Thermal Leptogenesis in the Minimal Gauged $U(1)_{L_{\mu}-L_{\tau}}$ Model

A<EMAIL_ADDRESS>K.
Hamaguchi${}^{c,d~{}}$†††<EMAIL_ADDRESS>N.
<EMAIL_ADDRESS>M. E. Ramirez-
<EMAIL_ADDRESS>and J.
<EMAIL_ADDRESS>

aDipartimento di Fisica e Astronomia, Università di Bologna, via Irnerio 46,
40126,
bINFN, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy,
cDepartment of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113–0033, Japan,

dKavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU),
University of Tokyo, Kashiwa 277–8583, Japan.
eDual CP Institute of High Energy Physics, C.P. 28045, Colima, México

We discuss the thermal leptogenesis mechanism within the minimal gauged
U(1)${}_{L_{\mu}-L_{\tau}}$ model to explain the observed baryon asymmetry of
the Universe (BAU). In such framework, the phases of the
Pontecorvo–Maki–Nakagawa–Sakata neutrino mixing matrix and the sum of the
Standard Model neutrino masses are predictable because of a restricted
neutrino mass matrix structure. Additionally, in the context of thermal
leptogenesis, the BAU can be computed in terms of the three remaining free
variables that parameterise the right-handed neutrino masses and their Yukawa
couplings to the Higgs and lepton doublets. We identify the ranges of such
parameters for which the correct BAU can be reproduced. We adopt the formalism
of the density matrix equations to fully account for flavour effects and
consider the decays of all the three right-handed neutrinos. Our analysis
reveals that thermal leptogenesis is feasible within a wide parameter space,
specifically for Yukawa couplings ranging from approximate unity to
$\mathcal{O}(0.03-0.05)$ and mass of the lightest right-handed neutrino
$M_{1}\gtrsim 10^{11-12}\,\text{GeV}$, setting a leptogenesis scale in the
considered model which is higher than that of the non-thermal scenario.

##  1 Introduction

There is astrophysical and cosmological evidence for the existence of a
matter-antimatter asymmetry in the present Universe. The baryon-to-photon
ratio parameterises the baryon asymmetry of the Universe (BAU),

$\eta_{B}=\frac{n_{B}-n_{\bar{B}}}{n_{\gamma}},$ (1)

where $n_{B}$, $n_{\bar{B}}$, and $n_{\gamma}$ are the number densities of
baryons, anti-baryons, and photons, respectively. The baryon-to-photon ratio
has been determined independently from observations of the Cosmic Microwave
Background (CMB) anisotropies and Big Bang Nucleosynthesis (BBN) estimates.
Both estimations are consistent with the best-fit value $\eta_{B}\simeq
6.1\times 10^{-10}$ [1, 2]. One compelling mechanism that explains the
observed BAU is leptogenesis [3] based on the existence of right-handed
neutrinos and their out-of-equilibrium decays in the early Universe. In the
simplest thermal leptogenesis scenario, the CP-violating, out-of-equilibrium
decays of the right-handed neutrinos generate a lepton asymmetry, which is
converted into a baryon asymmetry by the sphaleron processes predicted by the
SM [4]. The leptogenesis mechanism for the BAU generation can be studied in a
wide range of models beyond the SM, providing a valuable window into the study
of new physics (see, e.g., Ref. [5] for a recent review on the topic and
references therein).

In this paper, we focus on a model that exhibits a novel anomaly-free U(1)
gauge symmetry denoted by U(1)${}_{L_{\mu}-L_{\tau}}$[6, 7, 8, 9], where
$L_{e}$, $L_{\mu}$ and $L_{\tau}$ stand for the lepton number (flavour)
associated to the electron ($e$), muon ($\mu$) and tauon ($\tau$),
respectively. The model is implemented within the framework of the type-I
seesaw mechanism [10, 11, 12, 13, 14], hence providing an explanation for the
smallness of SM neutrino masses through the introduction of three heavy right-
handed neutrinos $N_{1,2,3}$ with masses $M_{1,\,2,\,3}$. The symmetry of the
model imposes constraints on the neutrino mass structure, since the second and
third-generation leptons, $\mu$ and $\tau$, are charged under
U(1)${}_{L_{\mu}-L_{\tau}}$ while the electron is not. Consequently, the Dirac
mass matrix has a simple diagonal structure, and only certain components of
the Majorana matrix are non-zero. This simple mass structure is insufficient
to explain neutrino oscillation data [15, 16], and hence, the
U(1)${}_{L_{\mu}-L_{\tau}}$ must be broken. This is typically achieved by
introducing a scalar field that has non-zero U(1)${}_{L_{\mu}-L_{\tau}}$
charge and a vacuum expectation value (VEV), which breaks the gauge symmetry
giving mass to the U(1)${}_{L_{\mu}-L_{\tau}}$ gauge bosons. We refer to this
model as the “minimal gauged U(1)${}_{L_{\mu}-L_{\tau}}$ model”. The model has
a strong predictive power [15, 16, 17, 18, 19, 20, 21, 22, 23, 24] because of
the so-called two-zero minor structure [17, 18, 19, 21, 22, 23] of the
neutrino mass matrix.

The aim of the present work is to study thermal leptogenesis in the minimal
gauged U(1)${}_{L_{\mu}-L_{\tau}}$ model.111For a study of the non-thermal
leptogenesis mechanism within the minimal gauged U(1)${}_{L_{\mu}-L_{\tau}}$
model, see Ref. [24]. Leptogenesis in (non-minimal) gauged
U(1)${}_{L_{\mu}-L_{\tau}}$ models is also discussed in Refs. [25, 26, 27].
Since the neutrino mass structure is highly restricted in the model, it is not
obvious whether the observed BAU is obtainable through thermal leptogenesis.
In a widely-considered setup for thermal leptogenesis [28], the right-handed
neutrinos mass spectrum is strongly hierarchical, $M_{1}\ll M_{2}\ll M_{3}$,
and the lepton asymmetries in the three flavors evolve equally. This scenario
has been extensively studied solving the unflavoured Boltzmann Equations (BEs)
and is subject to a model-independent bound on the mass scale of leptogenesis,
reading $M_{1}\gtrsim 10^{9}\,\text{GeV}$, below which the requisite CP-
asymmetry is too small to get the observed value of the BAU [29, 30, 31, 32,
33]. However, in our model, the condition of a strong mass hierarchy is not
satisfied in many parts of the parameter space, and we expect the decay of
each of the three right-handed neutrinos to contribute to the generation of
the BAU.

Charged lepton flavour effects can also play a crucial role in the generation
of the BAU [34, 35, 36, 37, 38, 39, 40, 41, 42, 43]. Specifically, the
unflavoured scenario is valid only in the single-flavour regime for
temperatures above $\sim 10^{12}\,\text{GeV}$, when the processes mediated by
the charged lepton SM Yukawa couplings are out-of-equilibrium. In the two-
flavour regime $10^{9}\ll T/\text{GeV}\ll 10^{12}$, processes induced by the
$\tau$-Yukawa coupling occur at a rate $\Gamma_{\tau}$ much larger than the
Hubble expansion rate $H$, which indicates that these processes are in thermal
equilibrium. Consequently, the asymmetries in the lepton charge $L_{\tau}$ and
$L_{\mu}+L_{e}$ evolve differently in this regime. Similarly, in the three-
flavour regime for $T\ll 10^{9}\,\text{GeV}$, processes mediated by the
$\mu$-Yukawa coupling with rate $\Gamma_{\mu}$ are also in thermal equilibrium
($\Gamma_{\mu}\gg H$), leading to the individual evolution of $L_{e}$,
$L_{\mu}$ and $L_{\tau}$. Given that the mass scales of interest cover
different flavour regimes, the simplest unflavoured scenario is not applicable
to our study, and we have to consider the impact of charged lepton flavour
effects on the BAU generation.

The study of thermal leptogenesis in the (three-) two-flavour regime can be
conducted using the formalism of (three-) two-flavoured BEs, provided that the
processes mediated by the ($\mu$\- and $\tau$-) $\tau$-Yukawa couplings are
sufficiently fast. In this formalism, the equations for the asymmetries in
($L_{\tau}$, $L_{\mu}$ and $L_{e}$) $L_{\tau}$ and $L_{\mu}+L_{e}$ are
different and solved separately. However, to accurately account for flavour
effects, it is more precise to trace the evolution of the elements of the
density matrix of the lepton flavour system with the Density Matrix Equations
(DMEs) [40, 41, 42, 43]. This approach is particularly useful when the
processes mediated by the charged lepton Yukawa couplings are neither
infinitely fast nor their effects negligible, such as at the transitions
between different flavour regimes, and has been shown to lead to different
predictions with respect to the BEs [40, 41, 42, 43, 44, 45].

