hep-ph/***

##  1 Introduction

The Standard Model (SM) successfully describes many experimental results.
However, important observations such as the smallness of neutrino masses and
the peculiar flavour structure in the lepton and quark sectors remain
unexplained.

Many extensions of the SM offer a generation mechanism for neutrino masses.
Among these are the well-known type-I [1, 2, 3, 4, 5], type-II [6, 7, 8, 9,
10, 11] and type-III seesaw mechanisms [12] which typically require new
particles with masses much larger than the electroweak scale, if couplings are
of order one. Alternatives such as the inverse seesaw (ISS) mechanism [13, 16,
14, 15] can reconcile new particles with masses in the TeV-range with not too
small couplings, while the smallness of neutrino masses is due to a small
breaking of lepton number (in the form of a Majorana mass term for the gauge
singlets).

The organising principle of the flavour sector responsible for the observed
pattern of fermion masses and mixing is yet to be found. Nevertheless,
discrete flavour symmetries, especially when combined with CP, that are broken
in a certain way among e.g. charged leptons and neutral states are promising
candidates for correctly describing the observed lepton mixing angles and
making testable predictions for the leptonic CP phases [17, 18, 19] (for
earlier works see [20, 21, 22, 23, 24, 25]); reviews can be found, for
example, in [26, 27, 28, 29, 30]. In this work, we employ the series of
discrete groups $\Delta(3\,n^{2})$ and $\Delta(6\,n^{2})$, $n$ integer, [31,
32] and CP symmetries, which correspond to automorphisms of these [17, 18,
19]. As is known [33], four different types of lepton mixing patterns are
found, called Case 1), Case 2), Case 3 a) and Case 3 b.1), if the flavour
symmetry $G_{f}$ and CP are broken to a residual $Z_{3}$ symmetry among
charged leptons and to $Z_{2}\times CP$ among the neutral states. For further
studies of lepton mixing from the groups $\Delta(3\,n^{2})$ and
$\Delta(6\,n^{2})$ and CP, see e.g. [34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44].

In the current study, we analyse a scenario in which light neutrino masses are
generated by the ISS mechanism with 3+3 gauge singlet fermions and the flavour
symmetries $\Delta(3\,n^{2})$ and $\Delta(6\,n^{2})$ and CP are broken non-
trivially. 111We do not specify the breaking mechanism here. Concretely, the
non-trivial flavour structure among the neutral states is encoded in the Dirac
neutrino Yukawa matrix, while the mass matrices (Dirac and Majorana) of the
gauge singlet fermions do neither break $G_{f}$ nor CP. This possibility has
been called option 2 in [45]. We scrutinise analytically and numerically the
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix and different charged
lepton flavour violating (cLFV) processes for Case 1) through Case 3 b.1). The
main findings are that the effects of non-unitarity of the PMNS mixing matrix
induced by the existence of the heavy sterile states have a more general form
than those found in the scenario analysed in [45] (called option 1) and that
signals of cLFV $\mu-e$ transitions can be sizeable in the studied parameter
space, while those involving the tau lepton are more suppressed.

The remainder of the paper is organised as follows: in section 2 we outline
the scenario, show the form of the different mass and coupling matrices,
comment on the masses of the different neutral states, discuss the mixing
matrices and the effects of the heavy sterile states on the light neutrinos.
Section 3 contains information on lepton mixing including the characteristics
of the mixing patterns of Case 1) through Case 3 b.1) and the effects of non-
unitarity due to the mixing of the light neutrinos and the heavy sterile
states. In section 4 we present analytic estimates as well as numerical
results for the branching ratios (BRs) $\mathrm{BR}(\mu\to e\gamma)$ and
$\mathrm{BR}(\mu\to 3\,e)$ and the $\mu-e$ conversion rate (CR) in aluminium,
$\mathrm{CR}(\mu-e,\mathrm{Al})$, for examples of all cases, Case 1) through
Case 3 b.1). We comment on the expected size of the signals of the cLFV tau
lepton decays $\tau\to\mu\gamma$, $\tau\to e\gamma$, $\tau\to 3\,\mu$ and
$\tau\to 3\,e$ in section 5 and summarise in section 6. In appendix A basics
of the group theory of $\Delta(3\,n^{2})$ and $\Delta(6\,n^{2})$ can be found,
while we collect the form of different relevant matrices in appendix B.
Appendix C comprises a concise description of the numerical scan and
supplementary plots displaying results of the studied $\mu-e$ transitions for
Case 3 b.1) are given in appendix D.

##  2 Scenario

In the following, we focus on the lepton sector. We add 3+3 gauge singlet
fermions $N_{i}$ and $S_{j}$, $i,j=1,2,3$, to the SM in order to generate
light neutrino masses via the ISS mechanism.

The flavour structure of the scenario is controlled by a flavour and a CP
symmetry that are assumed to be broken to different non-trivial subgroups in
the charged lepton, $G_{e}$, and in the neutral lepton sectors, $G_{\nu}$,
respectively. As flavour symmetry we choose one of the groups
$\Delta(3\,n^{2})$ and $\Delta(6\,n^{2})$ with $n\geq 2$, see appendix A for
relevant details of these groups. The CP symmetry corresponds to an
automorphism of the flavour group and is selected according to the findings of
[33].

Left-handed (LH) lepton doublets $L_{\alpha}$, $\alpha=e,\mu,\tau$, are
assigned to a triplet ${\bf 3}$. This representation is taken to be
irreducible, faithful and complex. The gauge singlets $N_{i}$ and $S_{j}$
instead each transform as an irreducible, unfaithful, real triplet ${\bf
3^{\prime}}$. The existence of such a representation requires that the flavour
group has an even index $n$. On the contrary, right-handed (RH) charged
leptons $\ell_{\alpha R}$ are singlets, all in ${\bf 1}$, under the flavour
group. In order to distinguish these, we employ an additional $Z_{3}$ symmetry
$Z_{3}^{\mathrm{(aux)}}$, i.e. $\ell_{eR}\sim 1$, $\ell_{\mu R}\sim\omega$ and
$\ell_{\tau R}\sim\omega^{2}$ with $\omega=e^{\frac{2\,\pi i}{3}}$. LH lepton
doublets and the gauge singlets are neutral under this additional group.

The group $G_{e}$ determines the structure of the charged lepton mass matrix
$m_{\ell}$. We use a residual $Z_{3}$ symmetry that is a diagonal subgroup of
the $Z_{3}$ symmetry, generated by the generator $a$ of the flavour group
(which is represented by a diagonal matrix in the chosen basis, compare
appendix A), and the additional $Z_{3}$ symmetry $Z_{3}^{\mathrm{(aux)}}$.
Consequently, $m_{\ell}$ is diagonal and contains three free parameters
corresponding to the masses of the electron, muon and tau lepton. We take
these to be ordered canonically such that the contribution $U_{\ell}$ to
lepton mixing is $U_{\ell}=\mathbb{1}$.

The masses of the neutral states arise from the terms222For simplicity, we do
not take into account a possible coupling between LH lepton doublets and the
gauge singlets $S_{j}$ nor do we consider a Majorana mass term for the
singlets $N_{i}$. Additional symmetries and/or an extension of the gauge group
can motivate this choice.

$-(y_{D})_{\alpha
i}\,\overline{L}^{c}_{\alpha}\,H\,N^{c}_{i}-(M_{NS})_{ij}\,\overline{N}_{i}\,S_{j}-\frac{1}{2}\,(\mu_{S})_{kl}\,\overline{S}^{c}_{k}\,S_{l}+\mathrm{h.c.}$
(1)

with $H$ being the Higgs field, $y_{D}$ the Dirac neutrino Yukawa matrix,
$M_{NS}$ the matrix connecting the gauge singlets $N_{i}$ and $S_{j}$ and
$\mu_{S}$ the Majorana mass matrix of the singlets $S_{i}$. From eq. (1), we
can define the Dirac neutrino mass matrix $m_{D}$ as $m_{D}=y_{D}\,\langle
H\rangle$ with $\langle H\rangle$ being the vacuum expectation value (VEV) of
the Higgs, $\langle H\rangle\approx 174\,\mbox{GeV}$. The masses of all
neutral states, light and heavy ones, come from the nine-by-nine mass matrix

${\cal
M}_{\mathrm{Maj}}=\left(\begin{array}[]{ccc}\mathbb{0}&m_{D}&\mathbb{0}\\\
m_{D}^{T}&\mathbb{0}&M_{NS}\\\
\mathbb{0}&M_{NS}^{T}&\mu_{S}\end{array}\right)\;,$ (2)

which is given in the basis $(\nu_{\alpha L},N_{i}^{c},S_{j})$. This matrix
can be diagonalised as

${\cal U}^{T}\,{\cal M}_{\mathrm{Maj}}\,{\cal U}={\cal
M}_{\mathrm{Maj}}^{\mathrm{diag}}$ (3)

with

${\cal U}=\left(\begin{array}[]{cc}\tilde{U}_{\nu}&S\\\
T&V\end{array}\right)\;\;\mbox{and}\;\;{\cal
M}_{\mathrm{Maj}}^{\mathrm{diag}}=\mathrm{diag}\left(m_{1},m_{2},m_{3},m_{4},\dots,m_{9}\right)\;,$
(4)

where $\tilde{U}_{\nu}$ is a three-by-three, $S$ a three-by-six, $T$ a six-by-
three and $V$ a six-by-six matrix. The masses $m_{1,2,3}$ are the light
neutrino masses and $m_{4}$ to $m_{9}$ those of the heavy states. The matrix
${\cal U}$ is unitary, while none of the matrices $\tilde{U}_{\nu}$, $S$, $T$
and $V$ has this property. We assume in the following
$|\mu_{S}|\ll|m_{D}|\ll|M_{NS}|$. As shown in [46], the light neutrino mass
matrix at leading, $m_{\nu}$, and at sub-leading order, $m_{\nu}^{1}$, in
$(|m_{D}|/|M_{NS}|)^{2}$ read

$m_{\nu}=m_{D}\,\Big{(}M_{NS}^{-1}\Big{)}^{T}\,\mu_{S}\,M_{NS}^{-1}\,m_{D}^{T}$
(5)

and

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!m_{\nu}^{1}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\,m_{D}\,\Big{(}M_{NS}^{-1}\Big{)}^{T}\,\left(\mu_{S}\,M_{NS}^{-1}\,m_{D}^{T}\,m_{D}^{\star}\,\Big{(}M_{NS}^{-1}\Big{)}^{\dagger}+\Big{(}M_{NS}^{-1}\Big{)}^{\star}\,m_{D}^{\dagger}\,m_{D}\,\Big{(}M_{NS}^{-1}\Big{)}^{T}\,\mu_{S}\right)\,M_{NS}^{-1}\,m_{D}^{T}$
(6) $\displaystyle=$
$\displaystyle-\left(m_{\nu}\,\eta+\eta^{\star}\,m_{\nu}\right)\;,$

using the matrix $\eta$ defined in eq. (10). The matrix $m_{\nu}$ is
(approximately) diagonalised by $\tilde{U}_{\nu}$,

$\tilde{U}_{\nu}^{T}\,m_{\nu}\tilde{U}_{\nu}\approx\mathrm{diag}(m_{1},m_{2},m_{3})\;.$
(7)

We can define the (non-unitary) lepton mixing matrix

$\tilde{U}_{\mathrm{PMNS}}=U_{\ell}^{\dagger}\,\tilde{U}_{\nu}=\tilde{U}_{\nu}\;,$
(8)

where the second equality takes into account that $U_{\ell}=\mathbb{1}$ in the
chosen basis. This non-unitarity is due to the mixing of the light neutrinos
with the heavy states and it can be encoded in the hermitean matrix $\eta$,

$\tilde{U}_{\mathrm{PMNS}}=\Big{(}\mathbb{1}-\eta\Big{)}\,U_{0}$ (9)

with $U_{0}$ being unitary. Note that in the chosen basis eq. (9) also holds
for $\tilde{U}_{\nu}$. The form of the matrix $\eta$ is at leading order given
by

$\eta=\frac{1}{2}\,m_{D}^{\star}\,\Big{(}M_{NS}^{-1}\Big{)}^{\dagger}\,M_{NS}^{-1}\,m_{D}^{T}\;.$
(10)

As $N_{i}$ and $S_{j}$ transform both as the real representation ${\bf
3^{\prime}}$, the matrices $M_{NS}$ and $\mu_{S}$ are non-vanishing in the
limit of unbroken flavour and CP symmetries. In particular, using the basis
given in appendix A, the form of the matrices is

$M_{NS}=M_{0}\,\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
0&0&1\end{array}\right)\;\;\;\mbox{and}\;\;\;\mu_{S}=\mu_{0}\,\left(\begin{array}[]{ccc}1&0&0\\\
0&0&1\\\ 0&1&0\end{array}\right)$ (11)

with $M_{0}$ and $\mu_{0}$ being positive and having dimension of mass. The
non-trivial flavour information is encoded in the Dirac neutrino Yukawa matrix
$y_{D}$ whose form is determined by the following equations

$Z({\bf 3})^{T}\,y_{D}\,Z({\bf
3^{\prime}})^{\star}=y_{D}\;\;\;\mbox{and}\;\;\;X({\bf 3})\,y_{D}\,X({\bf
3^{\prime}})^{\star}=y_{D}^{\star}\;,$ (12)

where the matrices $Z({\bf 3})$ and $Z({\bf 3^{\prime}})$ are the
representation matrices of the generator of the preserved $Z_{2}$ group,
belonging to the residual symmetry $G_{\nu}$, in the representations ${\bf 3}$
(for the LH lepton doublets) and ${\bf 3^{\prime}}$ (for the gauge singlets
$N_{i}$), respectively. The matrices $X({\bf 3})$ and $X({\bf 3^{\prime}})$
represent the CP transformation in these representations that also belongs to
the group $G_{\nu}$. The explicit form of these matrices can be found in [47].
As a result, the matrix $y_{D}$ depends on five real parameters in total:
three Yukawa couplings, $y_{1}$, $y_{2}$ and $y_{3}$, and two angles, called
$\theta_{L}$ and $\theta_{R}$. Its form is

$y_{D}=\Omega({\bf
3})^{\star}\,R_{ij}(\theta_{L})\,\mathrm{diag}(y_{1},y_{2},y_{3})\,P^{ij}_{kl}\,R_{kl}(-\theta_{R})\,\Omega({\bf
3^{\prime}})^{T}\;.$ (13)

The different matrices are fixed as follows: the unitary matrices $\Omega({\bf
3})$ and $\Omega({\bf 3^{\prime}})$ are determined by the form of the matrices
$X({\bf 3})$ and $X({\bf 3^{\prime}})$, respectively,

$\Omega({\bf 3})\,\Omega({\bf 3})^{T}=X({\bf
3})\;\;\;\mbox{and}\;\;\;\Omega({\bf 3^{\prime}})\,\Omega({\bf
3^{\prime}})^{T}=X({\bf 3^{\prime}})\;,$ (14)

the planes of the rotation matrices $R_{ij}(\theta_{L})$ and
$R_{kl}(\theta_{R})$ are given by the plane of the two degenerate eigenvalues
of the matrices $Z({\bf 3})$ and $Z({\bf 3^{\prime}})$ in the basis
transformed by $\Omega({\bf 3})$ and $\Omega({\bf 3^{\prime}})$, respectively.
If these two planes do not coincide, the permutation matrix $P^{ij}_{kl}$ is
necessary. The form of these matrices depends on the mixing pattern which we
consider, i.e. Case 1) through Case 3 b.1). It can be found in [47] and is
repeated for convenience in appendix B. This scenario has been first mentioned
in the context of the ISS mechanism in [45] and called option 2. However,
neither the results for fermion masses and mixing nor for the signal strength
of different cLFV processes have been discussed there. This analysis is
performed in the current study.

With the given information, we can compute both the masses of the neutral
states as well as their mixing. We begin with the mass matrix of the heavy
states and their mass spectrum, since this structure is very simple. It can be
derived (to very good approximation) from the matrix

$\left(\begin{array}[]{cc}\mathbb{0}&M_{NS}\\\
M_{NS}^{T}&\mu_{S}\end{array}\right)$ (15)

which is (approximately) diagonalised by the six-by-six matrix $V$, i.e.

$V^{T}\,\left(\begin{array}[]{cc}\mathbb{0}&M_{NS}\\\
M_{NS}^{T}&\mu_{S}\end{array}\right)\,V\approx\mathrm{diag}\left(m_{4},\dots,m_{9}\right)\;.$
(16)

For $M_{NS}$ and $\mu_{S}$ as in eq. (11), one finds for the masses of the
heavy states that these are all degenerate to very high degree and form three
pairs of pseudo-Dirac neutrinos, splitted by $\mu_{0}$,

$m_{4}\approx m_{5}\approx m_{6}\approx
M_{0}-\frac{\mu_{0}}{2}\;\;\;\mbox{and}\;\;\;m_{7}\approx m_{8}\approx
m_{9}\approx M_{0}+\frac{\mu_{0}}{2}\;,$ (17)

as well as the approximate form of $V$

$V=\frac{1}{\sqrt{2}}\,\left(\begin{array}[]{cc}i\,U_{S}^{\star}&U_{S}^{\star}\\\
-i\,U_{S}&U_{S}\end{array}\right)$ (18)

with $U_{S}$ being chosen as

$U_{S}=\frac{1}{\sqrt{2}}\,\left(\begin{array}[]{ccc}\sqrt{2}&0&0\\\ 0&1&i\\\
0&1&-i\end{array}\right)$ (19)

such that

$U_{S}^{T}\,\mu_{S}\,U_{S}=\mu_{0}\,\left(\begin{array}[]{ccc}1&0&0\\\
0&1&0\\\ 0&0&1\end{array}\right)\;,$ (20)

compare [45, 47]. The approximate form of the matrix $S$ is related to the
form of $V$, i.e.

$S=\left(\begin{array}[]{ccc}\mathbb{0}&,&m_{D}^{\star}\,\Big{(}M_{NS}^{-1}\Big{)}^{\dagger}\end{array}\right)\,V=\frac{\langle
H\rangle}{\sqrt{2}\,M_{0}}\,\left(\begin{array}[]{ccc}-i\,y_{D}^{\star}\,U_{S}&,&y_{D}^{\star}\,U_{S}\end{array}\right)\;,$
(21)

see also [46, 45].

Using the general formulae, found in [46, 45], we see that the light neutrino
mass matrix $m_{\nu}$ at leading order is of the form

$\displaystyle m_{\nu}$ $\displaystyle=$
$\displaystyle\left(\frac{\mu_{0}\,\langle
H\rangle^{2}}{M_{0}^{2}}\right)\,U_{0}(\theta_{L})^{\star}\,\mathrm{diag}(y_{1},y_{2},y_{3})$
(26)
$\displaystyle\;\,\times\left[P^{ij}_{kl}\,R_{kl}(-\theta_{R})\,\Omega({\bf
3^{\prime}})^{T}\,\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&1\\\
0&1&0\end{array}\right)\,\Omega({\bf
3^{\prime}})\,R_{kl}(\theta_{R})\,(P^{ij}_{kl})^{T}\right]$
$\displaystyle\;\;\;\,\times\,\mathrm{diag}(y_{1},y_{2},y_{3})\,U_{0}(\theta_{L})^{\dagger}$

with the definition of the unitary matrix $U_{0}(\theta)$

$U_{0}(\theta)=\Omega({\bf 3})\,R_{ij}(\theta)$ (27)

and the form of the hermitean matrix $\eta$ given by

$\eta=\eta_{0}^{\prime}\,U_{0}(\theta_{L})\,\mathrm{diag}(y_{1}^{2},y_{2}^{2},y_{3}^{2})\,U_{0}(\theta_{L})^{\dagger}\;\;\;\mbox{with}\;\;\;\eta_{0}^{\prime}=\frac{\langle
H\rangle^{2}}{2\,M_{0}^{2}}\;.$ (28)

The quantity $\eta_{0}^{\prime}$ is similar to the quantity $\eta_{0}$ defined
in [45].

