#  Dynamical scoto-seesaw mechanism with gauged $B-L$

Julio Leite<EMAIL_ADDRESS>AHEP Group, Institut de Física Corpuscular
– C.S.I.C./Universitat de València, Parc Científic de Paterna.
C/ Catedrático José Beltrán, 2 E-46980 Paterna (Valencia) - Spain  Soumya
Sadhukhan<EMAIL_ADDRESS>Ramakrishna Mission Residential College
(Autonomous), Vivekananda Centre for Research,
Narendrapur, Kolkata, India  AHEP Group, Institut de Física Corpuscular –
C.S.I.C./Universitat de València, Parc Científic de Paterna.
C/ Catedrático José Beltrán, 2 E-46980 Paterna (Valencia) - Spain  José W. F.
Valle<EMAIL_ADDRESS>AHEP Group, Institut de Física Corpuscular –
C.S.I.C./Universitat de València, Parc Científic de Paterna.
C/ Catedrático José Beltrán, 2 E-46980 Paterna (Valencia) - Spain

###### Abstract

We propose a dynamical scoto-seesaw mechanism using a gauged $B-L$ symmetry.
Dark matter is reconciled with neutrino mass generation, in such a way that
the atmospheric scale arises a la seesaw, while the solar scale is scotogenic,
arising radiatively from the exchange of “dark” states. This way we “explain”
the solar-to-atmospheric scale ratio. The TeV-scale seesaw mediator and the
two dark fermions carry different $B-L$ charges. Dark matter stability follows
from the residual matter parity that survives $B-L$ breaking. Besides having
collider tests, the model implies sizeable charged lepton flavour violating
(cLFV) phenomena, including Goldstone boson emission processes.

##  I Introduction

Despite the great success of the Standard Model (SM) [1], new physics is
required in order to account for the existence of neutrino masses [2, 3] as
well as dark matter [4]. A popular paradigm for neutrino mass generation is
the seesaw mechanism, while weakly interacting massive particle (WIMP) dark
matter candidates constitute a paradigm for explaining cold dark matter. Even
taking these paradigms for granted there are many ways to realize either. A
particularly interesting possibility is provided by the so-called scotogenic
approach [5, 6] in which WIMP dark matter mediates neutrino mass generation.
In the simplest schemes all neutrino masses arise at the one-loop level, with
a common overall scale, modulated only by Yukawa couplings.

Rather than invoking these paradigms separately, here we suggest a dynamical
mechanism to realize naturally the scoto-seesaw scenario [7, 8] that
reconciles these two paradigms together 111There are also attempts to realize
the scoto-seesaw picture by combining with family symmetries [9, 10, 11, 12].,
opening the possibility of having a loop-suppressed solar-to-atmospheric scale
ratio.

We first note that the $U(1)_{B-L}$ symmetry arises automatically in the SM
and is closely related to neutrino masses. However, the presence of non-
vanishing anomaly coefficients forbids us to promote $U(1)_{B-L}$ to a local
gauge symmetry. Adding 3 lepton singlets $\nu_{iR}\sim-1$ cancels the anomaly
coefficients, allowing for a local realisation. Then, Dirac neutrino mass
entries are generated, so that a $\nu_{iR}$-mediated (type-I) seesaw mechanism
can be triggered by allowing for Majorana mass terms, that break $B-L$ by 2
units. In such “canonical” construction the seesaw-mediating $\nu_{iR}$ carry
identical charges, so that all neutrino masses become proportional to a single
energy scale. As a result, this fails to account for the observed hierarchy
$\Delta m^{2}_{sol}/\Delta m^{2}_{atm}$ [13].

However, an alternative anomaly-free $U(1)_{B-L}$ can be obtained if instead
we introduce three neutral fermions with $B-L$ charges
$(f_{1R},f_{2R},N_{R})\sim(-4,-4,5)$ [14, 15].

We show next that, thanks to their unequal charges, $f_{aR}$ and $N_{R}$
couple differently to the active neutrinos and may trigger different mass
generation mechanisms, providing a natural dynamical setup for the scoto-
seesaw mechanism and an explanation for the smallness of the ratio $\Delta
m^{2}_{sol}/\Delta m^{2}_{atm}$. The neutrino-mass mediators $f_{aR}$ become
part of a dark sector whose stability originates from an unbroken matter
parity that survives the breaking of the gauged $U(1)_{B-L}$ symmetry.

Besides having interesting dark matter features and collider phenomenology, an
interesting implication of our model concerns charged lepton flavour
violation. Indeed, a characteristic feature to notice is the presence of
sizeable cLFV processes involving Goldstone boson emission [16, 17, 18], in
addition to conventional cLFV processes, such as those involving photon
emission, for instance $\mu\to e\gamma$.

##  II Gauged $B-L$ extension of the Standard Model

Here we describe the essential features of our proposal. The leptons and
scalars present in the model and their symmetry properties are shown in Table
1. In addition to the SM leptons, we introduce $f_{aR}$ ($a=1,2$) and $N_{R}$
– singlets under the SM group but charged under $U(1)_{B-L}$. Besides a SM-
like Higgs doublet $H$, the scalar sector contains two extra $SU(2)_{L}$
doublets ($\Phi$ and $\eta$) and four singlets ($\varphi_{1,2,3},\sigma$).
Note that all of the new fields are charged under $U(1)_{B-L}$. The specific
roles of the extra fields will become clear in the next sections when we
discuss neutrino mass generation, dark matter, and how these are linked to
each other in our model.

|  Fields |  $SU(2)_{L}\otimes U(1)_{Y}$ |  $U(1)_{B-L}$ $\to$ $M_{P}$
---|---|---|---
Leptons | $L_{iL}$ | ($\mathbf{2},{-1/2}$) |  ${-1}$ $\to$ $+1$
$e_{iR}$ | ($\mathbf{1},{-1}$) |  $-1$ $\to$ $+1$
$N_{R}$ | ($\mathbf{1},{0}$) |  $5$ $\to$ $+1$
Scalars | $H$ | ($\mathbf{2},{1/2}$) |  ${0}$ $\to$ $+1$
$\Phi$ | ($\mathbf{2},{1/2}$) |  ${6}$ $\to$ $+1$
$\varphi_{1}$ | ($\mathbf{1},{0}$) |  ${10}$ $\to$ $+1$
$\varphi_{2}$ | ($\mathbf{1},{0}$) |  ${8}$ $\to$ $+1$
| $\varphi_{3}$ | ($\mathbf{1},{0}$) |  ${2}$ $\to$ $+1$
Dark | $f_{aR}$ | ($\mathbf{1},{0}$) |  $-4$ $\to$ $-1$
$\eta$ | ($\mathbf{2},{1/2}$) |  ${-3}$ $\to$ $-1$
$\sigma$ | ($\mathbf{1},{0}$) |  ${9}$ $\to$ $-1$
Table 1:  Lepton, scalar and dark sector fields with their symmetry
transformation properties ($i=1,2,3$ and $a=1,2$).

First we note that, with this field content, we ensure that $B-L$ is free from
anomalies and hence can be promoted to local gauge symmetry. To see this,
first recall that in the SM the conventional $B-L$ appears as an automatic
global symmetry which is anomalous, due to the non-vanishing of the following
triangle anomaly coefficients

$A_{1}^{\mbox{SM}}[U(1)_{B-L}^{3}]=-3\quad\mbox{and}\quad
A_{2}^{\mbox{SM}}[\mbox{Grav}^{2}U(1)_{B-L}]=-3.$ (1)

By extending the SM with the fermions $f_{aR}$ and $N_{R}$, as given in Table
1, these coefficients vanish exactly [14, 15]

$\displaystyle A_{1}[U(1)_{B-L}^{3}]$ $\displaystyle=$ $\displaystyle
A_{1}^{\mbox{SM}}[U(1)_{B-L}^{3}]+A_{1}^{\mbox{new}}[U(1)_{B-L}^{3}]=-3-[(-4)^{3}+(-4)^{3}+(5)^{3}]=0,$
(2) $\displaystyle A_{2}[\mbox{Grav}^{2}U(1)_{B-L}]$ $\displaystyle=$
$\displaystyle
A_{2}^{\mbox{SM}}[\mbox{Grav}^{2}U(1)_{B-L}]+A_{2}^{\mbox{new}}[\mbox{Grav}^{2}U(1)_{B-L}]=-3-(-4-4+5)=0.$

Needless to say, since these new fermions are singlets under the SM gauge
group, they do not contribute to the other anomaly coefficients, i.e.
$U(1)_{Y}^{2}U(1)_{B-L}$, $U(1)_{Y}U(1)_{B-L}^{2}$, $SU(2)_{L}^{2}U(1)_{B-L}$
and $SU(3)_{C}^{2}U(1)_{B-L}$ which, therefore, remain zero.

With the fields in Table 1, we can write down the most general renormalisable
Yukawa Lagrangian as 222 For convenience, we have omitted the Yukawa
interactions involving quarks and $H$, which are standard.

$\displaystyle-\mathcal{L}_{Y}$ $\displaystyle=$ $\displaystyle
Y^{H}_{ij}\overline{L}_{iL}He_{jR}+Y^{\Phi}_{i}\overline{L}_{iL}\tilde{\Phi}N_{R}+Y^{\eta}_{ia}\overline{L}_{iL}\tilde{\eta}f_{aR}+\frac{Y^{N}}{2}\varphi_{1}^{*}\overline{(N_{R})^{c}}N_{R}+\frac{Y^{f}_{a}}{2}\varphi_{2}\overline{(f_{aR})^{c}}f_{aR}+h.c.,$
(3)

Analogous to the SM case, once the Higgs doublet $H$ acquires its vacuum
expectation value (VEV), the charged fermions become massive. In contrast,
neutrino mass generation will involve the other scalar bosons, as discussed in
Sec. V.

