{2}_{11})}\,\frac{\Gamma_{N_{1}}}{2m_{N_{1}}},$
(3.7)

which is irrelevant to the $N_{2}$ physics at the leading order. (This is the
approximate formula for $m_{\chi}\ll m_{N_{1}}\ll m_{N_{2}}$. The exact form
of asymmetry parameter is found in Appendix B.)

The non-vanishing “interference” from a single diagram (3.7) is a
characteristic feature of the present decay mode and its origin is understood
by the amplitude form. The three-body decay amplitude (3.1) contains the
factor $P_{R}(\not{q}+\bar{m}_{N})(\xi P_{R}+\xi^{*}P_{L})$ between the
initial and final spinor wavefunctions. The first projection $P_{R}$ comes
from the Yukawa (chirality-violating) vertex, the second factor
$(\not{q}+\bar{m}_{N})$ the propagator of heavy unstable fermion, and the
third one $(\xi P_{R}+\xi^{*}P_{L})$ means the Majorana fermion vertex. The
chirality structure implies this factor is divided into two pieces
$\bar{m}_{N}\xi$ and $\not{q}\xi^{*}$, and hence the squared amplitude
generally contains a cross term of these two pieces (Fig. 3).

Figure 3: CP asymmetry (cross term) from a single amplitude squared.

Further the corresponding decay width to anti-particles is obtained by the
replacement of couplings $\xi\leftrightarrow\xi^{*}$ (and
$y^{\nu}\leftrightarrow y^{\nu*}$, $P_{R}\leftrightarrow P_{L}$). In the end,
the interference of these two pieces leads to the CP asymmetry proportional to
$\mathop{\mathrm{Im}}(\xi^{2})\mathop{\mathrm{Im}}(\bar{m})$, appearing in the
numerator of (3.7). We thus find that this CP asymmetry from a single decay
process is generated in the presence of a chiral vertex, a unstable
intermediate state, and a complex decay coupling. The three-body decay of a RH
neutrino to the SM particles plus scalar is an interesting realization of all
these criteria satisfied.

The interference of different intermediate states also contributes to the CP
asymmetry as usual, and its approximate form is found

$\displaystyle\epsilon_{1}^{(12)}\approx\frac{m_{N_{1}}\Gamma_{N_{2}}}{8m_{N_{2}}^{2}}\frac{\mathop{\mathrm{Im}}\bigl{[}(y^{\nu}y^{\nu\dagger})_{12}\xi_{12}(2\xi_{11}+3\xi_{11}^{*})\bigr{]}}{(y^{\nu}y^{\nu\dagger})_{11}\big{[}|\xi_{11}|^{2}+\mathop{\mathrm{Re}}(\xi^{2}_{11})\big{]}\ln\bigl{(}\frac{m_{N_{1}}}{m_{\chi}}\bigr{)}}\,.$
(3.8)

Which contribution (3.7) or (3.8) is the dominant CP asymmetry depends on the
model parameters.

The $k=l=2$ part in the $N_{1}$ decay width only gives a subdominant CP
asymmetry in almost case due to the large $m_{N_{2}}$ suppression.

###  3.2 Heavier modes

A heavier RH neutrino than $N_{1}$ can also generate CP asymmetry at its
three-body decay. A lighter intermediate state, e.g., $N_{1}$ meets the
resonance around its mass $m_{12}^{2}\sim m_{N_{1}}^{2}$ and then the
amplitude is largely enhanced. There are two types of enhancement in the decay
width (3.2), the $k=l=1$ and cross terms. The enhanced contributions to the
decay width are evaluated with the narrow width approximation. For the $k=l=1$
part, we obtain

$\displaystyle\Gamma_{2}^{(11)}=$
$\displaystyle~{}\frac{m_{N_{1}}}{128\pi^{3}m_{N_{2}}^{3}\Gamma_{N_{1}}}(y^{\nu}y^{\nu\dagger})_{11}\sqrt{\lambda(m_{N_{1}}^{2},m_{N_{2}}^{2},m_{\chi}^{2})}$
$\displaystyle\quad\times\big{[}|\xi_{12}|^{2}(m_{N_{1}}^{2}+m_{N_{2}}^{2}-m_{\chi}^{2})+2\mathop{\mathrm{Re}}(\xi^{2}_{12})m_{N_{1}}m_{N_{2}}\big{]},$
(3.9)

where we have neglected $\mathcal{O}\bigl{(}(\Gamma_{N_{1}})^{2}\bigr{)}$
terms. In the limit $m_{\chi}\ll m_{N_{1}}\ll m_{N_{2}}$ and
$\Gamma_{N_{1}}\approx\tilde{\Gamma}_{1}$, the $N_{1}$ resonant contribution
(3.9) is equal to $\frac{1}{2}\Gamma_{N_{2}\to N_{1}\chi}$. (The prefactor
$\frac{1}{2}$ implies $\tilde{\Gamma}_{1}$ contains the decays both to
particles and anti-particles.) We also find the cross-term contribution is
relatively suppressed than the $k=l=1$ one. The enhanced on-shell decay (3.9)
is controlled by the off-diagonal coupling $\xi_{12}$. When $\xi_{12}$ is
negligibly small, the $N_{2}$ three-body decay width is dominantly given by
the non-resonant $k=l=2$ part and its approximate form is

$\displaystyle\Gamma_{2}^{(22)}\approx\frac{m_{N_{2}}}{64\pi^{3}}(y^{\nu}y^{\nu\dagger})_{22}\big{[}|\xi_{22}|^{2}+\mathop{\mathrm{Re}}(\xi^{2}_{22})\big{]}\ln\Bigl{(}\frac{m_{N_{2}}}{m_{\chi}}\Bigr{)}.$
(3.10)

The CP asymmetry of $N_{2}$ three-body decay is also generated by the resonant
and non-resonant parts. The asymmetry from the $N_{1}$ resonant part comes
from the last term in (3.9) and reads

$\displaystyle\epsilon_{2}^{(11)}=\frac{m_{N_{1}}}{128\pi^{2}m_{N_{2}}^{2}\Gamma_{2}}(y^{\nu}y^{\nu\dagger})_{11}\sqrt{\lambda(m_{N_{1}}^{2},m_{N_{2}}^{2},m_{\chi}^{2})}\mathop{\mathrm{Im}}(\xi_{12})^{2}.$
(3.11)

The usual interference of different diagrams also contributes to the asymmetry
and is approximately found

$\displaystyle\epsilon_{2}^{(12)}\approx\frac{-m_{N_{1}}}{128\pi^{2}\Gamma_{2}}\mathop{\mathrm{Im}}\big{[}(y^{\nu}y^{\nu\dagger})_{21}\xi_{22}^{*}\xi_{12}\big{]}$
(3.12)

in the limit $m_{N_{1}}\ll m_{N_{2}}$. This usual cross-term part gives a tiny
contribution to the decay width as mentioned above, but a possibly large one
to the CP asymmetry parameter. The exact form of asymmetry parameters from the
resonant contribution is found in Appendix B.

Similar to the $N_{1}$ decay, there is the CP asymmetry from a single diagram
where the unstable $N_{2}$ is the intermediate state. In this case, the
difference of the decay widths to particles and anti-particles is the same
form as in the $N_{1}$ decay, but the total three-body decay width has several
possibilities as described above. The resultant asymmetry parameter from the
$N_{2}$-intermediate diagram becomes

$\displaystyle\epsilon_{2}^{(22)}\approx\frac{\Gamma_{N_{2}}}{128\pi^{3}\Gamma_{2}}(y^{\nu}y^{\nu\dagger})_{22}\mathop{\mathrm{Im}}(\xi_{22}^{2})\ln\Bigl{(}\frac{m_{N_{2}}}{m_{\chi}}\Bigr{)}$
(3.13)

Among the above 3 types of $\epsilon_{2}$, the leading contribution is
generally given by the non-resonant part (3.13) except for a tiny $\xi_{22}$
(see Fig. 4). We thus find for the $N_{2}$ three-body decay that the width is
determined by the diagonal $k=l=1$ or $k=l=2$ part, and the CP asymmetry is
governed by the non-resonant $k=l=2$ part.

Figure 4: A typical comparison of various asymmetry parameters. The
approximate formulae of these parameters are given in the text. (Left)
$|\xi_{11}|=|\xi_{22}|=10^{-5}$. (Right) $|\xi_{11}|=|\xi_{22}|=1$. In these
figures, the neutrino Yukawa couplings are roughly replaced by the mass
eigenvalues of neutrinos as
$(y^{\nu}y^{\nu\dagger})_{ij}\sim(m_{N_{i}}m_{N_{j}}m_{\nu_{i}}m_{\nu_{j}})^{1/2}/v^{2}$,
and the complex phases of couplings are assumed such that the asymmetry
parameters take their maximal values.

##  4 Lepton asymmetry with $\chi$ scalar

###  4.1 Boltzmann equations

In this section, solving the Boltzmann equations in the present system, we
discuss the time evolution of yields and the washout effect of asymmetry. When
the temperature of the Universe is denoted by $T$, the Hubble parameter $H$
and the entropy density $s$ in the radiation-dominated Universe are defined by

$\displaystyle H(T)=\sqrt{\frac{\pi^{2}}{90}g_{*}}\frac{T^{2}}{M_{P}},\qquad
s(T)=\frac{2\pi^{2}}{45}g^{S}_{*}T^{3},$ (4.1)

where $M_{P}=2.44\times 10^{18}~{}\mathrm{GeV}$ is the reduced Planck scale,
$g_{*}$ ($g_{*}^{S}$) denotes the effective degrees of freedom for the energy
(entropy) density. The yield of particle $i$ is defined by $Y_{i}=n_{i}/s$
with the number density $n_{i}$, and $Y^{\text{eq}}_{i}$ is that of the
thermal equilibrium.