In order to continue we have to have knowledge about the form of the matrix
combination in square brackets in eq. (26). Indeed, it is known [47] that this
expression333In [47], the complex conjugate of this matrix combination is
found. is either diagonal or has a block-diagonal form that can be
diagonalised by a rotation in the $(ij)$-plane, i.e. the same plane in which
the rotation $R_{ij}(\theta_{L})$ acts. In the following, we first discuss the
situation in which this expression is diagonal and then turn to the instance
in which an additional rotation is necessary.

If the expression is diagonal, we find that the matrix diagonalising $m_{\nu}$
at leading order is of the form

$\tilde{U}_{\nu}\approx U_{0}(\theta_{L})\,K_{\nu}=\Omega({\bf
3})\,R_{ij}(\theta_{L})\;K_{\nu}\;,$ (29)

where the role of $K_{\nu}$ is to ensure that the light neutrino masses are
positive semi-definite.444We set $K_{\nu}$ to the identity matrix in analytic
results, but take it properly into account in the numerical study.
Interestingly, then the leading order values of the light neutrino masses
$m_{f}$ are simply given by

$m_{f}=\left(\frac{\mu_{0}\,\langle
H\rangle^{2}}{M_{0}^{2}}\right)\,y_{f}^{2}\;\;\;\mbox{with}\;\;\;f=1,2,3$ (30)

for Case 1) through Case 3 a), whereas for Case 3 b.1) we have to take into
account an additional permutation $P$ among the light neutrino masses in order
to produce the corresponding mixing pattern, i.e.

$m_{1}=\left(\frac{\mu_{0}\,\langle
H\rangle^{2}}{M_{0}^{2}}\right)\,y_{3}^{2}\;,\;\;m_{2}=\left(\frac{\mu_{0}\,\langle
H\rangle^{2}}{M_{0}^{2}}\right)\,y_{1}^{2}\;,\;\;m_{3}=\left(\frac{\mu_{0}\,\langle
H\rangle^{2}}{M_{0}^{2}}\right)\,y_{2}^{2}$ (31)

and thus

$\tilde{U}_{\nu}\approx U_{0}(\theta_{L})\,P\,K_{\nu}=\Omega({\bf
3})\,R_{ij}(\theta_{L})\,P\;K_{\nu}\;\;\mbox{with}\;\;P=\left(\begin{array}[]{ccc}0&1&0\\\
0&0&1\\\ 1&0&0\end{array}\right)\;,$ (32)

see e.g. [47]. Furthermore, we can express $\eta$ in terms of
$\tilde{U}_{\nu}$ and the light neutrino masses $m_{f}$

$\eta\approx\frac{1}{2\,\mu_{0}}\,\tilde{U}_{\nu}\,\mathrm{diag}(m_{1},m_{2},m_{3})\,\tilde{U}_{\nu}^{\dagger}\;.$
(33)

This has two immediate consequences: firstly, the matrix $m_{\nu}^{1}$ is
diagonalised by the same matrix as the leading order term $m_{\nu}$, since we
can write the expression in the second line of eq. (6) as

$m_{\nu}^{1}\approx-\frac{1}{\mu_{0}}\,\tilde{U}_{\nu}^{\star}\,\mathrm{diag}(m_{1}^{2},m_{2}^{2},m_{3}^{2})\,\tilde{U}_{\nu}^{\dagger}\;.$
(34)

Consequently, lepton mixing is not influenced by this sub-leading term, while
the light neutrino masses are (slightly) corrected. Secondly, we can estimate
the effect of non-unitarity of the lepton mixing matrix using eq. (9)

$\\!\\!\\!\\!\\!\\!\tilde{U}_{\nu}=U_{0}(\theta_{L})\,\left(\mathbb{1}-\eta_{0}^{\prime}\,\mathrm{diag}(y_{1}^{2},y_{2}^{2},y_{3}^{2})\right)\;\;\mbox{and}\;\;\tilde{U}_{\nu}=U_{0}(\theta_{L})\,P\,\left(\mathbb{1}-\eta_{0}^{\prime}\,\mathrm{diag}(y_{3}^{2},y_{1}^{2},y_{2}^{2})\right)$
(35)

for Case 1) through Case 3 a) and Case 3 b.1), respectively. This can also be
written in terms of the light neutrino masses $m_{f}$ as

$\\!\\!\\!\\!\\!\\!\tilde{U}_{\nu}=\tilde{U}_{\nu}^{0}\,\left(\mathbb{1}-\frac{1}{2\,\mu_{0}}\,\mathrm{diag}(m_{1},m_{2},m_{3})\right)\;,$
(36)

where $\tilde{U}_{\nu}^{0}$ indicates the unitary matrix in eq. (29) and (32)
for Case 1) through Case 3 a) and Case 3 b.1), respectively. A suppression of
the columns of the PMNS mixing matrix is induced that is in general different
for the different columns, since it depends on the Yukawa couplings
$y_{f}$/the light neutrino masses $m_{f}$. In particular, in case of strong
normal ordering (NO) or strong inverted ordering (IO), i.e. the lightest
neutrino mass $m_{0}$ vanishes, one of the columns remains unaffected. This is
slightly more general than the effect observed in [45], where a common
suppression factor for all entries of the mixing matrix is encountered. We
discuss the effect of the suppression in eq. (35) in more detail in section 3.

In case the expression in square brackets in eq. (26) is not diagonal, we have
to take into account an additional rotation through an angle determined by the
Yukawa couplings $y_{f}$ and the angle $\theta_{R}$ occurring in the
$(ij)$-plane such that the angle $\theta_{L}$ becomes replaced by an effective
angle $\overline{\theta}_{L}$,555The effective angle $\overline{\theta}_{L}$
is denoted by $\widetilde{\theta}_{L}$ in [47]. i.e. (disregarding the matrix
$K_{\nu}$)

$\tilde{U}_{\nu}\approx U_{0}(\overline{\theta}_{L})=\Omega({\bf
3})\,R_{ij}(\overline{\theta}_{L})\;.$ (37)

Only one of the three light neutrino masses, $m_{k}$ with $k\neq i$ and $k\neq
j$, depends on a single Yukawa coupling, as shown in eq. (30), while the
others are determined by the remaining Yukawa couplings and the angle
$\theta_{R}$, e.g. for Case 1) (and independent of the choice of the parameter
$s$, see section 3) we have

$m_{2}=\left(\frac{\mu_{0}\,\langle
H\rangle^{2}}{M_{0}^{2}}\right)\,y_{2}^{2}\;,\;\;m_{1,3}=\left(\frac{\mu_{0}\,\langle
H\rangle^{2}}{2\,M_{0}^{2}}\right)\,\left|(y_{1}^{2}-y_{3}^{2})\,\cos
2\,\theta_{R}\pm\sqrt{4\,y_{1}^{2}\,y_{3}^{2}+(y_{1}^{2}-y_{3}^{2})^{2}\,\cos^{2}2\,\theta_{R}}\right|\;,$
(38)

which in the case of strong NO ($m_{1}=0$) is reduced to

$m_{1}=0\;,\;\;m_{2}=\left(\frac{\mu_{0}\,\langle
H\rangle^{2}}{M_{0}^{2}}\right)\,y_{2}^{2}\;,\;\;m_{3}=\left(\frac{\mu_{0}\,\langle
H\rangle^{2}}{M_{0}^{2}}\right)\,y_{3}^{2}\,|\cos 2\,\theta_{R}|\;,$ (39)

and for strong IO ($m_{3}=0$) to

$m_{1}=\left(\frac{\mu_{0}\,\langle
H\rangle^{2}}{M_{0}^{2}}\right)\,y_{1}^{2}\,|\cos
2\,\theta_{R}|\;,\;\;m_{2}=\left(\frac{\mu_{0}\,\langle
H\rangle^{2}}{M_{0}^{2}}\right)\,y_{2}^{2}\;,\;\;m_{3}=0\;.$ (40)

The form of the matrix $\eta$ in eq. (28) does not change, but we can no
longer identify $U_{0}(\theta_{L})$ with $\tilde{U}_{\nu}$. Instead we have

$\eta\approx\eta_{0}^{\prime}\,\tilde{U}_{\nu}\,R_{ij}(\Delta\theta)\,\mathrm{diag}(y_{1}^{2},y_{2}^{2},y_{3}^{2})\,R_{ij}(-\Delta\theta)\,\tilde{U}_{\nu}^{\dagger}\,,$
(41)

using the definition of $U_{0}(\theta)$ in eq. (27) and setting
$\Delta\theta=\theta_{L}-\overline{\theta}_{L}$. Regarding the form of the
sub-leading term $m_{\nu}^{1}$ we note that it can be written as

$\displaystyle m_{\nu}^{1}$ $\displaystyle\approx$
$\displaystyle-\eta_{0}^{\prime}\,\tilde{U}_{\nu}^{\star}\,\left[\mathrm{diag}(m_{1},m_{2},m_{3})\,R_{ij}(\Delta\theta)\,\mathrm{diag}(y_{1}^{2},y_{2}^{2},y_{3}^{2})\,R_{ij}(-\Delta\theta)\right.$
$\displaystyle+$
$\displaystyle\left.R_{ij}(\Delta\theta)\,\mathrm{diag}(y_{1}^{2},y_{2}^{2},y_{3}^{2})\,R_{ij}(-\Delta\theta)\,\mathrm{diag}(m_{1},m_{2},m_{3})\right]\,\tilde{U}_{\nu}^{\dagger}\;.$
(42)

Hence, this sub-leading term is not diagonalised by $\tilde{U}_{\nu}$ (the
matrix that diagonalises the leading term $m_{\nu}$), but induces a (small)
additional rotation in the $(ij)$-plane. Also, the light neutrino masses get
slightly shifted. For the effects of non-unitarity of lepton mixing, we have
in this case

$\tilde{U}_{\nu}=U_{0}(\overline{\theta}_{L})\,\left(\mathbb{1}-\eta_{0}^{\prime}\,R_{ij}(\Delta\theta)\,\mathrm{diag}(y_{1}^{2},y_{2}^{2},y_{3}^{2})\,R_{ij}(-\Delta\theta)\right)\;.$
(43)

Thus, only one column receives a common suppression determined by one of the
light neutrino masses, while the effects on the other two are more involved,
see discussion in section 3. We note that the formulae given here are valid
for Case 1) through Case 3 a), while for Case 3 b.1) the permutation $P$ has
to be taken into account which slightly complicates the expressions.

##  3 Lepton mixing

In this section, we comment on the results for the lepton mixing parameters
derived from the different cases, Case 1) through Case 3 b.1), and how these
are affected by the presence of the heavy sterile states. There are two
effects, already mentioned in section 2: the sub-leading contribution
$m_{\nu}^{1}$ can lead to deviations from the results, determined by the
flavour and CP symmetries, and the non-unitarity of the lepton mixing matrix,
induced through the heavy sterile states.

Whether the sub-leading contribution $m_{\nu}^{1}$ gives rise to deviations
from the lepton mixing obtained at leading order or not depends on whether the
expression in square brackets in eq. (26) is diagonal or not. If it is
diagonal, then the lepton mixing parameters remain unaltered. Otherwise, the
form of $m_{\nu}^{1}$ leads to an additional rotation, compare eq. (42), whose
effect is suppressed by $\eta_{0}^{\prime}$. The typical size of
$\eta_{0}^{\prime}$ is expected to be of the order of the experimental bounds
on the entries of the matrix $\eta$, $|\eta_{\alpha\beta}|<1.0\times
10^{-5}\div 1.4\times 10^{-3}$ depending on the flavours $\alpha$ and $\beta$
[48]. This effect is, thus, in general small.

In order to analyse the non-unitarity of the lepton mixing matrix, we extract
the lepton mixing angles and CP invariants $J_{\mathrm{CP}}$, $I_{1}$ and
$I_{2}$ from a three-by-three mixing matrix $U$ with elements $U_{\alpha i}$,
$\alpha=e,\mu,\tau$ and $i=1,2,3$, as follows

$\displaystyle\sin^{2}\theta_{13}=|U_{e3}|^{2}\;,\;\;\sin^{2}\theta_{12}=\frac{|U_{e2}|^{2}}{1-|U_{e3}|^{2}}\;,\;\;\sin^{2}\theta_{23}=\frac{|U_{\mu
3}|^{2}}{1-|U_{e3}|^{2}}\;,$ $\displaystyle
J_{\mathrm{CP}}=\mathrm{Im}\left(U_{e1}\,U_{e3}^{\star}\,U_{\tau
1}^{\star}\,U_{\tau
3}\right)=\frac{1}{8}\,\sin(2\,\theta_{12})\,\sin(2\,\theta_{23})\,\sin(2\,\theta_{13})\,\cos\theta_{13}\,\sin\delta\;,$
$\displaystyle
I_{1}=\mathrm{Im}\left(U_{e2}^{2}(U_{e1}^{\star})^{2}\right)=\frac{1}{4}\,\sin^{2}(2\,\theta_{12})\,\cos^{4}\theta_{13}\,\sin\alpha\;,$
$\displaystyle
I_{2}=\mathrm{Im}\left(U_{e3}^{2}(U_{e1}^{\star})^{2}\right)=\frac{1}{4}\,\sin^{2}(2\,\theta_{13})\,\cos^{2}\theta_{12}\,\sin\beta\;.$
(44)

The size of these effects crucially depends on the parameter
$\eta_{0}^{\prime}$, the Yukawa couplings $y_{f}$ and potentially also the
angle $\theta_{R}$ (i.e. the difference between the angles $\theta_{L}$ and
$\overline{\theta}_{L}$).

Before commenting on the results for the different cases, we discuss the
generic effects using as form of the lepton mixing matrix either the one in
eq. (35) or (43), depending on whether the matrix in square brackets in eq.
(26) is diagonal or not. We define the deviations of the lepton mixing angles
and CP invariants in the ISS framework as

$\Delta\sin^{2}\theta_{ij}=\frac{(\sin^{2}\theta_{ij})_{\mathrm{ISS}}-(\sin^{2}\theta_{ij})_{\mathrm{FCS}}}{(\sin^{2}\theta_{ij})_{\mathrm{FCS}}}\;,\;\;\Delta
J_{\mathrm{CP}}=\frac{(J_{\mathrm{CP}})_{\mathrm{ISS}}-(J_{\mathrm{CP}})_{\mathrm{FCS}}}{(J_{\mathrm{CP}})_{\mathrm{FCS}}}\;,\;\;\Delta
I_{i}=\frac{(I_{i})_{\mathrm{ISS}}-(I_{i})_{\mathrm{FCS}}}{(I_{i})_{\mathrm{FCS}}}\;,$
(45)

as long as the result obtained from the flavour and CP symmetries (FCS) is
non-vanishing; this definition is identical to the one used in [45].

As already mentioned in section 2, from eq. (35) follows that the elements of
one column are suppressed by the same factor. We always observe a suppression,
since $\eta_{0}^{\prime}\,y_{f}^{2}$ is positive semi-definite. Using eq. (44)
and the result for Case 1) through Case 3 a) in eq. (35), the explicit
dependence on $\eta_{0}^{\prime}$ and $y_{f}$ of these deviations is

$\displaystyle\Delta\sin^{2}\theta_{13}\approx-2\,\eta_{0}^{\prime}\,y_{3}^{2}\;,\;\;\Delta\sin^{2}\theta_{12}\approx-2\,\eta_{0}^{\prime}\,y_{2}^{2}\;,\;\;\Delta\sin^{2}\theta_{23}\approx-2\,\eta_{0}^{\prime}\,y_{3}^{2}\;,$
$\displaystyle\Delta
J_{\mathrm{CP}}\approx-2\,\eta_{0}^{\prime}\,(y_{1}^{2}+y_{3}^{2})\;,\;\;\Delta
I_{1}\approx-2\,\eta_{0}^{\prime}\,(y_{1}^{2}+y_{2}^{2})\;,\;\;\Delta
I_{2}\approx-2\,\eta_{0}^{\prime}\,(y_{1}^{2}+y_{3}^{2})\;.$ (46)

In these approximations, we only take into account the leading order in
$\eta_{0}^{\prime}$ and neglect terms of order $|U_{e3}|^{2}$ or smaller.
Since $\eta_{0}^{\prime}\,y_{f}^{2}\geq 0$, all deviations are negative or
zero. Given that experimental data are compatible with at most one vanishing
light neutrino mass, only one of the couplings $y_{f}$ can be zero. We comment
on the consequences below. Obviously, in the (unrealistic) limit in which
$y_{f}$ are equal, we end up with the same expressions as in [45].

If the matrix in square brackets in eq. (26) is not diagonal, we encounter as
form of the non-unitary lepton mixing matrix the one mentioned in eq. (43), at
least for Case 1) through Case 3 a). Then, the deviations of the lepton mixing
angles and the CP invariants in the ISS framework also depend on the
difference, $\Delta\theta$, between the angles $\theta_{L}$ and
$\overline{\theta}_{L}$, where the latter depends on the Yukawa couplings
$y_{f}$ and the free angle $\theta_{R}$. For example for $(ij)=(13)$, i.e. the
additional rotation diagonalising the matrix in square brackets in eq. (26)
acts in the $(13)$-plane, we find at linear order in $\eta_{0}^{\prime}$ and
$|U_{e3}|$

$\displaystyle\Delta\sin^{2}\theta_{13}\approx\eta_{0}^{\prime}\,\left(\frac{\mathrm{Re}\left(U_{e1}^{\star}\,U_{e3}\right)}{|U_{e3}|^{2}}\,(y_{1}^{2}-y_{3}^{2})\,\sin
2\,\Delta\theta-2\,(y_{1}^{2}\,\sin^{2}\Delta\theta+y_{3}^{2}\,\cos^{2}\Delta\theta)\right)\;,$
$\displaystyle\Delta\sin^{2}\theta_{12}\approx\eta_{0}^{\prime}\,\left(-2\,y_{2}^{2}+\mathrm{Re}\left(U_{e1}^{\star}\,U_{e3}\right)\,(y_{1}^{2}-y_{3}^{2})\,\sin
2\,\Delta\theta\right)\;,$ (47)
$\displaystyle\Delta\sin^{2}\theta_{23}\approx\eta_{0}^{\prime}\,\left(-2\,(y_{1}^{2}\,\sin^{2}\Delta\theta+y_{3}^{2}\,\cos^{2}\Delta\theta)+\mathrm{Re}\left(U_{e1}^{\star}\,U_{e3}+\frac{U_{\mu
1}^{\star}\,U_{\mu 3}}{|U_{\mu 3}|^{2}}\right)\,(y_{1}^{2}-y_{3}^{2})\,\sin
2\,\Delta\theta\right)$

as well as

$\displaystyle\Delta
J_{\mathrm{CP}}\approx-2\,\eta_{0}^{\prime}\,(y_{1}^{2}+y_{3}^{2})$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;+\eta_{0}^{\prime}\,(y_{1}^{2}-y_{3}^{2})\,\frac{\left(\left(|U_{e1}|^{2}+|U_{e3}|^{2}\right)\,\mathrm{Im}(U_{\tau
1}^{\star}\,U_{\tau 3})+\left(|U_{\tau 1}|^{2}+|U_{\tau
3}|^{2}\right)\,\mathrm{Im}(U_{e1}\,U_{e3}^{\star})\right)}{2\,\mathrm{Im}\left(U_{e1}\,U_{e3}^{\star}\,U_{\tau
1}^{\star}\,U_{\tau 3}\right)}\,\sin 2\,\Delta\theta\;,$ $\displaystyle\Delta
I_{1}\approx-2\,\eta_{0}^{\prime}\,(y_{1}^{2}\,\cos^{2}\Delta\theta+y_{2}^{2}+y_{3}^{2}\,\sin^{2}\Delta\theta)+\eta_{0}^{\prime}\,(y_{1}^{2}-y_{3}^{2})\,\frac{\mathrm{Im}\left(U_{e2}^{2}\,U_{e1}^{\star}U_{e3}^{\star}\right)}{\mathrm{Im}\left(U_{e2}^{2}\,(U_{e1}^{\star})^{2}\right)}\,\sin
2\,\Delta\theta\;,$ $\displaystyle\Delta
I_{2}\approx-2\,\eta_{0}^{\prime}\,(y_{1}^{2}+y_{3}^{2})+\eta_{0}^{\prime}\,(y_{1}^{2}-y_{3}^{2})\,\frac{\left(|U_{e1}|^{2}+|U_{e3}|^{2}\right)}{2\,\mathrm{Re}\left(U_{e1}^{\star}\,U_{e3}\right)}\,\sin
2\,\Delta\theta\;,$ (48)

where we refer with $U_{\alpha i}$ to the elements of the unitary matrix, i.e.
the one obtained from flavour and CP symmetries. Clearly, in the limit in
which $\theta_{L}$ and $\overline{\theta}_{L}$ coincide, $\Delta\theta=0$, we
recover the formulae found in eq. (46). Given the more general form of the
deviation it is possible that the non-unitary lepton mixing matrix leads to
non-trivial CP violation, although flavour and CP symmetries predict certain
CP phases to be trivial.