In the scalar sector, we assume that only neutral fields with even $B-L$
charges acquire VEVs. Thus, a subgroup of $U(1)_{B-L}$ remains conserved after
spontaneous symmetry breaking takes place. The residual symmetry is matter-
parity and can be defined as

$M_{P}=(-1)^{3(B-L)+2s}.$ (4)

Therefore, only $f_{aR}$, $\eta$ and $\sigma$ are odd under $M_{P}$. Due to
$M_{P}$ conservation, the lightest of such fields is stable and, if
electrically neutral, will be our dark matter candidate.

##  III Scalar sector

For convenience, the most general renormalisable scalar potential is separated
in two parts $V=V_{1}+V_{2}$, where

$\displaystyle V_{1}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\left[\mu_{\mathcal{D}_{i}}^{2}\mathcal{D}_{i}^{\dagger}\mathcal{D}_{i}+\lambda_{\mathcal{D}_{i}}(\mathcal{D}_{i}^{\dagger}\mathcal{D}_{i})^{2}\right]+\sum^{i<j}_{i,j}\left[\lambda_{\mathcal{D}_{i}\mathcal{D}_{j}}(\mathcal{D}_{i}^{\dagger}\mathcal{D}_{i})(\mathcal{D}_{j}^{\dagger}\mathcal{D}_{j})+\lambda^{\prime}_{\mathcal{D}_{i}\mathcal{D}_{j}}(\mathcal{D}_{i}^{\dagger}\mathcal{D}_{j})(\mathcal{D}_{j}^{\dagger}\mathcal{D}_{i})\right]$
$\displaystyle+\sum_{k=1}^{4}\left[\mu_{\mathcal{S}_{k}}^{2}S_{k}^{\dagger}\mathcal{S}_{k}+\lambda_{\mathcal{S}_{k}}(\mathcal{S}_{k}^{\dagger}\mathcal{S}_{k})^{2}\right]+\sum^{k<l}_{k,l}\lambda_{\mathcal{S}_{k}\mathcal{S}_{l}}(\mathcal{S}_{k}^{\dagger}\mathcal{S}_{k})(\mathcal{S}_{l}^{\dagger}\mathcal{S}_{l})+\sum_{i,k}\lambda_{\mathcal{D}_{i}\mathcal{S}_{k}}(\mathcal{D}_{i}^{\dagger}\mathcal{D}_{i})(\mathcal{S}_{k}^{\dagger}\mathcal{S}_{k}),$
$\displaystyle-V_{2}$ $\displaystyle=$
$\displaystyle\frac{\mu_{1}}{\sqrt{2}}\,\Phi^{\dagger}\eta\sigma+\frac{\mu_{2}}{\sqrt{2}}\varphi_{1}^{*}\varphi_{2}\varphi_{3}+\lambda_{1}\,\varphi_{1}\varphi_{2}\sigma^{*\,2}+\lambda_{2}\Phi^{\dagger}H\varphi_{2}\varphi_{3}^{*}+\text{h.c.}\,.$
(6)

where $\mathcal{D}_{i}=H,\Phi,\eta$ ($i=1,2,3$) represent the $SU(2)_{L}$
doublets and $\mathcal{S}_{k}=\varphi_{1},\varphi_{2},\varphi_{3},\sigma$
($k=1,2,3,4$) the singlets; while $\mu_{H}^{2}$ and $\mu_{\varphi_{i}}^{2}$
are negative. The first part, $V_{1}$, contains only self-adjoint operators,
while the second, $V_{2}$, includes all non-self-adjoint operators allowed by
the symmetries.

The scalars are decomposed as follows

$\displaystyle H$ $\displaystyle=$ $\displaystyle\begin{pmatrix}H^{+}\\\
\frac{1}{\sqrt{2}}(v_{H}+S_{H}+iA_{H})\end{pmatrix},\quad\Phi=\begin{pmatrix}\Phi^{+}\\\
\frac{1}{\sqrt{2}}(v_{\Phi}+S_{\Phi}+iA_{\Phi})\end{pmatrix},\quad\eta=\begin{pmatrix}\eta^{+}\\\
\frac{1}{\sqrt{2}}(S_{\eta}+iA_{\eta})\end{pmatrix},$ (7)
$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(S_{\sigma}+iA_{\sigma}),\quad\varphi_{i}=\frac{1}{\sqrt{2}}(v_{\varphi_{i}}+S_{\varphi_{i}}+iA_{\varphi_{i}}),\quad\text{with}\,\,i=1,2,3,$

and we assume that the breaking of $U(1)_{B-L}$ takes place well above the
electro-weak scale: $v_{\varphi_{i}}^{2}\gg
v_{EW}^{2}=v_{H}^{2}+v_{\Phi}^{2}$. Notice that only the electrically neutral
scalars with even $U(1)_{B-L}$ charges acquire VEVs, so that matter-parity
($M_{P}$), defined in Eq. (4), remains conserved.

After replacing the field decompositions above into the potential, we obtain
the following tadpole equations

$\displaystyle
v_{H}\left[2\mu_{H}^{2}+2\lambda_{H}v_{H}^{2}+\lambda_{H\varphi_{1}}v_{\varphi_{1}}^{2}+\lambda_{H\varphi_{2}}v_{\varphi_{2}}^{2}+\lambda_{H\varphi_{3}}v_{\varphi_{3}}^{2}+v_{\Phi}^{2}(\lambda_{H\Phi}+\lambda_{H\Phi}^{\prime})\right]-\lambda_{2}v_{\Phi}v_{\varphi_{2}}v_{\varphi_{3}}$
$\displaystyle=$ $\displaystyle 0\,,$ (8) $\displaystyle
v_{\Phi}\left[2\mu_{\Phi}^{2}+2\lambda_{\Phi}v_{\Phi}^{2}+\lambda_{\Phi\varphi_{1}}v_{\varphi_{1}}^{2}+\lambda_{\Phi\varphi_{2}}v_{\varphi_{2}}^{2}+\lambda_{\Phi\varphi_{3}}v_{\varphi_{3}}^{2}+v_{H}^{2}(\lambda_{H\Phi}+\lambda_{H\Phi}^{\prime})\right]-\lambda_{2}v_{H}v_{\varphi_{2}}v_{\varphi_{3}}$
$\displaystyle=$ $\displaystyle 0\,,$ $\displaystyle
v_{\varphi_{1}}\left(2\mu_{\varphi_{1}}^{2}+2\lambda_{\varphi_{1}}v_{\varphi_{1}}^{2}+\lambda_{\Phi\varphi_{1}}v_{\Phi}^{2}+\lambda_{\varphi_{1}\varphi_{2}}v_{\varphi_{2}}^{2}+\lambda_{\varphi_{1}\varphi_{3}}v_{\varphi_{3}}^{2}+\lambda_{H\varphi_{1}}v_{H}^{2}\right)-\mu_{2}v_{\varphi_{2}}v_{\varphi_{3}}$
$\displaystyle=$ $\displaystyle 0\,,$ $\displaystyle
v_{\varphi_{2}}\left(2\mu_{\varphi_{2}}^{2}+2\lambda_{\varphi_{2}}v_{\varphi_{2}}^{2}+\lambda_{H\varphi_{2}}v_{H}^{2}+\lambda_{\Phi\varphi_{2}}v_{\Phi}^{2}+\lambda_{\varphi_{1}\varphi_{2}}v_{\varphi_{1}}^{2}\right)-v_{\varphi_{3}}\left(\lambda_{2}v_{H}v_{\Phi}+\mu_{2}v_{\varphi_{1}}\right)$
$\displaystyle=$ $\displaystyle 0\,,$ $\displaystyle
v_{\varphi_{3}}\left(2\mu_{\varphi_{3}}^{2}+2\lambda_{\varphi_{3}}v_{\varphi_{3}}^{2}+\lambda_{H\varphi_{3}}v_{H}^{2}+\lambda_{\Phi\varphi_{3}}v_{\Phi}^{2}+\lambda_{\varphi_{1}\varphi_{3}}v_{\varphi_{1}}^{2}\right)-v_{\varphi_{2}}\left(\lambda_{2}v_{H}v_{\Phi}+\mu_{2}v_{\varphi_{1}}\right)$
$\displaystyle=$ $\displaystyle 0\,.$

In the limit where $\mu_{\Phi}$ is much larger than the other mass scales in
the model, the second tadpole equation leads to the induced VEV

$v_{\Phi}\simeq\frac{\lambda_{2}v_{H}v_{\varphi_{2}}v_{\varphi_{3}}}{2\mu_{\Phi}^{2}}\equiv
v_{H}\epsilon\ll v_{H}.$ (9)

Therefore, the induced VEV of the doublet $\Phi$ is suppressed as in the type-
II seesaw mechanism [19, 20], and also models with a leptophilic Higgs doublet
[21, 22], a natural example of which emerges naturally within the linear
seesaw mechanism [23, 24, 25].

Due to matter-parity conservation, that survives as a remnant symmetry, fields
with different $M_{P}$ charges remain unmixed. Therefore, we can separate the
scalar spetrum into a sector with fields transforming trivially under $M_{P}$,
the $M_{P}$-even sector, and another with those that do not, the $M_{P}$-odd
sector. Moreover, for simplicity, we assume that all the VEVs as well as the
couplings in $V$ are real, so that CP is conserved.