The presence of multiple decaying right-handed neutrinos also has implications
for the generation of the BAU, specifically because of the effects of heavy
neutrino flavours. The right-handed neutrinos couple to different
superpositions of flavour states, whose interactions can induce additional
decoherence effects in the context of the DMEs [42] (see also Ref. [43]).
These effects can be particularly relevant when the right-handed neutrinos do
not have a strongly hierarchical mass spectrum, which is the case in certain
parts of the parameter space of the considered model where $M_{2}\lesssim
3M_{1}$ and $M_{3}\lesssim 3M_{2}$. In this regime, the different
superposition of flavour states, associated with the different right-handed
neutrinos, are simultaneously present in the Universe.

In this work, we consider the formalism of the DMEs with three decaying right-
handed neutrinos to fully account for all the relevant effects mentioned
above. By performing a numerical scan of the parameter space, we investigate
for which values of the parameters of the minimal gauge
U(1)${}_{L_{\mu}-L_{\tau}}$ model the DMEs are successful in reproducing the
observed BAU. We solve numerically the DMEs for thermal leptogenesis using the
Python package ULYSSES [46, 47], that is a freely accessible code for the
numerical evaluation of the BAU in the context of leptogenesis. Its major
features are the variety of equations available, allowing for comparisons
between the BEs and DMEs in the different regimes, and a rapid evaluation,
making scans of large parameter spaces feasible over relatively short periods
of time.

The paper is structured as follows. In Sec. 2, we present the minimal gauged
U(1)${}_{L_{\mu}-L_{\tau}}$ model and analyse the corresponding neutrino mass
structure. In Sec. 3, we discuss in more details the mechanism of thermal
leptogenesis and the formalism of DMEs. We introduce the DMEs with three
decaying right-handed neutrinos and the corresponding CP-asymmetry parameters,
which are crucial for understanding the generation of the lepton and baryon
asymmetries in the early Universe. The results of the scan of the parameter
space for viable leptogenesis is presented in Sec. 4, and we finally conclude
in Sec. 5.

##  2 The Minimal Gauged U(1)${}_{L_{\mu}-L_{\tau}}$ Model

In the minimal gauged U(1)${}_{L_{\mu}-L_{\tau}}$ model, the second and third-
generation leptons, namely those of lepton flavour $\mu$ and $\tau$, carry
charges of $+1$ and $-1$, respectively. Notably, the first-generation leptons
(of lepton flavour $e$), quarks, and the Higgs field in the Standard Model
(SM) are not charged under this particular gauge symmetry. Moreover, we
introduce three right-handed sterile neutrinos, $N_{e}$, $N_{\mu}$, and
$N_{\tau}$, each described by a Weyl spinor that transforms under the
$(0,\,\frac{1}{2})$ representation of the Lorentz group (thus right-handed),
which are singlets under the SM gauge group (thus sterile) and carry the
U(1)${}_{L_{\mu}-L_{\tau}}$ charges 0, $+1$, and $-1$, respectively.
Additionally, we introduce a scalar boson $\sigma$, which is a singlet under
the SM gauge group and carries the U(1)${}_{L_{\mu}-L_{\tau}}$ charge $+1$.
This scalar field develops a VEV that spontaneously breaks the
U(1)${}_{L_{\mu}-L_{\tau}}$ gauge symmetry. We summarise the charges of the
field content of this model in Table 1, denoting the left-handed SU(2) lepton
doublets of flavour $\alpha$ with $\ell_{\alpha}$ and the right-handed charged
leptons with $\alpha_{R}$, $\alpha=e,\,\mu,\,\tau$. We adopt a two-component
spinor notation (see, e.g., Ref. [48]) as in Ref. [21].

$L_{\mu}-L_{\tau}$ Charges of the Field Content
---
|  $\ell_{e},e_{R},N_{e}$ |  $\ell_{\mu},\mu_{R},N_{\mu}$ |  $\ell_{\tau},\tau_{R},N_{\tau}$ |  $\sigma$ | Others
${L_{\mu}-L_{\tau}}$ |  $0$ |  $+1$ |  $-1$ |  $+1$ | $0$
Table 1: The ${L_{\mu}-L_{\tau}}$ charges of the field content in the
considered minimal gauged U(1)${}_{L_{\mu}-L_{\tau}}$ model.

For the purposes of our discussion, we focus on the new leptonic interactions
involving the right-handed neutrinos and their Majorana mass terms, reading

$\displaystyle\Delta{\cal L}=$
$\displaystyle-\lambda_{e}N_{e}^{c}(\ell_{e}\cdot\Phi)-\lambda_{\mu}N_{\mu}^{c}(\ell_{\mu}\cdot\Phi)-\lambda_{\tau}N_{\tau}^{c}(\ell_{\tau}\cdot\Phi)$
$\displaystyle-\frac{1}{2}M_{ee}N_{e}^{c}N_{e}^{c}-M_{\mu\tau}N_{\mu}^{c}N_{\tau}^{c}-\lambda_{e\mu}\sigma
N_{e}^{c}N_{\mu}^{c}-\lambda_{e\tau}\sigma^{*}N_{e}^{c}N_{\tau}^{c}+\text{h.c.}~{},$
(2)

where the dots indicate the contraction of the SU(2) indices between the
lepton doublets and the Higgs doublet $\Phi$. Additionally,
$(N_{\alpha}^{c})_{a}\equiv\varepsilon_{ab}(N_{\alpha}^{*})_{b}$, where
$\alpha=e,\,\mu,\,\tau$ and $\varepsilon_{ab}$ is the antisymmetric tensor of
the spinor indices $a,\,b$. The interaction terms in Eq. (2) lead to neutrino
mass terms after the Higgs field $\Phi$ and singlet scalar $\sigma$ acquire
their VEVs, denoted as $\langle\Phi\rangle=v/\sqrt{2}$ and
$\langle\sigma\rangle$, respectively. These mass terms can be expressed as
follows:

$\displaystyle{\cal L}_{\rm mass}$
$\displaystyle=-(\nu_{e},\nu_{\mu},\nu_{\tau}){\cal
M}_{D}\begin{pmatrix}N_{e}^{c}\\\ N_{\mu}^{c}\\\
N_{\tau}^{c}\end{pmatrix}-\frac{1}{2}(N_{e}^{c},N_{\mu}^{c},N_{\tau}^{c}){\cal
M}_{R}\begin{pmatrix}N_{e}^{c}\\\ N_{\mu}^{c}\\\
N_{\tau}^{c}\end{pmatrix}+\text{h.c.}~{},$ (3)

where $\nu_{\alpha}$, are the left-handed Weyl spinors describing the SM
neutrinos of lepton flavour $\alpha$, ${\cal M}_{D}$ is the Dirac mass matrix
and ${\cal M}_{R}$ is the Majorana mass matrix given by,

${\cal M}_{D}=\frac{v}{\sqrt{2}}\begin{pmatrix}\lambda_{e}&0&0\\\
0&\lambda_{\mu}&0\\\ 0&0&\lambda_{\tau}\end{pmatrix}~{},\qquad{\cal
M}_{R}=\begin{pmatrix}M_{ee}&\lambda_{e\mu}\langle\sigma\rangle&\lambda_{e\tau}\langle\sigma\rangle\\\
\lambda_{e\mu}\langle\sigma\rangle&0&M_{\mu\tau}\\\
\lambda_{e\tau}\langle\sigma\rangle&M_{\mu\tau}&0\end{pmatrix}~{},$ (4)

respectively. Notably the Dirac mass matrix $\mathcal{M}_{D}$ is diagonal,
while the $(\mu,\mu)$ and $(\tau,\tau)$ components in the Majorana mass matrix
$\mathcal{M}_{R}$ are zero due to the U(1)${}_{L_{\mu}-L_{\tau}}$ gauge
symmetry. This particular structure leads to interesting predictions for
neutrino observables [49, 50, 19, 21, 22], as discussed below. For the
remainder of this discussion, we assume that the Dirac Yukawa couplings
$\lambda_{e}$, $\lambda_{\mu}$, and $\lambda_{\tau}$, as well as the VEVs $v$
and $\langle\sigma\rangle$ are real without loss of generality, with
$v=246\,\text{GeV}$; this can always be realised via field redefinition.