##### Case 1)

As has been shown in [33], the choice of residual symmetries is described by
one integer, called $s$, varying between $0$ and $n-1$, which corresponds to
the employed CP symmetry. Two of the three CP phases, $\beta$ and $\delta$,
are trivial in this case, i.e. $I_{2}=0$ and $J_{\mathrm{CP}}=0$, while the
third one, the Majorana phase $\alpha$, is determined by the choice of the CP
symmetry, $|\sin\alpha|=|\sin 6\,\phi_{s}|$ with
$\phi_{s}=\frac{\pi\,s}{n}$.666The sign in the relation for the Majorana phase
$\alpha$ depends on the explicit form of the matrix $K_{\nu}$. In contrast to
this, the three lepton mixing angles are independent of this choice and only
depend on the free angle $\theta_{L}$ (or in case the matrix combination in
square brackets in eq. (26) is not diagonal this angle is given by
$\overline{\theta}_{L}$). This angle is adjusted such that the reactor mixing
angle is very close to its experimental best-fit value [49] and, consequently,
the solar and atmospheric mixing angle are given by
$\sin^{2}\theta_{12}\approx 0.341$ and $\sin^{2}\theta_{23}\approx 0.604\,(5)$
for light neutrino masses with NO (IO).

As has been analysed in [47], only for $y_{1}=0$ or $y_{3}=0$ or $\theta_{R}$
chosen such that $\sin 2\,\theta_{R}=0$, the matrix combination in square
brackets in eq. (26) is diagonal. Then, we can apply the results found in eq.
(46). In particular, we see that in the case of light neutrino masses with
strong NO ($y_{1}=0$) the deviations of the CP invariants also only depend on
one coupling $y_{f}$, while for strong IO ($y_{3}=0$) the deviations of both
the reactor and the atmospheric mixing angle are suppressed. Otherwise, we can
use the formulae in eqs. (47) and (48), since the rotation always occurs in
the $(13)$-plane in this case. Although one may presume that one can achieve
non-trivial CP phases $\delta$ and $\beta$ then, this does not happen, as can
be checked by explicit computation.

##### Case 2)

From [33] it is known that the CP symmetry is fixed by two integer parameters,
$s$ and $t$, ranging from $0$ to $n-1$. Alternatively, one can define the
combinations $u$ and $v$ as

$u=2\,s-t\;\;\;\mbox{and}\;\;\;v=3\,t\;.$ (49)

The smallness of the reactor mixing angle can be achieved for $\frac{u}{n}$
($\phi_{u}=\frac{\pi\,u}{n}$) being small, $-0.1\lesssim\frac{u}{n}\lesssim
0.12$ ($-0.31\lesssim\phi_{u}\lesssim 0.37$), and the free angle, appearing in
the lepton mixing matrix, close to zero or $\pi$, see [33, 47]. The solar
mixing angle is, similar to Case 1), constrained to fulfil
$\sin^{2}\theta_{12}\gtrsim\frac{1}{3}$, while the atmospheric mixing angle
can lie in its experimentally preferred $3\,\sigma$ range [49]. Valid choices
of $u$ and $n$ can be found in section 4.4, where the numerical results are
presented, as well as in e.g. [33, 45, 47]. While the Dirac phase $\delta$ and
the Majorana phase $\beta$ depend like the lepton mixing angles on the
parameter $u$ and the free angle, the Majorana phase $\alpha$ is (mainly)
determined by the parameter $v$ and to very good approximation it holds
$|\sin\alpha|\approx|\sin\phi_{v}|$ with $\phi_{v}=\frac{\pi\,v}{n}$.777As for
Case 1), the sign in this relation depends on $K_{\nu}$.

In [47] it has been found that for $t$ even the matrix combination in square
brackets in eq. (26) is always diagonal, while for $t$ odd this only occurs
for $y_{1}=0$ or $y_{3}=0$ or $\theta_{R}$ chosen such that $\cos
2\,\theta_{R}=0$. We, thus, can apply in these situations the results given in
eq. (46). Like for Case 1), we can consider light neutrino masses with strong
NO ($y_{1}=0$) or strong IO ($y_{3}=0$) and arrive at the same conclusions.
Note that setting $y_{2}$ to zero does not lead to a viable light neutrino
mass spectrum, since $y_{2}$ is always associated with the light neutrino mass
$m_{2}$, compare eq. (30). For $t$ odd in general, a rotation in the
$(13)$-plane is necessary in order to diagonalise the matrix combination in
square brackets in eq. (26). Consequently, we can apply the formulae in eqs.
(47) and (48). As has been shown in [33], for the free angle being zero, one
can accommodate the lepton mixing angles well for $\frac{u}{n}\approx 0.12$,
while both $J_{\mathrm{CP}}$ and $I_{2}$ vanish. Taking into account the
effects of non-unitarity, we see that for $\sin 2\Delta\theta\neq 0$
$J_{\mathrm{CP}}$ as well as $I_{2}$ can be non-zero. On the other hand, for
$u=0$ the CP invariant $I_{2}$ is zero and the inclusion of the effects of
non-unitarity does not change this result.

##### Case 3 a)

In this case the residual $Z_{2}$ symmetry is characterised by the parameter
$m$ which can take values between $0$ and $n-1$, while the CP symmetry is
fixed by one integer parameter $s$, also varying between $0$ and $n-1$. For
Case 3 a) the ratio $\frac{m}{n}$ has to be either close to zero or to one (or
equivalently $\phi_{m}$, $\phi_{m}=\frac{\pi\,m}{n}$, small or close to $\pi$)
in order to accommodate well the reactor mixing angle. For this reason, $n$
should be at least $16$ and, consequently, $m=1$ or $m=15$.888Since the index
$n$ has to be even in the current scenario, $n=17$ and $m=1$ (or $m=16$) is
not eligible, but very similar results can be obtained for $n=34$ and $m=2$
(or $m=32$), see [47]. This choice also permits to study even values of $m$.
At the same time, fixing this ratio determines the value of the atmospheric
mixing angle. The free angle, contained in the lepton mixing matrix, can take,
depending on the choice of $s$ ($\phi_{s}=\frac{\pi\,s}{n}$), up to two
distinct values that allow for a good fit of the solar mixing angle. Further
details can be found in [33, 45, 47] as well as in section 4.5. The three CP
phases are in general non-trivial; numerical values and analytic estimates are
given in the cited works.

Using the results of [47] we know that for the parameters $m$ and $s$ both
even or both odd the matrix combination in square brackets in eq. (26) is
diagonal and the formulae in eq. (46) can be used. In contrast, for $m$ even
and $s$ odd or vice versa this matrix combination is only diagonal for
$y_{1}=0$ or $\sin 2\,\theta_{R}=0$. For $y_{1}=0$ we encounter the same
situation as for Case 1) and Case 2), since light neutrino masses then follow
strong NO and all deviations only depend on one Yukawa coupling $y_{f}$. If
both parameters $m$ and $s$ are even or both are odd, setting $y_{3}=0$
permits us to suppress the deviations of the reactor and the atmospheric
mixing angle, compare eq. (46). In case the matrix combination in square
brackets in eq. (26) is not diagonal, it is diagonalised by a rotation in the
$(12)$-plane. For this situation similar formulae as those in eqs. (47) and
(48) can be derived.

##### Case 3 b.1)

Like for Case 3 a), the integer parameters $m$ and $s$ with $0\leq m,s\leq
n-1$ describe the residual $Z_{2}$ symmetry and the CP symmetry, respectively.
From [33], we know that the requirement to accommodate the solar mixing angle
well constrains the parameter $m$ to be close to $\frac{n}{2}$. The free
angle, appearing in the lepton mixing matrix, has to be close to
$\frac{\pi}{2}$ such that the reactor mixing angle can be fitted well. For
$\frac{m}{n}=\frac{1}{2}$, usually two such values exist. Otherwise, it
depends on the choice of $m$ and $s$ how many distinct values of the free
angle admit a good fit to the experimental data of the lepton mixing angles.
The atmospheric mixing angle also depends on the parameter $s$. Likewise, all
CP phases are determined by the actual values of the ratios $\frac{m}{n}$ and
$\frac{s}{n}$ as well as the free angle. More information, in particular about
analytical estimates and numerical values of the CP phases, is given in [33,
45, 47]. Explicit choices of $m$ and $s$ are discussed in section 4.6.

Whether or not the matrix combination in square brackets in eq. (26) is
diagonal is subject to very similar constraints as for Case 3 a). The only
difference is due to the fact that the light neutrino masses are assigned
differently in Case 3 b.1) than Case 3 a), compare eqs. (31) and (30), i.e. in
order to have a diagonal matrix combination for $m$ even and $s$ odd or vice
versa we have to set $y_{2}=0$ which corresponds to light neutrino masses with
strong IO. For Case 3 b.1) the relevant rotation effectively appears in the
$(23)$-plane and expressions equivalent to those in eqs. (47) and (48) can be
obtained.

##  4 $\mu-e$ transitions

In the discussion of cLFV processes, we focus on different $\mu-e$
transitions, since they are subject to the strongest experimental bounds. As
we show in the following, current experimental limits do not constrain the
considered parameter space of this scenario, while prospective bounds, in
particular on $\mu-e$ conversion in aluminium, turn out to be very
constraining in general.

###  4.1 Prerequisites

We first list the experimental inputs and constraints we apply. Furthermore,
we outline the parameter space of the scenario that we scan for viable choices
of the group theory parameters (depending on the case) in terms of the free
angle $\theta_{R}$ and the two mass scales $\mu_{0}$ and $M_{0}$. The angle
$\theta_{L}$ is determined by the requirement to accommodate the measured
lepton mixing angles well and the couplings $y_{f}$ are fixed by the light
neutrino mass spectrum, i.e. the value of the lightest neutrino mass $m_{0}$
and the mass ordering, either NO or IO.

####  4.1.1 Experimental inputs and constraints

The lepton mixing angles are constrained by the global fits found in [49]
(version 5.3).999In the numerical analysis we use version 5.2, but check that
the fitted lepton mixing angles are also compatible with the results given in
version 5.3. Since the experimental preference for a certain value of the
Dirac phase $\delta$ is still rather weak, we do not include this information
here, when fitting the angle $\theta_{L}$ in order to accommodate well the
lepton mixing parameters.

Taking into account the experimental information on neutrino masses, in
particular the results for the mass squared differences from neutrino
oscillations [49] and the bound on the sum of the light neutrino masses,
derived from cosmology [50], we use the following benchmark values for the
lightest neutrino mass $m_{0}$: either $m_{0}=0$ (strong NO or strong IO) or

$m_{0}=0.03\,\mathrm{eV}\;\;\mbox{for NO
and}\;\;m_{0}=0.015\,\mathrm{eV}\;\;\mbox{for IO.}$ (50)

The mass squared differences $\Delta m^{2}_{\mathrm{sol}}$ and $\Delta
m^{2}_{\mathrm{atm}}$ are defined as

$\Delta m^{2}_{\mathrm{sol}}=m_{2}^{2}-m_{1}^{2}\;\;\mbox{and}\;\;\Delta
m^{2}_{\mathrm{atm}}=\left\\{\begin{array}[]{c}m_{3}^{2}-m_{1}^{2}\;\;\mbox{(NO)}\\\
\\!\\!m_{3}^{2}-m_{2}^{2}\;\;\mbox{(IO)}\end{array}\right.\;.$ (51)

These are always fixed to their experimental best-fit values [49]. The light
neutrino masses for NO are given by

$m_{1}=m_{0}\;,\;\;m_{2}=\sqrt{m_{1}^{2}+\Delta
m^{2}_{\mathrm{sol}}}\;,\;\;m_{3}=\sqrt{m_{1}^{2}+\Delta
m^{2}_{\mathrm{atm}}}\;,$ (52)

and for IO they read

$m_{1}=\sqrt{m_{0}^{2}+|\Delta m^{2}_{\mathrm{atm}}|-\Delta
m^{2}_{\mathrm{sol}}}\;,\;\;m_{2}=\sqrt{m_{0}^{2}+|\Delta
m^{2}_{\mathrm{atm}}|}\;,\;\;m_{3}=m_{0}\;.$ (53)

As experimental bounds on the parameters $\eta_{\alpha\beta}$, $\alpha$,
$\beta=e,\mu,\tau$, we use [48]

$|\eta_{\alpha\beta}|<\left(\begin{array}[]{ccc}1.3\,(1.4)\times
10^{-3}&1.2\times 10^{-5}&9.0\,(8.0)\times 10^{-4}\\\ 1.2\times
10^{-5}&1.1\,(1.0)\times 10^{-5}&5.7\,(1.8)\times 10^{-5}\\\ 9.0\,(8.0)\times
10^{-4}&5.7\,(1.8)\times 10^{-5}&1.0\,(0.81)\times
10^{-3}\end{array}\right)\;\;\mbox{at}\;\;95\%\;\mathrm{C.L.}$ (54)

for light neutrino masses with NO (IO). Additionally, the trace of $\eta$ is
constrained by

$\mathrm{tr}(\eta)<1.9\,(1.5)\times
10^{-3}\;\;\mbox{at}\;\;95\%\;\mathrm{C.L.}$ (55)

for NO (IO).

Furthermore, we employ as current bounds on the BRs of the cLFV decays $\mu\to
e\gamma$ and $\mu\to 3\,e$

$\displaystyle\mathrm{BR}(\mu\to e\gamma)<3.1\times 10^{-13}\;\;\,\mbox{at
90\% C.L.}\;\;\mbox{(MEG II\leavevmode\nobreak\
\cite[cite]{[\@@bibref{Number}{MEGII:2023ltw}{}{}]})}\;,$
$\displaystyle\mathrm{BR}(\mu\to 3\,e)<1.0\times 10^{-12}\;\;\mbox{at 90\%
C.L.}\;\;\,\mbox{(SINDRUM\leavevmode\nobreak\
\cite[cite]{[\@@bibref{Number}{SINDRUM:1987nra}{}{}]})}\;,$ (56)

as well as on the CR of $\mu-e$ conversion in gold and titanium

$\displaystyle\mathrm{CR}(\mu-e,\mathrm{Au})<7.0\times 10^{-13}\;\;\mbox{at
90\% C.L.}\;\;\;\;\mbox{(SINDRUMII\leavevmode\nobreak\
\cite[cite]{[\@@bibref{Number}{SINDRUMII:2006dvw}{}{}]})}\;,$
$\displaystyle\mathrm{CR}(\mu-e,\mathrm{Ti})<6.1\times 10^{-13}\;\;\;\mbox{at
90\% C.L.}\;\;\;\;\,\mbox{(SINDRUMII\leavevmode\nobreak\
\cite[cite]{[\@@bibref{Number}{SINDRUMIITi}{}{}]})}\;.$ (57)

Future bounds on the first two processes are

$\displaystyle\mathrm{BR}(\mu\to e\gamma)<6\times
10^{-14}\;\;\;\;\;\;\;\;\;\;\mbox{(MEG II\leavevmode\nobreak\
\cite[cite]{[\@@bibref{Number}{MEGII:2021fah}{}{}]})}\;,$
$\displaystyle\mathrm{BR}(\mu\to 3\,e)<20\,(1)\times
10^{-16}\;\;\,\mbox{(Phase 1(2) of Mu3E\leavevmode\nobreak\
\cite[cite]{[\@@bibref{Number}{Blondel:2013ia}{}{}]})}\;,$ (58)

and for the CR of $\mu-e$ conversion in aluminium we expect

$\displaystyle\mathrm{CR}(\mu-e,\mathrm{Al})<2.6\times
10^{-17}\;\;\mbox{(COMET\leavevmode\nobreak\
\cite[cite]{[\@@bibref{Number}{COMET:2018auw}{}{}]})}\;,$
$\displaystyle\mathrm{CR}(\mu-e,\mathrm{Al})<2.9\times
10^{-17}\;\;\mbox{(Mu2e\leavevmode\nobreak\
\cite[cite]{[\@@bibref{Number}{Mu2e:2014fns}{}{}]})}\;.$ (59)

####  4.1.2 Parameter space

The parameter space of this scenario is spanned by the following quantities:
the group theory parameters which depend on the considered case ($n$ and $s$
for Case 1), $n$, $s$ and $t$ (or $u$ and $v$) for Case 2), $n$, $m$ and $s$
for Case 3 a) and Case 3 b.1)), the free angles $\theta_{L}$ and $\theta_{R}$,
the three couplings $y_{f}$ and the two scales $\mu_{0}$ and $M_{0}$,
appearing in the matrices $y_{D}$, $\mu_{S}$ and $M_{NS}$, see eqs. (13) and
(11), respectively. The group theory parameters are fixed to certain values
that have been shown to lead to lepton mixing angles that agree well with the
experimental data, see [33, 45, 47]. The angle $\theta_{R}$ can be chosen
freely and different values are explored. In general, it is linearly varied in
the interval

$0\leq\theta_{R}\leq 2\,\pi\;.$ (60)

In several of the following figures, see e.g. figs. 1, 6 and 22, we only
display results for values of $\theta_{R}$ that lie in the range
$0\leq\theta_{R}\leq\frac{\pi}{4}$. Nevertheless, we have checked that we
obtain the same results for values outside this range. The angle $\theta_{L}$,
on the other hand, is determined by fitting the measured values of the lepton
mixing angles as best as possible. Depending on the case, Case 1) through Case
3 b.1), we find one or two such values for $\theta_{L}$, see section 3. The
couplings $y_{f}$ are fixed such that the neutrino masses $m_{i}$, $i=1,2,3$,
are correctly reproduced, while still being perturbative, i.e.
$y_{f}\leq\sqrt{8\,\pi}$ [59]. Indeed, in the presented scans these range in
the interval $4\times 10^{-5}\lesssim y_{f}\lesssim 1.2$, independent of the
ordering of the light neutrino masses. We choose the signs of $y_{f}$ to be
positive. The scales $\mu_{0}$ and $M_{0}$ are varied in certain ranges. In
particular, we take the lepton number violating parameter $\mu_{0}$ to be much
smaller than the electroweak scale, as expected in the ISS framework, and vary
it logarithmically as follows

$100\,\mbox{eV}\leq\mu_{0}\leq 100\,\mbox{keV}\;,$ (61)

while the scale $M_{0}$ is in the range

$150\,\mbox{GeV}\leq M_{0}\leq 10\,\mbox{TeV}\;.$ (62)

Heavy sterile states in such a mass range can have different observational
imprints, see e.g. [60, 61, 62, 63, 64, 65, 66, 67, 68, 69]. In order to
optimise the exploration of this range of $M_{0}$, we divide it into three
intervals, i.e. a lower mass interval, $150\,\mbox{GeV}\leq M_{0}\leq
1\,\mbox{TeV}$, an intermediate mass interval, $1\,\mbox{TeV}\leq M_{0}\leq
5\,\mbox{TeV}$, and a higher mass interval, $5\,\mbox{TeV}\leq M_{0}\leq
10\,\mbox{TeV}$. In each of these intervals we vary $M_{0}$ uniformly and
logarithmically. A concise description of the numerical scan is found in
appendix C.