In the following analysis, we apply for simplicity the single flavor
approximation to the lepton asymmetry, and write the yields of leptons and
anti-leptons as

$\displaystyle Y_{L}=Y^{\text{eq}}_{L}+\frac{1}{2}Y_{\Delta L},\qquad
Y_{\bar{L}}=Y^{\text{eq}}_{L}-\frac{1}{2}Y_{\Delta L},$ (4.2)

where $Y_{\Delta L}$ denotes the yield of the lepton asymmetry. If one would
like to include flavor-dependent effects, a further detailed analysis with,
e.g., the density matrix formalism, is needed [18], but that is beyond the
purpose of this work. The SM particles including the leptons and the Higgs
field are assumed to be in the thermal bath. In this setup, the Boltzmann
equations needed for the lepton asymmetry are given by

$\displaystyle Hx\frac{dY_{N_{1}}}{dx}\approx$
$\displaystyle\;\frac{K_{1}(m_{N_{1}}/T)}{K_{2}(m_{N_{1}}/T)}Y^{\text{eq}}_{N_{1}}\Biggl{[}\tilde{\Gamma}_{1}\biggl{(}1-\frac{Y_{N_{1}}}{Y^{\text{eq}}_{N_{1}}}\biggr{)}-\frac{1}{2}\tilde{\Gamma}_{1}\tilde{\epsilon}_{1}\frac{Y_{\Delta
L}}{Y^{\text{eq}}_{L}}+\Gamma_{1}\biggl{(}-\frac{Y_{N_{1}}}{Y^{\text{eq}}_{N_{1}}}+\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)}$
$\displaystyle\;-\frac{1}{2}\Gamma_{1}\epsilon_{1}\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\frac{Y_{\Delta
L}}{Y^{\text{eq}}_{L}}\Biggr{]}+\frac{K_{1}(m_{N_{2}}/T)}{K_{2}(m_{N_{2}}/T)}Y^{\text{eq}}_{N_{2}}\Gamma_{N_{2}\to
N_{1}\chi}\biggl{(}\frac{Y_{N_{2}}}{Y_{N_{2}}^{\text{eq}}}-\frac{Y_{N_{1}}}{Y^{\text{eq}}_{N_{1}}}\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)},$
(4.3) $\displaystyle Hx\frac{dY_{N_{2}}}{dx}\approx$
$\displaystyle\;\frac{K_{1}(m_{N_{2}}/T)}{K_{2}(m_{N_{2}}/T)}Y^{\text{eq}}_{N_{2}}\Biggl{[}\tilde{\Gamma}_{2}\biggl{(}1-\frac{Y_{N_{2}}}{Y^{\text{eq}}_{N_{2}}}\biggr{)}-\frac{1}{2}\tilde{\Gamma}_{2}\tilde{\epsilon}_{2}\frac{Y_{\Delta
L}}{Y^{\text{eq}}_{L}}+\Gamma_{2}\biggl{(}-\frac{Y_{N_{2}}}{Y^{\text{eq}}_{N_{2}}}+\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)}$
$\displaystyle\;-\frac{1}{2}\Gamma_{2}\epsilon_{2}\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\frac{Y_{\Delta
L}}{Y^{\text{eq}}_{L}}-\Gamma_{N_{2}\to
N_{1}\chi}\biggl{(}\frac{Y_{N_{2}}}{Y^{\text{eq}}_{N_{2}}}-\frac{Y_{N_{1}}}{Y^{\text{eq}}_{N_{1}}}\frac{Y_{\chi}}{Y_{\chi}^{\text{eq}}}\biggr{)}\Biggr{]},$
(4.4) $\displaystyle Hx\frac{dY_{\Delta L}}{dx}\approx$
$\displaystyle\;\sum_{i}\frac{K_{1}(m_{N_{i}}/T)}{K_{2}(m_{N_{i}}/T)}Y^{\text{eq}}_{N_{i}}\Biggl{[}\tilde{\Gamma}_{i}\tilde{\epsilon}_{i}\biggl{(}\frac{Y_{N_{i}}}{Y^{\text{eq}}_{N_{i}}}-1\biggr{)}-\frac{1}{2}\tilde{\Gamma}_{i}\frac{Y_{\Delta
L}}{Y^{\text{eq}}_{L}}$
$\displaystyle\;+\Gamma_{i}\epsilon_{i}\biggl{(}\frac{Y_{N_{i}}}{Y^{\text{eq}}_{N_{i}}}-\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)}-\frac{1}{2}\Gamma_{i}\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\frac{Y_{\Delta
L}}{Y^{\text{eq}}_{L}}\Biggr{]},$ (4.5) $\displaystyle
Hx\frac{dY_{\chi}}{dx}\approx$
$\displaystyle\;\sum_{i}\frac{K_{1}(m_{N_{i}}/T)}{K_{2}(m_{N_{i}}/T)}Y^{\text{eq}}_{N_{i}}\Biggl{[}\Gamma_{i}\biggl{(}\frac{Y_{N_{i}}}{Y^{\text{eq}}_{N_{i}}}-\frac{Y_{\chi}}{Y_{\chi}^{\text{eq}}}\biggr{)}+\frac{1}{2}\Gamma_{i}\epsilon_{i}\frac{Y_{\chi}}{Y_{\chi}^{\text{eq}}}\frac{Y_{\Delta
L}}{Y^{\text{eq}}_{L}}\Biggr{]}$
$\displaystyle\;+Y^{\text{eq}}_{N_{2}}\frac{K_{1}(m_{N_{2}}/T)}{K_{2}(m_{N_{2}}/T)}\Gamma_{N_{2}\to
N_{1}\chi}\biggl{(}\frac{Y_{N_{2}}}{Y^{\text{eq}}_{N_{2}}}-\frac{Y_{N_{1}}}{Y^{\text{eq}}_{N_{1}}}\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)},$
(4.6)

where the dimensionless parameter $x$ is introduced by $x=m_{N_{1}}/T$,
$K_{n}(z)$ denotes the modified Bessel function of second kind, and off-shell
scattering contributions are dropped for simplifying the analysis. For the
three-body decay process, the on-shell contributions of RH neutrinos are
deducted in order to avoid the double counting when deriving the Boltzmann
equations (4.3)–(4.6).

|  $N_{2}$ decay dominant |  $N_{1}$ decay dominant
---|---|---
weak washout | $\mathcal{Y}^{\mathrm{FI}}_{\Delta L}(z_{1})$ | $Y^{\mathrm{FI}}_{\Delta L}(\infty)$
strong washout | —– | $\mathcal{Y}^{\mathrm{WO}}_{\Delta L}(\infty)$
Table 1: Four possible patterns of dominant contributions to the lepton
asymmetry.

The RH neutrinos are first generated by their interactions to the thermal
bath, and the lepton asymmetry and the scalar $\chi$ are produced via the RH
neutrino decays. While the total amounts of the lepton asymmetry and $\chi$
are related, the washout (inverse decay) effect via the neutrino Yukawa
couplings deduces only the asymmetry. The Boltzmann equation for the lepton
asymmetry approximately becomes

$\displaystyle\frac{dY_{\Delta
L}}{dx}\approx\sum_{i}\frac{1}{Hx}\frac{K_{1}(m_{N_{i}}x/m_{N_{1}})}{K_{2}(m_{N_{i}}x/m_{N_{1}})}\biggl{[}\Gamma_{i}\epsilon_{i}\biggl{(}Y_{N_{i}}-\frac{Y^{\text{eq}}_{N_{i}}Y_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)}-\frac{1}{2}\biggl{(}\tilde{\Gamma}_{i}+\Gamma_{i}\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)}\frac{Y^{\text{eq}}_{N_{i}}}{Y^{\text{eq}}_{L}}Y_{\Delta
L}\biggr{]},$

where the $\tilde{\epsilon}_{i}$ terms have been dropped since we are
interested in the asymmetry produced by the three-body decay process. The
first term of RHS produces the asymmetry via the $N_{i}$ decay and the second
term denotes the washout term. Depending on the $N_{1}$ or $N_{2}$ decay
process being dominant to the production and the washout effect being strong
or weak, there are four possibilities of the main lepton asymmetry as listed
in Table 1. We also show in Fig. 5 the typical time evolution of the yields
corresponding to these four patterns, respectively. The top panels in Fig. 5
are the results for the weak washout, and the bottom panels are for the strong
one. Further, the $N_{2}$ decay process is the main contribution to the
asymmetry in the left panels in Fig. 5 and the $N_{1}$ process is dominant in
the right panels. Roughly speaking, the relic of lepton asymmetry in the
bottom line is given by that in the top line times the washout suppression
factor.

Figure 5: The time evolution of the yields. In the top panels, the washout
effect is weak and the asymmetry is produced by the freeze-in mechanism of the
RH neutrino decays. The washout works well in the bottom panels. In the left
panels, the $N_{2}$ decay is the dominant production channels and $N_{1}$
dominant in the right panels. The solid lines are numerical solutions of the
Boltzmann equations [Eqs. (4.3)–(4.6)]. The orange dashed line is evaluated by
solving the Boltzmann equations of $N_{1}$, $\chi$ and $\Delta L$ with
$Y_{N_{2}}=Y^{\text{eq}}_{N_{2}}$. The dot-dashed dark red, dark green and
dark blue lines are given by Eqs. (4.7), (4.14) and (4.15), respectively.

In the case that the freeze-in production from $N_{2}$ is dominant (top-left
panel), we have to take the washout effect into account (for the detail, see
Appendix C.2). The lepton asymmetry of this case is evaluated by