The seesaw master formula can naturally explain the smallness of SM neutrino
masses by assuming that the non-zero components in the Majorana mass matrix
are much larger than those in the Dirac matrix. The formula relates the light
neutrino masses to the masses of the heavy right-handed neutrinos, and their
couplings to the SM particles [10, 11, 12, 13, 14],

${\cal M}_{\nu_{L}}\simeq-{\cal M}_{D}{\cal M}_{R}^{-1}{\cal M}_{D}^{T}~{}.$
(5)

The mass matrix can be diagonalised, allowing us to express the flavour
eigenstates of the neutrinos as linear combinations of the mass eigenstates,

$U^{T}{\cal M}_{\nu_{L}}U=\text{diag}(m_{1},m_{2},m_{3})~{}.$ (6)

Here, $U$ is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix $U$ [51,
52, 53, 54] and can be parameterised as [55]

$U=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}&c_{23}c_{13}\end{pmatrix}\times\begin{pmatrix}1&0&0\\\
0&e^{i\frac{\alpha_{2}}{2}}&0\\\ 0&0&e^{i\frac{\alpha_{3}}{2}}\end{pmatrix},$
(7)

where $c_{ij}\equiv\cos\theta_{ij}$ and $s_{ij}\equiv\sin\theta_{ij}$ and
$\delta,\,\alpha_{2},\,\alpha_{3}\in[0,2\pi]$. The Normal Ordering (NO) of
neutrino masses, where $m_{1}<m_{2}<m_{3}$, is the only mass hierarchy that is
consistent with neutrino oscillation data in this model, as demonstrated in
Ref. [21].

Using Eqs. (5) and (6), it is possible to express the Majorana mass matrix
$\mathcal{M}_{R}$ as

${\cal M}_{R}=-{\cal
M}_{D}^{T}\,U\,\text{diag}(m_{1}^{-1},m_{2}^{-1},m_{3}^{-1})\,U^{T}{\cal
M}_{D}~{}.$ (8)

The $(\mu,\mu)$ and $(\tau,\tau)$ components of the right-hand side of the
above equation vanish. This is a direct consequence of the two-zero-minor
structure [49, 50] of ${\cal M}_{\nu_{L}}$ in this model.222 Note that this
structure is stable against the renormalisation-group effects [21].  The
vanishing conditions on complex quantities result in four real parameter
equations. These equations allow us to predict the values of the lightest
neutrino mass $m_{1}$, as well as the Dirac and Majorana CP phases $\delta$,
$\alpha_{2}$, and $\alpha_{3}$, as functions of the other neutrino oscillation
parameters. These parameters include the neutrino mixing angles $\theta_{12}$,
$\theta_{23}$, $\theta_{13}$, and the squared mass differences $\Delta
m_{21}^{2}\equiv m_{2}^{2}-m_{1}^{2}$, and $\Delta m_{31}^{2}\equiv
m_{3}^{2}-m_{1}^{2}$. For each set of input parameters, there exist two
possible sets of predictions, as shown in Ref. [21]. Specifically, if the set
$(m_{1},\,\delta,\,\alpha_{2},\,\alpha_{3})$ satisfies the two-vanishing
conditions, then the set
$(m_{1},\,2\pi-\delta,\,2\pi-\alpha_{2},\,2\pi-\alpha_{3})$ also satisfies
them.

For our numerical study, we fix the neutrino mixing angles and the squared
mass differences following the NuFit analyses [56], which are comprehensive
global fits that include data from all the relevant neutrino oscillation
experiments. The most recent NuFit analyses are conducted both with and
without incorporating data from the Super-Kamiokande (SK) experiment. These
two approaches yield somewhat different values for the neutrino mixing angles
and squared mass differences. We summarise the results obtained in the NuFit
5.2 analysis [56, 57] for the best-fit values, $1\sigma$ deviations and
$3\sigma$ ranges of the neutrino mixing angles and mass squared differences in
Table 2.

Neutrino Masses and Mixing Parameters
---
Parameters |  $\theta_{12}$ |  $\theta_{13}$ |  $\theta_{23}$ |  $\Delta m_{21}^{2}$ |  $\Delta m_{31}^{2}$
(units) |  (∘) |  (∘) |  (∘) |  ($10^{-5}\text{ eV}^{2}$) |  ($10^{-3}\text{ eV}^{2}$)
With SK |  $33.41^{+0.75}_{-0.72}$ |  $8.58^{+0.11}_{-0.11}$ |  $42.2^{+1.1}_{-0.9}$ |  $7.41^{+0.21}_{-0.20}$ |  $2.507^{+0.026}_{-0.027}$
$3\sigma$ range |  $[31.31,35.74]$ |  $[8.23,8.91]$ |  $[39.7,51.0]$ |  $[6.82,8.03]$ |  $[2.427,2.590]$
Without SK |  $33.41^{+0.75}_{-0.72}$ |  $8.54^{+0.11}_{-0.12}$ |  $49.1^{+1.0}_{-1.3}$ |  $7.41^{+0.21}_{-0.20}$ |  $2.511^{+0.028}_{-0.027}$
$3\sigma$ range |  $[31.31,35.74]$ |  $[8.19,8.89]$ |  $[39.6,51.9]$ |  $[6.82,8.03]$ |  $[2.427,2.590]$
Table 2:  Best-fit values, $1\sigma$ deviations and $3\sigma$ allowed ranges
of the neutrino mixing angles $\theta_{12}$, $\theta_{13}$, $\theta_{23}$, and
of the squared mass differences $\Delta m^{2}_{21}$ and $\Delta m^{2}_{31}$ in
the case of NO light neutrino mass spectrum, from the latest NuFit 5.2
analysis [56, 57]. The two lines of values correspond to the fit with (top)
and without (bottom) the inclusion of SK data.

We observe that, while the inclusion of the SK data substantially affects the
best-fit $\pm 1\sigma$ values of $\theta_{23}$, it does not change
dramatically its $3\sigma$ allowed range (the maximal value without SK data is
larger by $0.9^{\circ}$) and the results for $\theta_{12}$, $\theta_{13}$,
$\Delta m_{21}^{2}$, and $\Delta m_{31}^{2}$. Specifically, the best fit $\pm
1\sigma$ values for $\theta_{23}$ lie below (above) $\pi/4$ with (without) the
inclusion of SK data. Since the minimal gauged U(1)${}_{L_{\mu}-L_{\tau}}$
model predictions are highly dependent on the value of $\theta_{23}$, as the
sum of neutrino masses diverges for $\theta_{23}=\pi/4$ [21], we examine the
cases with and without the inclusion of SK data separately.333See Fig. 1(b) in
Ref. [24], in which the sum of neutrino masses as a function of $\theta_{23}$
is shown. For such figure, the authors of Ref. [24] adopted an older version
of the neutrino oscillation data (NuFit 4.1), but the behaviour of the same
function in our case remains basically the same. In both cases, we set
$\theta_{12}$, $\theta_{13}$, $\Delta m_{21}^{2}$, and $\Delta m_{31}^{2}$ to
their best-fit values, while we treat $\theta_{23}$ differently given the
implications it has on the sum of neutrino masses predicted by the considered
model.