###  4.2 Analytical considerations

In this subsection, we remark features of the discussed BRs and CR that are
common to all cases. In the following, we use the formulae found in [70, 71].
These can be simplified by taking into account the mass spectrum of the
neutral states and the fact that the heavy sterile states form three pairs of
pseudo-Dirac neutrinos, compare eqs. (18) and (21), see details in [45]. For
the radiative cLFV decays $\ell_{\beta}\to\ell_{\alpha}\,\gamma$, we have

$\mathrm{BR}(\ell_{\beta}\to\ell_{\alpha}\,\gamma)\approx\frac{\alpha_{w}^{3}\,s_{w}^{2}}{64\,\pi^{2}}\,\frac{m_{\beta}^{4}}{M_{W}^{4}}\,\frac{m_{\beta}}{\Gamma_{\beta}}\,\left|\eta_{\alpha\beta}\right|^{2}\,G_{\gamma}(x_{0})^{2}\,,$
(63)

where $\alpha_{w}=\frac{g_{w}^{2}}{4\,\pi}$ is the weak coupling, $s_{w}$ the
sine of the weak mixing angle, $M_{W}$ the mass of the $W$ boson, $m_{\beta}$
the mass of the charged lepton of flavour $\beta$ and $\Gamma_{\beta}$ the
corresponding total decay width as well as $G_{\gamma}(x)$ the relevant loop
function with the limit $G_{\gamma}(x)\approx\frac{1}{2}$ for $x\gg 1$ and
$x_{0}=\left(\frac{M_{0}}{M_{W}}\right)^{2}$. Similarly, we get for the BR of
the tri-lepton decays $\ell_{\beta}\to 3\,\ell_{\alpha}$

$\displaystyle\mathrm{BR}(\ell_{\beta}\to 3\,\ell_{\alpha})$ $\displaystyle=$
$\displaystyle\frac{\alpha_{w}^{4}}{24576\,\pi^{3}}\,\frac{m_{\beta}^{4}}{M_{W}^{4}}\,\frac{m_{\beta}}{\Gamma_{\beta}}\,\left(2\,\Big{|}\frac{1}{2}\,F^{\beta
3\alpha}_{\mathrm{box}}+F^{\beta\alpha}_{Z}-2\,s_{w}^{2}\,(F^{\beta\alpha}_{Z}-F^{\beta\alpha}_{\gamma})\Big{|}^{2}\right.$
(64)
$\displaystyle+\,4\,s_{w}^{4}\,|F^{\beta\alpha}_{Z}-F^{\beta\alpha}_{\gamma}|^{2}+16\,s_{w}^{2}\,\mathrm{Re}\left((F^{\beta\alpha}_{Z}-\frac{1}{2}\,F^{\beta
3\alpha}_{\mathrm{box}})\,(G^{\beta\alpha}_{\gamma})^{\star}\right)$
$\displaystyle\left.-\,48\,s_{w}^{4}\,\mathrm{Re}\left((F^{\beta\alpha}_{Z}-F^{\beta\alpha}_{\gamma})\,(G^{\beta\alpha}_{\gamma})^{\star}\right)+32\,s_{w}^{4}\,|G^{\beta\alpha}_{\gamma}|^{2}\,\left(\mathrm{log}\frac{m_{\beta}^{2}}{m_{\alpha}^{2}}-\frac{11}{4}\right)\right)$

with the form factors reading approximately

$\displaystyle G^{\beta\alpha}_{\gamma}$ $\displaystyle\approx$ $\displaystyle
2\,\eta_{\alpha\beta}\,G_{\gamma}(x_{0})\;,\;\;F^{\beta\alpha}_{\gamma}\approx
2\,\eta_{\alpha\beta}\,F_{\gamma}(x_{0})\;,$ $\displaystyle
F^{\beta\alpha}_{Z}$ $\displaystyle\approx$ $\displaystyle
2\,\eta_{\alpha\beta}\,F_{Z}(x_{0})+4\,(\eta_{\alpha\beta}-2\,\eta^{2}_{\alpha\beta})\,G_{Z}(x_{0},0)+4\,\eta^{2}_{\alpha\beta}\,G_{Z}(x_{0},x_{0})\;,$
$\displaystyle F^{\beta 3\alpha}_{\mathrm{box}}$ $\displaystyle\approx$
$\displaystyle-4\,(\eta_{\alpha\beta}-2\,\eta_{\alpha\alpha}\,\eta_{\alpha\beta})-8\,\eta_{\alpha\alpha}\,\eta_{\alpha\beta}\,F_{\mathrm{Xbox}}(x_{0},x_{0})$
(65)

with the limits for $x\gg 1$ being
$F_{\gamma}(x)\approx-\frac{7}{12}-\frac{1}{6}\,\mbox{log}\,x$,
$F_{Z}(x)\approx\frac{5}{2}-\frac{5}{2}\,\mbox{log}\,x$,
$G_{Z}(x,0)=G_{Z}(0,x)\approx\frac{1}{2}\,\mbox{log}\,x$,
$G_{Z}(x,x)\approx-\frac{x}{2}-\frac{x\,\mathrm{log}\,x}{1-x}$ and
$F_{\mathrm{Xbox}}(x,x)\approx-\frac{x}{4}+\frac{1}{4}\,(13-6\,\mbox{log}\,x)$.
Furthermore, the CR for $\mu-e$ conversion in nuclei is

$\mathrm{CR}(\mu-e,\mathrm{N})=\frac{2\,G_{F}^{2}\,\alpha_{w}^{2}\,m_{\mu}^{5}}{(4\,\pi)^{2}\,\Gamma_{\mathrm{capt}}}\,\Big{|}4\,V^{(p)}\,\left(2\,\widetilde{F}^{\mu
e}_{u}+\widetilde{F}^{\mu e}_{d}\right)+4\,V^{(n)}\,\left(\widetilde{F}^{\mu
e}_{u}+2\,\widetilde{F}^{\mu e}_{d}\right)+s_{w}^{2}\,\frac{G^{\mu
e}_{\gamma}\,D}{2\,e}\Big{|}^{2}\;,$ (66)

where $G_{F}$ is the Fermi constant, $\Gamma_{\mathrm{capt}}$ the muon capture
rate and $D$, $V^{(p)}$ and $V^{(n)}$ are nuclear form factors. The
expressions for the form factors are

$\displaystyle\widetilde{F}^{\mu e}_{d}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\,s_{w}^{2}\,F^{\mu
e}_{\gamma}-\left(\frac{1}{4}-\frac{1}{3}\,s_{w}^{2}\right)\,F^{\mu
e}_{Z}+\frac{1}{4}\,F^{\mu edd}_{\mathrm{box}}\;,$
$\displaystyle\widetilde{F}^{\mu e}_{u}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\,s_{w}^{2}\,F^{\mu
e}_{\gamma}+\left(\frac{1}{4}-\frac{2}{3}\,s_{w}^{2}\right)\,F^{\mu
e}_{Z}+\frac{1}{4}\,F^{\mu euu}_{\mathrm{box}}$ (67)

with $F^{\mu e}_{\gamma}$ and $F^{\mu e}_{Z}$ found in eq. (65), and

$F^{\mu edd}_{\mathrm{box}}\approx 2\,\eta_{e\mu}\;\;\;\mbox{and}\;\;\;F^{\mu
euu}_{\mathrm{box}}\approx-8\,\eta_{e\mu}$ (68)

in the limit $x_{0}\gg 1$. Moreover, one makes use of the unitarity of the
quark mixing matrix and takes into account the smallness of $|V_{td}|$. These
formulae can be used to show the following three statements: all these
observables are mostly proportional to $|\eta_{\alpha\beta}|^{2}$, both the
tri-lepton decays and $\mu-e$ conversion in nuclei are dominated by the
contribution due to the $Z$ penguin (in particular for larger values of
$x_{0}$), see also [73, 72], as well as $\mathrm{CR}(\mu-e,\mathrm{N})$ is
very suppressed for a specific value of $M_{0}$ which depends on the actual
nucleus $\mathrm{N}$. In the case of aluminium and using the (approximate)
formula found in [70] together with the numerical values of the nuclear form
factors $D$, $V^{(p)}$ and $V^{(n)}$, $D=0.0362$, $V^{(p)}=0.0161$,
$V^{(n)}=0.0173$, and the capture rate $\Gamma_{\mathrm{capt}}=0.7054\times
10^{6}\,\mbox{s}^{-1}$, all given in [74], we arrive at

$x_{0}\approx 6470\;\;\;\mbox{corresponding to}\;\;\;M_{0}\approx
6.5\,\mbox{TeV}$ (69)

for $s_{w}^{2}=0.23$. This cancellation is clearly visible in e.g. fig. 2.

###  4.3 Results for Case 1)

We discuss in detail the results for Case 1). For this, we choose as exemplary
values

$n=26\;\;\;\mbox{and}\;\;\;0\leq s\leq 25\;,$ (70)

see also [45]. We inspect the form of the matrix $U_{0}(\theta)$ in eq. (27)
for Case 1), using $\Omega(s)({\bf 3})$ as given in eq. (112) in appendix B
and that the rotation is in the $(13)$-plane, and see that $U_{0}(\theta)$ can
be written as

$U_{0}(\theta)=U_{0}(\theta,s=0)\,\mathrm{diag}(e^{i\,\phi_{s}},e^{-2\,i\,\phi_{s}},e^{i\,\phi_{s}})\;,$
(71)

where $U_{0}(\theta,s=0)$ refers to the matrix $U_{0}(\theta)$ for $s=0$. With
this, it follows that the matrix $\eta$, see eq. (28), does not depend on the
parameter $s$. Thus, we can set $s$ without loss of generality to $s=1$. We
have also numerically studied the results for different values of $s$ (both
even and odd) and no dependence on this parameter is encountered.

Figure 1: Case 1). Predictions for $\mathrm{BR}(\mu\to e\gamma)$,
$\mathrm{BR}(\mu\to 3\,e)$ and $\mathrm{CR}(\mu-e,\mathrm{Al})$ as a function
of the angle $\theta_{L}$ in the upper, middle and lower row. This angle is
strongly constrained by the measured lepton mixing angles, compare the
coloured bands in the plots for $\mathrm{CR}(\mu-e,\mathrm{Al})$. The index
$n$ of the flavour symmetry is fixed to $n=26$ and the CP symmetry is chosen
such that $s=1$. However, the obtained results do not depend on $s$. Left
(right) plots are for light neutrino masses with NO (IO) and
$m_{0}=0.03\,(0.015)\,\mathrm{eV}$. Different values of the free angle
$\theta_{R}$, see the different colours, are explored. The scales $\mu_{0}$
and $M_{0}$ are set to $\mu_{0}=1\,\mbox{keV}$ and $M_{0}=3\,\mbox{TeV}$,
respectively. Grey regions are excluded by the prospective bounds on the
different BRs and CR with the darker (lighter) grey referring to Phase 1 (2)
of Mu3E in the plots in the middle row.

A dependence of the size of $\mathrm{BR}(\mu\to e\gamma)$, $\mathrm{BR}(\mu\to
3\,e)$ and $\mathrm{CR}(\mu-e,\mathrm{Al})$ on the angle $\theta_{R}$ is in
general expected, since the matrix combination in square brackets in eq. (26)
is not diagonal and the light neutrino mass spectrum is either given by eq.
(38), (39) or (40). We first explore the impact of this free angle by fixing
the two scales $\mu_{0}$ and $M_{0}$ to $\mu_{0}=1\,\mbox{keV}$ and
$M_{0}=3\,\mbox{TeV}$. Light neutrino masses either follow NO or IO with the
lightest neutrino mass being set to $m_{0}=0.03\,\mbox{eV}$ or
$m_{0}=0.015\,\mbox{eV}$, respectively. The results for the BRs of $\mu\to
e\gamma$ and $\mu\to 3\,e$ as well as the CR of $\mu-e$ conversion in
aluminium are displayed in fig. 1 for different values of $\theta_{R}$, as
indicated by the different colours, and for both light neutrino mass orderings
in the left and right plots, respectively. The small vertical coloured bands
in the plots for $\mathrm{CR}(\mu-e,\mathrm{Al})$ show the interval of
$\theta_{L}$ which leads to an acceptable fit of the lepton mixing angles with
$\chi^{2}\leq 27$, as defined in eq. (164) in appendix C. As mentioned, none
of the current bounds on $\mu-e$ transitions can constrain the considered
parameter space. Thus, we present in fig. 1 the exclusion due to the
prospective limits as grey area, see eqs. (58) and (59); in darker and lighter
grey in the case of Phase 1 and Phase 2 of the experiment Mu3E in the plots in
the middle. The constraining power of the future bound on $\mathrm{BR}(\mu\to
e\gamma)$ is limited to the larger values of $\theta_{R}$ that are shown,
while the bound on $\mathrm{BR}(\mu\to 3\,e)$ expected from Phase 1 of Mu3E
can exclude all used values apart from vanishing $\theta_{R}$. Phase 2 of Mu3E
and even more the prospective bound on $\mathrm{CR}(\mu-e,\mathrm{Al})$ have
the potential to test all the displayed values of $\theta_{R}$ for
$\mu_{0}=1\,\mbox{keV}$ and $M_{0}=3\,\mbox{TeV}$. We note that for these
scales the obtained BRs and CR are for most of the studied values of
$\theta_{R}$ slightly larger for light neutrino masses with IO than for a NO
light neutrino mass spectrum. We can estimate their size easily for
$\theta_{R}=0$, since for this choice the matrix combination in square
brackets in eq. (26) is automatically diagonal. We use that the BRs and CR are
mainly proportional to $|\eta_{e\mu}|^{2}$, as argued in section 4.2. The
quantity $\eta_{e\mu}$ is calculated from eq. (28). It turns out to depend on
$\theta_{L}$ (which can be written in terms of $\sin\theta_{13}$, see [33]) as
well as on the differences of the squares of the Yukawa couplings $\Delta
y_{ij}^{2}$, $\Delta y_{ij}^{2}=y_{i}^{2}-y_{j}^{2}$,

$\eta_{e\mu}=\frac{\eta_{0}^{\prime}}{6}\,\left(2\,\Delta
y_{21}^{2}-3\,(\sqrt{2-3\,\sin^{2}\theta_{13}}+\sin\theta_{13})\,\sin\theta_{13}\,\Delta
y_{31}^{2}\right)\;.$ (72)

The differences $\Delta y_{ij}^{2}$ can be expressed as the differences of the
light neutrino masses, i.e. $\Delta
y_{ij}^{2}=\left(\frac{M_{0}^{2}}{\mu_{0}\,\langle
H\rangle^{2}}\right)\,(m_{i}-m_{j})$. With this information, we find

$\mathrm{BR}(\mu\to e\gamma)\approx 5.9\times
10^{-15}\;,\;\;\mathrm{BR}(\mu\to 3\,e)\approx 9.5\times
10^{-16}\;,\;\;\mathrm{CR}(\mu-e,\mathrm{Al})\approx 4.2\times 10^{-16}$ (73)

for light neutrino masses with NO and $m_{0}=0.03\,\mbox{eV}$ as well as

$\mathrm{BR}(\mu\to e\gamma)\approx 1.4\times
10^{-14}\;,\;\;\mathrm{BR}(\mu\to 3\,e)\approx 2.2\times
10^{-15}\;,\;\;\mathrm{CR}(\mu-e,\mathrm{Al})\approx 9.9\times 10^{-16}$ (74)

for IO light neutrino masses with $m_{0}=0.015\,\mbox{eV}$. As one can see,
these estimates agree well with the numerical results displayed in fig. 1.

Figure 2: Case 1). Predictions for $\mathrm{BR}(\mu\to e\gamma)$,
$\mathrm{BR}(\mu\to 3\,e)$ and $\mathrm{CR}(\mu-e,\mathrm{Al})$ as a function
of $M_{0}$ in the upper, middle and lower row. Note that the range of the
scale $M_{0}$ is larger in these plots, $150\,\mbox{GeV}\lesssim M_{0}\lesssim
50\,\mbox{TeV}$, than in the numerical scans. For the group theoretical
parameters and light neutrino mass orderings compare fig. 1. The parameter
$\mu_{0}$ is set to $\mu_{0}=1\,\mbox{keV}$ and $\theta_{L}$ to its best-
fitting value for the lepton mixing angles. Six different values of
$\theta_{R}$ between $0$ and $\frac{\pi}{4}$ are explored, see different
colours. The grey regions indicate exclusions of future experiments, see fig.
1.  Figure 3: Case 1). Results of numerical scan for $\mathrm{BR}(\mu\to
e\gamma)$, $\mathrm{BR}(\mu\to 3\,e)$ and $\mathrm{CR}(\mu-e,\mathrm{Al})$
varying $M_{0}$, $\mu_{0}$ and $\theta_{R}$ in the ranges in eqs. (62), (61)
and (60), respectively. The parameters $n$ and $s$ are the same as in figs. 1
and 2. Lepton mixing angles are always accommodated well. The light neutrino
mass ordering is fixed to NO and $m_{0}$ to $m_{0}=0.03\,\mbox{eV}$. The
average Yukawa coupling $\overline{y}$ is defined in eq. (75). Points in grey,
orange and red are excluded by different (current/future) experimental bounds,
see text for details. Points in other colours correspond to a certain value of
$\overline{y}$, see colour bar, and pass all the imposed limits.

In fig. 2, we vary in addition to $\theta_{R}$ also the scale $M_{0}$ (up to
$50\,\mbox{TeV}$), while still fixing $\mu_{0}=1\,\mbox{keV}$. The value of
$\theta_{L}$ is always fitted such that the lepton mixing angles are
accommodated best. Furthermore, light neutrino masses are taken as in fig. 1.
We observe that the smaller values of $\theta_{R}$ that we show give rise to
smaller BRs and CR and that the dependence on the angle $\theta_{R}$ appears
less pronounced for light neutrino masses with IO and than with NO.
Furthermore, we see that for the larger values of $\theta_{R}$ slightly
smaller values of the BRs and CR are achieved for light neutrino masses with
IO. The dependence on $M_{0}$ is in general mild, in particular for
$\mathrm{BR}(\mu\to e\gamma)$, if not for the cancellation occurring in the
CR, see comments in section 4.2. As already shown in fig. 1, the future bound
on $\mathrm{BR}(\mu\to e\gamma)$ is only effective for the larger values of
$\theta_{R}$, while the limit expected from Phase 1 of Mu3E reduces the
available parameter space for all $\theta_{R}$ to $M_{0}\lesssim
10\,(3)\,\mbox{TeV}$ for light neutrino masses with NO (IO). Imposing the
prospective bound of Mu3E Phase 2 on $\mathrm{BR}(\mu\to 3\,e)$ leaves for
$\mu_{0}=1\,\mbox{keV}$ only small values of $M_{0}$, $M_{0}\lesssim
200\,\mbox{GeV}$, for light neutrino masses with NO unconstrained, if
$\theta_{R}=0$. In contrast to this, the forecasts from COMET and Mu2e clearly
exclude this possibility and would only allow for viable parameter space close
to $M_{0}\approx 6.5\,\mbox{TeV}$, where the cancellation occurs.