Starting with the $M_{P}$-even sector, we have a $5\times 5$ squared-mass
matrix for the CP-even scalars
$(S_{H},S_{\Phi},S_{\varphi_{1}},S_{\varphi_{2}},S_{\varphi_{3}})$, given in
Eqs. (46) and (47) of the appendix. The associated physical states
$S_{1,2,3,4,5}$ all become massive, and one of them is identified as the $125$
GeV Higgs boson, $S_{1}\equiv h$ discovered at CERN [26, 27]. The other
scalars are expected to be heavier, three of them, $S_{2,3,4}$, with masses
proportional to the $B-L$-breaking scale $v_{\varphi_{i}}$. The mass of the
heaviest state, $S_{5}$, will be governed by the largest scale in the model,
i.e. $\mu_{\Phi}$. Thus, in the limit of interest, $h\simeq S_{H}$ and
$S_{5}\simeq S_{\Phi}$.

Concerning the CP-odd fields
$(A_{H},A_{\Phi},A_{\varphi_{1}},A_{\varphi_{2}},A_{\varphi_{3}})$, there is
another $5\times 5$ squared-mass matrix, given in Eq. (53) of the appendix.
Three of the mass eigenstates are massless, two of them can be written as

$\displaystyle G_{Z}$ $\displaystyle=$ $\displaystyle
v_{EW}^{-1}\left(v_{H}A_{H}+v_{\Phi}A_{\Phi}\right)$ (10) $\displaystyle
G_{Z^{\prime}}$ $\displaystyle=$ $\displaystyle
v_{BL}^{-1}\left(6v_{\Phi}A_{\Phi}+10v_{\varphi_{1}}A_{\varphi_{1}}+8v_{\varphi_{2}}A_{\varphi_{2}}+2v_{\varphi_{3}}A_{\varphi_{3}}\right),$
$\displaystyle\mbox{with}\quad v_{BL}$ $\displaystyle=$
$\displaystyle\sqrt{6^{2}v_{\Phi}^{2}+10^{2}v_{\varphi_{1}}^{2}+8^{2}v_{\varphi_{2}}^{2}+2^{2}v_{\varphi_{3}}^{2}}.$
(11)

and are absorbed by the neutral gauge bosons $Z$ and $Z^{\prime}$.

On the other hand, two states, $A_{1}$ and $A_{2}$, become massive 333The
exact mass expressions are given in Eq. (54) of the appendix. and, in the
limit $v_{\varphi}\equiv v_{\varphi_{i}}\gg v_{H}\gg v_{\Phi}$, their masses
can be written as

$m^{2}_{A_{1}}\simeq\frac{3\mu_{2}v_{\varphi}}{2}\quad\mbox{and}\quad
m^{2}_{A_{2}}\simeq\mu_{\Phi}^{2}.$ (12)

Finally, there is a remaining massless field, a physical Nambu-Goldstone
boson, $G$, analogous to the Majoron [28, 29, 30]. In order to obtain its
profile we make use of the fact that it must be orthogonal to would-be
Goldstone bosons $G_{Z,Z^{\prime}}$, as well as to the massive $A_{1,2}$
fields, see appendix, Eq. (A). Within the same limit as above, we find that
this physical Goldstone lies mainly along the $SU(2)_{L}$ singlet directions,
while its projections along the doublets are suppressed by VEV ratios

$G\simeq\frac{1}{\sqrt{14}}\left(5\frac{v_{\Phi}^{2}}{v_{H}v_{\varphi}}A_{H}-5\frac{v_{\Phi}}{v_{\varphi}}A_{\Phi}+A_{\varphi_{1}}-2A_{\varphi_{2}}+3A_{\varphi_{3}}\right).$
(13)

As we will comment below in Sec. VI there are stringent limits on the flavour-
conserving Goldstone boson couplings to charged leptons, mainly electrons.
Moreover, as discussed in Sec. VII, there are also bounds on the flavour-
violating Goldstone boson couplings to charged leptons.

Turning now to the charged fields, $(H^{\pm},\Phi^{\pm})$, we have

$\displaystyle
M^{2}_{\pm}=\frac{1}{2}\left(\begin{array}[]{cc}\lambda_{2}\frac{v_{\varphi_{2}}v_{\varphi_{3}}v_{\Phi}}{v_{H}}-\lambda_{H\Phi}^{\prime}v_{\Phi}^{2}&-\lambda_{2}v_{\varphi_{2}}v_{\varphi_{3}}+\lambda_{H\Phi}^{\prime}v_{H}v_{\Phi}\\\
-\lambda_{2}v_{\varphi_{2}}v_{\varphi_{3}}+\lambda_{H\Phi}^{\prime}v_{H}v_{\Phi}&\lambda_{2}\frac{v_{\varphi_{2}}v_{\varphi_{3}}v_{H}}{v_{\Phi}}-\lambda_{H\Phi}^{\prime}v_{H}^{2}\\\
\end{array}\right).$ (16)

After diagonalising the above matrix we find that one eigenstate is massless
(absorbed by the $W^{\pm}$ gauge boson), whereas the other, $\phi^{\pm}$, gets
a mass

$\displaystyle\phi^{\pm}=\frac{1}{\sqrt{v_{H}^{2}+v_{\Phi}^{2}}}\left(-v_{\Phi}H^{\pm}+v_{H}\Phi^{\pm}\right)\quad\text{with}\quad
m_{\phi^{\pm}}^{2}=\frac{(\lambda_{2}v_{\varphi_{2}}v_{\varphi_{3}}-\lambda_{H\Phi}^{\prime}v_{H}v_{\Phi})\left(v_{H}^{2}+v_{\Phi}^{2}\right)}{2v_{H}v_{\Phi}}\,.$
(17)

Notice that the dominant component of $\phi^{\pm}$ lies along the doublet
$\Phi$, which gets a large mass $m_{\phi^{\pm}}\simeq\mu_{\Phi}$, see Eq. (9).

Turning now to the $M_{P}$-odd sector, the CP-even and CP-odd scalars have the
following mass matrices, respectively,

$\displaystyle M^{2}_{a(s)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{cc}X&\kappa_{a(s)}\mu_{1}v_{\Phi}\\\
\kappa_{a(s)}\mu_{1}v_{\Phi}&Y_{a(s)}\\\ \end{array}\right),$ (20)
$\displaystyle\text{with}\quad X$ $\displaystyle=$ $\displaystyle
2\mu_{\eta}^{2}+(\lambda_{H\eta}+\lambda_{H\eta}^{\prime})v_{H}^{2}+(\lambda_{\Phi\eta}+\lambda_{\Phi\eta}^{\prime})v_{\Phi}^{2}+\lambda_{\eta\varphi_{1}}v_{\varphi_{1}}^{2}+\lambda_{\eta\varphi_{2}}v_{\varphi_{2}}^{2}+\lambda_{\eta\varphi_{3}}v_{\varphi_{3}}^{2},$
$\displaystyle Y_{a(s)}$ $\displaystyle=$ $\displaystyle
2\mu_{\sigma}^{2}+\lambda_{H\sigma}v_{H}^{2}+\lambda_{\Phi\sigma}v_{\Phi}^{2}+\lambda_{\varphi_{1}\sigma}v_{\varphi_{1}}^{2}+\lambda_{\varphi_{2}\sigma}v_{\varphi_{2}}^{2}+\lambda_{\varphi_{3}\sigma}v_{\varphi_{3}}^{2}+\kappa_{a(s)}2\lambda_{1}v_{\varphi_{1}}v_{\varphi_{2}}\,$
$\displaystyle\mbox{and}\quad\kappa_{a(s)}$ $\displaystyle=$
$\displaystyle+1(-1).$

when expressed in the bases $(S_{\eta},S_{\sigma})$ and
$(A_{\eta},A_{\sigma})$. The matrices can be diagonalised by performing the
following rotations

$\displaystyle\begin{pmatrix}a(s)_{1}\\\
a(s)_{2}\end{pmatrix}=\begin{pmatrix}\cos{\theta_{a(s)}}&-\sin{\theta_{a(s)}}\\\
\sin{\theta_{a(s)}}&\cos{\theta_{a(s)}}\end{pmatrix}\begin{pmatrix}A(S)_{\eta}\\\
A(S)_{\sigma}\end{pmatrix},\quad\mbox{where}\quad\tan(2\theta_{a(s)})=\frac{\kappa_{a(s)}2\mu_{1}v_{\Phi}}{Y_{a,s}-X}.$
(21)

The corresponding eigenvalues are then given by

$\displaystyle
m^{2}_{a_{i}(s_{i})}=\frac{X+Y_{a(s)}\pm\mbox{sign}(X-Y_{a(s)})\sqrt{(X-Y_{a(s)})^{2}+16\mu_{1}^{2}v_{\Phi}^{2}}}{4}.$
(22)

Finally, the only charged scalar in the dark sector gets the following mass

$\displaystyle
m_{\eta^{\pm}}^{2}=\mu_{\eta}^{2}+\frac{1}{2}\left(\lambda_{H\eta}v_{H}^{2}+\lambda_{\Phi\eta}v_{\Phi}^{2}+\lambda_{\eta\varphi_{1}}v_{\varphi_{1}}^{2}+\lambda_{\eta\varphi_{2}}v_{\varphi_{2}}^{2}+\lambda_{\eta\varphi_{3}}v_{\varphi_{3}}^{2}\right)\,.$
(23)

##  IV Gauge sector: $Z-Z^{\prime}$ mixing

As usual, to find the gauge sector spectrum, we expand the covariant
derivative terms in the scalar sector, i.e.
$\mathcal{L}\supset(D_{\mu}\phi_{i})^{\dagger}(D^{\mu}\phi^{i})$, where
$\phi_{i}$ denote the scalars in the model and 444Given that the model
contains “large” $B-L$ charges, $q_{BL}$, to ensure perturbativity of the
associated gauge interactions, one may adopt the conservative limit
$q_{BL}^{max}\times g^{\prime}=10\times g^{\prime}\lesssim 1$, leading to
$g^{\prime}\lesssim 0.1$.