$\displaystyle\mathcal{Y}^{\mathrm{FI}}_{\Delta
L}(z_{1})=\int_{0}^{z_{1}}dx^{\prime}\,\mathcal{F}_{2}(x^{\prime})\exp\biggl{[}-\int_{x^{\prime}}^{z_{1}}dx^{\prime\prime}\,W_{2}(x^{\prime\prime})\biggr{]},$
(4.7)

where

$\displaystyle W_{2}$
$\displaystyle:=\frac{1}{2Hx}\frac{K_{1}(m_{N_{2}}x/m_{N_{1}})}{K_{2}(m_{N_{2}}x/m_{N_{1}})}\biggl{(}\tilde{\Gamma}_{2}+\Gamma_{2}\frac{\bm{Y}_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)}\frac{Y^{\text{eq}}_{N_{2}}}{Y^{\text{eq}}_{L}}+\frac{1}{2Hx}\frac{K_{1}(x)}{K_{2}(x)}\biggl{(}\tilde{\Gamma}_{1}+\Gamma_{1}\frac{\bm{Y}_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)}\frac{Y^{\text{eq}}_{N_{2}}}{Y^{\text{eq}}_{L}},$
(4.8) $\displaystyle\mathcal{F}_{2}=$
$\displaystyle\;\frac{1}{Hx}\frac{K_{1}(m_{N_{2}}x/m_{N_{1}})}{K_{2}(m_{N_{2}}x/m_{N_{1}})}\Gamma_{2}\epsilon_{2}\biggl{(}\bm{Y}_{N_{2}}-\frac{Y^{\text{eq}}_{N_{2}}}{Y^{\text{eq}}_{\chi}}\bm{Y}_{\chi}\biggr{)}+\frac{1}{Hx}\frac{K_{1}(x)}{K_{2}(x)}\Gamma_{1}\epsilon_{1}\biggl{(}\bm{Y}_{N_{1}}-\frac{Y^{\text{eq}}_{N_{1}}}{Y^{\text{eq}}_{\chi}}\bm{Y}_{\chi}\biggr{)}.$
(4.9) $\displaystyle\bm{Y}_{N_{2}}$
$\displaystyle=\frac{45\sqrt{10}}{2\pi^{5}g^{1/2}_{*}g^{S}_{*}}\frac{M_{P}m_{N_{2}}\tilde{\Gamma}_{2}}{m_{N_{1}}^{3}}x^{3},$
(4.10) $\displaystyle\bm{Y}_{\chi}$
$\displaystyle=\frac{135}{2\pi^{6}g_{*}g_{*}^{S}}\frac{M_{P}^{2}m_{N_{2}}^{2}\tilde{\Gamma}_{2}(\Gamma_{N_{2}\to
N_{1}\chi}+\Gamma_{2})}{m_{N_{1}}^{6}}x^{6},$ (4.11)
$\displaystyle\bm{Y}_{N_{1}}$
$\displaystyle=\frac{45\sqrt{10}}{2\pi^{5}g^{1/2}_{*}g^{S}_{*}}\frac{M_{P}\tilde{\Gamma}_{1}}{m_{N_{1}}^{2}}x^{3}+\frac{135}{2\pi^{6}g_{*}g_{*}^{S}}\frac{M_{P}^{2}m_{N_{2}}^{2}\tilde{\Gamma}_{2}(\Gamma_{N_{2}\to
N_{1}\chi}+\Gamma_{2})}{m_{N_{1}}^{6}}x^{6}.$ (4.12)

$z_{1}$ is given by using these $\bm{Y}_{N_{2}}$ and $\bm{Y}_{\chi}$ as

$\displaystyle
z_{1}=\mathop{\mathrm{min}}(z_{N_{2}},z_{\chi}),\quad\bm{Y}_{N_{2}}(z_{N_{2}})=\frac{45}{\pi^{4}g^{S}_{*}},\quad\bm{Y}_{\chi}(z_{\chi})=\frac{45}{2\pi^{4}g^{S}_{*}},$
(4.13)

and $z_{N_{2}}$ ($z_{\chi}$) is regarded as the time of $N_{2}$ ($\chi$) being
close to the thermal equilibrium. The washout gives the exponential
suppression through $e^{-\int dx^{\prime}\,W_{2}(x^{\prime})}$. On the other
hand, if the $N_{1}$ process is dominant, the asymmetry is mainly produced by
the ordinary freeze-in mechanism via $N_{1}$ decay such as the FIMP dark
matter [19] (top-right panel in Fig. 5). The relic of the lepton asymmetry is
written as

$\displaystyle Y^{\mathrm{FI}}_{\Delta
L}(\infty)\approx\frac{135\sqrt{10}}{2\pi^{5}g^{1/2}_{*}g_{*}^{S}}\frac{M_{P}\Gamma_{1}\epsilon_{1}}{m_{N_{1}}^{2}}\int_{0}^{\infty}dx\,x^{3}K_{1}(x)=\frac{405\sqrt{10}}{4\pi^{4}g^{1/2}_{*}g_{*}^{S}}\frac{M_{P}\Gamma_{1}\epsilon_{1}}{m_{N_{1}}^{2}}.$
(4.14)

If the washout is strong and dilutes the relic from the $N_{1}$ decay (bottom
right panel in Fig. 5), the lepton asymmetry is evaluated as

$\displaystyle\mathcal{Y}^{\mathrm{WO}}_{\Delta
L}(\infty)=\int_{0}^{\infty}dx^{\prime}\,\mathcal{F}_{1}(x^{\prime})\exp\biggl{[}-\int_{x^{\prime}}^{\infty}dx^{\prime\prime}\,W_{1}(x^{\prime\prime})\biggr{]},$
(4.15)

where the washout function of this case is given by

$\displaystyle
W_{1}:=\frac{1}{2Hx}\frac{K_{1}(m_{N_{2}}x/m_{N_{1}})}{K_{2}(m_{N_{2}}x/m_{N_{1}})}(\tilde{\Gamma}_{2}+\Gamma_{2})\frac{Y^{\text{eq}}_{N_{2}}}{Y^{\text{eq}}_{L}}+\frac{1}{2Hx}\frac{K_{1}(x)}{K_{2}(x)}(\tilde{\Gamma}_{1}+\Gamma_{1})\frac{Y^{\text{eq}}_{N_{1}}}{Y^{\text{eq}}_{L}},$
(4.16)

and we introduce the following function

$\displaystyle\mathcal{F}_{1}:=$
$\displaystyle\frac{1}{Hx}\frac{K_{1}(m_{N_{2}}x/m_{N_{1}})}{K_{2}(m_{N_{2}}x/m_{N_{1}})}\Gamma_{2}\epsilon_{2}\biggl{(}\frac{\tilde{\Gamma}_{2}+\Gamma_{2}}{m_{N_{2}}}\biggr{)}Y^{\text{eq}}_{N_{2}}+\frac{1}{Hx}\frac{K_{1}(x)}{K_{2}(x)}\Gamma_{1}\epsilon_{1}\biggl{(}\frac{\tilde{\Gamma}_{1}+\Gamma_{1}}{m_{N_{1}}}\biggr{)}Y^{\text{eq}}_{N_{1}}.$
(4.17)

###  4.2 Parameter space

Based on the analysis of the Boltzmann equations in the previous subsection,
we discuss the parameter spaces of the model where the lepton asymmetry is
properly generated. The Majorana masses of the RH neutrinos are fixed to
$m_{N_{1}}=10^{10}$ GeV, $m_{N_{2}}=10^{12}$ GeV, and $m_{N_{3}}=10^{14}$ GeV.
$N_{3}$ is heavy so that it is assumed not to join the initial and final
states of our considering processes. The scalar $\chi$ is light and its mass
is assumed in this work to be $m_{\chi}=10^{-3}$ GeV, but its detailed value
does not largely affect the asymmetry. As for the CI parameters (the degrees
of freedom of neutrino Yukawa couplings), we consider the following two
typical patterns of parameters:

  1. (I)

Real CI parameters

$\displaystyle\omega_{12}=\frac{1}{4},\quad\omega_{23}=\frac{3}{7},\quad\omega_{13}=0$
(4.18)

  2. (II)

Pure imaginary CI parameters

$\displaystyle\omega_{12}=2i,\quad\omega_{23}=i,\quad\omega_{13}=0.$ (4.19)

With these parameter sets, the asymmetry from the SM particle loops vanish,
$\tilde{\epsilon}^{(LH)}_{i}=0$, found from Eq. (2.8).

Figure 6: The density plots of the lepton asymmetry in the
$(\mathop{\mathrm{Re}}\xi_{12},\mathop{\mathrm{Im}}\xi_{12})$ plane. The left
panel is the parameter space for the normal mass hierarchy and all the CI
parameters real-valued. In other two panels, the inverted mass hierarchy is
assumed and the middle (right) panel shows the parameter space with all the CI
parameters being real (pure imaginary). The light blue regions properly
produce the observed baryon asymmetry in the Universe.

Fig. 6 shows the relic of the lepton asymmetry $Y_{\Delta L}(\infty)$ for

$\displaystyle\mathop{\mathrm{Re}}\xi=\begin{pmatrix}\frac{1}{20}&\mathop{\mathrm{Re}}\xi_{12}\\\
\mathop{\mathrm{Re}}\xi_{12}&\frac{1}{10}\end{pmatrix},\qquad\mathop{\mathrm{Im}}\xi=\begin{pmatrix}\frac{1}{2}&\mathop{\mathrm{Im}}\xi_{12}\\\
\mathop{\mathrm{Im}}\xi_{12}&1\end{pmatrix}.$ (4.20)

The light blue regions properly produce the observed baryon asymmetry. The CP
asymmetry parameters Eqs. (3.7), (3.8) and (3.11)–(3.13) are roughly
determined by the large values in $\xi^{2}$, which then lead to the contours
seen in Fig. 6.

The left panel of Fig. 6 is the parameter space for the normal mass hierarchy
of neutrinos and real CI parameters, where the washout effect is less
dominant. In this case, larger values of couplings lead larger lepton
asymmetry through the freeze-in like production of $N_{1}$. Then the lepton
asymmetry reduces due to weaker couplings, but the asymmetry becomes large
again when $\xi_{12}$ is small enough. This is because the freeze-in like
production of $N_{2}$ is dominant. If the washout effect works well as in the
inverted mass hierarchy, the lepton asymmetry is suppressed as shown in the
middle and right panels of Fig. 6. When the CI parameters are real, the final
value of the lepton asymmetry tends to converge to a value slightly smaller
than $10^{-10}$ if $\xi$ are feeble couplings, which means the relic asymmetry
is determined by the neutrino Yukawa couplings. However the washout is
relatively stronger if $\xi$ is larger, and the relic is more suppressed. On
the other hand, the washout effect of neutrino Yukawa couplings works well and
the lepton asymmetry can only take a tiny value. In this case, an enough large
coupling is needed for a large amount of $Y_{\Delta L}$ produced in the early
Universe to explain the observed asymmetry even when it is washed out. From
these observations, the tree-level leptogenesis from the three-body decay of
RH neutrinos with a flavorful scalar does not favor the inverted mass
hierarchy of neutrinos.

Figure 7: The density plots of the lepton asymmetry in the
$(\mathop{\mathrm{Im}}\xi_{12},\mathop{\mathrm{Re}}\xi_{11})$,
$(\mathop{\mathrm{Im}}\xi_{12},\mathop{\mathrm{Re}}\xi_{22})$ and
$(\mathop{\mathrm{Re}}\xi_{11},\mathop{\mathrm{Re}}\xi_{22})$ planes with the
parameterization (4.21). The normal mass hierarchy and real CI parameters are
adopted for realizing $Y_{\Delta L}(\infty)\sim 10^{-10}$.