Figure 4: Case 1). Comparison of results for $\mathrm{BR}(\mu\to e\gamma)$,
$\mathrm{BR}(\mu\to 3\,e)$ and $\mathrm{CR}(\mu-e,\mathrm{Al})$ for $n=26$,
$s=1$ and light neutrino masses with NO and $m_{0}=0.03\,\mathrm{eV}$. The
colour-coding is the same as in fig. 3. For $\mathrm{BR}(\mu\to e\gamma)$ the
black line indicates the future bound from MEG II, for $\mathrm{BR}(\mu\to
3\,e)$ the solid (dashed) black line corresponds to the expected limit from
Mu3E Phase 1 (2) and for $\mathrm{CR}(\mu-e,\mathrm{Al})$ the black line
represents the prospective bound from COMET.

To comprehensively analyse the parameter space that we have scanned, we
display in fig. 3 the results for the BRs and CR either as function of the
scale $M_{0}$, $\mu_{0}$ or the free angle $\theta_{R}$. The different colour-
coding indicates, on the one hand, which experimental constraint can/does
exclude a data point, i.e. for grey points at least one of the prospective
bounds on $\mathrm{BR}(\mu\to e\gamma)$, $\mathrm{BR}(\mu\to 3\,e)$ and
$\mathrm{CR}(\mu-e,\mathrm{Al})$ is violated as well as the current limit on
at least one of the elements of the matrix $\eta$, see eq. (54), while the
orange points correspond to the situation in which at least one of the
prospective bounds is violated, but all current constraints on the magnitude
of the matrix elements $\eta_{\alpha\beta}$ are fulfilled and red points,
eventually, represent data points that are excluded by the current limits on
$\eta$, but not by the future bounds on the three studied $\mu-e$
transitions.101010We remark that not in all scans data points corresponding to
red points are found. On the other hand, the different colours which can be
read off from the colour bar signal the average value of the Yukawa coupling,
$\overline{y}$, which is defined as

$\overline{y}=\frac{1}{3}\,\left(y_{1}+y_{2}+y_{3}\right)$ (75)

for each viable data point that satisfies both the expected limits on the
mentioned cLFV processes and the current bounds on
$\eta_{\alpha\beta}$.111111Note that first the grey points, then the orange
ones, then the viable points which are coloured according to the colour bar
and eventually the red points are plotted. We notice that the areas of the
grey and the orange points can overlap, whereas the areas occupied by the
viable points and by the grey and orange points usually hardly have an
overlap. We clearly see that a minimum value of $\mu_{0}$ around
$2\,\mathrm{keV}$ is required in order to pass all bounds, while values of
$M_{0}$ in the entire studied range, see eq. (62), can lead to viable points
(with corresponding size of $\overline{y}$). As mentioned, the behaviour of
$\eta_{e\mu}$ and, thus, of the BRs and CR with respect to $\theta_{R}$ is
determined by the behaviour of the Yukawa couplings. In particular, we observe
enhanced values of the cLFV signals for $\cos 2\,\theta_{R}$ being close to
zero, since it entails at least one large Yukawa coupling, which can easiest
be seen from eqs. (39) and (40). Smaller values of these signals are obtained,
if the magnitude of $\cos 2\,\theta_{R}$ is maximised, as this allows for
smaller Yukawa couplings. As we see, the prospective bounds on
$\mathrm{BR}(\mu\to 3\,e)$ and $\mathrm{CR}(\mu-e,\mathrm{Al})$ are both
effective, whereas all viable points correspond to $\mathrm{BR}(\mu\to
e\gamma)\lesssim 6\times 10^{-16}$, which is two orders of magnitude smaller
than the future limit, see eq. (58). The results in fig. 3 are for light
neutrino masses with NO and $m_{0}=0.03\,\mbox{eV}$. Those obtained for
smaller $m_{0}$ (in particular, $m_{0}=0$) and the same light neutrino mass
ordering look similar. Also light neutrino masses with IO and larger values of
$m_{0}$, i.e. the benchmark value $m_{0}=0.015\,\mbox{eV}$, exhibit the same
behaviour. However, if $m_{0}$ is sufficiently small (we have checked
$m_{0}=0$ and $m_{0}=10^{-5}\,\mbox{eV}$ numerically), the dependence of the
BRs and CR on $\theta_{R}$ differs, since a cancellation among the terms of
$\eta_{e\mu}$ becomes possible for certain values of $\theta_{R}$.
Nevertheless, the expected size of the BRs and CR is the same as for larger
$m_{0}$ and light neutrino masses with NO.

Lastly, we display possible correlations among $\mathrm{BR}(\mu\to e\gamma)$,
$\mathrm{BR}(\mu\to 3\,e)$ and $\mathrm{CR}(\mu-e,\mathrm{Al})$ in fig. 4.
Since all three observables mainly depend on $|\eta_{e\mu}|^{2}$, see section
4.2, we expect them to be proportional to each other. This is to a certain
extent also visible in fig. 4. The shown results confirm the exclusion
potential of the future experiments for the three different observables:
limited for $\mathrm{BR}(\mu\to e\gamma)$ alone, considerable for
$\mathrm{BR}(\mu\to 3\,e)$ and strongest for $\mathrm{CR}(\mu-e,\mathrm{Al})$.
In particular, we see that more data points still compatible with the
prospective limit on $\mathrm{BR}(\mu\to 3\,e)$ from Mu3E Phase 2 can be
excluded by the bound expected from $\mu-e$ conversion in aluminium than vice
versa, compare plot in the middle of fig. 4.

###  4.4 Results for Case 2)

In the discussion of the results for Case 2), it is decisive whether $t$ is
odd (equivalent to $u$ odd, see eq. (49)) or even (meaning $u$ even), since
for $t$ odd the matrix combination in square brackets in eq. (26) is not
automatically diagonal and the light neutrino masses depend on the angle
$\theta_{R}$. Indeed, the formulae for them are the same as those in eqs.
(38)-(40) for Case 1) with $\cos 2\,\theta_{R}$ being replaced by $\sin
2\,\theta_{R}$. Consequently, we in general observe a dependence of the
flavour observables on this angle if $t$ is odd, while this does not happen
for $t$ even, because the mentioned matrix combination is then always diagonal
and the light neutrino masses do not depend on $\theta_{R}$, see eq. (30). As
expected and observed in the numerical analysis, the results do not depend on
whether $s$ is even or odd.

Before focussing on values of $u$ (or equivalently of the combination
$2\,s-t$) that allow for a good fit of the experimental data on the lepton
mixing angles, see comment below eq. (49), we study the general dependence of
the results for the three different $\mu-e$ transitions on the parameters of
this scenario. We first remark that, similar to the parameter $s$ for Case 1),
see eq. (71), the parameter $v$ for Case 2) does not appear in the expression
for $\eta$ as given in eq. (28), since the matrix $U_{0}(\theta)$ can be
written as

$U_{0}(\theta)=U_{0}(\theta,v=0)\,\mathrm{diag}(e^{i\,\phi_{v}/6},e^{-i\,\phi_{v}/3},e^{i\,\phi_{v}/6})\;,$
(76)

where $U_{0}(\theta,v=0)$ refers to the matrix $U_{0}(\theta)$ with $v=0$.
Here we have used the form of $\Omega({\bf 3})$ shown in eq. (122) in appendix
B and that the relevant rotation occurs in the (13)-plane.

We, then, display for the choice $t$ even ($u$ even) the results for
$\mathrm{BR}(\mu\to e\gamma)$, $\mathrm{BR}(\mu\to 3\,e)$ and
$\mathrm{CR}(\mu-e,\mathrm{Al})$ in the
$\frac{u}{n}-\theta_{L}$-plane121212According to the definition of $u$ in eq.
(49) it varies between $-(n-1)$ and $2\,(n-1)$ and should be an integer. In
this analysis, we treat $\frac{u}{n}$ as continuous parameter in the interval
$-1\leq\frac{u}{n}\leq 2$. for a fixed value of $\mu_{0}$ and $M_{0}$,
$\mu_{0}=1\,\mbox{keV}$ and $M_{0}=3\,\mbox{TeV}$, and light neutrino masses
with NO (IO) and non-zero $m_{0}$, see eq. (50), in the left (right) of fig.
5.

Figure 5: Case 2), $t$ even ($u$ even). Predictions for $\mathrm{BR}(\mu\to
e\gamma)$, $\mathrm{BR}(\mu\to 3\,e)$ and $\mathrm{CR}(\mu-e,\mathrm{Al})$ in
the $\frac{u}{n}-\theta_{L}$-plane in the upper, middle and lower row. Left
(right) plots are for light neutrino masses with NO (IO) and
$m_{0}=0.03\,(0.015)\,\mathrm{eV}$. The scales $\mu_{0}$ and $M_{0}$ are set
to $\mu_{0}=1\,\mbox{keV}$ and $M_{0}=3\,\mbox{TeV}$, respectively. Dark
(light) blue and red regions indicate the regions of parameter space
compatible at the $1\,\sigma$ ($3\,\sigma$) level with different prospective
bounds on the BRs and CR, see text for details. In the plots for
$\mathrm{CR}(\mu-e,\mathrm{Al})$ grey regions indicate the parameter space in
which the lepton mixing angles can be accommodated, i.e. dark (light) grey for
$\chi^{2}\leq 100\,(300)$ and in the insets $\chi^{2}\leq 27$.

As mentioned, the angle $\theta_{R}$ is irrelevant and thus not specified. For
$\mathrm{BR}(\mu\to e\gamma)$, shown in the upper row, the dark (light) blue
regions indicate the parameter space in which the prospective bound from the
experiment MEG II is passed at the $1\,\sigma$ ($3\,\sigma$) level, see eq.
(58). As one can see, the impact of this future limit is mild on the
considered parameter space, while being slightly stronger for light neutrino
masses with IO. Similarly, for $\mathrm{BR}(\mu\to 3\,e)$, see plots in the
middle row, the dark (light) blue regions indicate the parameter space
compatible with the expected bound from Mu3E Phase 1 at the $1\,\sigma$
($3\,\sigma$) level, see eq. (58). We observe that about half of the parameter
space in the shown plane can be disfavoured; again, for light neutrino masses
with IO the exclusion reach is slightly larger. The impact of the prospective
limit from Mu3E Phase 2 at the $1\,\sigma$ ($3\,\sigma$) level on the
parameter space is displayed with the dark (light) red regions. We clearly see
that the viable parameter space is considerably reduced. The lower row of fig.
5 contains the corresponding plots for $\mathrm{CR}(\mu-e,\mathrm{Al})$. As
expected, the future bounds from COMET and Mu2e, respectively, see eq. (59),
can only be passed in small regions of the parameter space. These are shown in
dark (light) blue for the bound from COMET at the $1\,\sigma$ ($3\,\sigma$)
level and in dark (light) red for Mu2e at the same $\sigma$ levels. Since
these bounds are very similar, the regions are largely super-imposed. As
explained in section 3, the ratio $\frac{u}{n}$ and the free angle
$\theta_{L}$ are also constrained by the requirement to accommodate the lepton
mixing angles well. For this reason, we show in the plots for
$\mathrm{CR}(\mu-e,\mathrm{Al})$ in dark (light) grey the parameter space
which leads to $\chi^{2}\leq 100\,(300)$, when using the $\chi^{2}$-function
as defined in eq. (164) in appendix C. These large values are chosen in order
to ensure the visibility of the corresponding parameter space in the
$\frac{u}{n}-\theta_{L}$-plane. In the insets the regions with $\chi^{2}\leq
27$ are displayed in grey. While compatible with the future bound from MEG II
and mostly with the one from Mu3E Phase 1, this part of the parameter space
will be excluded by Mu3E Phase 2 and the upcoming searches for $\mu-e$
conversion in aluminium for the chosen values of the two scales $\mu_{0}$ and
$M_{0}$.

Figure 6: Case 2), $t$ odd ($u$ odd). Predictions for $\mathrm{BR}(\mu\to
e\gamma)$, $\mathrm{BR}(\mu\to 3\,e)$ and $\mathrm{CR}(\mu-e,\mathrm{Al})$ in
the $\frac{u}{n}-\theta_{L}$-plane in the upper, middle and lower row. Light
neutrino masses, $\mu_{0}$ and $M_{0}$ are chosen as in fig. 5. For six
different values of $\theta_{R}$, the parameter space that is compatible at
the $1\,\sigma$ level with the future bound on $\mathrm{BR}(\mu\to e\gamma)$
from MEG II, on $\mathrm{BR}(\mu\to 3\,e)$ from Mu3E Phase 1 and on
$\mathrm{CR}(\mu-e,\mathrm{Al})$ from COMET, respectively, is displayed. In
the plots for $\mathrm{CR}(\mu-e,\mathrm{Al})$, different light-coloured
regions (with dashed contours) indicate the parameter space in which the
lepton mixing angles can be accommodated (corresponding to $\chi^{2}\leq 100$
and in the insets to $\chi^{2}\leq 27$).

In fig. 6 the corresponding results for $t$ odd ($u$ odd) are displayed for
several fixed values of $\theta_{R}$. Again, $\mu_{0}$ and $M_{0}$ are set to
$\mu_{0}=1\,\mbox{keV}$ and $M_{0}=3\,\mbox{TeV}$. As for $t$ even, we study
the three different $\mu-e$ transitions for both light neutrino mass
orderings, see left and right plots, respectively. In the upper row the
parameter space which is compatible with the prospective bound on
$\mathrm{BR}(\mu\to e\gamma)$ from MEG II at the $1\,\sigma$ level is shown in
different colours corresponding to the six different values of $\theta_{R}$.
As we can see, the smaller the shown value of $\theta_{R}$ is, the smaller the
allowed parameter space in the $\frac{u}{n}-\theta_{L}$-plane is. Furthermore,
the viable parameter space for light neutrino masses with IO is slightly
reduced compared to the one for NO. The same observations can be made when
analysing the parameter space allowed by the future limit on
$\mathrm{BR}(\mu\to 3\,e)$ from Mu3E Phase 1 at the $1\,\sigma$ level. We note
that, as expected, the resulting allowed portion of parameter space is
considerably smaller than in the case of $\mathrm{BR}(\mu\to e\gamma)$.
Applying the prospective bound on $\mu-e$ conversion in aluminium from COMET
at the $1\,\sigma$ level to the parameter space, we find that only tiny
regions remain allowed in the $\frac{u}{n}-\theta_{L}$-plane, see plots in the
lower row in fig. 6. In these plots, we also display the areas in which the
lepton mixing angles are accommodated to a certain degree, i.e. $\chi^{2}\leq
100$, see eq. (164) in appendix C, in different light colours (and with dashed
contours) depending on the value of $\theta_{R}$.131313Again, a larger value
of $\chi^{2}$ is used in order to increase visibility. In the insets, instead
the parameter space with $\chi^{2}\leq 27$ is displayed. Overall, we see that
it is very challenging, if not impossible, for all shown values of
$\theta_{R}$ to reconcile the requirement to fit the lepton mixing angles well
with passing the expected strong bounds on $\mu-e$ conversion in aluminium. We
remark that for $\theta_{R}\approx\frac{\pi}{4}$ we obtain areas very similar
those for $t$ even. This is consistent, since in this case the matrix
combination in square brackets in eq. (26) is (nearly) diagonal and the light
neutrino masses are (almost) proportional to the square of the Yukawa
couplings, see eq. (30), as it always happens for $t$ even.

In order to further study the results for Case 2), we specify the values of
$n$ and $u$, corresponding to certain combinations of $s$ and $t$, such that
the lepton mixing angles can be accommodated well. As has been shown in [33,
45, 47], a viable choice is

$n=14\;\;\;\mbox{together with}\;\;\;u=-1,0,1\;,$ (77)

since then $\frac{u}{n}$ is small enough. In terms of the parameters $s$ and
$t$, see eq. (49), the mentioned values of $u$ correspond to e.g. $s=0$ and
$t=0$ as well as $s=1$ and $t=2$ for $u=0$, $s=0$ and $t=1$ for $u=-1$ and
$s=1$ and $t=1$ for $u=1$.141414The choice $s=0$ and $t=0$ leads to trivial
Majorana phases, since $I_{1}$ and $I_{2}$ are zero, see e.g. [33]. In the
following, we numerically explore these combinations of $s$ and $t$. As is
known [33], two different values of the free angle lead to a good fit of the
lepton mixing parameters and we consider both of these. Like for Case 1), we
first analyse the behaviour of the BRs and CR for varying $M_{0}$, while
fixing $\mu_{0}$ to $\mu_{0}=1\,\mbox{keV}$, and either assuming light
neutrino masses with NO or IO.

Figure 7: Case 2), $t$ even ($u$ even). Predictions for $\mathrm{BR}(\mu\to
e\gamma)$, $\mathrm{BR}(\mu\to 3\,e)$ and $\mathrm{CR}(\mu-e,\mathrm{Al})$ as
a function of $M_{0}$ in the upper, middle and lower row. The group theory
parameters are chosen as $n=14$, $s=1$ and $t=2$, leading to $u=0$. Light
neutrino masses with NO (IO) correspond to the plots in the left (right).
Solid and dashed lines (almost always super-imposed) refer to the two
different values of the angle $\theta_{L}$ that allow for best-fitting the
lepton mixing angles. For the range of $M_{0}$, the value of $\mu_{0}$ and the
grey regions, see fig. 2 of Case 1).

For $t$ even, we take $s=1$ and $t=2$ (leading to $u=0$), as shown in fig. 7.
This leads to $\sin^{2}\theta_{13}\approx 0.022$, $\sin^{2}\theta_{12}\approx
0.34$ and $\sin^{2}\theta_{23}=0.5$, in agreement with [33, 45, 47]. Clearly,
there is no dependence on the angle $\theta_{R}$. Also the two different
values of $\theta_{L}$ that permit a good fit to lepton mixing data lead to
nearly identical results, since the solid and dashed curves are almost always
super-imposed. Here the solid curve represents the results for the smaller
value of the angle $\theta_{L}$, while the dashed curve corresponds to those
obtained for the larger value of $\theta_{L}$; explicit values of $\theta_{L}$
can be found in e.g. [33, 47]. In fig. 7 we display the same experimental
constraints for the BRs of $\mu\to e\,\gamma$ and $\mu\to 3\,e$ and
$\mathrm{CR}(\mu-e,\mathrm{Al})$ as in fig. 2 for Case 1). The conclusions
which can be drawn regarding the parameter space compatible with these future
limits are very similar to those found for Case 1) and $\theta_{R}=0$, namely
the future bound on $\mathrm{BR}(\mu\to e\gamma)$ is passed for the shown
interval of $M_{0}$, while the prospective limit from Mu3E Phase 1 can only
constrain larger masses, i.e. $M_{0}\gtrsim 10\,(3)\,\mbox{TeV}$, for light
neutrino masses with NO (IO). In contrast to this, Mu3E Phase 2 is expected to
reduce the allowed parameter space to $M_{0}\lesssim 200\,\mbox{GeV}$ for
light neutrino masses with NO and the future bound on
$\mathrm{CR}(\mu-e,\mathrm{Al})$ forces $M_{0}$ to lie in the interval
$5\,\mbox{TeV}\lesssim M_{0}\lesssim 7\,\mbox{TeV}$ for both light neutrino
mass orderings – close to the value of $M_{0}$ which leads to a (complete)
cancellation in the CR, see eq. (69).