$D_{\mu}\phi_{i}=\left(\partial_{\mu}-ig_{L}\frac{\sigma_{a}}{2}W^{a}_{\mu}-ig_{Y}YB^{Y}_{\mu}-ig^{\prime}q_{BL}B^{\prime}_{\mu}\right)\phi_{i}.$
(24)

When the scalars acquire VEVs, the neutral gauge bosons acquire the following
squared-mass matrix, written in the basis
$(W^{3}_{\mu},B^{Y}_{\mu},B^{\prime}_{\mu}),$555We are assuming, for
simplicity, that the kinetic mixing between the Abelian fields vanishes.

$M^{2}=\frac{1}{4}\left(\begin{array}[]{ccc}g_{L}^{2}v_{EW}^{2}&-g_{L}g_{Y}v_{EW}^{2}&-12g_{L}g^{\prime}v_{\Phi}^{2}\\\
-g_{L}g_{Y}v_{EW}^{2}&g_{Y}^{2}v_{EW}^{2}&12g^{\prime}g_{Y}v_{\Phi}^{2}\\\
-12g_{L}g^{\prime}v_{\Phi}^{2}&12g^{\prime}g_{Y}v_{\Phi}^{2}&4g^{\prime
2}v_{\varphi_{1}}^{2}\alpha\\\ \end{array}\right),$ (25)

where
$\alpha=(2/v_{\varphi_{1}})^{2}\left(9v_{\Phi}^{2}+25v_{\varphi_{1}}^{2}+16v_{\varphi_{2}}^{2}+v_{\varphi_{3}}^{2}\right)$.
In order to determine the mixing among the fields, we diagonalise this matrix
in two steps. First, we single out the photon ($A_{\mu}$) by making use of the
mixing matrix $R_{1}$, and then we diagonalise the resulting $Z-Z^{\prime}$
matrix with the help of $R_{2}$, namely

$R=R_{2}\times R_{1}=\left(\begin{array}[]{ccc}1&0&0\\\
0&\cos{\delta}&-\sin{\delta}\\\
0&\sin{\delta}&\cos{\delta}\end{array}\right)\left(\begin{array}[]{ccc}\frac{g_{Y}}{g}&\frac{g_{L}}{g}&0\\\
-\frac{g_{L}}{g}&\frac{g_{Y}}{g}&0\\\ 0&0&1\\\
\end{array}\right),\quad\mbox{with}\quad\tan(2\delta)=\frac{12gg^{\prime}v_{\Phi}^{2}}{4g^{\prime
2}v_{\varphi_{1}}^{2}\alpha-g^{2}v_{EW}^{2}},$ (26)

and $g^{2}=g_{L}^{2}+g_{Y}^{2}$. It is easy to see that the $Z-Z^{\prime}$
mixing, parametrised by $\delta$, is rather suppressed since
$v_{\Phi}/v_{\varphi_{1}}\ll 1$. Thus, the mass eigenstates are given by

$\displaystyle A_{\mu}$ $\displaystyle=$
$\displaystyle\frac{g_{Y}}{g}W^{3}_{\mu}+\frac{g_{L}}{g}B^{Y}_{\mu},$ (27)
$\displaystyle Z_{\mu}$ $\displaystyle=$
$\displaystyle\frac{\cos{\delta}}{g}\left(g_{L}W^{3}_{\mu}+g_{Y}B^{Y}_{\mu}\right)-\sin{\delta}B^{\prime}_{\mu},$
$\displaystyle Z^{\prime}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{\sin{\delta}}{g}\left(g_{L}W_{3\mu}+g_{Y}B_{\mu}\right)+\cos{\delta}B^{\prime}_{\mu},$

and the non-vanishing eigenvalues are

$m^{2}_{Z,Z^{\prime}}=\frac{1}{8}\left[4g^{\prime
2}v_{\varphi_{1}}^{2}\alpha+g^{2}v_{EW}^{2}\mp\sqrt{(4g^{\prime
2}v_{\varphi_{1}}^{2}\alpha-g^{2}v_{EW}^{2})^{2}+(24gg^{\prime}v_{\Phi}^{2})^{2}}\right],$
(28)

which, upon assuming $v_{\varphi}\equiv v_{\varphi_{i}}\gg v_{H}\gg v_{\Phi}$,
give $m_{Z}\simeq gv_{EW}/2$ while $m_{Z^{\prime}}\simeq
13g^{\prime}v_{\varphi}$ is the mass of the heavy ${Z^{\prime}}$.

##  V Dynamical scoto-seesaw mechanism

Neutrino masses arise from the operators in Eqs. (3) and (6) and are generated
by the diagrams in Fig. 1, where the blue and the black legs denote the
exchange of $M_{P}$-even and $M_{P}$-odd/dark fields, respectively.

Fig. 1:  Seesaw and scotogenic contributions to neutrino masses. The fields
which are even under $M_{P}$ are depicted in blue, whereas the black legs
represent the $M_{P}$-odd (dark) fields.

The tree-level contribution, in the basis $(\nu_{iL},(N_{R})^{c})$, leads to
the following mass matrix

$\displaystyle
M^{\nu,N}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&0&Y^{\Phi}_{1}v_{\Phi}\\\
0&0&0&Y^{\Phi}_{2}v_{\Phi}\\\ 0&0&0&Y^{\Phi}_{3}v_{\Phi}\\\
Y^{\Phi}_{1}v_{\Phi}&Y^{\Phi}_{2}v_{\Phi}&Y^{\Phi}_{3}v_{\Phi}&Y^{N}v_{\varphi_{1}}\\\
\end{pmatrix}.$ (29)

Upon diagonalisation we find the seesaw-suppressed mass matrix for the active
neutrinos to be

$M^{\nu(ss)}_{ij}\simeq-\frac{Y^{\Phi}_{i}Y^{\Phi}_{j}}{2}\frac{v_{\Phi}^{2}}{m_{N}},$
(30)

in the limit $v_{\varphi_{i}}\gg v_{H}\gg v_{\Phi}$. Here $m_{N}\simeq
v_{\varphi_{1}}Y^{N}/\sqrt{2}$ is the mass of $N_{R}$ and
$v_{\Phi}=v_{H}\epsilon$ is the small induced VEV defined in Eq. (9). The
light neutrino mass matrix has only one non-vanishing eigenvalue
$\sim-\frac{v_{\Phi}^{2}}{m_{N}}\sum_{i}(Y^{\Phi}_{i})^{2}$.

One sees that, in contrast to the conventional type-I seesaw mechanism, in
which neutrino mass suppression follows from the large size of $m_{N}$ with
respect to the electroweak scale ($v_{EW}\sim v_{H}$), here, there is an
additional suppression from the small induced VEV of $\Phi$, characterized by
$\epsilon$ given in Eq. (9). This feature allows us to have moderate values
for $m_{N}$, or $v_{\varphi_{1}}$, say around the TeV scale, without the need
for appealing to tiny Yukawa couplings.

At this point we also stress that, in contrast to low-scale inverse [31, 30]
or linear seesaw schemes [32, 33, 34], the presence of TeV-scale neutrino-
mass-mediators does not require the addition of extra gauge singlets beyond
the “right-handed neutrinos”. Moreover, since $m_{N}$ is directly associated
with the spontaneous breaking of the gauged $B-L$, the mass of the associated
gauge boson, $Z^{\prime}$, can also be naturally within experimental reach,
leading to rich phenomenological implications, as we discuss below.

Concerning the other two neutrinos, their masses are generated at 1-loop
through the scotogenic mechanism, namely

$\displaystyle M^{\nu(sc)}_{ij}$ $\displaystyle=$
$\displaystyle\sum_{c=1}^{2}Y^{\eta}_{ic}\mathcal{M}_{c}\,Y^{\eta}_{jc},\quad\quad\mbox{with}$
(31) $\displaystyle\mathcal{M}_{c}$ $\displaystyle=$
$\displaystyle\frac{m_{f_{c}}}{16\pi^{2}}\left[\frac{\cos^{2}\theta_{s}m^{2}_{s_{1}}}{m^{2}_{s_{1}}-m_{f_{c}}^{2}}\ln\frac{m^{2}_{s_{1}}}{m_{f_{c}}^{2}}-\frac{\cos^{2}\theta_{a}m^{2}_{a_{1}}}{m^{2}_{a_{1}}-m_{f_{c}}^{2}}\ln\frac{m^{2}_{a_{1}}}{m_{f_{c}}^{2}}+\frac{\sin^{2}\theta_{s}m^{2}_{s_{2}}}{m^{2}_{s_{2}}-m_{f_{c}}^{2}}\ln\frac{m^{2}_{s_{2}}}{m_{f_{c}}^{2}}-\frac{\sin^{2}\theta_{a}m^{2}_{a_{2}}}{m^{2}_{a_{2}}-m_{f_{c}}^{2}}\ln\frac{m^{2}_{a_{2}}}{m_{f_{c}}^{2}}\right],$

where $m_{f_{c}}=v_{\varphi_{2}}Y^{f}_{c}/\sqrt{2}$ are the masses of the two
dark fermions $f_{cR}$. One can easily check that when either $\lambda_{1}\to
0$ or $\mu_{1}\to 0$ the loop-generated masses vanish. When $\lambda_{1}\to
0$, $m_{s_{i}}\to m_{a_{i}}$, with $\cos^{2}\theta_{s}\to\cos^{2}\theta_{a}$
and $\sin^{2}\theta_{s}\to\sin^{2}\theta_{a}$, leading to a cancellation
between the first and the second, as well as the third and the forth terms in
Eq. (31). On the other hand, when $\mu_{1}\to 0$, we have
$\theta_{s},\theta_{a}\to 0$ so that only the first and the second terms in
Eq. (31) survive; nevertheless, these cancel out since, in this limit,
$m_{s_{1}}\to m_{a_{1}}$.