We now discuss a singlet scalar $\chi$ and its general form of couplings to
the RH neutrinos. If $\chi$ is something like a pseudo-Nambu-Goldstone boson
associated with the lepton number symmetry, the imaginary parts of the
diagonal couplings $\xi_{ii}$ tend to be dominant. Keeping in mind this fact,
we take the coupling $\xi$ as

$\displaystyle\mathop{\mathrm{Re}}\xi=\begin{pmatrix}\mathop{\mathrm{Re}}\xi_{11}&0\\\
0&\mathop{\mathrm{Re}}\xi_{22}\end{pmatrix},\qquad\mathop{\mathrm{Im}}\xi=\begin{pmatrix}10^{-2}&\mathop{\mathrm{Im}}\xi_{12}\\\
\mathop{\mathrm{Im}}\xi_{12}&1\end{pmatrix}.$ (4.21)

The parameter spaces are shown in Fig. 7 when $\mathop{\mathrm{Re}}\xi_{11}$,
$\mathop{\mathrm{Re}}\xi_{22}$ and $\mathop{\mathrm{Im}}\xi_{12}$ are assumed
to be the modifications to the imaginary diagonal elements. In this situation,
the lepton asymmetry is produced by the freeze-in mechanism of $N_{1}$ or
$N_{2}$ as shown in the top panels of Fig. 5, and there exist the allowed
regions between the parameter spaces of over- and under-productions of the
asymmetry.

On the other hand, the parameter spaces of the imaginary diagonal elements are
shown in Fig. 8. For evaluating the lepton asymmetry in these planes,
$\mathop{\mathrm{Im}}\xi_{12}$ and one of $\mathop{\mathrm{Im}}\xi_{12}$ or
$\mathop{\mathrm{Re}}\xi_{22}$ are fixed to $10^{-1}$ or $10^{-5}$, and the
normal mass hierarchy for the neutrino masses is assumed. As seen from these
figures, larger couplings lead to larger relic asymmetry, similar to the above
cases of the normal mass hierarchy, and $\mathop{\mathrm{Im}}\xi_{11}$,
$\mathop{\mathrm{Im}}\xi_{22}$ $\sim 10^{-(1\text{--}2)}$ are favored.

Figure 8: The density plots of the lepton asymmetry in the
$(\mathop{\mathrm{Im}}\xi_{11},\mathop{\mathrm{Im}}\xi_{22})$ plane for the
normal mass hierarchy. (Left) $\mathop{\mathrm{Im}}\xi_{12}=10^{-1}$,
$\mathop{\mathrm{Re}}\xi_{12}=10^{-1}$. (Middle)
$\mathop{\mathrm{Im}}\xi_{12}=10^{-5}$,
$\mathop{\mathrm{Re}}\xi_{12}=10^{-1}$. (Right)
$\mathop{\mathrm{Im}}\xi_{12}=10^{-5}$,
$\mathop{\mathrm{Re}}\xi_{22}=10^{-1}$.

###  4.3 Property of $\chi$ scalar

In the present work, the scalar field $\chi$ has two important property that
(i) it has the complex coupling $\xi_{ij}$ to the parent decay particles and
(ii) its remnant $Y_{\chi}$ in the present Universe is large. That is a
general result to have an appropriate order of baryon asymmetry generated from
the three-body decay including $\chi$.

Let us first discuss about the coupling $\xi_{ij}$. The analysis in the above
shows that $\xi_{ij}$ should be complex-valued so that the CP asymmetry is
properly produced through the three-body decay. Further the flavor-changing
components $\xi_{ij}$ ($i\neq j$) can play an important role for the
leptogenesis. These nature of couplings put some constraint on the property of
the field $\chi$, for example, a dynamical completion in high-energy regime.

  * •

Nambu-Goldstone boson :
xx A simple example for scalar couplings to neutrinos is the one whose vacuum
expectation value (VEV) gives the masses of RH neutrinos $N_{i}$. If such a
scalar is complex, the coupling has the lepton number symmetry (the chiral
rotation of $N_{i}$) and the scalar VEV breaks it. As a result of symmetry
breaking, a Nambu-Goldstone boson (NGB) appears [20, 21, 22], which couples to
$N_{i}$ and can be identified to $\chi$. This is a typical and the simplest
dynamical realization of the present model. It is however noticed that, if the
NGB nature is exact, $\chi$ is massless and its couplings to $N_{i}$ are real
and flavor diagonal, i.e.,
$\xi_{ij}=c_{i}\,\delta_{ij},~{}c_{i}\in\mathbb{R}$. As mentioned above, such
a too simple form of scalar couplings is not suitable for generating the
asymmetry.

  * •

Flavor-dependent interaction :
xx There are several ways to ameliorate the NGB problem, too simple
$\xi_{ij}$, found in the above simplest setup. The first is to introduce multi
scalars which couple to the RH neutrinos $N_{i}$. These scalars generally
couple to each other and develop nonzero VEVs. The resultant NGB, which is
identified to $\chi$, is a linear combination of (the phases of) these
scalars. As a result, the flavor structure becomes different between the
masses and the scalar couplings of RH neutrinos.
xx A more interesting realization is to assign generation-dependent charges to
leptons under some flavor symmetry. The assignment is chosen so that the
masses (and Yukawa couplings) are forbidden and then induced by some VEV of
complex scalar, charged under the same flavor symmetry [23]. In this case,
$\chi$ is possibly the NGB of flavor symmetry and its coupling to leptons is
determined by the flavor structure of lepton masses and symmetry charges.
Unless the masses and charges have exactly the same structure, the $\chi$
couplings are generally flavor dependent [24] (and complex valued). Typical
examples may be constructed with flavor $U(1)$ symmetry for fermion mass
hierarchy, and non-universal anomaly-free lepton numbers such as
$L_{\mu}-L_{\tau}$ and others [25, 26].

  * •

Radiative corrections :
xx The NGB property is not exactly hold if the original symmetry is somehow
violated via explicit breaking terms. A well-known example is the scalar mass
term which does not respect (global) symmetry, and hence the NGB acuires its
nonzero mass (the “pion” mass) at classical level. With such explicit
breaking, one generally expects to have non-vanishing radiative corrections to
unusual NGB couplings and resolve the NGB problem mentioned above. There are
phenomenological analysis for large breaking, e.g., a heavy NGB of the lepton
number symmetry [27, 28], where suitably size of corrections might be
obtained.

The large remnant of $\chi$ is another characteristic result of the model. We
here discuss two approaches to this issue: (i) additional scalar interactions.
(ii) a very light $\chi$. The first resolution is to introduce additional
interaction such that it suppresses the scalar abundance somewhere in the
thermal history. However, when $\chi$ is the NGB of high-scale symmetry, its
large decay constant generally suppresses the $\chi$ couplings to the SM
fields and cannot give an enough suppression to the $\chi$ abundance. For
example, when $\chi$ is the NGB of lepton number symmetry which couples to the
RH neutrinos, a typical ratio $\Gamma/H(T=m_{\chi})$ is
$10^{-20}\,\xi^{2}(m_{\chi}/\text{GeV})$. We are therefore lead to the
situation that the scalar including three-body decay is not a NGB-like pseudo-
scalar $\chi$ but some heavy real scalar $\rho$ (heavier than the electroweak
scale). As long as $\rho$ is rather lighter than $N_{i}$, the analysis of
leptogenesis from three-body decay discussed in this paper is unchanged even
for $N_{i}\to L_{j}H\rho$. Dynamical examples of $\rho$ are a partner of NGB
(a radial fluctuation around a VEV), a massive scalar in the multi scalar
scenario, and so on. A main difference between $\chi$ and $\rho$ is the
interaction to other (SM) fields. In particular, for the Higgs portal
interaction, $\rho$ interacts with the portal quartic coupling $\lambda$, but
$\chi$ has a suppressed amplitude given by $\lambda E^{2}/(\text{VEV})^{2}$
where $E$ is the energy scale considered. Upon the decoupling $E\sim m_{\rho}$
or $m_{\chi}$, the latter interaction (for $\chi$) is too tiny to reduce the
abundance, and the former one (for $\rho$) can be used for the suppression.
Whether $\rho$ is the dark matter component in the Universe or not depends on
the model parameters, though both are possible to realize.

The second option is to assume that $\chi$ is very light and does not
contribute much to the mass density of the present Universe. The relic
abundance of $\chi$ is given by
$\Omega_{\chi}h^{2}=m_{\chi}s_{0}Y_{\chi,0}/(\varepsilon_{c,0}/h^{2})$ with
the today entropy and critical energy densities,
$s_{0}=2891~{}\mathrm{cm}^{-3}$ and $\varepsilon_{c,0}=5.16(h/0.7)^{2}$
$\mathrm{GeV}/\mathrm{m}^{3}$. If this abundance is required to be less than
the observed dark matter density times a small ratio $\delta$, that implies
the upper bound on $m_{\chi}$,

$\displaystyle
m_{\chi}<\delta\,\frac{\Omega_{\text{DM}}\varepsilon_{c,0}}{s_{0}Y_{\chi}^{\text{eq}}}=1.89\times
10^{-7}\,\delta\;[\text{GeV}]$ (4.22)

where we have assumed $Y_{\chi,0}$ is equal to the equilibrium value
$Y_{\chi}^{\text{eq}}$. For example, $\chi$ has a sub-eV mass for
$\delta=10^{-3}$ but weighs more than the current temperature. Furthermore the
interaction of $\chi$ to the SM sector is generally suppressed by
$\mathcal{O}(m_{\nu})$. Such a very light and feebly-interacting scalar may be
harmlessly floating in the present Universe.

##  5 Summary

We have studied the lepton asymmetry produced by the tree-level RH neutrino
three-body decays with a singlet scalar field. In order to cover the general
case, we considered the scalar and pseudo-scalar couplings between the scalar
field and RH neutrinos. We evaluated the RH neutrino two-body and three-body
decay widths with the scalar field and the asymmetry parameters. For the decay
widths, the $k=l=1$ contribution tends to be dominant in the lightest $N_{1}$
decay, and the contribution of the resonant $k=l=1$ or the non-resonant
$k=l=2$ is dominant depending on $\xi$ in the heavier $N_{2}$ decay. For the
asymmetry parameters, not only usual cross terms but also the $k=l=1$ can be
dominant to $\epsilon_{1}$, and the $k=l=2$ term is the leading contribution
to $\epsilon_{2}$. In the latter two contributions, the CP asymmetry comes
from a single decay process, which is characteristic of the existence of the
flavorful scalar. We derived the Boltzmann equations and discussed there are
four typical patterns of the lepton asymmetry production depending on the
$N_{1}$ or $N_{2}$ decay process being dominant to the production and the
washout effect being strong or weak. Based on these analysis, we show the
parameter space of the model where the lepton asymmetry is properly generated.
For example, we find the tree-level leptogenesis via the three-body decay with
the flavorful scalar does not favor the inverted mass hierarchy due to the
strong washout suppression, without a help of the two-body decay asymmetry.