Figure 8: Case 2), $t$ odd ($u$ odd). Predictions for $\mathrm{BR}(\mu\to
e\gamma)$, $\mathrm{BR}(\mu\to 3\,e)$ and $\mathrm{CR}(\mu-e,\mathrm{Al})$ as
a function of $M_{0}$ in the upper, middle and lower row. The group theory
parameters are chosen as $n=14$, $s=1$ and $t=1$, corresponding to $u=1$. The
conventions in these plots are the same as in figs. 2 and 7.

For $t$ odd, we employ as example $s=1$ and $t=1$ which give rise to $u=1$.
For this value of $u$ and $n=14$, the lepton mixing angles read
$\sin^{2}\theta_{13}\approx 0.022$, $\sin^{2}\theta_{12}\approx 0.34$ and
$\sin^{2}\theta_{23}\approx 0.56$, see also [33, 45, 47]. Using the same
conventions as in figs. 2 and 7, we display the results in fig. 8 for six
different values of the angle $\theta_{R}$ in different colours. While for $t$
even, we hardly see any difference in the results for the two different values
of $\theta_{L}$ that lead to a good fit of the lepton mixing angles, the solid
and dashed curves can be clearly distinguished in fig. 8. We also observe that
the larger of the shown values of $\theta_{R}$ result in smaller values of the
BRs and CR, except for $\theta_{R}=\frac{\pi}{4}$. For
$\theta_{R}=\frac{\pi}{4}$ we instead find very similar results as for the
combination $s=1$, $t=2$ ($u=0$), see fig. 7. This is expected, since for
$\theta_{R}=\frac{\pi}{4}$ the matrix combination in square brackets in eq.
(26) is automatically diagonal and the light neutrino masses are proportional
to the squares of the Yukawa couplings even for $t$ odd. Comparing the results
for light neutrino masses with NO and IO, they appear to be very similar,
compare plots in the left and the right in fig. 8. For the larger values of
$\theta_{R}$ that we show the prospective bound on $\mathrm{BR}(\mu\to
e\gamma)$ is passed, while the expected limit from Mu3E Phase 1 is more
constraining, especially for larger values of $M_{0}$. The future bound from
Mu3E Phase 2 can exclude all parameter space for $\mu_{0}=1\,\mbox{keV}$,
unless $M_{0}\lesssim 600\,(300)\,\mbox{GeV}$ for light neutrino masses with
NO (IO) and certain larger values of $\theta_{R}$ are chosen. The limit
expected from COMET (and also Mu2e), on the other hand, restricts the allowed
parameter space to values of $M_{0}$ close to $M_{0}\approx 6.5\,\mbox{TeV}$,
for which a cancellation occurs in the CR. Furthermore, we note that the
choice $s=0$ and $t=1$, corresponding to $u=-1$, leads to similar results as
shown in fig. 8.151515For this value of $u$ and $n=14$, the atmospheric mixing
angle is best-fitted to $\sin^{2}\theta_{23}\approx 0.44$, while the reactor
and the solar mixing angles are like for $u=0$ and $u=1$.

Before moving on to the results of the numerical scans, we estimate the size
of the BRs and CR with the help of the analytic approximations, found in
section 4.2. Similar to Case 1), see eq. (72), we first express the most
relevant element of the matrix $\eta$, $\eta_{e\mu}$, in terms of $\Delta
y_{ij}^{2}$ and lepton mixing observables which replace the combinations of
$\theta_{L}$ and $\frac{u}{n}$ that show up

$\eta_{e\mu}=\frac{\eta_{0}^{\prime}}{6}\,\left(2\,\Delta
y_{21}^{2}+3\,(1-2\,(\sin^{2}\theta_{13}+\sin^{2}\theta_{23}-\sin^{2}\theta_{13}\,\sin^{2}\theta_{23})-6\,i\,J_{\mathrm{CP}})\,\Delta
y_{31}^{2}\right)\;.$ (78)

This formula holds for $t$ even as well as $t$ odd and
$\theta_{R}=\frac{\pi}{4}$. Then, fixing $\mu_{0}=1\,\mbox{keV}$ and
$M_{0}=3\,\mbox{TeV}$, we find for $u=0$ ($u=1$) and light neutrino masses
with NO and $m_{0}=0.03\,\mbox{eV}$

$\mathrm{BR}(\mu\to e\gamma)\approx 6.2\,(5.7)\times
10^{-15}\;,\;\;\mathrm{BR}(\mu\to 3\,e)\approx 9.9\,(9.1)\times
10^{-16}\;,\;\;\mathrm{CR}(\mu-e,\mathrm{Al})\approx 4.3\,(4.0)\times
10^{-16}$ (79)

as well as for IO light neutrino masses and $m_{0}=0.015\,\mbox{eV}$

$\mathrm{BR}(\mu\to e\gamma)\approx 1.1\,(1.2)\times
10^{-14}\;,\;\;\mathrm{BR}(\mu\to 3\,e)\approx 1.7\,(1.9)\times
10^{-15}\;,\;\;\mathrm{CR}(\mu-e,\mathrm{Al})\approx 7.4\,(8.4)\times
10^{-16}\;.$ (80)

These estimates are consistent with the results displayed in figs. 7 and 8.

Figure 9: Case 2), $t$ even ($u$ even). Results of numerical scan for
$\mathrm{BR}(\mu\to e\gamma)$, $\mathrm{BR}(\mu\to 3\,e)$ and
$\mathrm{CR}(\mu-e,\mathrm{Al})$ varying $M_{0}$, $\mu_{0}$ and $\theta_{R}$
in the ranges in eqs. (62), (61) and (60), respectively. As example of the
group theory parameters $n=14$, $s=1$ and $t=2$, corresponding to $u=0$, is
chosen. For conventions, see fig. 3.

In figs. 9 and 10 we provide results of the numerical scans for $t$ even
corresponding to $u$ even ($s=1$, $t=2$ and thus $u=0$) and for $t$ odd/$u$
odd ($s=1$, $t=1$, meaning $u=1$), respectively. The colour-coding is the same
as in fig. 3 and the definition of the quantity $\overline{y}$ can be found in
eq. (75). We see that the same orders of magnitude for the BRs and CR are
predicted as in Case 1) and that also for Case 2) the prospective limits for
both processes $\mu\to 3\,e$ and $\mu-e$ conversion in aluminium can be
reached, while $\mathrm{BR}(\mu\to e\gamma)$ is expected to be smaller than
$6\times 10^{-16}$ in the portion of parameter space that is allowed.
Furthermore, this part of the parameter space is characterised by $\mu_{0}$
being larger than $2\,\mbox{keV}$, whereas the scanned intervals of $M_{0}$
and $\theta_{R}$ are not constrained. Nevertheless, we also observe certain
differences compared to the results for Case 1). In particular, both BRs
reveal lower limits, $\mathrm{BR}(\mu\to e\gamma)\gtrsim 10^{-19}$ and
$\mathrm{BR}(\mu\to 3\,e)\gtrsim 10^{-21}$, and the CR is predicted to be
larger than $5\times 10^{-22}$ for values of $M_{0}$ which are not too close
to $M_{0}\approx 6.5\,\mbox{TeV}$, where the cancellation occurs.161616 We
actually exclude all values of the CR which are obtained for
$4.5\,\mbox{TeV}\lesssim M_{0}\lesssim 7.5\,\mbox{TeV}$. As expected, for $t$
even, see fig. 9, no dependence of the flavour observables on the angle
$\theta_{R}$ is seen. Consequently, we observe less spread in the data points.
For $t$ odd, see fig. 10, a clear dependence on $\theta_{R}$ is observed. In
contrast to the results for Case 1), the BRs and CR are now enhanced for $\sin
2\,\theta_{R}$ close to zero, since then at least one Yukawa coupling becomes
large (compare the results found in [47]), while for $|\sin 2\,\theta_{R}|$
being maximal the Yukawa couplings and, hence, the size of the cLFV signals
are suppressed. This behaviour is very similar for light neutrino masses with
IO and $m_{0}=0.015\,\mbox{eV}$ as well as in situations in which the lightest
neutrino mass $m_{0}$ is set to zero.

Figure 10: Case 2), $t$ odd ($u$ odd). Results of numerical scan for
$\mathrm{BR}(\mu\to e\gamma)$, $\mathrm{BR}(\mu\to 3\,e)$ and
$\mathrm{CR}(\mu-e,\mathrm{Al})$ varying $M_{0}$, $\mu_{0}$ and $\theta_{R}$
in the ranges in eqs. (62), (61) and (60), respectively. As example of the
group theory parameters $n=14$, $s=1$ and $t=1$, meaning $u=1$, is chosen. For
conventions, see fig. 3.

Eventually, we compare the results of the numerical scans for the four
different combinations $\\{s=0,t=0\\}$, $\\{s=1,t=2\\}$ (corresponding to
$u=0$), $\\{s=0,t=1\\}$ ($u=-1$) and $\\{s=1,t=1\\}$ ($u=1$) in fig. 11. In
order to do so, we divide the scanned range of $\mu_{0}$ in 30 logarithmically
uniform bins and also the data points accordingly. Then, we take the average
of the results for the two BRs and the CR in each bin and plot it with respect
to $\mu_{0}$.171717In fig. 11, we take as value representing a bin the central
value of $\mu_{0}$ in this bin. We see that the two combinations corresponding
to $u=0$ lead to (nearly) the same averages. This is supported by the
observation that the matrix element $\eta_{e\mu}$ in eq. (78) can be written
in terms of the lepton mixing angles and $J_{\mathrm{CP}}$ that all depend on
$s$ and $t$ only through $u$, compare [33]. Furthermore, it can be observed
that the resulting averages are smaller by up to a factor 10 to 25, if $u=0$.
We note that also the two combinations corresponding to $u=-1$ and $u=1$,
respectively, lead to very similar results for the cLFV signals. The reason
why the results for $u=-1$ and $u=1$ are larger than those for $u=0$ (mainly)
lies in the fact that the displayed averages also take into account the part
of the parameter space in which $\theta_{R}$ fulfils $\sin
2\,\theta_{R}\approx 0$, entailing at least one large Yukawa coupling and,
thus, larger BRs and CR. This can also explain the similarity of the results
for $u=-1$ and $u=1$. Moreover, we note that the analogue of fig. 11 looks
very similar for light neutrino masses with IO and $m_{0}=0.015\,\mbox{eV}$.

Figure 11: Case 2). Comparison of results for $\mathrm{BR}(\mu\to e\gamma)$,
$\mathrm{BR}(\mu\to 3\,e)$ and $\mathrm{CR}(\mu-e,\mathrm{Al})$ for different
combinations of $s$ and $t$ (choices of $u$). The group index $n$ is fixed to
$n=14$. Light neutrino masses follow NO and $m_{0}=0.03\,\mathrm{eV}$. The
shown averages are made over the results of the scans in logarithmically
uniform bins of the scale $\mu_{0}$. For details see text.

###  4.5 Results for Case 3 a)

Figure 12: Case 3 a), $m$ and $s$ both even (or both odd). Results of
numerical scan for $\mathrm{BR}(\mu\to e\gamma)$, $\mathrm{BR}(\mu\to 3\,e)$
and $\mathrm{CR}(\mu-e,\mathrm{Al})$ varying $M_{0}$, $\mu_{0}$ and
$\theta_{R}$ in the ranges in eqs. (62), (61) and (60), respectively. As
example of the group theory parameters $n=34$, $m=2$ and $s=2$ is chosen. For
conventions, see fig. 3.

Like for Case 2), also for Case 3 a) it is relevant which residual symmetries,
encoded in the parameters $m$ and $s$, are preserved in the neutral lepton
sector in order to determine whether the results depend in general on the
angle $\theta_{R}$ or not. Concretely, for $m$ and $s$ being both even or
being both odd no dependence on $\theta_{R}$ is expected, while for the
combination $m$ even and $s$ odd or vice versa such a dependence is
anticipated, because the matrix combination in square brackets in eq. (26) is
not diagonal and the light neutrino masses depend on $\theta_{R}$, see section
3. It is, thus, important to consider numerical examples which represent these
different possibilities. For this reason, we choose, as in [47], one example
with $m$ odd,

$n=16\;\;\;\mbox{and}\;\;\;m=1\;,$ (81)

leading to $\sin^{2}\theta_{13}\approx 0.025$ and $\sin^{2}\theta_{23}\approx
0.61$,181818We note that the value of the reactor mixing angle is slightly
outside the experimentally preferred $3\,\sigma$ range [49]. and another one
with $m$ even,

$n=34\;\;\;\mbox{and}\;\;\;m=2\;,$ (82)

giving rise to $\sin^{2}\theta_{13}\approx 0.022$ and
$\sin^{2}\theta_{23}\approx 0.61$. The solar mixing angle is fixed by the
choice of the CP symmetry, i.e. the parameter $s$, and the free angle and all
values of $s$, $0\leq s\leq n-1$, allow for a good fit of the lepton mixing
angles, usually for two different values of the free angle. For examples of
viable fits, see [33] and more recently [47].

Figure 13: Case 3 a), $m$ even and $s$ odd (or vice versa). Results of
numerical scan for $\mathrm{BR}(\mu\to e\gamma)$, $\mathrm{BR}(\mu\to 3\,e)$
and $\mathrm{CR}(\mu-e,\mathrm{Al})$ varying $M_{0}$, $\mu_{0}$ and
$\theta_{R}$ in the ranges in eqs. (62), (61) and (60), respectively. As
example of the group theory parameters $n=34$, $m=2$ and $s=1$ is chosen. For
conventions, see fig. 3.

With this information, we can study the signal strength of the BRs and CR for
$n$ and $m$ given in the $\frac{s}{n}-\theta_{L}$-plane (treating
$\frac{s}{n}$ as continuous parameter between $0$ and $1$, but still keeping
track of whether $s$ is even or odd). We fix the values of $\mu_{0}$ and
$M_{0}$, e.g. $\mu_{0}=1\,\mbox{keV}$ and $M_{0}=3\,\mbox{TeV}$, and the light
neutrino mass spectrum. Then, we superimpose the results of the fit to the
solar mixing angle, similar to the analysis for Case 2) found in figs. 5 and
6. We find for all possible combinations of $m$ and $s$ being even and odd as
well as for light neutrino masses with NO (IO) and
$m_{0}=0.03\,(0.015)\,\mbox{eV}$ that no constraint on the parameter space
arises from the prospective bound on $\mu\to e\gamma$ from MEG II and on
$\mu\to 3\,e$ from Mu3E Phase 1 at the $3\,\sigma$ level, while the future
limit of Mu3E Phase 2 and the expected reach of the experiments COMET and Mu2e
exclude the entire parameter space. Thus, we do not display any corresponding
plots.

We instead continue with giving an estimate for $m$ and $s$ both even or both
odd as well as for $m$ even and $s$ odd or vice versa for $\theta_{R}=0$
(since then the matrix combination in square brackets in eq. (26) is diagonal)
that is based on the analytic form of $\eta_{e\mu}$. For this, we replace the
terms depending on $\frac{m}{n}$ with the sine of the reactor mixing angle and
take the limit $\theta_{L}$ being zero (using $\theta_{L}\approx\pi$ leads to
the same result), compare section 3 and [33]. Then, we have

$\eta_{e\mu}\approx\frac{\eta_{0}^{\prime}}{6}\,e^{\frac{4\,\pi\,i}{3}}\,\left(2\,\Delta
y_{21}^{2}-3\,(\sqrt{2-3\,\sin^{2}\theta_{13}}+\sin\theta_{13})\,\sin\theta_{13}\,\Delta
y_{31}^{2}\right)\;.$ (83)

This expression coincides with the one obtained for Case 1), see eq. (72), up
to the overall phase. Consequently, we refer to the estimates of the BRs and
CR found in eqs. (73) and (74) for light neutrino masses with NO and IO and
non-vanishing $m_{0}$, respectively.

We perform numerical scans for the concrete choices $n=16$, $m=1$ and $s=1$ as
well as $s=2$ and for $n=34$, $m=2$ and $0\leq s\leq 2$. The results are
presented in figs. 12-14 for light neutrino masses with NO and non-zero
$m_{0}$.

As expected, in fig. 12 no dependence on the free angle $\theta_{R}$ is
observed. Although these plots assume $m$ and $s$ both even (since we use
$n=34$), the results for $m$ and $s$ both odd (for $n=16$) look very similar.
For light neutrino masses with IO and $m_{0}=0.015\,\mbox{eV}$ we also find
comparable results. We have verified that taking $m_{0}=0$ leads to very
similar plots as well. Like for Case 2), $t$ even ($u$ even), the viable data
points can be characterised by $\mu_{0}\gtrsim 3\,\mbox{keV}$, while $M_{0}$
and $\theta_{R}$ are not restricted. Furthermore, the future limit on
$\mathrm{BR}(\mu\to 3\,e)$ as well as on $\mathrm{CR}(\mu-e,\mathrm{Al})$ can
be saturated for the viable points, whereas the maximum value of
$\mathrm{BR}(\mu\to e\gamma)$ is around $6\times 10^{-16}$ and thus much lower
than the prospective bound from MEG II. We also remark that both BRs appear to
be bounded from below, $\mathrm{BR}(\mu\to e\gamma)\gtrsim 10^{-19}$ and
$\mathrm{BR}(\mu\to 3\,e)\gtrsim 10^{-20}$. The CR of $\mu-e$ conversion in
aluminium reveals – for $M_{0}\lesssim 4.5\,\mbox{TeV}$ or $M_{0}\gtrsim
7.5\,\mbox{TeV}$ – also a lower limit, $\mathrm{CR}(\mu-e,\mathrm{Al})\gtrsim
10^{-21}$.

Figure 14: Case 3 a). Comparison of results for $\mathrm{BR}(\mu\to e\gamma)$,
$\mathrm{BR}(\mu\to 3\,e)$ and $\mathrm{CR}(\mu-e,\mathrm{Al})$ for different
combinations of $m$ and $s$. The group index $n$ is either fixed to $n=16$ or
to $n=34$. For further conventions and details see fig. 11 for Case 2).

For the choice $m$ even and $s$ odd (for $n=34$) we expect the results to
depend on $\theta_{R}$ and this dependence should be similar to the one found
for Case 1), compare [47]. As can be observed in fig. 13, the dependence is
much milder (for light neutrino masses with NO and $m_{0}=0.03\,\mbox{eV}$)
than in the other studied cases, Case 1) (see fig. 3), Case 2) for $t$ odd
(see fig. 10) and Case 3 b.1) for $m$ even and $s$ odd or vice versa (see fig.
19). We have confirmed that for $m$ odd and $s$ even (employing $n=16$) a very
similar result is obtained. Assuming light neutrino masses with IO and
$m_{0}=0.015\,\mbox{eV}$ also leads to plots similar to those shown in fig.
13. For NO light neutrino masses and $m_{0}$ being very small ($m_{0}=0$ or
$m_{0}=10^{-5}\,\mbox{eV}$) we find instead a larger dependence on
$\theta_{R}$ like for Case 1) in fig. 3, when considering e.g. $n=16$, $m=1$
and $s=2$. Regarding the experimental prospects to detect cLFV signals
involving the muon and the electron for Case 3 a) and $m$ even and $s$ odd or
vice versa, these are similar to those for $m$ and $s$ both even or both odd,
namely for the studied examples we obtain $4\times
10^{-21}\lesssim\mathrm{BR}(\mu\to e\gamma)\lesssim 6\times 10^{-16}$,
$\mathrm{BR}(\mu\to 3\,e)\gtrsim 10^{-22}$ and
$\mathrm{CR}(\mu-e,\mathrm{Al})\gtrsim 10^{-21}$, if the cancellation region,
see footnote 16, is avoided.