As usual in scotogenic setups, the loop mediators – black fields in Fig. 1 –
are part of a dark sector from which the lightest field is stable. This can
play the role of a WIMP dark matter candidate. In our case the viable dark
matter candidates are the neutral fermions $f_{cR}$ and the neutral scalars
$s_{i},a_{i}$. The stability of the lightest of them is ensured by the
residual subgroup of our gauged $B-L$ symmetry, i.e. the conserved matter
parity ($M_{P}$), defined in Eq. (4).

Lastly, since one neutrino mass is generated at tree level, while the other
two masses arise through the scotogenic diagram at the one-loop level, we
expect these masses to be loop-suppressed with respect to the former [7, 8].
Therefore, this framework favours the normal ordering of neutrino masses –
also preferred experimentally [13] – as well as an understanding of the origin
of the smallness of the ratio $\Delta m^{2}_{sol}/\Delta m^{2}_{atm}$.

##  VI Goldstone couplings to fermions

The effective interaction between the Goldstone boson, $G$, and the fermions,
$F_{i}$, can parametrised as

$\mathcal{L}_{GF_{i}F_{j}}=G\,\overline{F_{j}}\left(\Sigma_{L}^{GF\,ji}P_{L}+\Sigma_{R}^{GF\,ji}P_{R}\right)F_{i}+\text{h.c.},$
(32)

where $P_{L,R}$ are the usual chiral projectors, and $\Sigma^{GF}_{L,R}$ are
model-dependent dimensionless coefficients. As seen from Eq. (13), the
Goldstone boson has projections along of the $M_{P}$-even scalars, including
the SM-like Higgs doublet $H$. As a result, $G$ couples to all of the fermions
at tree-level.

For quarks and charged leptons, which get their tree-level masses exclusively
from their interactions with $H$, the tree-level couplings to $G$ arise from
the projection of $G$ into the $H$, $\left\langle G|H\right\rangle$. This can
be expressed as

$\displaystyle|\Sigma_{R}^{GF\,ji}|=|\left\langle
G|H\right\rangle|\frac{m_{F_{i}}}{v_{H}}\delta_{ij},$ (33)

with $F_{i}=e,d,u$ representing the charged leptons, down- and up-type quarks,
repectively. Notice that when we diagonalize the charged-fermion mass matrices
the G-couplings also become diagonal. In other words, flavour-violating
couplings to charged fermions are absent at tree-level.

Of particular interest are the couplings to charged leptons, for example
electrons. Depending on the strength of such couplings, Goldstone bosons can
be over-produced in Compton-like processes $e+\gamma\to e+G$ and emitted by
stars leading to excessive stellar cooling. Recently it was noted that also
the pseudoscalar couplings to the muon can be restricted by SN1987A cooling
rates [35]. The bounds for the couplings to electrons and muons are
$|\Sigma^{G\,ee}|\lesssim 2.1\times 10^{-13}$ and
$|\Sigma^{G\,\mu\mu}|\lesssim 2.1\times 10^{-10}$, respectively [36, 35].
Assuming $v_{\varphi}=v_{\varphi_{i}}\gg v_{H}\gg v_{\Phi}$ Eq. (13) leads to
$v_{\Phi}^{2}/(v_{H}v_{\varphi})<7.5\times 10^{-8}$ or, equivalently,

$\frac{v_{\Phi}^{2}}{v_{\varphi}}\lesssim 1.9\times 10^{-5}\,\text{GeV}.$ (34)

One sees that both constraints can be satisfied through the above restriction
on the Goldstone projection into the SM-like Higgs doublet. It is worth
noticing that this effective mass scale, $v_{\Phi}^{2}/v_{\varphi}$ also
appears in the seesaw-suppressed neutrino mass in Eq. (30), and should
therefore be small.

Note also that the Goldstone couplings to charged leptons also receive
radiative contributions, mediated mainly by the dark sector, via the diagram
in Fig. 2 (right). Such contributions can be calculated using the results of
the next section by taking the diagonal entries of the 1-loop couplings
$\Sigma^{G}_{L,R}$ in Eqs. (35) and (38).

The tree-level Goldstone couplings to neutrinos emerge from the second term in
Eq. (3) involving the leptophilic doublet $\Phi$, i.e.
$Y^{\Phi}_{i}\overline{L_{iL}}\tilde{\Phi}N_{R}$. As a consequence, they are
suppressed by the product of the Goldstone projection into $\Phi$, of order
$v_{\Phi}/v_{\varphi}$, and the small $\nu-N$ mixing, also of order
$v_{\Phi}/v_{\varphi}$. Dark-mediated loop contributions are also present,
however, these are expected to be very small as they are proportional to
neutrino masses, a feature reminiscent of Majoron models [29, 30].

##  VII Charged lepton flavour violation

In this section, we discuss the most relevant charged lepton flavour violating
(cLFV) processes occurring in our model, involving the first two families. For
theoretical and experimental reviews, see, for instance, Refs. [37, 38, 39].

An interesting characteristic feature to notice is the presence of charged
lepton flavour violation involving Goldstone boson emission. The projected
$\mu\to e$ conversion at COMET [40] can also probe $\mu\to eG$. There are good
prospects for improving the sensitivities on Goldstone boson searches with the
COMET experiment, where the sensitivity for the cLFV process $\mu\to eG$ may
improve from the current limit ${BR}(\mu\to eG)=2.3\times 10^{-5}$ in Phase-I
to a sensitivity $\hbox{$\cal O$ }(10^{-8})$ in Phase-II, under somewhat
optimistic assumptions [18] 666 In a similar fashion, the Mu3e experiment,
which aims at substantially improving the bounds on the decay $\mu\to 3e$, can
also probe $\mu\to eG$ [41], with sensitivity one order of magnitude worse
than the above one for COMET. . In what follows we pay special attention not
only to radiative cLFV processes involving photon emission, but also to
Goldstone boson emitting processes.

###  VII.1 cLFV through photon and Goldstone boson emission

We focus on cLFV decays of the form $e_{i}\to e_{j}X$ – with $X=\gamma,G$ –
which, here, receive their main contributions from diagrams mediated by the
$M_{P}$-odd fields in the “dark” sector, as shown in Fig. 2.

Other diagrams, although present, give rise only to sub-leading contributions.
For instance, analogously to Fig. 2, diagrams mediated by the $M_{P}$-even
fields $\phi^{+}$ and $N$, instead of the $M_{P}$-odd fields $\eta^{+}$ and
$f_{a}$, exist but their contributions are significantly suppressed by the
large mass of $\phi^{+}$. Likewise, charged-current contributions mediated by
the $W^{+}$ gauge boson and $N$ are present; however, they are also suppressed
due to the small $\nu-N$ mixing.

Fig. 2:  Leading contributions to the charged lepton flavour violating decays
$e_{i}\to e_{j}\gamma$ and $e_{i}\to e_{j}G$.

The effective operators associated with the decays $e_{i}\to e_{j}X$, with
$X=\gamma,G$, can be expressed, respectively, as follows [42, 43]

$\mathcal{L}_{Xe_{i}e_{j}}=\frac{F_{\mu\nu}}{2}\overline{e_{j}}\sigma^{\mu\nu}\left(\Sigma_{L}^{\gamma\,ji}P_{L}+\Sigma_{R}^{\gamma\,ji}P_{R}\right)e_{i}+G\,\overline{e_{j}}\left(\Sigma_{L}^{G\,ji}P_{L}+\Sigma_{R}^{G\,ji}P_{R}\right)e_{i}+\text{h.c.},$
(35)

where $\sigma^{\mu\nu}=[\gamma^{\mu},\gamma^{\nu}]$ and $P_{L,R}$ are the
usual chiral projectors. Here $\Sigma^{\gamma}_{L,R}$ ($\Sigma^{G}_{L,R}$) are
model-dependent coefficients with dimension $\mathrm{mass}^{-1}$
($\mathrm{mass}^{0}$).

The branching ratios for the processes of interest, assuming $m_{j}/m_{i}\ll
1$, are given by [42, 44, 38, 45, 43, 46]

$\displaystyle\mathrm{BR}\left(e_{i}\to
e_{j}\gamma\right)\simeq\frac{m_{e_{i}}^{3}}{16\pi\Gamma_{e_{i}}}\left(|\Sigma_{L}^{\gamma\,ji}|^{2}+|\Sigma_{R}^{\gamma\,ji}|^{2}\right)\quad\mbox{and}\quad\mathrm{BR}\left(e_{i}\to
e_{j}G\right)\simeq\frac{m_{e_{i}}}{32\pi\Gamma_{e_{i}}}\left(|\Sigma_{L}^{G\,ji}|^{2}+|\Sigma_{R}^{G\,ji}|^{2}\right)\,.$
(36)

In particular, we shall focus on the most constraining processes, i.e. $\mu\to
e\gamma$ and $\mu\to eG$, whose current limits are $\mathrm{BR}(\mu\to
e\gamma)\lesssim 4.2\times 10^{-13}$ [47] and $\mathrm{BR}(\mu\to eG)\lesssim
10^{-5}$ [48, 17, 49], while future experiments are expected to improve these
bounds to $\mathrm{BR}(\mu\to e\gamma)\lesssim 6\times 10^{-14}$ [50] and
$\mathrm{BR}(\mu\to eG)\lesssim 10^{-8}$ [18].