In this paper, we consider a simplified model with a single flavorful scalar
interacting to the RH neutrinos and investigate the possibility of the
leptogenesis via the three-body decay with specific values of couplings.
Pursuing the UV origin of this additional scalar such as a pseudo-Nambu-
Goldstone boson and a more detailed analysis of flavor dynamics are important
and left for future study.

## Acknowledgments

The authors thank to Takashi Toma and Takahiro Yoshida for useful comments and
discussions. This work is supported by JSPS Grant-in-Aid for Scientific
Research KAKENHI Grant No. JP20J11901 (YA), JP18H01214 and JP20K03949 (KY).

##  Appendix A Three-body decay widths

We consider for simplicity the two-generation case $N_{1,2}$. The
generalization to more generations is straightforward. We assume $m_{\chi}\ll
m_{N_{i}}$ and no large hierarchy among $m_{N_{i}}$. The masses of charged
leptons and Higgs boson are dropped in the following formulae. The three-body
decay width of the lightest RH neturino $N_{1}$ is found from the full form
(3.2) after the phase space integral as

$\displaystyle\Gamma_{N_{1}\to
L_{j}H\chi}=\sum_{k,\,l}y^{\nu}_{lj}y^{\nu*}_{kj}\Big{(}\xi_{1l}^{*}\xi_{1k}G_{kl}^{A}+\xi_{1l}\xi_{1k}^{*}G_{kl}^{B}+\xi_{1l}^{*}\xi_{1k}^{*}G_{kl}^{C}+\xi_{1l}\xi_{1k}G_{kl}^{D}\Big{)}\,=\,\sum_{k,\,l}\Gamma_{1}^{(kl)},$
(A.1)

where $k,\,l$ indicate the contributions to the amplitudes which contain the
intermediate states are $N_{k,l}$, respectively. For example, $k\neq l$ means
the cross term in the amplitude squared. Each piece $G^{A,B,C,D}$ reads from
the full form (3.2) and not explicitly given here. We instead show the exact
form of $\Gamma_{1}^{(kl)}$, corresponding
$\mathcal{M}_{k}^{*}\mathcal{M}_{l}$, where $\mathcal{M}_{m}$ is the $N_{1}$
three-body decay amplitude with the intermediate state $N_{m}$. The result is
found in the limit of light $m_{\chi}$ as

$\displaystyle\Gamma_{1}^{(11)}=$
$\displaystyle~{}\frac{1}{128\pi^{3}}y^{\nu}_{1j}y^{\nu*}_{1j}\big{[}\,|\xi_{11}|^{2}m_{N_{1}}+\mathop{\mathrm{Re}}(\xi^{2}_{11}\bar{m}_{N_{1}})\big{]}\ln\Bigl{(}\frac{m_{N_{1}}}{m_{\chi}}\Bigr{)},$
(A.2) $\displaystyle\Gamma_{1}^{(12)}=$
$\displaystyle~{}\frac{1}{512\pi^{3}m_{N_{1}}^{3}}y^{\nu}_{2j}y^{\nu*}_{1j}\bigg{[}\xi_{11}\xi_{12}^{*}\bar{m}_{N_{1}}\bar{m}_{N_{2}}^{*}\Big{[}\Big{(}m_{N_{1}}^{2}+m_{N_{2}}^{2}\Big{)}\ln\Big{(}\frac{m_{N_{2}}^{2}}{m_{N_{2}}^{2}-m_{N_{1}}^{2}}\Big{)}-m_{N_{1}}^{2}\Big{]}$
$\displaystyle\quad+2m_{N_{1}}(\xi_{11}\xi_{12}\bar{m}_{N_{1}}+\xi_{11}^{*}\xi_{12}^{*}\bar{m}_{N_{2}}^{*})\Big{[}m_{N_{2}}^{2}\ln\Big{(}\frac{m_{N_{2}}^{2}}{m_{N_{2}}^{2}-m_{N_{1}}^{2}}\Big{)}-m_{N_{1}}^{2}\Big{]}$
$\displaystyle\quad+\xi_{11}^{*}\xi_{12}\Big{[}m_{N_{2}}^{2}(m_{N_{1}}^{2}+m_{N_{2}}^{2})\ln\Big{(}\frac{m_{N_{2}}^{2}}{m_{N_{2}}^{2}-m_{N_{1}}^{2}}\Big{)}-m_{N_{1}}^{2}m_{N_{2}}^{2}-\frac{3}{2}m_{N_{1}}^{4}\Big{]}\,\bigg{]},$
(A.3) $\displaystyle\Gamma_{1}^{(21)}=$ $\displaystyle~{}\Gamma_{1}^{(12)*},$
(A.4) $\displaystyle\Gamma_{1}^{(22)}=$
$\displaystyle~{}\frac{1}{512\pi^{3}m_{N_{1}}^{3}}y^{\nu}_{2j}y^{\nu*}_{2j}\bigg{[}|\xi_{12}|^{2}\Big{[}(5m_{N_{2}}^{4}-m_{N_{1}}^{4})\ln\Big{(}\frac{m_{N_{2}}^{2}}{m_{N_{2}}^{2}-m_{N_{1}}^{2}}\Big{)}-5m_{N_{1}}^{2}m_{N_{2}}^{2}-\frac{5}{2}m_{N_{1}}^{4}\Big{]}$
$\displaystyle\quad+4\mathop{\mathrm{Re}}(\xi_{12}^{2}\bar{m}_{N_{2}})m_{N_{1}}\Big{[}(2m_{N_{2}}^{2}-m_{N_{1}}^{2})\ln\Big{(}\frac{m_{N_{2}}^{2}}{m_{N_{2}}^{2}-m_{N_{1}}^{2}}\Big{)}-2m_{N_{1}}^{2}\Big{]}\,\bigg{]}.$
(A.5)

When $N_{2}$ is much heavier than $N_{1}$, the expressions are reduced to

$\displaystyle\Gamma_{1}^{(12)}=\Gamma_{1}^{(21)*}\approx$
$\displaystyle~{}\frac{m_{N_{1}}\bar{m}_{N_{2}}^{*}}{1024\pi^{3}m_{N_{2}}^{2}}y^{\nu}_{2j}y^{\nu*}_{1j}\,\xi_{12}^{*}(3\xi_{11}\bar{m}_{N_{1}}+2\xi_{11}^{*}m_{N_{1}}),$
(A.6) $\displaystyle\Gamma_{1}^{(22)}\approx$
$\displaystyle~{}\frac{m_{N_{1}}^{3}}{768\pi^{3}m_{N_{2}}^{2}}y^{\nu}_{2j}y^{\nu*}_{2j}|\xi_{12}|^{2}.$
(A.7)

The last equation implies that the $k=l=2$ mode does not induce CP asymmetry
at the leading order and then highly suppressed for the $N_{1}$ decay.

For a heavier RH neutrino, its three-body decay can meet the resonance around
the mass of a lighter intermediate state, and then the width is largely
enhanced. In the present case, the $N_{2}$ decay width $\Gamma_{N_{2}\to
L_{j}H\chi}$ has the enhancement both for $k=l=1$ and the sum of cross terms,
namely, $\Gamma_{2}^{(11)}$ and $\Gamma_{2}^{(12)}+\Gamma_{2}^{(21)}$ in the
similar notation as (A.1). These on-shell contributions to the decay width are
evaluated from the full form (3.2) with the narrow width approximation. For
the $k=l=1$ part, we obtain

$\displaystyle\Gamma_{2}^{(11)}=\frac{m_{N_{1}}}{256\pi^{2}}\frac{m_{N_{2}}^{2}-m_{N_{1}}^{2}}{m_{N_{2}}^{3}\Gamma_{N_{1}}}\,y^{\nu}_{1j}y^{\nu*}_{1j}\Big{[}|\xi_{12}|^{2}(m_{N_{1}}^{2}+m_{N_{2}}^{2})+2\mathop{\mathrm{Re}}(\xi_{12}^{2}\bar{m}_{N_{1}})m_{N_{2}}\Big{]},$
(A.8)

where $\Gamma_{N_{1}}$ in the denominator indicates $N_{1}$ is the real
intermediate state. Compared with the decay width for $N_{2}\to N_{1}\chi$
given in (2.6), the on-shell $k=l=1$ contribution is found

$\displaystyle\Gamma_{2}^{(11)}\approx\frac{m_{N_{1}}}{16\pi\Gamma_{N_{1}}}y^{\nu}_{1j}y^{\nu*}_{1j}\,\Gamma_{N_{2}\to
N_{1}\chi}\,\approx\,\frac{1}{2}\Gamma_{N_{2}\to N_{1}\chi}.$ (A.9)

For the cross-term part, a similar evaluation leads to the on-shell resonant
contribution,

$\displaystyle\Gamma_{2}^{(12)}+\Gamma_{2}^{(21)}=$
$\displaystyle~{}\frac{m_{N_{1}}}{256\pi^{2}m_{N_{2}}^{3}(m_{N_{2}}^{2}-m_{N_{1}}^{2})}\bigg{[}\Gamma_{N_{2}}\Big{[}\tfrac{1}{2}\mathop{\mathrm{Re}}(y^{\nu}_{2j}y^{\nu*}_{1j}\xi_{22}^{*}\xi_{12})(m_{N_{1}}^{2}+m_{N_{2}}^{2})^{2}$
$\displaystyle\quad-4\mathop{\mathrm{Im}}(y^{\nu}_{2j}y^{\nu*}_{1j}\xi_{12}^{*})\mathop{\mathrm{Im}}(\xi_{22})m_{N_{1}}m_{N_{2}}(m_{N_{1}}^{2}+m_{N_{2}}^{2})+2\mathop{\mathrm{Re}}(y^{\nu}_{2j}y^{\nu*}_{1j}\xi_{22}\xi_{12})m_{N_{1}}^{2}m_{N_{2}}^{2}\Big{]}$
$\displaystyle\quad-\big{[}\mathop{\mathrm{Im}}(y^{\nu}_{2j}y^{\nu*}_{1j}\xi_{22}\xi_{12}^{*})m_{N_{1}}+\mathop{\mathrm{Im}}(y^{\nu}_{2j}y^{\nu*}_{1j}\xi_{22}^{*}\xi_{12})m_{N_{2}}\big{]}(m_{N_{2}}^{4}-m_{N_{1}}^{4})$
$\displaystyle\quad-2\big{[}\mathop{\mathrm{Im}}(y^{\nu}_{2j}y^{\nu*}_{1j}\xi_{22}\xi_{12})m_{N_{1}}+\mathop{\mathrm{Im}}(y^{\nu}_{2j}y^{\nu*}_{1j}\xi_{22}^{*}\xi_{12}^{*})m_{N_{2}}\big{]}m_{N_{1}}m_{N_{2}}(m_{N_{2}}^{2}-m_{N_{1}}^{2})\,\bigg{]}.$
(A.10)