Lastly, we would like to compare the different choices of $n$, $m$ and $s$
that we have analysed numerically. Like for Case 2), see fig. 11, we make use
of averages corresponding to a certain binning of the data in the parameter
$\mu_{0}$. We find that the results of the BRs and CR are nearly independent
of the choice of $n$, $m$ and $s$, as can be clearly seen in fig. 14, where we
show the results for light neutrino masses with NO and non-zero $m_{0}$. This
behaviour can be explained with the fact that no enhancement of the BRs and CR
is observed for certain values of the angle $\theta_{R}$, compare fig. 13. The
corresponding plots for light neutrino masses with IO and
$m_{0}=0.015\,\mbox{eV}$ reveal the same features, as expected from the above
discussion.

Figure 15: Case 3 b.1), $m$ and $s$ both even. Predictions for
$\mathrm{BR}(\mu\to e\gamma)$, $\mathrm{BR}(\mu\to 3\,e)$ and
$\mathrm{CR}(\mu-e,\mathrm{Al})$ in the $\frac{s}{n}-\theta_{L}$-plane in the
upper, middle and lower row. The group theory parameters are chosen as $n=20$
and $m=10$. The conventions are like in fig. 5 of Case 2) and $t$ even ($u$
even). Plots reflecting the features of examples with $m$ and $s$ both odd can
be found in figs. 16 and 17, since these are displayed for the choice
$\theta_{R}=0$.

###  4.6 Results for Case 3 b.1)

The relevant parameters for Case 3 b.1) are the same as for Case 3 a). Given
the different assignment of the light neutrino masses, see e.g. eq. (31), the
values of the parameters $n$, $m$ and $s$ that can lead to a good agreement
with the experimental data on lepton mixing angles differ for Case 3 b.1) from
those for Case 3 a). As mentioned in section 3, see also [33], the parameter
$m$ is preferred to be close to $\frac{n}{2}$ in order to correctly
accommodate the measured value of the solar mixing angle. Unlike for Case 3
a), the smallest value of the index $n$ that permits a good fit can be small,
e.g. even $n=2$. Nevertheless, we study numerically the example

$n=20\;\;\;\mbox{and}\;\;\;m=9,10,11\;,$ (84)

since a larger $n$ allows to analyse, on the one hand, values of $m$ which are
not equal to $\frac{n}{2}$ and, on the other hand, several different values of
the parameter $s$ (corresponding to different CP symmetries), because $s$
varies between $0$ and $n-1$. Results for the lepton mixing angles can be
found in [33, 47]. Like for Case 3 a), we expect that the results for $m$ and
$s$ both even or both odd are independent of the angle $\theta_{R}$, whereas
those for $m$ even and $s$ odd or vice versa reveal a dependence on this
parameter. This expectation is confirmed in the following.

Figure 16: Case 3 b.1), $m$ odd, $s$ even and $\theta_{R}=0$. Predictions for
$\mathrm{BR}(\mu\to e\gamma)$, $\mathrm{BR}(\mu\to 3\,e)$ and
$\mathrm{CR}(\mu-e,\mathrm{Al})$ in the $\frac{s}{n}-\theta_{L}$-plane in the
upper, middle and lower row. The group theory parameters are chosen as $n=20$
and $m=9$. For the remaining conventions see fig. 15. The dependence on
$\theta_{R}$ is illustrated with six different values of $\theta_{R}$ in
different colours in fig. 22 in appendix D. Results for an example of the
combination $m$ even and $s$ odd and different values of $\theta_{R}$ are
found in fig. 24 in the same appendix.

We begin the analysis of the BRs and CR by evaluating their sizes in the
$\frac{s}{n}-\theta_{L}$-plane191919The ratio $\frac{s}{n}$ is taken to be
continuous, $0\leq\frac{s}{n}\leq 1$, and $s$ is either assumed to be even or
odd. for a fixed ratio of $\frac{m}{n}$, $\frac{m}{n}=\frac{9}{20}$,
$\frac{m}{n}=\frac{10}{20}$ and $\frac{m}{n}=\frac{11}{20}$, as well as for
fixed values of the two scales $\mu_{0}$ and $M_{0}$, $\mu_{0}=1\,\mbox{keV}$
and $M_{0}=3\,\mbox{TeV}$, see figs. 15-17. The colour-coding in these figures
is the same as in fig. 5 for Case 2). In particular, the dark (light) grey
regions in the plots for $\mathrm{CR}(\mu-e,\mathrm{Al})$ indicate the
parameter space in which the lepton mixing angles are accommodated to a
certain degree, i.e. $\chi^{2}\leq 100\,(300)$. These regions are found to be
consistent with those shown in corresponding plots in [33, 47]. The plots in
the left in figs. 15-17 assume that light neutrino masses follow NO with
$m_{0}=0.03\,\mbox{eV}$, while the plots in the right in these figures are
based on an IO light neutrino mass spectrum with $m_{0}=0.015\,\mbox{eV}$.
Fig. 15 represents the combination $m$ and $s$ both even and, thus, no
dependence on $\theta_{R}$ is expected, and also observed, while in figs. 16
and 17 $m$ is odd and $s$ even such that the results depend on $\theta_{R}$.
Here, they are shown for $\theta_{R}=0$, for which the matrix combination in
square brackets in eq. (26) is diagonal. We complete the set of plots by figs.
22-24 in appendix D in which we display for $\frac{m}{n}=\frac{9}{20}$ and
$\frac{m}{n}=\frac{11}{20}$ and $s$ even as well as for
$\frac{m}{n}=\frac{10}{20}$ and $s$ odd the results for six different values
of the angle $\theta_{R}$ in different colours, similar to e.g. fig. 6 for
Case 2) and $t$ odd. We note that the results for $\frac{m}{n}=\frac{9}{20}$
($\frac{m}{n}=\frac{11}{20}$) and $s$ even agree for the choice $\theta_{R}=0$
with those for $\frac{m}{n}=\frac{9}{20}$ ($\frac{m}{n}=\frac{11}{20}$) and
$s$ odd and, similarly, the choice $\frac{m}{n}=\frac{10}{20}$ and $s$ odd
with $\theta_{R}=0$ reproduces the results of $\frac{m}{n}=\frac{10}{20}$ and
$s$ even. Consequently, the results for all possible combinations of $m$ and
$s$ even and odd are covered with the six figures, figs. 15-17 and figs.
22-24. We see for $m$ and $s$ both even or both odd as well as for $m$ even
and $s$ odd or vice versa and $\theta_{R}=0$ that in most situations the
prospective bound on $\mathrm{BR}(\mu\to e\gamma)$ is passed in the entire
parameter space – at least at the $3\,\sigma$ level, whereas the limit
expected from Mu3E Phase 1 excludes about half of the parameter space at the
$3\,\sigma$ level and the future bound from Mu3E Phase 2 considerably reduces
the viable regions of parameter space. Employing the expected limits from
COMET and Mu2e only leaves a small portion of parameter space in the
$\frac{s}{n}-\theta_{L}$-plane allowed, which in general is hardly compatible
with the regions preferred by the measured values of the lepton mixing angles.
In addition, we observe that the future bounds are slightly more likely to be
satisfied for light neutrino masses with NO and $m_{0}=0.03\,\mbox{eV}$ than
in the case of an IO light neutrino mass spectrum with
$m_{0}=0.015\,\mbox{eV}$ (compare plots in the left with those in the right in
figs. 15-17). When considering the dependence on $\theta_{R}$ for the
combinations $m$ even and $s$ odd or vice versa, see especially figs. 22-24 in
appendix D, we note that the shape and size of the parameter space passing a
certain future bound depends on the actual value of $\theta_{R}$ and for the
larger ones among the shown values it is usually smaller. Furthermore, the
regions of parameter space that lead to an agreement of the lepton mixing
angles with experimental data shift as function of $\theta_{R}$. It is, thus,
necessary to investigate each value of $\theta_{R}$ separately in order to
make more quantitative statements about the constraints imposed by the future
limits on the different $\mu-e$ transitions and their compatibility with the
restrictions coming from fitting the lepton mixing angles well.

Figure 17: Case 3 b.1), $m$ odd, $s$ even and $\theta_{R}=0$. Predictions for
$\mathrm{BR}(\mu\to e\gamma)$, $\mathrm{BR}(\mu\to 3\,e)$ and
$\mathrm{CR}(\mu-e,\mathrm{Al})$ in the $\frac{s}{n}-\theta_{L}$-plane in the
upper, middle and lower row. The group theory parameters are chosen as $n=20$
and $m=11$. For the remaining conventions see fig. 15. The dependence on
$\theta_{R}$ is illustrated with six different values of $\theta_{R}$ in
different colours in fig. 23 in appendix D. Results for an example of the
combination $m$ even and $s$ odd and different values of $\theta_{R}$ are
found in fig. 24 in the same appendix.

To numerically estimate the BRs and CR, we first give an approximate formula
for $\eta_{e\mu}$ for $m$ and $s$ both even or both odd or $m$ even and $s$
odd or vice versa, assuming $\theta_{R}=0$. For this, we take $m=\frac{n}{2}$,
express $\theta_{L}$ in terms of the reactor mixing angle, compare [33, 47],
and then arrive at

$\eta_{e\mu}=\frac{\eta_{0}^{\prime}}{3\,\sqrt{2}}\,e^{\frac{4\,\pi\,i}{3}}\,\left(3\,(e^{-3\,i\,\phi_{s}}\,\sqrt{1-3\,\sin^{2}\theta_{13}}+\sqrt{2}\,\sin\theta_{13})\,\sin\theta_{13}\,\Delta
y_{21}^{2}-\sqrt{2}\,\Delta y_{31}^{2}\right)\,\;,$ (85)

where $\Delta y_{21}^{2}$ and $\Delta y_{31}^{2}$ are related to the light
neutrino masses as follows $\Delta
y_{21}^{2}=\left(\frac{M_{0}^{2}}{\mu_{0}\,\langle
H\rangle^{2}}\right)\,(m_{3}-m_{2})$ and $\Delta
y_{31}^{2}=\left(\frac{M_{0}^{2}}{\mu_{0}\,\langle
H\rangle^{2}}\right)\,(m_{1}-m_{2})$, since the identification of Yukawa
couplings and light neutrino masses is altered compared to Case 1) through
Case 3 a) due to the permutation, c.f. eq. (31). Using that $-0.6\lesssim\cos
3\,\phi_{s}\lesssim 0.71$, see [33], we obtain as intervals for the BRs and CR

$\displaystyle 3.9\times 10^{-15}\lesssim$ $\displaystyle\mathrm{BR}(\mu\to
e\gamma)$ $\displaystyle\lesssim 9.3\times 10^{-15}\;,$ $\displaystyle
6.3\times 10^{-16}\lesssim$ $\displaystyle\mathrm{BR}(\mu\to 3\,e)$
$\displaystyle\lesssim 1.5\times 10^{-15}\;,$ $\displaystyle 2.7\times
10^{-16}\lesssim$ $\displaystyle\mathrm{CR}(\mu-e,\mathrm{Al})$
$\displaystyle\lesssim 6.6\times 10^{-16}\;$ (86)

for light neutrino masses with NO and $m_{0}=0.03\,\mbox{eV}$, while for IO
light neutrino masses wtih $m_{0}=0.015\,\mbox{eV}$ we get

$\displaystyle 8.8\times 10^{-15}\lesssim$ $\displaystyle\mathrm{BR}(\mu\to
e\gamma)$ $\displaystyle\lesssim 1.3\times 10^{-14}\;,$ $\displaystyle
1.4\times 10^{-15}\lesssim$ $\displaystyle\mathrm{BR}(\mu\to 3\,e)$
$\displaystyle\lesssim 2.1\times 10^{-15}\;,$ $\displaystyle 6.2\times
10^{-16}\lesssim$ $\displaystyle\mathrm{CR}(\mu-e,\mathrm{Al})$
$\displaystyle\lesssim 9.1\times 10^{-16}\;.$ (87)

Again, we have fixed the two scales $\mu_{0}$ and $M_{0}$ to
$\mu_{0}=1\,\mbox{keV}$ and $M_{0}=3\,\mbox{TeV}$, respectively. From these
estimates we see that the future bound on $\mathrm{BR}(\mu\to e\gamma)$ is
usually passed as well as the limit on $\mathrm{BR}(\mu\to 3\,e)$ expected
from Mu3E Phase 1, whereas Mu3E Phase 2 can exclude this scenario and also the
experiments COMET and Mu2e have this potential, since
$\mathrm{CR}(\mu-e,\mathrm{Al})$ is larger than $2\times 10^{-16}$ for the
chosen values of the scales $\mu_{0}$ and $M_{0}$ and the assumed light
neutrino mass spectra.

Figure 18: Case 3 b.1), $m$ and $s$ both odd (or both even). Results of
numerical scan for $\mathrm{BR}(\mu\to e\gamma)$, $\mathrm{BR}(\mu\to 3\,e)$
and $\mathrm{CR}(\mu-e,\mathrm{Al})$ varying $M_{0}$, $\mu_{0}$ and
$\theta_{R}$ in the ranges in eqs. (62), (61) and (60), respectively. As
example of the group theory parameters $n=20$, $m=9$ and $s=1$ is chosen. For
conventions, see fig. 3.

In a next step, we scrutinise the parameter space by performing numerical
scans for different values of $m$ and $s$: for $m=10$ we have studied $s=2$,
$s=3$ and $s=10$, for $m=9$ the choices of $s$ are $s=1$, $s=2$, $s=18$ and
$s=19$, while for $m=11$ we have fixed $s$ to $s=0$, $s=1$ and $s=2$. We show
examples of these scans in figs. 18 and 19 for the concrete choices $m=9$ and
$s=1$ as well as $m=9$ and $s=2$, respectively, assuming that light neutrino
masses follow NO and $m_{0}=0.03\,\mbox{eV}$. Since $m=9$ and $s=1$ are both
odd, no dependence on $\theta_{R}$ is revealed, while for $m=9$ and $s=2$ a
clear dependence on $\theta_{R}$ is observed, as expected. Like for Case 1),
at least one Yukawa coupling is enhanced for $\cos 2\,\theta_{R}\approx 0$
and, consequently, also the BRs and CR, whereas for $|\cos
2\,\theta_{R}|\approx 1$ the signal strength of the studied cLFV processes is
suppressed, compare [47] for the dependence on $\theta_{R}$. We have checked
that the other choices of $m$ and $s$ both even or both odd as well as light
neutrino masses with IO and $m_{0}=0.015\,\mbox{eV}$ lead to plots that are
very similar to those found in fig. 18. This holds analogously for the other
choices of $m$ and $s$ with one of them being even and the other one odd and
the results displayed in fig. 19. Regarding the ranges of the BRs and CR, we
usually find for the viable data points (the ones coloured according to the
colour bar in figs. 18 and 19) $10^{-19}\lesssim\mathrm{BR}(\mu\to
e\gamma)\lesssim 6\times 10^{-16}$, $\mathrm{BR}(\mu\to 3\,e)\gtrsim 10^{-21}$
and $\mathrm{CR}(\mu-e,\mathrm{Al})\gtrsim 10^{-22}$ (excluding
$4.5\,\mbox{TeV}\lesssim M_{0}\lesssim 7.5\,\mbox{TeV}$) with
$\mathrm{BR}(\mu\to 3\,e)$ and $\mathrm{CR}(\mu-e,\mathrm{Al})$ saturating the
expected limits from Mu3E Phase 2 and COMET, respectively.

Figure 19: Case 3 b.1), $m$ odd and $s$ even (or vice versa). Results of
numerical scan for $\mathrm{BR}(\mu\to e\gamma)$, $\mathrm{BR}(\mu\to 3\,e)$
and $\mathrm{CR}(\mu-e,\mathrm{Al})$ varying $M_{0}$, $\mu_{0}$ and
$\theta_{R}$ in the ranges in eqs. (62), (61) and (60), respectively. As
example of the group theory parameters $n=20$, $m=9$ and $s=2$ is chosen. For
conventions, see fig. 3.

We summarise and compare the results of all the performed numerical scans for
light neutrino masses with NO and non-vanishing $m_{0}$ in fig. 20 using the
same strategy of binning and averaging as in fig. 11. We observe that the
combinations of $m$ even and $s$ odd or vice versa lead to larger BRs and CR
in average (up to a factor $25$) than those with $m$ and $s$ both even or both
odd. Analogous to Case 2), this is due to the enhancement of the BRs and CR
for certain values of the angle $\theta_{R}$ for $m$ even and $s$ odd or vice
versa, entailing larger averages, which does not occur for $m$ and $s$ both
even or both odd. For light neutrino masses with IO and
$m_{0}=0.015\,\mbox{eV}$ the plots shown in fig. 20 look very similar.

Figure 20: Case 3 b.1). Comparison of results for $\mathrm{BR}(\mu\to
e\gamma)$, $\mathrm{BR}(\mu\to 3\,e)$ and $\mathrm{CR}(\mu-e,\mathrm{Al})$ for
different combinations of $m$ and $s$. The group index $n$ is set to $n=20$.
For further conventions and details see fig. 11 for Case 2).

##  5 Comments on cLFV $\tau$ decays

We also consider the following cLFV decays of the tau lepton
$\tau\to\mu\gamma$, $\tau\to e\gamma$, $\tau\to 3\,\mu$ and $\tau\to 3\,e$. As
current experimental bounds on the BRs of these decays we use (all at 90%
C.L.)