The sum of the neutrino masses is constrained by various cosmological and
astrophysical measurements, yielding an upper bound of
$\sum_{i}m_{i}<(0.12-0.69)\,\text{eV}$ (95$\%$ C.L.), which depends on the
adopted model and level of statistical complexity [58, 59] (see also Refs.
[60, 61, 62, 63]). By adopting the best-fit values for $\theta_{23}$ in the
two cases with and without SK data, we obtain $\sum_{i}m_{i}\simeq
0.241\,\text{eV}$ and $0.173\,\text{eV}$, respectively. However, in order to
evade the aforementioned limitations, some level of complexity and assumptions
is required. We instead set $\theta_{23}$ to its $-\,(+)\,3\sigma$ minimal
(maximal) limit for the case with (without) SK data; specifically,
$\theta_{23}=39.7^{\circ}\,(51.9^{\circ})$, with the other input parameters at
their best-fit values. This yields
$\sum_{i}m_{i}=0.142\,(0.117)~{}\mathrm{eV}$, which minimises the sum of
neutrino masses and reduces the tension with the cosmological bounds. To
summarise, in our numerical analysis, we consider the following two sets of
input parameters:

Set I | Set II
---|---
$\theta_{12}=33.41^{\circ}$ | $\theta_{12}=33.41^{\circ}$
$\theta_{13}=8.58^{\circ}$ | $\theta_{13}=8.54^{\circ}$
$\theta_{23}=39.7^{\circ}$ | $\theta_{23}=51.9^{\circ}$
$\Delta m^{2}_{21}=7.41\times 10^{-5}\,\text{eV}^{2}$ | $\Delta m^{2}_{21}=7.41\times 10^{-5}$
$\Delta m^{2}_{31}=2.507\times 10^{-3}\,\text{eV}^{2}$ |  $\Delta m^{2}_{31}=2.511\times 10^{-3}\,\text{eV}^{2}$.

The minimal gauged U(1)${}_{L_{\mu}-L_{\tau}}$ model, when combined with the
neutrino mixing angles and squared mass differences in Set I (II), predicts a
value of $m_{1}=0.039\,(0.029)\,\text{eV}$.444The model also makes predictions
about the effective Majorana mass $\langle m_{\beta\beta}\rangle$ which
determines the rate of the neutrinoless double-beta decay. The parameter set I
(II) predicts $\langle m_{\beta\beta}\rangle\simeq 0.025$ eV (0.016 eV), which
is below the current constraint given by the KamLAND-Zen experiment, $\langle
m_{\beta\beta}\rangle<0.036$-0.156 eV [64], and may be probed by future
experiments with sensitivities of $\langle
m_{\beta\beta}\rangle\simeq{\mathcal{O}}(0.01)$ [65, 66].  In addition, the
PMNS phases are determined to be $\delta\simeq 301^{\circ}\,(228^{\circ})$,
$\alpha_{2}=116^{\circ}\,(225^{\circ})$ and
$\alpha_{3}=269^{\circ}\,(70^{\circ})$, or equivalently,
$\delta=59^{\circ}\,(132^{\circ})$, $\alpha_{2}=244^{\circ}\,(135^{\circ})$
and $\alpha_{3}=91^{\circ}\,(290^{\circ})$. The Dirac phase $\delta$ has also
been estimated by the NuFit analysis, which reports a $3\sigma$ range of
$[144^{\circ},350^{\circ}]$ when including SK data, and
$[0,44^{\circ}]\cup[108^{\circ},360^{\circ}]$ when not including it.
Consequently, the set of parameters of Set I with $\delta=59^{\circ}$ is
disfavoured for more than $3\sigma$.

By specifying the three additional input parameters in the Dirac mass matrix,
$\mathcal{M}_{D}=(v/\sqrt{2})\rm{diag}(\lambda_{e},\lambda_{\mu},\lambda_{\tau})$,
it is possible to obtain the right-handed neutrino mass matrix
$\mathcal{M}_{R}$. These Yukawa couplings are parameterised, according to Ref.
[21], as

$(\lambda_{e},\lambda_{\mu},\lambda_{\tau})=\lambda(\cos\theta,\sin\theta\cos\phi,\sin\theta\sin\phi),$
(9)

where we consider the range of $0\leq\theta\,\rm{and}\,\phi\leq\pi/2$. We also
limit the value of $\lambda$ to $\lambda\lesssim 1$ to ensure that the Yukawa
couplings remain perturbative. The masses of the right-handed neutrinos are
obtained by performing the Takagi diagonalisation on the complex symmetric
matrix ${\cal M}_{R}$

${\cal M}_{R}=\Omega^{\ast}\text{diag}(M_{1},M_{2},M_{3})\Omega^{\dagger},$
(10)

where $\Omega$ is a unitary matrix and $M_{1,\,2,\,3}\geq 0$. Once the mass
matrix of the right-handed neutrinos $\mathcal{M}_{R}$ is diagonalised, the
terms in Eq. (2) lead to

$\displaystyle{\Delta\cal L}=$
$\displaystyle-\hat{\lambda}_{j\alpha}\hat{N}_{j}^{c}(\ell_{\alpha}\cdot
H)-\frac{1}{2}M_{j}\hat{N}_{j}^{c}\hat{N}_{j}^{c}+\text{h.c.}~{},$ (11)

where the sum over equal indices is implicit and

$\displaystyle\hat{N}_{j}^{c}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}\Omega_{\alpha j}^{\ast}N_{\alpha}^{c}~{},$ (12)
$\displaystyle\hat{\lambda}_{j\alpha}$ $\displaystyle=$
$\displaystyle\Omega_{\alpha j}\lambda_{\alpha}~{}(\text{not~{}summed}).$ (13)

The Weyl spinor $\hat{N}_{j}$, $j=1,\,2,\,3$, with
$(\hat{N}_{j}^{c})_{a}=\varepsilon_{ab}(\hat{N}_{j}^{*})_{b}$, describes a
right-handed neutrino $N_{j}$ with mass $M_{j}$.

Upon examining Eq. (8) and using the Yukawa coupling parameterisation provided
in Eq. (9), it becomes transparent that the masses of the right-handed
neutrinos are proportional to $\lambda^{2}$. This dependence can be expressed
more precisely as follows,

$M_{1,\,2,\,3}=\frac{v^{2}\lambda^{2}}{2m_{1}}\,\beta_{1,\,2,\,3}(\theta,\phi)\simeq
6\times
10^{14}\,\text{GeV}\left(\frac{0.05\,\text{eV}}{m_{1}}\right)\lambda^{2}\beta_{1,\,2,\,3}(\theta,\phi),$
(14)

where $\beta_{1,\,2,\,3}(\theta,\phi)$, satisfying
$\beta_{1,\,2,\,3}(\theta,\phi)\lesssim\mathcal{O}(1)$, are some real numbers
that only depend on the ratios $m_{1}/m_{2}<1$, $m_{1}/m_{3}<1$, the PMNS
parameters and on trigonometric functions of $\theta$, $\phi$.555More
precisely, $\beta_{1,\,2,\,3}(\theta,\phi)$ are defined by
$\text{diag}(\beta_{1},\beta_{2},\beta_{3})=-\Omega^{T}D_{\theta,\phi}UD_{m}U^{T}D_{\theta,\phi}\Omega$,
where
$D_{\theta,\phi}\equiv\text{diag}(\cos\theta,\sin\theta\cos\phi,\sin\theta\sin\phi)$
and $D_{m}\equiv\text{diag}(1,m_{1}/m_{2},m_{1}/m_{3})$.

Figure 1: The contour plots for the quantities $\beta_{1}$ (top panel),
$\Delta M_{21}/M_{1}$ (bottom-left panel) and $\Delta M_{32}/M_{2}$ (bottom-
right panel) in the $\phi-\theta$ plane. The red stars in the plots mark the
point of coordinates $\phi\sim 50^{\circ}$ and $\theta\sim 62^{\circ}$ for
which $\beta_{1}$ is maximised ($\Delta M_{21}/M_{1}$ is minimised). The plot
is obtained for the neutrino mixing angles, and squared mass differences as in
Set I. Choosing the input parameters as in Set II would lead to a
qualitatively similar figure.