Finally the non-resonant part, $k=l=2$, gives

$\displaystyle\Gamma_{2}^{(22)}=\frac{1}{128\pi^{3}}y^{\nu}_{2j}y^{\nu*}_{2j}\big{[}\,|\xi_{22}|^{2}m_{N_{2}}+\mathop{\mathrm{Re}}(\xi^{2}_{22}\bar{m}_{N_{2}})\big{]}\ln\Bigl{(}\frac{m_{N_{2}}}{m_{\chi}}\Bigr{)}.$
(A.11)

##  Appendix B CP asymmetry (width differences)

For a real particle (a real scalar, a Majorana fermion, etc), the decay to
anti-particle final states is obtained by replacing all coupling constants
with their complex conjugates in the corresponding decay amplitude to particle
final states. In the present case, the three-body decay of $N_{i}$ to the
anti-lepton $\bar{L}_{j}$, the conjugate of Higgs boson $\bar{H}$, and the
real scalar $\chi$ is described by $y^{\nu}\leftrightarrow y^{\nu*}$ and
$\xi\leftrightarrow\xi^{*}$ (and $P_{R}\leftrightarrow P_{L}$) everywhere in
the amplitude. The CP asymmetry is induced at the decay, proportionally to the
difference of decay widths to particles and corresponding anti-particles. As a
result, the asymmetry is originated from the pieces in the decay widths which
are not real with respect to coupling constants.

The decay width to anti-particles is defined in a similar way to the decay to
particles discussed in the previous section, namely,

$\displaystyle\Gamma_{N_{1}\to
L_{j}^{c}H^{\dagger}\chi}=\sum_{k,\,l}\bar{\Gamma}_{1}^{(kl)}.$ (B.1)

The differences of partial widths are written as
$\Delta\Gamma_{1}^{(kl)}=\Gamma_{1}^{(kl)}-\bar{\Gamma}_{1}^{(kl)}$. From the
explicit forms for $\Gamma_{1}^{(kl)}$ and the replacement rule mentioned
above, we obtain

$\displaystyle\Delta\Gamma_{1}^{(11)}=$
$\displaystyle~{}\frac{\Gamma_{N_{1}}}{128\pi^{3}}y^{\nu}_{1j}y^{\nu*}_{1j}\mathop{\mathrm{Im}}(\xi_{11}^{2})\ln\Bigl{(}\frac{m_{N_{1}}}{m_{\chi}}\Bigr{)},$
(B.2) $\displaystyle\Delta\Gamma_{1}^{(12)}+\Delta\Gamma_{1}^{(21)}=$
$\displaystyle~{}\frac{1}{256\pi^{3}m_{N_{1}}^{3}}\bigg{[}\mathop{\mathrm{Im}}(y^{\nu}_{2j}y^{\nu*}_{1j}\xi_{11}\xi_{12}^{*})(\Gamma_{N_{1}}m_{N_{2}}-\Gamma_{N_{2}}m_{N_{1}})$
$\displaystyle\qquad\times\Big{[}\Big{(}m_{N_{1}}^{2}+m_{N_{2}}^{2}\Big{)}\ln\Big{(}\frac{m_{N_{2}}^{2}}{m_{N_{2}}^{2}-m_{N_{1}}^{2}}\Big{)}-m_{N_{1}}^{2}\Big{]}$
$\displaystyle\quad+m_{N_{1}}\big{[}\mathop{\mathrm{Im}}(y^{\nu}_{2j}y^{\nu*}_{1j}\xi_{11}\xi_{12})\Gamma_{N_{1}}-\mathop{\mathrm{Im}}(y^{\nu}_{2j}y^{\nu*}_{1j}\xi_{11}^{*}\xi_{12}^{*})\Gamma_{N_{2}}\big{]}$
$\displaystyle\qquad\times\Big{[}m_{N_{2}}^{2}\ln\Big{(}\frac{m_{N_{2}}^{2}}{m_{N_{2}}^{2}-m_{N_{1}}^{2}}\Big{)}-m_{N_{1}}^{2}\Big{]}\,\bigg{]},$
(B.3) $\displaystyle\Delta\Gamma_{1}^{(22)}=$
$\displaystyle~{}\frac{\Gamma_{N_{2}}}{128\pi^{3}m_{N_{1}}^{2}}y^{\nu}_{2j}y^{\nu*}_{2j}\mathop{\mathrm{Im}}(\xi_{12}^{2})\Big{[}(2m_{N_{2}}^{2}-m_{N_{1}}^{2})\ln\Big{(}\frac{m_{N_{2}}^{2}}{m_{N_{2}}^{2}-m_{N_{1}}^{2}}\Big{)}-2m_{N_{1}}^{2}\Big{]}.$
(B.4)

When $N_{2}$ is much heavier than $N_{1}$, the expressions are reduced to

$\displaystyle\Delta\Gamma_{1}^{(12)}+\Delta\Gamma_{1}^{(21)}\approx$
$\displaystyle~{}\frac{-m_{N_{1}}^{2}\Gamma_{N_{2}}}{512\pi^{3}m_{N_{2}}^{2}}\mathop{\mathrm{Im}}\big{[}y^{\nu}_{2j}y^{\nu*}_{1j}\xi_{12}^{*}(3\xi_{11}+2\xi_{11}^{*})\big{]},$
(B.5) $\displaystyle\Delta\Gamma_{1}^{(22)}\approx$
$\displaystyle~{}\frac{m_{N_{1}}^{4}\Gamma_{N_{2}}}{768\pi^{3}m_{N_{2}}^{4}}y^{\nu}_{2j}y^{\nu*}_{2j}\mathop{\mathrm{Im}}(\xi_{12}^{2}).$
(B.6)

In a similar way, we have the width differences for the $N_{2}$ resonant and
non-resonant three-body decay,

$\displaystyle\Delta\Gamma_{2}^{(11)}=$
$\displaystyle~{}\frac{m_{N_{1}}}{128\pi^{2}}\frac{m_{N_{2}}^{2}-m_{N_{1}}^{2}}{m_{N_{2}}^{2}}\,y^{\nu}_{1j}y^{\nu*}_{1j}\mathop{\mathrm{Im}}(\xi_{12}^{2}),$
(B.7) $\displaystyle\Delta\Gamma_{2}^{(12)}+\Delta\Gamma_{2}^{(21)}=$
$\displaystyle~{}\frac{-m_{N_{1}}}{128\pi^{2}m_{N_{2}}^{3}}\bigg{[}\big{[}\mathop{\mathrm{Im}}(y^{\nu}_{2j}y^{\nu*}_{1j}\xi_{22}\xi_{12}^{*})m_{N_{1}}+\mathop{\mathrm{Im}}(y^{\nu}_{2j}y^{\nu*}_{1j}\xi_{22}^{*}\xi_{12})m_{N_{2}}\big{]}(m_{N_{1}}^{2}+m_{N_{2}}^{2})$
$\displaystyle\quad+2\big{[}\mathop{\mathrm{Im}}(y^{\nu}_{2j}y^{\nu*}_{1j}\xi_{22}\xi_{12})m_{N_{1}}+\mathop{\mathrm{Im}}(y^{\nu}_{2j}y^{\nu*}_{1j}\xi_{22}^{*}\xi_{12}^{*})m_{N_{2}}\big{]}m_{N_{1}}m_{N_{2}}\,\bigg{]},$
(B.8) $\displaystyle\Delta\Gamma_{2}^{(22)}=$
$\displaystyle~{}\frac{\Gamma_{N_{2}}}{128\pi^{3}}y^{\nu}_{2j}y^{\nu*}_{2j}\mathop{\mathrm{Im}}(\xi^{2}_{22})\ln\Bigl{(}\frac{m_{N_{2}}}{m_{\chi}}\Bigr{)}.$
(B.9)