Taking into account that $m_{e}/m_{\mu}\ll 1$, the leading contribution to the
first process can be approximated to

$\mathrm{BR}\left(\mu\to
e\gamma\right)\simeq\frac{\alpha_{\mathrm{em}}m_{\mu}^{5}}{4\,m_{\eta^{+}}^{4}\Gamma_{\mu}}\left|\left(Y^{\eta}F_{1}Y^{\eta\dagger}\right)^{e\mu}\right|^{2}\,\quad\text{with}\quad(F_{1})_{ab}=\frac{1-6x_{a}+3x_{a}^{2}+2x_{a}^{3}-6x_{a}^{2}\log
x_{a}}{6(4\pi)^{2}(1-x_{a})^{4}}\delta_{ab}\,,$ (37)

where $x_{a}=(m_{f_{a}}/m_{\eta^{+}})^{2}$, $\alpha_{\mathrm{em}}$ is the
fine-structure constant, $m_{\mu}\simeq 105.658$ MeV is the muon mass, Here
$\Gamma_{\mu}=\Gamma_{\mu}^{SM}+\Gamma_{\mu}^{new}\simeq 3\times
10^{-19}\,\text{GeV}+\Gamma_{\mu}^{new}$ is the total decay width of the muon,
with $\Gamma_{\mu}^{new}$ representing the new contributions to the muon decay
width calculated in this section. Similarly, for the process involving the
Goldstone boson emmission, we have

$\mathrm{BR}\left(\mu\to eG\right)\simeq\frac{m_{\mu}^{3}\,|\langle
G|\varphi_{2}\rangle|^{2}}{32\pi\,v_{\varphi_{2}}^{2}\Gamma_{\mu}}\left|\left(Y^{\eta}F_{2}Y^{\eta\dagger}\right)^{e\mu}\right|^{2},\quad\mathrm{with}\quad(F_{2})_{ab}=\frac{x_{a}\left(x_{a}-1-\log
x_{a}\right)}{2(2\pi)^{2}(x_{a}-1)^{2}}\delta_{ab},$ (38)

where $\left\langle G|\varphi_{2}\right\rangle$ is the Goldstone projection
along $\varphi_{2}$.

We would like to point out also that, despite the presence of several new
neutral scalars as well as an extra neutral gauge boson ($Z^{\prime}$),
flavour-changing neutral currents involving charged leptons only take place at
loop level. Indeed, notice that, at tree level, two charged leptons can only
couple to a scalar via the first term in Eq. (3), governed by the $Y^{H}$
matrix. The very same term leads to the charged lepton mass matrix:
$M^{e}=Y^{H}\left\langle H\right\rangle$. Therefore, by diagonalising $M^{e}$,
the charged lepton couplings to neutral scalars are also automatically
diagonalised. Lastly, since all the charged leptons couple to $Z^{\prime}$
with the same $B-L$ charge ($-1$), and there is no non-standard charged
lepton, the tree-level couplings between the charged leptons and $Z^{\prime}$
are also diagonal.

###  VII.2 Numerical results

Having discussed the main cLFV processes, as well as the relevant bounds, we
now proceed to calculate numerically the relevant contributions. Note that the
Yukawa couplings controlling the cLFV processes are also involved in the
generation of neutrino masses. Thus, in order to estimate the cLFV
contributions, we need to choose as inputs benchmarks satisfying neutrino
oscillation data. Although we perform this task numerically, it is useful to
develop an analytical form for extracting the Yukawa couplings from measured
paramenters that satisfy neutrino oscillation restrictions, à la Casas-Ibarra
[51], as this will optimise our subsequent scanning procedure. For this
purpose, we write the sum of the tree-level (seesaw) and the one-loop
(scotogenic) neutrino masses in Eqs. (30) and (31) in the following compact
form

$\displaystyle M^{\nu}$ $\displaystyle=$ $\displaystyle
M^{\nu(ss)}+M^{\nu(sc)}=-Y\mathcal{X_{M}}\,Y^{T},\quad\text{with}\quad
Y_{3\times 3}=\begin{pmatrix}Y^{\eta}_{3\times 2}&Y^{\Phi}_{3\times
1}\end{pmatrix}\quad\text{and}\quad\mathcal{X_{M}}_{ij}=0\,\,(i\neq j),$ (39)
$\displaystyle\mathcal{X_{M}}_{cc}$ $\displaystyle=$
$\displaystyle-\mathcal{M}_{c},\quad\text{for}\quad
c=1,2,\quad\mbox{and}\quad\mathcal{X_{M}}_{33}=\frac{v_{\Phi}^{2}}{2m_{N}},$

with $\mathcal{M}_{c}$ defined in Eq. (31). This way the ansatz below can be
used to obtain our benchmarks

$Y=iU_{\nu}\sqrt{m_{\nu}}~{}\rho~{}\sqrt{\mathcal{X_{M}}}^{-1},$ (40)

where $U_{\nu}$ is the lepton mixing matrix, $m_{\nu}$ is a diagonal matrix
containing the active neutrino masses, and $\rho$ is a generic $3\times 3$
orthogonal matrix. For the oscillation parameters that enter $U_{\nu}$ and
$m_{\nu}$, i.e. the neutrino mixing angles, CP phase and neutrino mass
splittings, we take the values obtained in Ref. [13] within a $3\sigma$ range,
for the case of normal neutrino mass ordering. Moreover, we fix some of the
free parameters present in $\mathcal{X_{M}}$ by assuming that all the relevant
dimensionless scalar couplings are $\lambda_{i}=0.1$, the bare masses for the
dark scalars $\mu_{\eta}=\mu_{\sigma}=1$ GeV, while $\mu_{1}=5$ TeV, and the
Yukawa couplings are $Y^{N}=Y^{f}_{a}=0.5$.

Fig. 3: Neutrino mass hierarchies versus the lightest neutrino mass, $m_{1}$,
for normal ordering.

As for the lightest neutrino mass, $m_{1}$, we vary it in the range
$[0.005,0.03]$ eV. The upper value comes from taking the upper limit on the
sum of the neutrino masses: $\Sigma_{i}m_{i}<0.12$ eV [52] together with the
normal ordering assumption [13]. On the other hand, the lower bound comes from
the radiative origin of the lighter neutrino masses, i.e. $m_{1},m_{2}$. In
general we expect no strong $m_{2}/m_{1}$ hierarchy or, if present, such a
hierarchy would be typically milder than the $m_{3}/m_{2}$ hierarchy, since
$m_{3}$ has a tree-level origin. This is seen from Fig. 3, where we show the
$m_{3}/m_{2}$ and $m_{2}/m_{1}$ mass ratios versus $m_{1}$ for normal
ordering. One sees that this is indeed the case as long as $m_{1}\geq 0.005$
eV.

In what follows, we consider two scenarios. In the first, the induced VEV
$v_{\Phi}$, defined in Eq. (9), varies with varying $v_{\varphi}\equiv
v_{\varphi_{1,2,3}}$, which describes the $B-L$-breaking scale. Whereas in the
second case, $v_{\Phi}$ remains constant although $v_{\varphi}$ varies. Our
results for the first scenario are presented in Fig. 4. In this case,
$v_{\varphi}$ varies randomly between $1$ and $15$ TeV, and the induced VEV
$v_{\Phi}$ varies with $v_{\varphi}$, according to Eq. (9), which leads to
$v_{\Phi}$ values roughly between $1$ and $277$ MeV. The red lines at
$10^{-5}$ (horizontal) and $4.2\times 10^{-13}$ (vertical) indicate the
experimental bounds for the branching ratios of $\mu\to eG$ [48, 17, 49] and
$\mu\to e\gamma$ [47], respectively, so that the points within the shaded red
areas are excluded. Moreover, the gray points are excluded by the
astrophysical constraints on the flavour-diagonal Goldstone couplings
$\Sigma^{G\,ee}$ and/or $\Sigma^{G\,\mu\mu}$. The dashed orange lines
represent future experimental sensitivities for both processes, i.e.
$BR(\mu\to e\gamma)\lesssim 6\times 10^{-14}$ from the MEG II experiment [50]
and $BR(\mu\to eG)\lesssim\mathcal{O}(10^{-8})$ from the COMET experiment
[40], as derived in Ref. [18]. The remaining points are coloured according to
the “size” of the corresponding Yukawas, defined as
$Y_{\mathrm{eff}}=\sqrt{|\sum_{a}Y^{\eta}_{ea}Y^{\eta*}_{\mu a}|}$, as shown
in the colour bar on the right.

Fig. 4:  Left: branching ratios for $\mu\to eG$ vs $\mu\to e\gamma$. Right:
branching ratio for $\mu\to eG$ vs the masses of the dark mediators,
$m_{\eta^{+}},m_{f_{a}}$, given as functions of $v_{\varphi}$. The shaded red
bands are excluded by the current limits on the branching ratios $BR(\mu\to
eG)$ and $BR(\mu\to e\gamma)$, whereas the gray points are excluded by the
limits on the diagonal couplings ($\Sigma^{G\,ee},\Sigma^{G\,\mu\mu}$). The
dashed orange lines are the expected experimental sensitivities, as discussed
in the text. The colour bar denotes the effective Yukawa defined as
$Y_{\mathrm{eff}}=\sqrt{|\sum_{a}Y^{\eta}_{ea}Y^{\eta*}_{\mu a}|}$. In this
scenario, $v_{\varphi}\in[1,15]$ TeV and $v_{\Phi}\in[1,277]$ MeV.

Fig. 5: Left: branching ratios for $\mu\to eG$ vs $\mu\to e\gamma$. Right:
branching ratio for $\mu\to eG$ vs the masses of the dark mediators, which are
both functions of $v_{\varphi}$. Same colour code as in Fig. 4. The relevant
Yukawas follow the colour bar on the right, with
$Y_{\mathrm{eff}}=\sqrt{|\sum_{a}Y^{\eta}_{ea}Y^{\eta*}_{\mu a}|}$. In this
scenario, we assume $v_{\varphi}\in[1,15]$ TeV, whereas the induced VEV is
fixed: $v_{\Phi}=50$ MeV.