Fig. 1 shows the contours for three parameters:
$\beta_{1}=2m_{1}v^{-2}\lambda^{-2}M_{1}$ (top), $\beta_{2}/\beta_{1}-1=\Delta
M_{21}/M_{1}$ (bottom-left), and $\beta_{3}/\beta_{2}-1=\Delta M_{32}/M_{2}$
(bottom-right); here, $\Delta M_{21(32)}\equiv M_{2(3)}-M_{1(2)}$. These
contours are obtained using the neutrino mixing angles and squared mass
differences as in Set I, with $\phi$ and $\theta$ varied in the $\pi/2$ range
with $0\leq\theta,\,\phi\leq\pi/2$. We observe that the parameter $\beta_{1}$
exhibits a maximum value of $\beta_{1}^{\text{max}}\sim 0.24$ at the location
with coordinates $\phi=\tilde{\phi}\sim 50^{\circ}$ and
$\theta=\tilde{\theta}\sim 62^{\circ}$, which is marked with a red star. At
the same coordinates, we find numerically that $\Delta M_{21}/M_{1}$ is
minimised, while $\Delta M_{32}/M_{2}$ is locally maximised. Moreover, we find
that the overall trend of the parameter space remains consistent if we choose
to work with the values for $\theta_{12}$, $\theta_{13}$, $\Delta m^{2}_{21}$
and $\Delta m_{31}^{2}$ as in Set II. However, the maximum value of
$\beta_{1}$ is found for different values of $\phi$ and $\theta$.
Specifically, the maximum value of $\beta_{1}$ is found at
$\phi=\tilde{\phi}\sim 38^{\circ}$ and $\theta=\tilde{\theta}\sim 65^{\circ}$,
taking a value of $\sim 0.21$. Upon examining the bottom panels of Fig. 1, it
is evident that our model predicts a nearly degenerate spectrum for the right-
handed neutrinos, with $M_{2}\lesssim 3M_{1}$ and $M_{3}\lesssim 3M_{2}$, over
a large part of the parameter space. However, we do not observe all three
right-handed neutrinos to be simultaneously nearly-degenerate in mass in any
region of the parameter space. There exists a tiny region where $N_{1}$ and
$N_{2}$ are nearly-degenerate with $M_{2}\lesssim 1.05M_{1}$. For $N_{2}$ and
$N_{3}$, there exists a larger region nearly degenerate, with $M_{3}\lesssim
1.05M_{2}$. These regions are shown in yellow-ish colour in the bottom panels
of Fig. 1.666We suspect that a specific point for which $M_{1}=M_{2}$ exists
around $(\tilde{\phi},\tilde{\theta})$ (although finding such precise point
would require infinite numerical resolution). It should be possible to
consider $M_{1}\simeq M_{2}$ with arbitrary precision around that point, but
this would require a certain amount of fine-tuning in the choice of the
parameters $\theta$ and $\phi$, much more severe than, e.g., the uncertainties
in the neutrino parameters we have considered.

##  3 Thermal leptogenesis

In this section, we present the equations necessary to investigate the thermal
leptogenesis mechanism in the minimal gauged U(1)${}_{L_{\mu}-L_{\tau}}$
model. To simplify our analysis, we make the following assumptions: i) the
U(1)${}_{L_{\mu}-L_{\tau}}$ gauge symmetry is never restored after the
reheating; ii) the masses of the U(1)${}_{L_{\mu}-L_{\tau}}$ gauge boson and
the singlet scalar field associated with $\sigma$ are larger than the
reheating temperature $T_{R}$ so that these fields are always absent from the
thermal bath; iii) the masses of all three right-handed neutrinos are smaller
than the reheating temperature. The first two can be realised by taking the
U(1)${}_{L_{\mu}-L_{\tau}}$ gauge coupling and the self coupling of the
$\sigma$ field sufficiently large and $\langle\sigma\rangle\gg T_{R}$. The
third assumption indicates
$|M_{ee,\mu\tau}|,|\lambda_{e\mu,e\tau}\langle\sigma\rangle|<T_{R}$.

To take the impact of charged lepton and heavy neutrino flavours into account,
we employ the formalism of DMEs instead of the simpler BEs.777Charged lepton
and heavy neutrino flavour effects can be particularly relevant when the
right-handed neutrino mass is not strongly hierarchical, as in our model (see
Fig. 1). In addition, as we will show later, to explain the observed BAU, the
mass of the lightest right-handed neutrino can be as low as $M_{1}\sim
10^{11-12}\,\text{GeV}$, with leptogenesis occurring at the transition between
the single- and two-flavour regimes, where BEs fail [67, 68]. Therefore, it is
necessary to solve the DMEs with three decaying right-handed neutrinos and
fully take flavour effects into account. Specifically, in the case of the
three decaying right-handed neutrinos, we can express the DMEs in the
following compact form [40, 41, 42, 44, 67, 45]:

$\displaystyle\frac{dN_{N_{j}}}{dz}$ $\displaystyle=$ $\displaystyle-
D_{j}(N_{N_{j}}-N^{\text{eq}}_{N_{j}})\,,$ (15)
$\displaystyle\frac{dN_{\alpha\beta}}{dz}$ $\displaystyle=$
$\displaystyle\begin{aligned}
&\sum_{j=1}^{3}\left[\epsilon^{(j)}_{\alpha\beta}D_{j}(N_{N_{j}}-N^{\text{eq}}_{N_{j}})-\frac{1}{2}W_{j}\left\\{P^{0(j)},N\right\\}_{\alpha\beta}\right]\\\
-&\frac{\Gamma_{\tau}}{Hz}\left[I_{\tau},\left[I_{\tau},N\right]\right]_{\alpha\beta}-\,\frac{\Gamma_{\mu}}{Hz}\left[I_{\mu},\left[I_{\mu},N\right]\right]_{\alpha\beta}\,,\end{aligned}$
(16)

where the indices $j=1,\,2,\,3$ refer to the three decaying right-handed
neutrinos, while $\alpha,\,\beta=e,\,\mu,\,\tau$ to the charged lepton
flavours. The variable $z$ is defined as $z\equiv M_{1}/T$. The quantity
$N_{N_{j}}$ represents the number of heavy neutrinos $N_{j}$ in a comoving
volume, which we normalise using the same method as in Refs. [44, 67, 45],
such that it contains a single photon when $z\ll 1$, i.e.,
$N^{\text{eq}}_{N_{j}}(0)=3/4$. We approximate the equilibrium number density
of heavy neutrinos as
$N^{\text{eq}}_{N_{j}}(z)=(3/8)x_{j}z^{2}K_{2}(\sqrt{x_{j}}z)$, where
$x_{j}\equiv(M_{j}/M_{1})^{2}$ and $K_{n}(z)$, $n=1\,,2\,,\,...$, is the
modified $n^{\text{th}}$ Bessel function of the second kind [33, 69]. The
diagonal entries $N_{\alpha\alpha}$ of the density matrix $N$ correspond to
the comoving number densities for the $(1/3)B-L_{\alpha}$ asymmetry, such that
$N_{B-L}=\sum_{\alpha=e,\,\mu,\,\tau}N_{\alpha\alpha}$. The off-diagonal
elements $N_{\alpha\beta}$, where $\alpha\neq\beta$, represent the degree of
coherence between the flavour states.

The projection matrices in the anti-commutator term in the first line of Eq.
(16) is defined as,

$P^{0(j)}_{\alpha\beta}\equiv\frac{\hat{\lambda}_{j\alpha}^{*}\hat{\lambda}_{j\beta}}{(\hat{\lambda}\hat{\lambda}^{\dagger})_{jj}}\,,$
(17)

generalising the notion of the projection probability. Furthermore, the double
commutator structure in the second line of Eq. (16) involves $3\times 3$
matrices $I_{\tau}$ and $I_{\mu}$ defined such that
$(I_{\tau})_{\alpha\beta}=\delta_{\alpha\tau}\delta_{\beta\tau}$ and
$(I_{\mu})_{\alpha\beta}=\delta_{\alpha\mu}\delta_{\beta\mu}$. Additionally,
we use the following analytical expressions for $D_{j}$ and $W_{j}$ [33, 44],

$\displaystyle D_{j}(z)$ $\displaystyle=$
$\displaystyle\kappa_{j}x_{j}z\frac{K_{1}(\sqrt{x_{j}}z)}{K_{2}(\sqrt{x_{j}}z)}\,,$
(18) $\displaystyle W_{j}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\kappa_{j}x_{j}^{2}z^{3}K_{1}(\sqrt{x_{j}}z)\,,$ (19)

where the decay parameter $\kappa_{j}$ quantifies the strength of the wash-out
processes in erasing the asymmetry and is defined as the ratio between the
total decay rate of $N_{j}$ at zero temperature,
$\Gamma_{j}=(\hat{\lambda}\hat{\lambda}^{\dagger})_{jj}M_{j}/8\pi$, and the
Hubble expansion rate $H$ at $T=M_{j}$.