##  Appendix C Boltzmann equations and asymmetry formulae

###  C.1 Boltzmann equations

The Boltzmann equations for the system of Eq. (2.1) are given by

$\displaystyle Hx\frac{dY_{N_{1}}}{dx}=$
$\displaystyle\;\frac{K_{1}(m_{N_{1}}/T)}{K_{2}(m_{N_{1}}/T)}Y_{N_{1}}^{\text{eq}}\biggl{[}\sum_{j}\tilde{\Gamma}_{1j}\biggl{(}1-\frac{Y_{N_{1}}}{Y_{N_{1}}^{\text{eq}}}\biggr{)}-\frac{1}{2}\sum_{j}\tilde{\Gamma}_{1j}\tilde{\epsilon}_{1j}\frac{Y_{\Delta
L_{j}}}{Y^{\text{eq}}_{L}}$
$\displaystyle\;+\sum_{j}\Gamma_{1j}\biggl{(}\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}-\frac{Y_{N_{1}}}{Y^{\text{eq}}_{N_{1}}}\biggr{)}-\frac{1}{2}\sum_{j}\Gamma_{1j}\epsilon_{1j}\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\frac{Y_{\Delta
L_{j}}}{Y^{\text{eq}}_{L}}\biggr{]}$
$\displaystyle\;+\frac{K_{1}(m_{N_{2}}/T)}{K_{2}(m_{N_{2}}/T)}Y^{\text{eq}}_{N_{2}}\Gamma_{N_{2}\to
N_{1}\chi}\biggl{(}\frac{Y_{N_{2}}}{Y^{\text{eq}}_{N_{2}}}-\frac{Y_{N_{1}}}{Y^{\text{eq}}_{N_{1}}}\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)}+C_{\text{scat.}},$
(C.1) $\displaystyle Hx\frac{dY_{N_{2}}}{dx}=$
$\displaystyle\;\frac{K_{1}(m_{N_{2}}/T)}{K_{2}(m_{N_{2}}/T)}Y^{\text{eq}}_{N_{2}}\biggl{[}\sum_{j}\tilde{\Gamma}_{2j}\biggl{(}1-\frac{Y_{N_{2}}}{Y^{\text{eq}}_{N_{2}}}\biggr{)}-\frac{1}{2}\sum_{j}\tilde{\Gamma}_{2j}\tilde{\epsilon}_{2j}\frac{Y_{\Delta
L_{j}}}{Y^{\text{eq}}_{L}}$
$\displaystyle\;+\sum_{j}\Gamma_{2j}\biggl{(}\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}-\frac{Y_{N_{2}}}{Y^{\text{eq}}_{N_{2}}}\biggr{)}-\frac{1}{2}\sum_{j}\Gamma_{2j}\epsilon_{2j}\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\frac{Y_{\Delta
L_{j}}}{Y^{\text{eq}}_{L}}$ $\displaystyle\;-\Gamma_{N_{2}\to
N_{1}\chi}\biggl{(}\frac{Y_{N_{2}}}{Y^{\text{eq}}_{N_{2}}}-\frac{Y_{N_{1}}}{Y^{\text{eq}}_{N_{1}}}\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)}\biggr{]}+C_{\text{scat.}},$
(C.2) $\displaystyle Hx\frac{dY_{\Delta L_{i}}}{dx}=$
$\displaystyle\;\sum_{j}\frac{K_{1}(m_{N_{j}}/T)}{K_{2}(m_{N_{j}}/T)}Y^{\text{eq}}_{N_{j}}\Biggl{[}\tilde{\Gamma}_{ji}\tilde{\epsilon}_{ji}\biggl{(}\frac{Y_{N_{j}}}{Y^{\text{eq}}_{N_{j}}}-1\biggr{)}-\frac{1}{2}\tilde{\Gamma}_{ji}\frac{Y_{\Delta
L_{i}}}{Y^{\text{eq}}_{L}}$
$\displaystyle\;+\Gamma_{ji}\epsilon_{ji}\biggl{(}\frac{Y_{N_{j}}}{Y^{\text{eq}}_{N_{j}}}-\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)}-\frac{1}{2}\Gamma_{ji}\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\frac{Y_{\Delta
L_{j}}}{Y^{\text{eq}}_{L}}\Biggr{]}$ $\displaystyle\;+C_{\text{scat.}},$ (C.3)
$\displaystyle Hx\frac{dY_{\chi}}{dx}=$
$\displaystyle\;\sum_{i,j}\frac{K_{1}(m_{N_{i}}/T)}{K_{2}(m_{N_{i}}/T)}Y^{\text{eq}}_{N_{i}}\biggl{[}\Gamma_{ij}\biggl{(}\frac{Y_{N_{i}}}{Y^{\text{eq}}_{N_{i}}}-\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)}+\frac{1}{2}\Gamma_{ij}\epsilon_{ij}\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\frac{Y_{\Delta
L}}{Y^{\text{eq}}_{L}}\biggr{]}$
$\displaystyle\;+\frac{K_{1}(m_{N_{2}}/T)}{K_{2}(m_{N_{2}}/T)}Y^{\text{eq}}_{N_{2}}\Gamma_{N_{2}\to
N_{1}\chi}\biggl{(}\frac{Y_{N_{2}}}{Y^{\text{eq}}_{N_{2}}}-\frac{Y_{N_{1}}}{Y^{\text{eq}}_{N_{1}}}\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)}+C_{\text{scat.}}.$
(C.4)

$C_{\text{scat.}}$ denotes the collision terms of the scattering divided by
the entropy density with the on-shell contribution removed. In deriving these
equations, the SM particles, in particular the leptons and the Higgs bosons,
are assumed to be in thermal equilibrium:

$\displaystyle Y_{L_{j}}=Y^{\text{eq}}_{L}+\frac{1}{2}Y_{\Delta L_{j}},\quad
Y_{\bar{L}_{j}}=Y^{\text{eq}}_{L}-\frac{1}{2}Y_{\Delta L_{j}},\quad
Y_{H}=Y_{\bar{H}}=Y^{\text{eq}}_{H},$ (C.5)

with the lepton asymmetry of $j$-th generation $Y_{\Delta L_{j}}$. In this
paper, we analyze the Boltzmann equations with the single-flavored
approximation. The total lepton asymmetry is defined by

$\displaystyle Y_{\Delta L}:=\sum_{i}Y_{\Delta L_{i}},$ (C.6)

and the collision terms are approximated as in the following forms:

$\displaystyle\sum_{i}\Gamma_{ji}\frac{Y_{\Delta
L_{i}}}{Y^{\text{eq}}_{L_{i}}}\sim\Gamma_{j}\frac{Y_{\Delta
L}}{Y^{\text{eq}}_{L}},\quad\sum_{i}\Gamma_{ji}\epsilon_{ji}\frac{Y_{\Delta
L_{i}}}{Y^{\text{eq}}_{L_{i}}}\sim\Gamma_{j}\epsilon_{j}\frac{Y_{\Delta
L}}{Y^{\text{eq}}_{L}},$ (C.7)
$\displaystyle\sum_{i}\tilde{\Gamma}_{ji}\frac{Y_{\Delta
L_{i}}}{Y^{\text{eq}}_{L_{i}}}\sim\tilde{\Gamma}_{j}\frac{Y_{\Delta
L}}{Y^{\text{eq}}_{L}},\quad\sum_{i}\tilde{\Gamma}_{ji}\tilde{\epsilon}_{ji}\frac{Y_{\Delta
L_{i}}}{Y^{\text{eq}}_{L_{i}}}\sim\tilde{\Gamma}_{j}\tilde{\epsilon}_{j}\frac{Y_{\Delta
L}}{Y^{\text{eq}}_{L}}.$ (C.8)

Using these equations, the Boltzmann equations (C.1)–(C.4) are rewritten as

$\displaystyle Hx\frac{dY_{N_{1}}}{dx}\approx$
$\displaystyle\;\frac{K_{1}(m_{N_{1}}/T)}{K_{2}(m_{N_{1}}/T)}Y^{\text{eq}}_{N_{1}}\Biggl{[}\tilde{\Gamma}_{1}\biggl{(}1-\frac{Y_{N_{1}}}{Y^{\text{eq}}_{N_{1}}}\biggr{)}-\frac{1}{2}\tilde{\Gamma}_{1}\tilde{\epsilon}_{1}\frac{Y_{\Delta
L}}{Y^{\text{eq}}_{L}}+\Gamma_{1}\biggl{(}-\frac{Y_{N_{1}}}{Y^{\text{eq}}_{N_{1}}}+\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)}$
$\displaystyle\;-\frac{1}{2}\Gamma_{1}\epsilon_{1}\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\frac{Y_{\Delta
L}}{Y^{\text{eq}}_{L}}\Biggr{]}+\frac{K_{1}(m_{N_{2}}/T)}{K_{2}(m_{N_{2}}/T)}Y^{\text{eq}}_{N_{2}}\Gamma_{N_{2}\to
N_{1}\chi}\biggl{(}\frac{Y_{N_{2}}}{Y_{N_{2}}^{\text{eq}}}-\frac{Y_{N_{1}}}{Y^{\text{eq}}_{N_{1}}}\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)}+C_{\text{scat.}},$
(4.3) $\displaystyle Hx\frac{dY_{N_{2}}}{dx}\approx$
$\displaystyle\;\frac{K_{1}(m_{N_{2}}/T)}{K_{2}(m_{N_{2}}/T)}Y^{\text{eq}}_{N_{2}}\Biggl{[}\tilde{\Gamma}_{2}\biggl{(}1-\frac{Y_{N_{2}}}{Y^{\text{eq}}_{N_{2}}}\biggr{)}-\frac{1}{2}\tilde{\Gamma}_{2}\tilde{\epsilon}_{2}\frac{Y_{\Delta
L}}{Y^{\text{eq}}_{L}}+\Gamma_{2}\biggl{(}-\frac{Y_{N_{2}}}{Y^{\text{eq}}_{N_{2}}}+\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)}$
$\displaystyle\;-\frac{1}{2}\Gamma_{2}\epsilon_{2}\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\frac{Y_{\Delta
L}}{Y^{\text{eq}}_{L}}-\Gamma_{N_{2}\to
N_{1}\chi}\biggl{(}\frac{Y_{N_{2}}}{Y^{\text{eq}}_{N_{2}}}-\frac{Y_{N_{1}}}{Y^{\text{eq}}_{N_{1}}}\frac{Y_{\chi}}{Y_{\chi}^{\text{eq}}}\biggr{)}\Biggr{]}+C_{\text{scat.}},$
(4.4) $\displaystyle Hx\frac{dY_{\Delta L}}{dx}\approx$
$\displaystyle\;\sum_{i}\frac{K_{1}(m_{N_{i}}/T)}{K_{2}(m_{N_{i}}/T)}Y^{\text{eq}}_{N_{i}}\Biggl{[}\tilde{\Gamma}_{i}\tilde{\epsilon}_{i}\biggl{(}\frac{Y_{N_{i}}}{Y^{\text{eq}}_{N_{i}}}-1\biggr{)}-\frac{1}{2}\tilde{\Gamma}_{i}\frac{Y_{\Delta
L}}{Y^{\text{eq}}_{L}}$
$\displaystyle\;+\Gamma_{i}\epsilon_{i}\biggl{(}\frac{Y_{N_{i}}}{Y^{\text{eq}}_{N_{i}}}-\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)}-\frac{1}{2}\Gamma_{i}\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\frac{Y_{\Delta
L}}{Y^{\text{eq}}_{L}}\Biggr{]}+C_{\text{scat.}},$ (4.5) $\displaystyle
Hx\frac{dY_{\chi}}{dx}\approx$
$\displaystyle\;\sum_{i}\frac{K_{1}(m_{N_{i}}/T)}{K_{2}(m_{N_{i}}/T)}Y^{\text{eq}}_{N_{i}}\Biggl{[}\Gamma_{i}\biggl{(}\frac{Y_{N_{i}}}{Y^{\text{eq}}_{N_{i}}}-\frac{Y_{\chi}}{Y_{\chi}^{\text{eq}}}\biggr{)}+\frac{1}{2}\Gamma_{i}\epsilon_{i}\frac{Y_{\chi}}{Y_{\chi}^{\text{eq}}}\frac{Y_{\Delta
L}}{Y^{\text{eq}}_{L}}\Biggr{]}$
$\displaystyle\;+Y^{\text{eq}}_{N_{2}}\frac{K_{1}(m_{N_{2}}/T)}{K_{2}(m_{N_{2}}/T)}\Gamma_{N_{2}\to
N_{1}\chi}\biggl{(}\frac{Y_{N_{2}}}{Y^{\text{eq}}_{N_{2}}}-\frac{Y_{N_{1}}}{Y^{\text{eq}}_{N_{1}}}\frac{Y_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)}+C_{\text{scat.}}.$
(4.6)