$\displaystyle\mathrm{BR}(\tau\to\mu\gamma)<4.2\times
10^{-8}\;\;\mbox{(Belle\leavevmode\nobreak\
\cite[cite]{[\@@bibref{Number}{Belle:2021ysv}{}{}]})}\;,\;\;\;\;\;\;\mathrm{BR}(\tau\to
e\gamma)<3.3\times 10^{-8}\;\;\,\mbox{(Belle\leavevmode\nobreak\
\cite[cite]{[\@@bibref{Number}{ParticleDataGroup:2020ssz}{}{}]})}\;,$
$\displaystyle\mathrm{BR}(\tau\to 3\,\mu)<1.9\times 10^{-8}\;\;\mbox{(Belle
II\leavevmode\nobreak\ \cite[cite]{[\@@bibref{Number}{Belle-
II:2024sce}{}{}]})}\;,\;\;\mathrm{BR}(\tau\to 3\,e)<2.7\times
10^{-8}\;\;\,\mbox{(Belle\leavevmode\nobreak\
\cite[cite]{[\@@bibref{Number}{ParticleDataGroup:2020ssz}{}{}]})}\;.$ (88)

The prospective limits are

$\displaystyle\mathrm{BR}(\tau\to\mu\gamma)<6.9\times
10^{-9}\;\;\;\mbox{(Belle II\leavevmode\nobreak\
\cite[cite]{[\@@bibref{Number}{Banerjee:2022xuw}{}{}]})}\;,\;\;\mathrm{BR}(\tau\to
e\gamma)<9\times 10^{-9}\,\;\;\;\;\;\mbox{(Belle II\leavevmode\nobreak\
\cite[cite]{[\@@bibref{Number}{Banerjee:2022xuw}{}{}]})}\,,$
$\displaystyle\mathrm{BR}(\tau\to 3\,\mu)<3.6\times 10^{-10}\;\,\mbox{(Belle
II\leavevmode\nobreak\
\cite[cite]{[\@@bibref{Number}{Banerjee:2022xuw}{}{}]})}\;,\;\,\mathrm{BR}(\tau\to
3\,e)<4.7\times 10^{-10}\;\mbox{(Belle II\leavevmode\nobreak\
\cite[cite]{[\@@bibref{Number}{Banerjee:2022xuw}{}{}]})}\,.$ (89)

Before discussing the results of the numerical scans, we give simple formulae
for $\eta_{\mu\tau}$ and $\eta_{e\tau}$ for Case 1) that are relevant for the
cLFV decays $\tau\to\mu\gamma$, $\tau\to 3\,\mu$ and $\tau\to e\gamma$,
$\tau\to 3\,e$, respectively, similar to $\eta_{e\mu}$ dominating the $\mu-e$
transitions. They read

$\eta_{\mu\tau}=\frac{\eta_{0}^{\prime}}{6}\,\left(2\,\Delta
y_{21}^{2}-3\,(1-2\,\sin^{2}\theta_{13})\,\Delta y_{31}^{2}\right)$ (90)

and

$\eta_{e\tau}=\frac{\eta_{0}^{\prime}}{6}\,\left(2\,\Delta
y_{21}^{2}+3\,(\sqrt{2-3\,\sin^{2}\theta_{13}}-\sin\theta_{13})\,\sin\theta_{13}\,\Delta
y_{31}^{2}\right)\;,$ (91)

respectively, with $\Delta y_{ij}^{2}$ being defined in the vicinity of eq.
(72). We see that, in particular, $\eta_{e\tau}$ has a form similar to
$\eta_{e\mu}$. Numerical estimates for the corresponding BRs can be derived
with the help of these formulae and the approximations found in section 4.2.
For $\mu_{0}=1\,\mbox{keV}$ and $M_{0}=3\,\mbox{TeV}$ we get

$\displaystyle\mathrm{BR}(\tau\to\mu\gamma)\approx 2.3\times 10^{-14}\;,$
$\displaystyle\mathrm{BR}(\tau\to 3\,\mu)\approx 3.6\times 10^{-15}\;,$
$\displaystyle\mathrm{BR}(\tau\to e\gamma)\approx 1.2\times 10^{-15}\;,$
$\displaystyle\mathrm{BR}(\tau\to 3\,e)\approx 2.0\times 10^{-16}$ (92)

for light neutrino masses with NO and $m_{0}=0.03\,\mbox{eV}$, while for IO
light neutrino masses with $m_{0}=0.015\,\mbox{eV}$ we find

$\displaystyle\mathrm{BR}(\tau\to\mu\gamma)\approx 4.2\times 10^{-14}\;,$
$\displaystyle\mathrm{BR}(\tau\to 3\,\mu)\approx 6.6\times 10^{-15}\;,$
$\displaystyle\mathrm{BR}(\tau\to e\gamma)\approx 1.4\times 10^{-15}\;,$
$\displaystyle\mathrm{BR}(\tau\to 3\,e)\approx 2.3\times 10^{-16}\;.$ (93)

These already indicate that the presented scenario is most likely not
constrained by the current nor by the prospective limits on the studied cLFV
tau lepton decays, see eqs. (88) and (89). This observation is confirmed by
the numerical scans that we have performed. As example, the results for
$n=26$, $s=1$ and light neutrino masses with NO and $m_{0}=0.03\,\mbox{eV}$
are shown in fig. 21. The colour-coding is the same as in fig. 3 for the
corresponding $\mu-e$ transitions. The largest values obtained for the
different BRs for the viable data points are

$\displaystyle\mathrm{BR}(\tau\to\mu\gamma)\lesssim 7\times
10^{-15}\;,\;\;\mathrm{BR}(\tau\to 3\,\mu)\lesssim 2\times 10^{-15}\;,\;\;$
$\displaystyle\mathrm{BR}(\tau\to e\gamma)\lesssim 3\times
10^{-15}\;\,,\;\;\mathrm{BR}(\tau\to 3\,e)\lesssim 2\times 10^{-16}$ (94)

and, thus, clearly inaccessible by the mentioned facilities. The observed
dependence on the angle $\theta_{R}$ is driven by $\cos 2\,\theta_{R}$, like
for the $\mu-e$ transitions, and is due to the fact that for $\cos
2\,\theta_{R}\approx 0$ at least one Yukawa coupling is large, compare eqs.
(38)-(40), while for $|\cos 2\,\theta_{R}|$ large smaller values of the Yukawa
couplings are accessible and, consequently, also smaller values of the BRs can
be obtained. The results for light neutrino masses with IO and
$m_{0}=0.015\,\mbox{eV}$ are similar and the upper limits obtained for the BRs
are only slightly larger than those given in eq. (94). We have also performed
numerical scans for $m_{0}=0$ and light neutrino masses with NO and IO,
respectively, and obtain results that are qualitatively similar to those
displayed in fig. 21.

Figure 21: Case 1). Results of numerical scan for
$\mathrm{BR}(\tau\to\mu\gamma)$, $\mathrm{BR}(\tau\to 3\,\mu)$,
$\mathrm{BR}(\tau\to e\gamma)$ and $\mathrm{BR}(\tau\to 3\,e)$ varying
$M_{0}$, $\mu_{0}$ and $\theta_{R}$ in the ranges in eqs. (62), (61) and (60),
respectively. The parameters $n$ and $s$ are the same as in fig. 1. For
conventions see fig. 3.

Also for the other cases, Case 2) through Case 3 b.1), the values of the BRs
are of similar order as those given in eq. (94) for the viable points in the
scans and, thus, cannot be tested by current and near-future experiments. We
have checked that for Case 2) the BRs of the different studied cLFV tau lepton
decays do not depend on the angle $\theta_{R}$ for $t$ even that corresponds
to $u$ even, whereas the expected dependence on $\theta_{R}$ is found for $t$
odd equivalent to $u$ odd. Furthermore, the exact value of the parameter $s$
is not directly relevant. For Case 3 a), we observe that the BRs of the cLFV
tau lepton decays also have similar features as the studied $\mu-e$
transitions, i.e. for $m$ and $s$ being both even or both odd there is no
dependence on the angle $\theta_{R}$, while for $m$ even and $s$ odd or vice
versa such a dependence is expected and usually a milder one is found in the
numerical scans. Lastly, we have checked that statements analogous to those
made for the $\mu-e$ transitions also hold for the cLFV tau lepton decays for
Case 3 b.1).

##  6 Summary

We have studied a scenario in which light neutrino masses are generated via
the ISS mechanism and a flavour and CP symmetry determine the flavour
structure. The 3+3 gauge singlet fermions form three pairs of pseudo-Dirac
states that are (nearly) degenerate in mass, because in the neutral lepton
sector only the Dirac neutrino Yukawa matrix breaks $G_{f}$ and CP to
$G_{\nu}$ which is the direct product of a $Z_{2}$ symmetry and CP. Among
charged leptons the residual symmetry is $G_{e}=Z_{3}$ and in the chosen basis
the charged lepton mass matrix is diagonal with three free parameters
corresponding to the charged lepton masses. There are four cases, Case 1)
through Case 3 b.1), that lead to four different types of lepton mixing
patterns. In the neutral lepton sector, the parameter space is spanned by the
three Yukawa couplings $y_{f}$, the two angles $\theta_{L}$ and $\theta_{R}$,
all contained in the Dirac neutrino Yukawa matrix, as well as the two scales
$M_{0}$ and $\mu_{0}$. The couplings $y_{f}$ are fixed by the light neutrino
masses, the angle $\theta_{L}$ is adjusted such that the lepton mixing angles
are accommodated well and the two scales $M_{0}$ and $\mu_{0}$ determine the
mass of the heavy sterile states and the size of lepton number breaking,
respectively. The angle $\theta_{R}$ is a free parameter and varied in its
entire range, $0\leq\theta_{R}\leq 2\,\pi$.

We have focussed on the BRs of the cLFV decays $\mu\to e\gamma$ and $\mu\to
3\,e$ and $\mu-e$ conversion in nuclei, both analytically and numerically. To
study these processes numerically, we have chosen examples for each of the
cases which can accommodate the lepton mixing angles well. While current
limits on these processes do not constrain the studied parameter space, future
bounds can exclude part of it. In particular, the expected limit on $\mu\to
3\,e$ from Mu3E Phase 2 and the prospective bounds on $\mu-e$ conversion in
aluminium from COMET and Mu2e set relevant restrictions. If these are passed,
the size of $\mathrm{BR}(\mu\to e\gamma)$ turns out to be about two orders of
magnitude smaller than the future limit from MEG II.

We have also analysed the signal strength of four cLFV tau lepton decays,
$\tau\to\mu\gamma$, $\tau\to e\gamma$, $\tau\to 3\,\mu$ and $\tau\to 3\,e$. If
the lepton mixing angles are accommodated well and the future bounds on the
three most relevant $\mu-e$ transitions are passed, the BRs of these decays
are rather suppressed, below $10^{-14}$, meaning these are much lower than the
expected limits from Belle II.

We have observed that the BRs and CR reveal different dependences on the free
angle $\theta_{R}$, depending on the choice of the case, Case 1) through Case
3 b.1), and the parameters determining the residual symmetries: for Case 1)
they always have such a dependence, while for Case 2) it is decisive whether
the parameter $t$ is even or odd, i.e. for $t$ even there is no such
dependence, whereas for $t$ odd a dependence is observed. For Case 3 a) and
Case 3 b.1) we have found that for the parameters $m$ and $s$ both even or
both odd no dependence exists, while for one of them being even and the other
one odd a dependence similar to the one observed for Case 1) is seen.
Nevertheless, it is usually milder for Case 3 a) than for the other cases.

The phenomenology of this option, option 2, is very different from that of the
already studied option 1 [45], where only the Majorana mass matrix $\mu_{S}$
of the gauge singlets $S_{j}$, $j=1,2,3$, carries non-trivial flavour
structure in the neutral lepton sector, since in the case of option 1 all cLFV
signals are highly suppressed. For this reason, it can be interesting to
consider further options and variants of this scenario.

## Acknowledgements

We thank Miguel G. Folgado for help with the local computer cluster. This work
is supported by the Spanish MINECO through the Ramón y Cajal programme
RYC2018-024529-I, the FPI fellowship PRE2021-098730, by the national grant
PID2020-113644GB-I00, by the Generalitat Valenciana through PROMETEO/2021/083
as well as by the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement No. 860881
(HIDDe$\nu$ network) and under the Marie Skłodowska-Curie Staff Exchange grant
agreement No. 101086085 (ASYMMETRY).

##  Appendix A Basics of group theory of $\Delta(3\,n^{2})$ and
$\Delta(6\,n^{2})$

For completeness, the basics of the discrete groups $\Delta(3\,n^{2})$ and
$\Delta(6\,n^{2})$ are presented. The groups $\Delta(3\,n^{2})$, $n\geq 2$
integer, can be described with three generators $a$, $c$ and $d$ that satisfy
the relations

$a^{3}=e\;,\;\;c^{n}=e\;,\;\;d^{n}=e\;,\;\;c\,d=d\,c\;,\;\;a\,c\,a^{-1}=c^{-1}d^{-1}\;,\;\;a\,d\,a^{-1}=c\;,$
(95)

where $e$ is the neutral element of the group [31]. As detailed in [32], the
groups $\Delta(6\,n^{2})$, $n\geq 2$ integer, can be generated by adding a
further generator $b$. The relations involving this generator read

$b^{2}=e\;,\;\;(a\,b)^{2}=e\;,\;\;b\,c\,b^{-1}=d^{-1}\;,\;\;b\,d\,b^{-1}=c^{-1}\,.$
(96)

For the trivial representation ${\bf 1}$ all elements of the group are
represented by the character $1$. The representation matrices of the
generators, $a({\bf 3})$, $b({\bf 3})$, $c({\bf 3})$ and $d({\bf 3})$, are
taken in one irreducible, faithful, complex three-dimensional representation,
called ${\bf 3}$, of $\Delta(6\,n^{2})$ as202020The similarity transformation
$U=\frac{1}{\sqrt{3}}\,\left(\begin{array}[]{ccc}1&1&1\\\
\omega^{2}&\omega&1\\\ \omega&\omega^{2}&1\end{array}\right)$ has to be
applied to the generators given in eq. (103) in order to obtain the form of
the representation matrices as found in [32].

$\displaystyle a({\bf 3})=\left(\begin{array}[]{ccc}1&0&0\\\ 0&\omega&0\\\
0&0&\omega^{2}\end{array}\right)\;\;,\;\;b({\bf
3})=\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&\omega^{2}\\\
0&\omega&0\end{array}\right)\;\;,$ (103) $\displaystyle c({\bf
3})=\frac{1}{3}\,\left(\begin{array}[]{ccc}1+2\cos\phi_{n}&1-\cos\phi_{n}-\sqrt{3}\sin\phi_{n}&1-\cos\phi_{n}+\sqrt{3}\sin\phi_{n}\\\
1-\cos\phi_{n}+\sqrt{3}\sin\phi_{n}&1+2\cos\phi_{n}&1-\cos\phi_{n}-\sqrt{3}\sin\phi_{n}\\\
1-\cos\phi_{n}-\sqrt{3}\sin\phi_{n}&1-\cos\phi_{n}+\sqrt{3}\sin\phi_{n}&1+2\cos\phi_{n}\end{array}\right)$
(107)

with $\omega=e^{\frac{2\pi i}{3}}$ and $\phi_{n}=\frac{2\pi}{n}$. The form of
$d({\bf 3})$ can be calculated using $d({\bf 3})=a({\bf 3})^{2}c({\bf
3})a({\bf 3})$. We note that the representation ${\bf 3}$ corresponds to ${\bf
3}_{\bf{1}\,(1)}$ in the nomenclature of [32].

If the index $n$ of the group $\Delta(6\,n^{2})$ is even, there exists an
irreducible, unfaithful, real three-dimensional representation ${\bf
3^{\prime}}$. The form of the representation matrices $a({\bf 3^{\prime}})$,
$b({\bf 3^{\prime}})$ and $c({\bf 3^{\prime}})$ is then

$a({\bf 3^{\prime}})=a({\bf 3})\;\;,\;\;b({\bf 3^{\prime}})=b({\bf
3})\;\;,\;\;c({\bf
3^{\prime}})=\frac{1}{3}\,\left(\begin{array}[]{ccc}-1&2&2\\\ 2&-1&2\\\
2&2&-1\end{array}\right)$ (108)

and $d({\bf 3^{\prime}})=a({\bf 3^{\prime}})^{2}c({\bf 3^{\prime}})a({\bf
3^{\prime}})$. Note that they do not depend on the index $n$ of the group. The
group generated by the representation matrices $g({\bf 3^{\prime}})$ has 24
elements and corresponds to the group $\Delta(24)$. The representation ${\bf
3^{\prime}}$ together with the one generated by the representation matrices
$a({\bf 3^{\prime}})$, $c({\bf 3^{\prime}})$, $d({\bf 3^{\prime}})$ and
$-b({\bf 3^{\prime}})$ are the only real three-dimensional representations in
a generic group $\Delta(6\,n^{2})$ with even $n$ and $3\nmid n$. They are
denoted as ${\bf 3}_{\bf{1}\,(n/2)}$ and ${\bf 3}_{\bf{2}\,(n/2)}$ in [32],
respectively.

##  Appendix B Relevant matrices $\Omega({\bf 3})$, $\Omega({\bf 3^{\prime}})$
and $R_{ij}(\theta)$

As supplementary material the different forms of the matrices $\Omega({\bf
3})$ and $\Omega({\bf 3^{\prime}})$ for the cases, Case 1) through Case 3
b.1), as well as the form of the rotation matrices $R_{ij}(\theta_{L})$ and
$R_{kl}(\theta_{R})$ for each combination are collected in this appendix; for
further details see [47].

For Case 1) the following form of $\Omega({\bf 3})$ and $\Omega({\bf
3^{\prime}})$ is used

$\displaystyle\Omega(s)({\bf
3})=e^{i\,\phi_{s}}\,U_{\mbox{\scriptsize{TB}}}\,\left(\begin{array}[]{ccc}1&0&0\\\
0&e^{-3\,i\,\phi_{s}}&0\\\ 0&0&-1\end{array}\right)\;,$ (112)
$\displaystyle\Omega(s\,\mbox{even})({\bf
3^{\prime}})=U_{\mbox{\scriptsize{TB}}}\;\;\mbox{and}\;\;\Omega(s\,\mbox{odd})({\bf
3^{\prime}})=U_{\mbox{\scriptsize{TB}}}\,\left(\begin{array}[]{ccc}i&0&0\\\
0&1&0\\\ 0&0&i\end{array}\right)\;,$ (116)

where the matrix $U_{\mbox{\scriptsize{TB}}}$ encodes tri-bimaximal (TB)
mixing,

$U_{\mbox{\scriptsize{TB}}}=\left(\begin{array}[]{ccc}\sqrt{2/3}&\sqrt{1/3}&0\\\
-\sqrt{1/6}&\sqrt{1/3}&\sqrt{1/2}\\\
-\sqrt{1/6}&\sqrt{1/3}&-\sqrt{1/2}\end{array}\right)\;,$ (117)

and $\phi_{s}$ is defined as $\phi_{s}=\frac{\pi\,s}{n}$. The rotation
matrices $R_{ij}(\theta_{L})$ and $R_{kl}(\theta_{R})$ read

$R_{13}(\theta_{L})=\left(\begin{array}[]{ccc}\cos\theta_{L}&0&\sin\theta_{L}\\\
0&1&0\\\
-\sin\theta_{L}&0&\cos\theta_{L}\end{array}\right)\;\;\mbox{and}\;\;R_{13}(\theta_{R})=\left(\begin{array}[]{ccc}\cos\theta_{R}&0&\sin\theta_{R}\\\
0&1&0\\\ -\sin\theta_{R}&0&\cos\theta_{R}\end{array}\right)\;.$ (118)

For Case 2) we use as matrices $\Omega({\bf 3})$ and $\Omega({\bf
3^{\prime}})$

$\displaystyle\Omega(s,t)({\bf 3})=\Omega(u,v)({\bf
3})=e^{i\phi_{v}/6}\,U_{\mbox{\scriptsize{TB}}}\,R_{13}\left(-\frac{\phi_{u}}{2}\right)\,\left(\begin{array}[]{ccc}1&0&0\\\
0&e^{-i\phi_{v}/2}&0\\\ 0&0&-i\end{array}\right)\;,$ (122)
$\displaystyle\Omega(s\,\mbox{even},t\,\mbox{even})({\bf
3^{\prime}})=U_{\mbox{\scriptsize{TB}}}\,\left(\begin{array}[]{ccc}1&0&0\\\
0&1&0\\\ 0&0&i\end{array}\right)\;,$ (126)
$\displaystyle\Omega(s\,\mbox{even},t\,\mbox{odd})({\bf
3^{\prime}})=e^{-i\pi/4}\,U_{\mbox{\scriptsize{TB}}}\,R_{13}\left(\frac{\pi}{4}\right)\,\left(\begin{array}[]{ccc}-i&0&0\\\
0&e^{-i\pi/4}&0\\\ 0&0&1\end{array}\right)\;,$ (130)
$\displaystyle\Omega(s\,\mbox{odd},t\,\mbox{even})({\bf
3^{\prime}})=U_{\mbox{\scriptsize{TB}}}\,\left(\begin{array}[]{ccc}i&0&0\\\
0&1&0\\\ 0&0&1\end{array}\right)\;,$ (134)
$\displaystyle\Omega(s\,\mbox{odd},t\,\mbox{odd})({\bf
3^{\prime}})=e^{-3\,i\,\pi/4}\,U_{\mbox{\scriptsize{TB}}}\,R_{13}\left(\frac{\pi}{4}\right)\,\left(\begin{array}[]{ccc}-i&0&0\\\
0&e^{i\,\pi/4}&0\\\ 0&0&1\end{array}\right)\;.$ (138)

Here, $\phi_{u}$ and $\phi_{v}$ are $\phi_{u}=\frac{\pi\,u}{n}$ and
$\phi_{v}=\frac{\pi\,v}{n}$, respectively. The rotation matrices