Note that the decay rate $\Gamma_{j}$ and the Hubble expansion rate
$H(T=M_{j})\propto M_{j}^{2}$ scale as $\lambda^{4}$. As a result, the decay
parameter $\kappa_{j}$ depends only on the angles $\theta$ and $\phi$.
Numerical calculations show that, for any $0\leq\theta,\,\phi\leq\pi/2$ and
$j=1,\,2,\,3$, $\kappa_{j}\gg 1$ and leptogenesis occurs in the strong wash-
out regime. In such a regime, the initial abundance of right-handed neutrinos
typically does not appreciably affect leptogenesis [70]. We thus focus on the
case for which the initial abundances of the right-handed neutrinos are zero.
It has also been estimated that, in the strong wash-out regime and in the case
of a hierarchical right-handed neutrino mass spectrum, the $\Delta L=1$
scattering processes and related wash-outs would only contribute to the final
BAU at most by $\mathcal{O}(10\%)$ [70, 71]. We find that this is still true
in most of the parameter space of our model, except for certain fine-tuned
choices of the parameters. Therefore, we do not consider the effects of
scatterings in the main analysis.

The CP-asymmetry $\epsilon_{\alpha\beta}^{(j)}$ appearing in the DMEs can be
separated into contributions from vertex and self-energy diagrams, namely
$\epsilon_{\alpha\beta}^{(j)}\equiv\epsilon_{\alpha\beta}^{\text{V}\,(j)}+\epsilon_{\alpha\beta}^{\text{S}\,(j)}$.
The two contributions are given by [72, 73, 74, 75, 38, 40, 42, 76],

$\displaystyle\epsilon^{\text{V}\,(j)}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi\left(\hat{\lambda}\hat{\lambda}^{\dagger}\right)_{jj}}\sum_{k\neq
j}\left\\{i\left[\hat{\lambda}_{j\alpha}^{*}\hat{\lambda}_{k\beta}(\hat{\lambda}\hat{\lambda}^{\dagger})_{kj}-\hat{\lambda}_{j\beta}\hat{\lambda}_{k\alpha}^{*}(\hat{\lambda}\hat{\lambda}^{\dagger})_{jk}\right]\xi\left(x_{k}/x_{j}\right)\right\\}\,,$
(20) $\displaystyle\epsilon^{\text{S}\,(j)}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi\left(\hat{\lambda}\hat{\lambda}^{\dagger}\right)_{jj}}\sum_{k\neq
j}\Bigg{\\{}i\left[\hat{\lambda}_{j\alpha}^{*}\hat{\lambda}_{k\beta}(\hat{\lambda}\hat{\lambda}^{\dagger})_{kj}-\hat{\lambda}_{j\beta}\hat{\lambda}_{k\alpha}^{*}(\hat{\lambda}\hat{\lambda}^{\dagger})_{jk}\right]\frac{\sqrt{x_{k}/x_{j}}}{x_{k}/x_{j}-1}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+i\left[\hat{\lambda}_{j\alpha}^{*}\hat{\lambda}_{k\beta}(\hat{\lambda}\hat{\lambda}^{\dagger})_{jk}-\hat{\lambda}_{j\beta}\hat{\lambda}^{*}_{k\alpha}(\hat{\lambda}\hat{\lambda}^{\dagger})_{kj}\right]\frac{1}{x_{k}/x_{j}-1}\Bigg{\\}}\,,$

where $\xi(x)\equiv\sqrt{x}\left[(1+x)\log\left(1+1/x\right)-1\right]$. In the
case of a degeneracy in the right-handed neutrino mass spectrum, the self-
energy contribution to the CP-asymmetry is resonantly enhanced and can
dominate over the vertex contribution. However, the self-energy contribution
in Eq. (3) becomes ill-defined when two neutrinos, say $N_{j}$ and $N_{k}$,
are quasi-degenerate in mass, as $1/(x_{k}/x_{j}-1)\to\infty$ when
$x_{k}/x_{j}\to 1$. To address this non-physical behaviour, the full-resummed
Yukawa couplings must be considered in the calculations, and the self-energy
contribution to the CP-asymmetry can be regularised by performing the
following substitution [77, 78, 79, 80],

$\frac{1}{x_{k}/x_{j}-1}\to\frac{(M_{k}^{2}-M_{j}^{2})M_{j}^{2}}{(M_{k}^{2}-M_{j}^{2})^{2}+M_{j}^{4}\Gamma_{k}^{2}/M_{k}^{2}}\,.$
(22)

The regularised CP-asymmetry, obtained by using the prescription in Eq.(22),
does not suffer from divergences and vanishes for equal right-handed neutrino
masses. However, the self-energy contribution to the CP-asymmetry, even after
regularisation, still exhibits a resonance. In particular, the self-energy
contribution is maximised when $|\Delta M_{jk}|\simeq 0.5\,\Gamma_{j}$, with
$\Delta M_{jk}\equiv M_{j}-M_{k}$. This resonant behaviour has been
extensively studied in the context of resonant leptogenesis, especially in the
efforts to avoid the Davidson-Ibarra bound [29] and extend the scenario of
thermal leptogenesis down to the electroweak scale [77, 78, 81, 79, 82, 80,
83, 84, 85, 86, 87, 88, 89, 68]. A comprehensive review can be found in Ref.
[90]. The ratios $|\Delta M_{jk}|/\Gamma_{j}$, $j\neq k$, quantify the
importance of resonance effects. Far away from the resonance, when
$|M_{j}-M_{k}|\gg\Gamma_{j}$, the effects of the enhancement are sub-leading,
and the regularised CP-asymmetry obtained with the prescription in Eq. (22)
resembles the form given in Eq. (3). We perform a scan of $|\Delta
M_{12}|/\Gamma_{1}$ and $|\Delta M_{23}|/\Gamma_{2}$ over the $\phi-\theta$
plane for various choices of $\lambda$. We found that resonance effects are
significant only in certain small regions of the parameter space. Furthermore,
these regions become even smaller with decreasing values of $\lambda$ and are
effectively negligible when $\lambda\lesssim 0.5$. The details of this
analysis can be found in Appendix A. Nevertheless, we have included the
effects of resonance as in Eq. (22) in our calculations.888Different
approaches to regularise the self-energy contribution to the CP-asymmetry have
been proposed [91, 92, 90]. However, in our case, the mass splittings between
the right-handed neutrinos are much larger than their decay widths and we
expect that a different estimation of the resonant effects would not
significantly affect our results. It is worth noting that in Ref. [79] (and in
the recent study in Ref. [93]), a more sophisticated regularisation taking
into account the mass degeneracy between all the three right-handed neutrinos
was found. However, our analysis presented in Appendix A shows no region of
the parameter space of our model for which $|M_{1}-M_{2}|/\Gamma_{1}\lesssim
10$ and $|M_{2}-M_{3}|/\Gamma_{2}\lesssim 10$ simultaneously. Hence, for our
purposes, it is sufficient to consider the mass degeneracy between two right-
handed neutrinos at a time and use the simplified regularisation in Eq. (22).

Finally, we numerically solve the DMEs in Eqs. (15) and (16) with the Python
package ULYSSES [46, 47]. The code computes
$N_{B-L}=N_{ee}+N_{\mu\mu}+N_{\tau\tau}$ and relates it to the present baryon-
to-photon ratio via

$\eta_{B}=\frac{c_{s}}{f}N_{B-L}\approx 0.013N_{B-L}\,,$ (23)

where $c_{s}$ is the SM sphaleron conversion coefficient and the $f$ factor
comes from the dilution of the baryon asymmetry due to the change in the
photon density between leptogenesis and recombination [33].

##  4 Results of the Parameter Scan of Viable Leptogenesis

In this section, we discuss the results of our parameter scan aimed at
identifying the viable space for thermal leptogenesis by solving the set of
DMEs introduced in Sec. 3.

Figure 2: The contour plots in the $\phi-\theta$ plane of the BAU (in modulus)
predicted by the DMEs with three decaying right-handed neutrinos. The plots
are obtained for $\lambda=0.5$ (top panels), $0.3$ (central panels) and $0.1$
(bottom panels), with the input parameters $\theta_{12}$, $\theta_{13}$,
$\theta_{23}$, $\Delta m^{2}_{21}$ and $\Delta m^{2}_{31}$ as in Set I with
$(\delta,\,\alpha_{2},\,\alpha_{3})=(301^{\circ},\,116^{\circ},\,269^{\circ})$
(left panels) and Set II with
$(\delta,\,\alpha_{2},\,\alpha_{3})=(228^{\circ},\,225^{\circ},\,70^{\circ})$
(right panels). As indicated in the bar legends, darker (lighter) regions
correspond to larger (smaller) values of the BAU, with the red solid contours
representing the points for which the predicted BAU equals the observed value
$\eta_{B}\simeq 6.1\times 10^{-10}$. For the red dotted contours, the BAU
equals the observed value in modulus, but the sign is negative. The dotted
vertical and horizontal lines mark the benchmark points BMPa (top panels),
BMPb (central panels), BMPcI and BMPcII (bottom panels).