###  C.2 Phenomenological formulae for lepton asymmetry

The Boltzmann equation of the lepton asymmetry (e.g. Eq. (4.5)) has following
form:

$\displaystyle\frac{dY_{\Delta L}(x)}{dx}=\mathcal{F}(x)-W(x)Y_{\Delta L}(x),$
(C.9)

where $\mathcal{F}(x)$ is the $Y_{\Delta L}$ independent function describing
the asymmetry production from decays and scatterings of $N_{i}$ and $W(x)$ is
the washout function. As discussed in Refs. [29, 30, 16], the solution of this
equation can be expressed by

$\displaystyle Y_{\Delta
L}(x)=\int_{x_{i}}^{x}dx^{\prime}\,\mathcal{F}(x^{\prime})\exp\biggl{[}-\int_{x^{\prime}}^{x}dx^{\prime\prime}\,W(x^{\prime\prime})\biggr{]}+Y_{\Delta
L}(x_{i})\exp\biggl{[}-\int_{x_{i}}^{x}dx^{\prime\prime}\,W(x^{\prime\prime})\biggr{]}.$
(C.10)

Intuitively, the integral of $\mathcal{F}$ means the freeze-in like production
[19] and $e^{-\int dx^{\prime}W(x^{\prime})}$ denotes the washout suppression.
Using this equation, we will show the approximated values of the lepton
asymmetry, $\mathcal{Y}^{\mathrm{FI}}_{\Delta L}(z_{1})$ and
$\mathcal{Y}^{\mathrm{WO}}_{\Delta L}(\infty)$ in the following part of this
section.

####  C.2.1 Case for freeze-in from $N_{2}$

$N_{2}$ is mainly produced by the first term of the RHS in Eq. (4.4). Using
the approximation for the modified Bessel function

$\displaystyle K_{n}(x)\underset{x\sim
0}{\sim}\frac{(n-1)!}{2}\biggl{(}\frac{x}{2}\biggr{)}^{-n},$ (C.11)

the Boltzmann equation is approximately written as

$\displaystyle\frac{dY_{N_{2}}}{dx}\approx\frac{1}{Hx}\frac{K_{1}\left(\frac{m_{N_{2}}}{m_{N_{1}}}x\right)}{K_{2}\left(\frac{m_{N_{2}}}{m_{N_{1}}}x\right)}\tilde{\Gamma}_{2}Y^{\text{eq}}_{N_{2}}\sim\frac{135\sqrt{10}}{2\pi^{5}g_{*}^{1/2}g_{*}^{S}}\frac{M_{P}m_{N_{2}}\tilde{\Gamma}_{2}}{m_{N_{1}}^{3}}x^{2},$
(C.12)

and the yield of $N_{2}$ is scaled by

$\displaystyle\bm{Y}_{N_{2}}=\frac{45\sqrt{10}}{2\pi^{5}g^{1/2}_{*}g^{S}_{*}}\frac{M_{P}m_{N_{2}}\tilde{\Gamma}_{2}}{m_{N_{1}}^{3}}x^{3}.$
(C.13)

This $N_{2}$ produces $\chi$ and $N_{1}$ via $N_{2}\to N_{1}\chi$ process, and
their yields are also written as

$\displaystyle\bm{Y}_{\chi}$
$\displaystyle=\frac{135}{2\pi^{6}g_{*}g_{*}^{S}}\frac{M_{P}^{2}m_{N_{2}}^{2}\tilde{\Gamma}_{2}(\Gamma_{N_{2}\to
N_{1}\chi}+\Gamma_{2})}{m_{N_{1}}^{6}}x^{6},$ (C.14)
$\displaystyle\bm{Y}_{N_{1}}$
$\displaystyle=\frac{45\sqrt{10}}{2\pi^{5}g^{1/2}_{*}g^{S}_{*}}\frac{M_{P}\tilde{\Gamma}_{1}}{m_{N_{1}}^{2}}x^{3}+\frac{135}{2\pi^{6}g_{*}g_{*}^{S}}\frac{M_{P}^{2}m_{N_{2}}^{2}\tilde{\Gamma}_{2}(\Gamma_{N_{2}\to
N_{1}\chi}+\Gamma_{2})}{m_{N_{1}}^{6}}x^{6}.$ (C.15)

where the contribution of the first term of the RHS in Eq. (4.3) is included,
which is scaled by $x^{3}$ as in the same way to $N_{2}$. Using these
functions, we will evaluate the lepton asymmetry. From Eq. (4.5), The washout
function

$\displaystyle
W_{2}:=\frac{1}{2Hx}\frac{K_{1}(m_{N_{2}}x/m_{N_{1}})}{K_{2}(m_{N_{2}}x/m_{N_{1}})}\biggl{(}\tilde{\Gamma}_{2}+\Gamma_{2}\frac{\bm{Y}_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)}\frac{Y^{\text{eq}}_{N_{2}}}{Y^{\text{eq}}_{L}}+\frac{1}{2Hx}\frac{K_{1}(x)}{K_{2}(x)}\biggl{(}\tilde{\Gamma}_{1}+\Gamma_{1}\frac{\bm{Y}_{\chi}}{Y^{\text{eq}}_{\chi}}\biggr{)}\frac{Y^{\text{eq}}_{N_{2}}}{Y^{\text{eq}}_{L}},$
(C.16)

is read, and the yield is evaluated by using (C.10) as

$\displaystyle\mathcal{Y}_{\Delta
L}^{\mathrm{FI}}(x)=\int_{0}^{x}dx^{\prime}\,\mathcal{F}_{2}(x^{\prime})\exp\biggl{[}-\int_{x^{\prime}}^{x}dx^{\prime\prime}\,W_{2}(x^{\prime\prime})\biggr{]},$
(C.17)

with

$\displaystyle\mathcal{F}_{2}=$
$\displaystyle\;\frac{1}{Hx}\frac{K_{1}(m_{N_{2}}x/m_{N_{1}})}{K_{2}(m_{N_{2}}x/m_{N_{1}})}\Gamma_{2}\epsilon_{2}\biggl{(}\bm{Y}_{N_{2}}-\frac{Y^{\text{eq}}_{N_{2}}}{Y^{\text{eq}}_{\chi}}\bm{Y}_{\chi}\biggr{)}+\frac{1}{Hx}\frac{K_{1}(x)}{K_{2}(x)}\Gamma_{1}\epsilon_{1}\biggl{(}\bm{Y}_{N_{1}}-\frac{Y^{\text{eq}}_{N_{1}}}{Y^{\text{eq}}_{\chi}}\bm{Y}_{\chi}\biggr{)}.$
(C.18)

If the washout suppression is weak enough, which is the case of the neutrino
Yukawa couplings being small, the lepton asymmetry is mainly produced by the
RH neutrinos before getting into the thermal bath and the total amount is
evaluated as

$\displaystyle Y_{\Delta L}\approx\mathcal{Y}^{\mathrm{FI}}_{\Delta
L}(z_{1}),$ (C.19)

where

$\displaystyle
z_{1}=\mathop{\mathrm{min}}(z_{N_{2}},z_{\chi}),\quad\bm{Y}_{N_{2}}(z_{N_{2}})=\frac{45}{\pi^{4}g^{S}_{*}},\quad\bm{Y}_{\chi}(z_{\chi})=\frac{45}{2\pi^{4}g^{S}_{*}}.$
(C.20)

This is regarded as the asymmetry by the freeze-in production including the
weak washout effects.

####  C.2.2 Case for strong $N_{1}$ washout

If the washout of $N_{1}$ is strong, the lepton asymmetry is mainly determined
by this, and the contribution from $\mathcal{Y}^{\mathrm{FI}}_{\Delta
L}(z_{1})$ is exponentially suppressed. The washout function of this case is
given by

$\displaystyle
W_{1}:=\frac{1}{2Hx}\frac{K_{1}(m_{N_{2}}x/m_{N_{1}})}{K_{2}(m_{N_{2}}x/m_{N_{1}})}(\tilde{\Gamma}_{2}+\Gamma_{2})\frac{Y^{\text{eq}}_{N_{2}}}{Y^{\text{eq}}_{L}}+\frac{1}{2Hx}\frac{K_{1}(x)}{K_{2}(x)}(\tilde{\Gamma}_{1}+\Gamma_{1})\frac{Y^{\text{eq}}_{N_{1}}}{Y^{\text{eq}}_{L}},$
(C.21)

where $Y_{\chi}\approx Y^{\text{eq}}_{\chi}$ is assumed. Then, the lepton
asymmetry is written as

$\displaystyle\mathcal{Y}^{\mathrm{WO}}_{\Delta
L}=\int_{0}^{x}dx^{\prime}\,\mathcal{F}_{1}(x^{\prime})\exp\biggl{[}-\int_{x^{\prime}}^{x}dx^{\prime\prime}\,W_{1}(x^{\prime\prime})\biggr{]},$
(C.22)

where we introduce the following function

$\displaystyle\mathcal{F}_{1}:=$
$\displaystyle\frac{1}{Hx}\frac{K_{1}(m_{N_{2}}x/m_{N_{1}})}{K_{2}(m_{N_{2}}x/m_{N_{1}})}\Gamma_{2}\epsilon_{2}\biggl{(}\frac{\tilde{\Gamma}_{2}+\Gamma_{2}}{m_{N_{2}}}\biggr{)}Y^{\text{eq}}_{N_{2}}+\frac{1}{Hx}\frac{K_{1}(x)}{K_{2}(x)}\Gamma_{1}\epsilon_{1}\biggl{(}\frac{\tilde{\Gamma}_{1}+\Gamma_{1}}{m_{N_{1}}}\biggr{)}Y^{\text{eq}}_{N_{1}}.$
(C.23)

Using these functions, the lepton asymmetry is given by

$\displaystyle Y_{\Delta L}\approx\mathcal{Y}^{\mathrm{WO}}_{\Delta
L}(\infty).$ (C.24)

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