We now turn to a scenario in which the induced VEV $v_{\Phi}$ is a constant,
i.e. $v_{\Phi}=50$ MeV, but $v_{\varphi}$ is allowed to vary, as previously,
between $1$ and $15$ TeV. The plots in Fig. 5 show very different behaviours
when compared to those in the first scenario. Here, the branching ratios
increase as the mediator masses, $m_{\eta^{+}},m_{f_{a}}$, increase, whereas
the opposite happens in the first scenario.

To understand this, notice first that the branching ratios in Eqs. (37) and
(38) vary roughly as $Y_{\mathrm{eff}}^{4}(m_{\mu}/v_{\varphi})^{4}$ and
$Y_{\mathrm{eff}}^{4}(m_{\mu}/v_{\varphi})^{2}$ 777In the region of interest,
the dark mediator masses, $m_{f_{1,2}}$ (assumed to be degenerate) and
$m_{\eta^{+}}$, vary linearly with $v_{\varphi}$ in such a way that the loop
functions $F_{1,2}(m_{f_{a}}/m_{\eta^{+}})$ are approximately constants.. The
entries of $Y_{\mathrm{eff}}=\sqrt{|\sum_{a}Y^{\eta}_{ea}Y^{\eta*}_{\mu a}|}$
contribute to (scotogenic) solar neutrino mass generation which depend on
$v_{\Phi}^{2}$, see Fig. 1. Thus, once we impose neutrino oscillation data,
via Eqs. (39) and (40), $Y_{\mathrm{eff}}$ can also be described as a function
of the induced VEV $v_{\Phi}$. In the first scenario (Fig. 4), as $v_{\Phi}$
increases with $v_{\varphi}$, following Eq. (9), the scotogenic loop functions
$\mathcal{M}_{c}$ in Eq. (31) also increase. Consequently, to “neutralise”
this increase in $\mathcal{M}_{c}$, so as to satisfy neutrino oscillation
data, the relevant Yukawas should decrease accordingly. Therefore, since both
$Y_{\mathrm{eff}}$ and $(m_{\mu}/v_{\varphi})$ decrease with increasing
$v_{\varphi}$, the branching ratios also decrease, as shown in Fig. 4. In
contrast, in the scenario where $v_{\Phi}$ is constant, the scotogenic loop
functions $\mathcal{M}_{c}$ decrease with increasing $v_{\varphi}$, and, in
turn, the Yukawas increase to satisfy oscillation data. Therefore, while
$(m_{\mu}/v_{\varphi})$ decreases, $Y_{\mathrm{eff}}$ ends up increasing, with
$v_{\varphi}$. Nevertheless, it turns out that the depence of
$Y_{\mathrm{eff}}$ on $v_{\varphi}$ is stronger than linear, so that the
branching ratios, given as functions of
$Y_{\mathrm{eff}}^{4}(m_{\mu}/v_{\varphi})^{4}$ and
$Y_{\mathrm{eff}}^{4}(m_{\mu}/v_{\varphi})^{2}$ end up increasing with
increasing $v_{\varphi}$, as shown in Fig. 5.

##  VIII Collider signatures

###  VIII.1 $pp\to Z^{\prime}\to NN$ Drell-Yann Production

In the simplest scoto-seesaw model [7, 8] the neutrino mass mediators are
unlikely to be produced at colliders. For example, the atmospheric neutrino
mass mediator, i.e. the heavy neutral lepton $N$ (right-handed neutrino), is
inaccessible through the Drell-Yan mechanism. Indeed, $N$ is typically very
heavy. Even if (artificially) taken to lie in the TeV scale, its production
would be suppressed by a tiny doublet-singlet $\nu-N$ mixing.

In the present case, however, both the neutral lepton $N$ and the vector boson
$Z^{\prime}$ can lie in the TeV-scale. The existence of the $Z^{\prime}$
provides a Drell-Yan pair-production portal for the heavy neutrino. Its
subequent decays into the SM states could potentially lead to verifiable
signals. Due to enhanced gauge charge (5) of the right-handed fermion $N$, its
$Z^{\prime}$ phenomenology can differ from that of other $U(1)_{B-L}$ models.

We remark that the neutrino mass suppression mechanism of our scoto-seesaw
model involving the induced VEV of the leptophilic doublet $\Phi$, Figs. 1,
requires a rich scalar sector compared to a vanilla $U(1)_{B-L}$ model. We
have an extended visible scalar sector with four more CP-even scalars apart
from the Higgs boson. Among them, the scalar predominantly coming from the
leptophilic doublet $\Phi$ is relatively heavy and beyond the LHC reach, while
the others are lighter, possibly around the TeV scale. Moreover, there are two
massive CP-odd scalars. While one of them is heavy, as it comes mostly from
$\Phi$, the other can potentially lie at the TeV scale.

The $pp\to NN$ collision process can also proceed through the exchange of
neutral scalars. These couple to the initial-state quarks and final-state
neutral fermion pair through mixing of the Higgs boson with other CP even
scalars. Nonetheless, for simplicity we assume that this mixing is not
significant so that such decays ($S_{i}\to NN$) are not kinematically allowed,
and these contributions can be neglected. In Table 2 we give estimates of the
expected $Z^{\prime}$ mediated production cross section.

Benchmark Points |  $\sigma(pp\to NN)$ (in fb)
---|---
BP-I: $m_{Z^{\prime}}$ = 1.5 TeV, $m_{N}=708$ GeV ($Z^{\prime}$ decays to dark sector are allowed) | 0.489
BP-II: $m_{Z^{\prime}}$ = 1.5 TeV, $m_{N}=708$ GeV ($Z^{\prime}$ decay to dark sector negligibly small) | 4.76
BP-III: $m_{Z^{\prime}}$ = 2.0 TeV, $m_{N}=708$ GeV | 5.14
BP-IV: $m_{Z^{\prime}}$ = 2.5 TeV, $m_{N}=708$ GeV | 1.44
BP-V: $m_{Z^{\prime}}$ = 3.0 TeV, $m_{N}=708$ GeV | 0.415
Table 2:  Drell-Yan neutral lepton pair production through the $Z^{\prime}$
portal. The cross section is computed with $\lambda_{ii}\sim
0.1,\lambda_{ij}\sim 0.01,Y_{i}\sim 0.5,g^{\prime}=0.1$. The dominant
contribution is through the $Z^{\prime}$ mediated process $\sigma(pp\to
Z^{\prime})\times BR(Z^{\prime}\to NN)$.

Note that the neutral fermion mass is taken around 700 GeV, allowing the
$Z^{\prime}$ boson to decay to it in on-shell. For a $Z^{\prime}$ mass of 1.5
TeV the branching ratio to $Z^{\prime}\to NN$ is relatively low, while it
peaks near 2 TeV $Z^{\prime}$ mass, and saturates at around $25\%$. Therefore
$\sigma(pp\to Z^{\prime})\times BR(Z^{\prime}\to NN)$ increases intitally up
to 2 TeV, then it gradually decreases with increasing $Z^{\prime}$ mass.
Notice that the heavy neutrino $N$ will decay to SM states, such as leptons,
through $\nu-N$ mixing in the electroweak currents. This leads to leptonic
final-state signatures such as di-leptons plus missing energy 888 Dark sector
fermions can also be produced through Drell-Yann processes like $\sigma(pp\to
f_{i}f_{i})$, with less striking signatures. . It is not in our scope here to
present a dedicated numerical simulation of these signatures.

###  VIII.2 $Z^{\prime}$ Phenomenology: Dilepton search

ATLAS and CMS experiments are looking for a heavy BSM gauge boson in different
channels. The leading LHC production mode for a $Z^{\prime}$ with couplings to
quarks (as is the case for all $U(1)_{B-L}$ models) is $q\bar{q}$ fusion. In
our model we have many decay modes of $Z^{\prime}\to S_{i}A_{j}$ and
$Z^{\prime}\to S_{i}Z$, where $S_{i}$ and $A_{j}$ are the CP-even and CP-odd
scalars, in both visible and dark scalar ($s_{i},a_{i}$) final states. As a
consequence, our $Z^{\prime}$ decay rates to the SM channels can be relatively
small.

Despite such smaller branching ratio to the SM channels, the $Z^{\prime}$
decay to a pair of charged leptons places strong constraints on the
$Z^{\prime}$ mass and coupling, see Fig. 6.

Fig. 6:  The left panel gives the expected $\sigma\times BR$ of the dilepton
signal in our dynamical scoto-seesaw model, obtained when all exotic channels
are accessible (red line) or kinematically forbidden (blue line), see text.
The black line represents the dilepton channel measurement from ATLAS [53]
(with similar results also from CMS [54]), while the green and yellow bands
give $1\sigma$ and $2\sigma$ regions, obtained from the LHC data repository
HEPData [55]. Exclusion limits on the $Z^{\prime}$ mass within these two
extreme benchmarks are readily obtained from the blue and red curves. The
right panel compares the agressive constraints (blue line) obtained in our
model and in a reference $\chi$ model.

The exclusion bounds on the parameters of our model at the LHC are derived by
comparing our theoretical predictions with the experimental limits obtained
from the non-observation of any BSM excess in a particular mode. Here we fix
the gauge coupling at a value $g^{\prime}=0.1$, and compare the theoretical
cross section $\sigma(pp\to Z^{\prime})\times BR(Z^{\prime}\to\ell\ell)$ in
our model with the upper bound obtained from the LHC dilepton search. This way
we provide limits on the $Z^{\prime}$ mass, $M_{Z^{\prime}}$. The $Z^{\prime}$
limit in the sequential standard model (SSM) obtained by CMS [54] excludes any
new boson below 4.4 TeV. In a similar $U(1)_{B-L}$ model with $5,-4,-4$ gauged
lepton number charges [56], in the presence of higher dimensional operators,
the lower bound on $Z^{\prime}$ hovers around $m_{Z^{\prime}}\approx 2.5-3$
TeV.