Fig. 2 shows the results of the scan in the $\phi-\theta$ plane for three
benchmark values of $\lambda$, namely $\lambda=0.5$ (top panels),
$\lambda=0.3$ (central panels) and $\lambda=0.1$ (bottom panels). The plots in
the left and right panels are obtained with the neutrino mixing angles and
squared mass differences as in Set I with
$(\delta,\,\alpha_{2},\,\alpha_{3})=(301^{\circ},\,116^{\circ},\,269^{\circ})$
and Set II with
$(\delta,\,\alpha_{2},\,\alpha_{3})=(228^{\circ},\,225^{\circ},\,70^{\circ})$,
respectively. The regions of viable leptogenesis are those surrounded by the
solid red contours, where the predicted BAU matches the observed value
$\eta_{B}=6.1\times 10^{-10}$. The points corresponding to the dotted red
contours result in a predicted BAU of $-6.1\times 10^{-10}$, indicating the
correct magnitude but wrong sign. To darker (lighter) regions correspond
larger (smaller) values of the predicted BAU, in modulus.

Figure 3: The BAU as a function of $\lambda$ predicted by the DMEs with three
decaying right-handed neutrinos. The input parameters in the left and right
panels are as in Set I with
$(\delta,\,\alpha_{2},\,\alpha_{3})=(301^{\circ},\,116^{\circ},\,269^{\circ})$
and Set II with
$(\delta,\,\alpha_{2},\,\alpha_{3})=(228^{\circ},\,225^{\circ},\,70^{\circ})$,
respectively. The different curves are obtained for BMPa (blue), BMPb
(orange), BMPcI and BMPcII (green), with the solid (dashed) style indicating
the positive (negative) sign of the predicted baryon-to-photon ratio. Note the
overall sign flip between the curves obtained in the two different sets of
input parameters.

In Fig. 2, it is clear that there are numerous regions in the parameter space
where leptogenesis within the minimal gauged U(1)${}_{L_{\mu}-L_{\tau}}$ model
predicts a BAU equal to or greater than the observed value, in modulus.
However, the sign of the BAU alternates between positive and negative values
in different regions, and only those regions where the BAU is positive
correspond to successful leptogenesis. It is worth noting that the sign of the
BAU can always be changed without changing its magnitude by switching to the
equivalent solution with
$(m_{1},\,2\pi-\delta,\,2\pi-\alpha_{2},\,2\pi-\alpha_{3})$ [21]. This can be
shown as follows. First, note that switching from a set of solutions
$(m_{1},\,\delta,\,\alpha_{2},\,\alpha_{3})$ to the other one
$(m_{1},\,2\pi-\delta,\,2\pi-\alpha_{2},\,2\pi-\alpha_{3})$ yields $U\to
U^{*}$, and thus $\mathcal{M}_{\nu_{L}}\to\mathcal{M}_{\nu_{L}}^{*}$,
$\mathcal{M}_{R}\to\mathcal{M}_{R}^{*}$, $\Omega\to\Omega^{*}$, and
$\hat{\lambda}\to\hat{\lambda}^{*}$. Then, $\epsilon^{(j)}_{\alpha\beta}$ and
$P^{0(j)}_{\alpha\beta}$ in Eq. (16) transform as
$\epsilon^{(j)}_{\alpha\beta}\to-\epsilon^{(j)}_{\beta\alpha}$ and
$P^{0(j)}_{\alpha\beta}\to P^{0(j)}_{\beta\alpha}$, respectively. This
indicates that if $N_{N_{j}}$ and $N_{\alpha\beta}$ are solutions of Eqs. (15)
and (16) for a given set of $(m_{1},\,\delta,\,\alpha_{2},\,\alpha_{3})$, then
the solutions for $(m_{1},\,2\pi-\delta,\,2\pi-\alpha_{2},\,2\pi-\alpha_{3})$
are given by $N_{N_{j}}$ and $-N_{\beta\alpha}$. By noting that
$\eta_{B}\propto(N_{ee}+N_{\mu\mu}+N_{\tau\tau})$, we conclude that if the
baryon-to-photon ratio for $(m_{1},\,\delta,\,\alpha_{2},\,\alpha_{3})$ is
given by $\eta_{B}$, that for
$(m_{1},\,2\pi-\delta,\,2\pi-\alpha_{2},\,2\pi-\alpha_{3})$ is given by
$-\eta_{B}$. It is also important to note that there is an overall sign
difference between the parameter scans obtained for the two sets of input
parameters, Set I and II. This is because $\theta_{23}$ varies from values
below $45^{\circ}$ in Set I to values above $45^{\circ}$ in Set II, resulting
in a shift in the PMNS phases and an opposite sign of the BAU.999The behaviour
is not exactly symmetric in $\theta_{23}$, see Fig. 1(b) of Ref. [24].
Consequently, only precise combinations of the phase $\delta$ and mixing angle
$\theta_{23}$ can lead to the correct sign of the BAU in a particular point of
the $\phi-\theta$ plane. Determining whether $\theta_{23}<$ or $>\pi/4$ and/or
$\delta<$ or $>\pi$ in future experiments would rule out certain regions of
the parameter space of viable leptogenesis in our model only based on the sign
of the predicted baryon-to-photon ratio.

In addition, the size of the allowed regions for successful leptogenesis in
the $\phi-\theta$ plane is dependent on the mass scale of leptogenesis, which
goes as $\lambda^{2}$. As $\lambda$ decreases (increases), the allowed ranges
of $\phi$ and $\theta$ for successful leptogenesis become smaller (larger). To
determine the minimal values of $\lambda$ at which the viable regions shrink
to points, we search for the local maxima of the predicted BAU. We identify
four benchmark points (BMPs) in the $\phi-\theta$ plane, around which the
symmetry is locally maximised. The BMPs are located at the coordinates BMP a)
$\theta=29.06^{\circ}$ and $\phi=47.28^{\circ}$; BMP b) $\theta=29.06^{\circ}$
and $\phi=19.94^{\circ}$; BMP cI) $\theta=56.39^{\circ}$ and
$\phi=56.39^{\circ}$ for Set I, and BMP cII) $\theta=56.39^{\circ}$ and
$\phi=48.19^{\circ}$ for Set II. These four BMPs are indicated in Fig. 2 by
horizontal and vertical dotted grid lines.

Finally, Fig. 3 illustrates the dependence of the BAU on the scale $\lambda$
for the four different BMPs. The left and right panels correspond to input
parameters as in Set I with
$(\delta,\,\alpha_{2},\,\alpha_{3})=(301^{\circ},\,116^{\circ},\,269^{\circ})$
and Set II with
$(\delta,\,\alpha_{2},\,\alpha_{3})=(228^{\circ},\,225^{\circ},\,70^{\circ})$,
respectively. Our numerical analysis yields the following minimal values of
$\lambda$ for which leptogenesis is viable: for Set I, $\lambda\simeq 0.35$
($M_{1}\simeq 10^{12.8}\,\text{GeV}$) for BMPa and
$(\delta,\,\alpha_{2},\,\alpha_{3})=(301^{\circ},\,116^{\circ},\,269^{\circ})$;
$\lambda\simeq 0.25$ ($M_{1}\\!\simeq\\!10^{12.2}\,\text{GeV}$) for BMPb and
$(\delta,\alpha_{2},\alpha_{3})\\!=\\!(59^{\circ},\,244^{\circ},\,91^{\circ})$;
$\lambda\simeq 0.05$ ($M_{1}\simeq 10^{11.6}\,\text{GeV}$) for BMPcI and
$(\delta,\,\alpha_{2},\,\alpha_{3})=(301^{\circ},\,116^{\circ},\,269^{\circ})$.
For Set II, we find: $\lambda\simeq 0.305$ ($M_{1}\simeq
10^{12.8}\,\text{GeV}$) for BMPa and
$(\delta,\,\alpha_{2},\,\alpha_{3})=(132^{\circ},\,135^{\circ},\,290^{\circ})$;
$\lambda\simeq 0.25$ ($M_{1}\simeq 10^{12.3}\,\text{GeV}$) for BMPb and
$(\delta,\,\alpha_{2},\,\alpha_{3})=(228^{\circ},\,225^{\circ},\,70^{\circ})$;
$\lambda\simeq 0.03$ ($M_{1}\simeq 10^{11.2}\,\text{GeV}$) for BMPcII and
$(\delta,\,\alpha_{2},\,\alpha_{3})=(228^{\circ},\,225^{\circ},\,70^{\circ})$.
Overall, we find that leptogenesis is viable for $M_{1}\gtrsim
10^{11}\,\text{GeV}$ across the entire parameter space.

##  5 Conclusions

In this work, we have studied thermal leptogenesis in the context of the
minimal gauged U(1)${}_{L_{\mu}-L_{\tau}}$ model. Given the light neutrino
squared mass differences and the PMNS mixing angles as input, the model
predicts the values of the lightest neutrino mass $m_{1}$, as well as the PMNS
phases $\delta$, $\alpha_{2}$, and $\alpha_{3}$. The model remains with three
free parameters, which we have identified as $\lambda$, $\theta$, and $\phi$
according to the parameterisation of the Yukawa couplings as in Eq. (9). We
have then performed a numerical scan of the parameter space and searched for
the allowed ranges of $\lambda$, $\theta$ and $\phi$ for which leptogenesis is
viable in reproducing the observed value of the baryon asymmetry of the
Universe. To fully account for the effects of the charged lepton and heavy
neutrino flavours, we have solved numerically the sets of density matrix
equations instead of the simpler Boltzmann equations, and took the decays of
all three right-handed neutrinos into account. Noting that thermal
leptogenesis proceeds in the strong wash-out regime in the entire parameter
space of the model so that the effects of scatterings and different initial
conditions are typically sub-leading, we have focused only on direct and
inverse decays and on the case of vanishing initial right-handed neutrino
abundances. To avoid the non-physical behaviour in the CP-asymmetry due to
degeneracy in the right-handed neutrino mass spectrum, we have included the
full-resummed Yukawa couplings and resonant effects in the calculation, even
though we found those to be relevant only in secluded regions of the parameter
space.

We have found numerous regions in the parameter space where thermal
leptogenesis within the minimal gauged U(1)${}_{\mu-L_{\tau}}$ model predicts
the correct baryon asymmetry of the Universe. This result was not obvious
given the restrictions in the neutrino mass structure imposed by the model.
The size of the allowed regions for successful leptogenesis in the
$\phi-\theta$ plane decreases with the mass scale of leptogenesis, which goes
as $\lambda^{2}$, implying minimal values of the mass scale and $\lambda$ for
which leptogenesis can be successful. We found that thermal leptogenesis is
viable for $M_{1}\gtrsim 10^{11-12}\,\text{GeV}$ across the entire parameter
space, with $\lambda$ taking values from order unity down to
$\mathcal{O}(0.03-0.05)$. These values are larger than those obtained in the
context of non-thermal leptogenesis within the minimal gauged
U(1)${}_{L_{\mu}-L_{\tau}}$ model [24]. The difference is mostly due to the
wash-out effects that are present in the thermal leptogenesis mechanism but
not in the non-thermal scenario, implying relatively heavier right-handed
neutrinos to satisfy the out-of-equilibrium condition.

The sign of the baryon asymmetry can be either positive or negative in the
various regions, but leptogenesis is successful only where the baryon
asymmetry is positive. However, after specifying the neutrino squared mass
differences and the PMNS mixing angles, the minimal gauged
U(1)${}_{L_{\mu}-L_{\tau}}$ model predicts two distinct sets of PMNS phases,
$(\delta,\,\alpha_{2},\,\alpha_{3})$ and
$(2\pi-\delta,\,2\pi-\alpha_{2},\,2\pi-\alpha_{3})$, each corresponding to
opposite signs of the predicted baryon asymmetry. At present, the uncertainty
on the estimate of $\delta$ obtained in the global analysis [57] is relatively
large so that, in general, both predictions are plausible (to a certain level
of confidence). We are then allowed to change the sign of the predicted
asymmetry by switching from one set of PMNS phases to the other. The baryon
asymmetry also has an opposite sign depending on whether $\theta_{23}$ lies
above or below $\pi/4$. According to the $3\sigma$ ranges for $\theta_{23}$
obtained in the global analysis [57], see Table 2, both cases are still valid,
and it is, therefore, impossible to discriminate the sign of the baryon
asymmetry in a given region of the parameter space. Of course, with future
more accurate measurements of the Dirac phase $\delta$ and of the mixing angle
$\theta_{23}$ from, e.g., T2K [94], NO$\nu$A [95], JUNO [96], DUNE [97], and
Hyper-Kamiokande [98] (see also Ref. [99]), we would be able to rule out the
certain region of the parameter space of leptogenesis in the considered model
only on the basis of the sign of the baryon asymmetry. We stress that the
results of this work are obtained for two different extreme choices of
$\theta_{23}$ corresponding to the $\pm 3\sigma$ limit values of the global
analysis, so to minimize the sum of neutrino masses, $\sum_{i}m_{i}$, and the
tension with the cosmological bounds. In the future, more precise measurements
of $\theta_{23}$ and/or $\sum_{i}m_{i}$ will likely impose even more stringent
constraints on the model presented here.

## Acknowledgements

A.G. wishes to thank the Kavli IPMU and the Department of Physics of the
University of Tokyo at Hongo Campus for the kind hospitality offered during
the first part of this project. A.G. acknowledges the use of computational
resources from the parallel computing cluster of the Open Physics Hub
(https://site.unibo.it/openphysicshub/en) at the Physics and Astronomy
Department in Bologna. The work of A.G. has received funding from the European
Union’s Horizon Europe research and innovation programme under the Marie
Skłodowska-Curie Staff Exchange grant agreement No. 101086085 - ASYMMETRY. The
work of K.H., N.N., and M.R.Q. was supported in part by the Grant-in-Aid for
Innovative Areas (No. 19H05810 [K.H.], No. 19H05802 [K.H.], No. 18H05542
[N.N.]), Scientific Research B (No. 20H01897 [K.H., N.N., and M.R.Q.]), and
Young Scientists (No. 21K13916 [N.N.]). The work of J.W. is supported by the
JSPS KAKENHI Grant (No. 22J21260).

##  Appendix A The Impact of Resonance Effects

In this appendix, we estimate the relevance of the resonance effects of the
CP-asymmetry in our numerical analysis. We first note that $\Delta
M_{kj}/\Gamma_{j}\propto\lambda^{-2}$ meaning that, for a given point in the
$\phi-\theta$ plane, the resonance effects are depleted if $\lambda$ is
sufficiently small. We show in Fig. 4 the points of our scan for which we find
that $\Delta M_{21}/\Gamma_{1}\lesssim 10$ (left panel) and $\Delta
M_{32}/\Gamma_{2}\lesssim 10$ (right panel) for $\lambda=1$ (top panels) and
$0.5$ (bottom panels). The figure is obtained for the neutrino mixing angles
and squared mass differences as in Set I. The results remain qualitatively
similar for the input parameters in Set II and are thus not shown here. The
plots in the top panels show that, when $\lambda=1$, we have $\Delta
M_{21}/\Gamma_{1}\lesssim 10$ and $\Delta M_{32}/\Gamma_{2}\lesssim 10$ in the
region of the parameter space corresponding to $M_{2}\lesssim 1.5M_{1}$ and
$M_{3}\lesssim 1.5M_{2}$, respectively. When $\lambda=0.5$, these regions are
reduced, and it becomes difficult to find points satisfying the two
inequalities for $\lambda<0.5$. This indicates that resonance effects are
negligible in most of the parameter space when $\lambda\lesssim 0.5$. By
comparing the plots in the left and right panels, it follows that there is no
region where the three right-handed neutrinos are simultaneously in the
resonant regime, thus justifying the use of the regularisation in Eq. (22) in
the main analysis.

Figure 4: The red points in the left and right panels are those for which we
find $\Delta M_{21}/\Gamma_{1}\lesssim 10$ and $\Delta
M_{32}/\Gamma_{2}\lesssim 10$, respectively. The points in the top (bottom)
panels are obtained for $\lambda=1\,(0.5)$. The figure is obtained for the
input parameters as in Set I. The remaining details of the plots are as in the
bottom panels of Fig. 1.

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