Considering the left plot, the blue line gives the strongest constraint,
excluding any $Z^{\prime}$ mass up to $m_{Z^{\prime}}\sim 3.9$ TeV. This is
the case where all scalar quartic couplings, except for $\lambda_{H}$, chosen
to fit the SM Higgs boson mass, are taken large $\lambda_{i}\sim 0.75$,
leading to a heavier scalar spectrum which restricts $Z^{\prime}$ decays to
BSM scalar modes. Likewise all BSM fermion Yukawa couplings are taken large
$Y_{a}=1.5$, so that also the $N,f_{a}$ fermions are too heavy to be produced
by $Z^{\prime}$ decays, leading to smaller branching ratio for $Z^{\prime}$ to
charged leptons, i.e. $BR(Z^{\prime}\to\ell_{i}\ell_{i})\sim 15\%$. Our second
benchmark corresponding to the red line leads to weaker constraints. In this
case the scalar quartics take values $\lambda_{i}\sim 0.10$, making the scalar
sector relatively light, so that $Z^{\prime}$ can decay to some dark scalar-
pseudoscalar pair. Even in this limit $Z^{\prime}$ decays to visible sector
scalars are not favored, mainly due to phase space. As for the BSM Yukawa
couplings, we assume $Y_{a}=0.5$, which opens up decay modes like
$Z^{\prime}\to NN,f_{a}f_{a}$ with BRs in the order of 30% and 20% each,
respectively. These new decay modes suppress the di-lepton BR to
$BR(Z^{\prime}\to\ell_{i}\ell_{i})\sim 2-4\%$, leading to a weaker limit on
the $Z^{\prime}$ mass, $m_{Z^{\prime}}\geq 2.7~{}$TeV. Finally, in the right
plot, we compare our “agressive” model prediction to that of the $U(1)_{\chi}$
model, an $E6$-motivated Grand Unification construction [57]. We observe that
the lower bound for the $Z^{\prime}$ mass of the $U(1)_{\chi}$ model lies
around $4.8$ TeV, roughly $1$ TeV above the limit for $m_{Z^{\prime}}$ in our
model. For details regarding the collider aspects of $Z^{\prime}$
phenomenology, we use Ref. [56].

##  IX Dark matter

In our gauged scoto-seesaw model, dark matter can be either the lightest
“dark” scalar or fermion, stabilized through the residual matter-parity
symmetry defined in Eq. (4). Radiative exchange of the “dark” particles
generates the solar neutrino mass scale, as illustrated in the right panel of
Fig. 1. The dark sector includes scalar dark matter candidates that are
mixtures of neutral scalars of the $SU(2)_{L}$ doublet $\eta$ and the singlet
$\sigma$, plus the charged scalar $\eta^{\pm}$. On the other hand, the
fermionic dark sector contains two neutral Majorana fermions $f_{a}$. In
addition to the standard Higgs portal mediated DM annihilation, we can have a
novel $Z^{\prime}$ mediated annihilation, along with some new scalar mediated
annihilation.

  * •

Scalar dark matter

The scalar dark matter sector of our scoto-seesaw model includes, besides the
doublet $\eta$, an additional singlet $\sigma$. The neutral components of the
singlet $S_{\sigma},A_{\sigma}$ and doublet $S_{\eta},A_{\eta}$ mix so as to
produce the CP-even ($s_{1},s_{2}$) and CP-odd ($a_{1},a_{2}$) DM candidates,
respectively. (In contrast, the dark scalar sector of the simplest scoto-
seesaw scenario, analysed in Ref. [7, 8], consists of a single scalar doublet
$\eta$.) As the doublet-singlet mixing is small, see Eq. (21), two of the
scalar DM candidates, one CP-even $s_{1}$ and one CP-odd $a_{1}$, are mainly
doublet DM candidates. The neutral components of the doublet $\eta$ resonantly
annihilate through the Higgs portal. In our setup, we can have additional
resonant annihilation through non-dark CP-even Higgs bosons ($S\equiv
S_{\Phi},S_{\varphi_{i}}$). This may lead to enhanced annihilation and under-
abundant relic at different DM masses $m_{\rm DM}\sim m_{S}/2$.

In addition to the $\eta$ mediated t-channel annihilation to SM states
$\ell\ell,WW,ZZ,hh$, present in the simplest scoto-seesaw (Ref. [7, 8]), in
our new model there are other t-channel contributions to scalar DM
annihilation, which can enhance the annihilation cross section, leading to
more parameter space with under-abundant relic.

The dark singlet $\sigma$ present in our model annihilates through similar
channels as the doublet-like dark matter, though the relative dominance of
different contributions will depend on the mixing of the CP-even scalars
$h,S_{i}$. For example, singlet $s_{2}$ and $a_{2}$ annihilation through
quartic couplings like $SSVV$ with $S\equiv s_{2},a_{2}$ and $\rm V\equiv
W^{\pm},Z$ are absent, whereas same vertices with $V\equiv Z^{\prime}$ and the
corresponding annihilation channel will be present.

Moreover, the scalar dark sector phenomenology of our model will be modified
due to the presence of a $Z^{\prime}$ portal. This happens because, if CP-even
($s_{1,2}$) and CP-odd ($a_{1,2}$) dark scalars are almost degenerate, co-
annihilation happens through a $Z^{\prime}$ mediated s-channel process. This
coannihilation adds up to the existing DM annihilation modes, opening up new
parameter space consistent with measured relic density.

  * •

Fermionic singlet dark matter

In our gauged scoto-seesaw model, two fermionic dark matter candidates $f_{a}$
mediate the loop-induced solar mass scale, instead of a single dark fermion in
the simplest scoto-seesaw [7, 8]. These dark fermions $f_{a}$ have a new
$Z^{\prime}$ portal to annihilate. In constrast to the simplest non-gauged
scoto-seesaw model in Ref. [7, 8], this brings a significant enhancement in
the DM annihilation, including a resonant dip in the relic abundance at
$m_{\rm DM}\sim m_{Z^{\prime}}/2$.

Moreover, in the simplest scoto-seesaw mechanism, the dark fermion mass term
is a bare mass term. As a result, there is no Higgs-mediated s-channel
fermionic dark matter annihilation channel. In contrast, in our gauged scoto-
seesaw model there is a singlet $\varphi_{2}$ which provides mass to the dark
Majorana fermions $f_{a}$, and can act as a portal of DM annihilation,
creating a resonant dip in the relic density.

  * •

Constraints and dark matter detection

In the standard scoto-seesaw (Ref. [7, 8]), light doublet-like dark matter is
constrained from the LEP Z decay measurement. In vanilla scotogenic models,
including the original one [5] and its simplest triplet extension [6], nearly
degenerate neutral CP-even and CP-odd scalars from $\eta$ is typically assumed
in order to implement scotogenic neutrino mass generation. In our present
model, however, instead of relying on the quasi-degeneracy of the scalar
mediators, the smallness of neutrino masses follows mainly from their
dependence on the (small) induced VEV $v_{\Phi}$, Fig. 1. Consequently, here
neutrinos can be light enough even when one has a light CP-even DM with a
heavier CP-odd counterpart, or vice versa, avoiding constraints from the $Z\to
S_{\eta}A_{\eta}$ decay. Therefore, in our gauged scoto-seesaw a light
doublet-like scalar dark matter is not as constrained as in vanilla scotogenic
models [5, 6] or the simplest scoto-seesaw scenario [7, 8]. The LEP constraint
can be completely avoided in our proposed scheme.

Our dynamical scoto-seesaw model can have light dark scalars, free from
constraints that hold for a generic doublet-only dark sector, allowing it to
harbor a viable light DM candidate. The latter can be probed through dedicated
experiments like CDMS-lite and CRESST, along with existing experiments, such
as LZ, Xenon-nT, and PandaX.

Note that in our model we have DM coannihilation through a $Z^{\prime}$
portal, the DM relic density being fixed by $Z^{\prime}$ related parameters
and dark sector mass degeneracy. Concerning direct detection, the scalar DM-
nucleon scattering dominantly happens through the Higgs portal, as in generic
scotogenic models. This relaxes the inter-dependence between the DM
annihilation and detection, as they are not controlled by the same
interaction.

In the simplest scoto-seesaw scenario, for instance, there is no mediator for
fermionic DM detection through DM-nucleon scattering at the tree level. This
results in a possible loop-suppressed scattering cross section. The presence
of DM-nucleon scattering mediated at tree-level by $Z^{\prime}$ provides a
significant advantage to our gauged scoto-seesaw model in this regard, as it
enhances the possibility of direct detection of fermionic DM. The fermionic
DM-nucleon scattering cross section then may reach the current sensitivity of
the DM direct detection experiments.

##  X Conclusions

We have proposed a scheme where the scoto-seesaw mechanism has a dynamical
origin, associated to a gauged $B-L$ symmetry. “Dark” states mediate solar
neutrino mass generation radiatively, while the atmospheric scale arises a la
seesaw, see Fig. 1. Indeed the origin of the solar scale is scotogenic, its
radiative nature explaining the solar-to-atmospheric scale ratio. The two dark
fermions and the TeV-scale seesaw mediator carry different dynamical $B-L$
charges. Dark matter stability follows from the residual matter-parity that
survives the breaking of $B-L$ gauge symmetry. Apart from the possibility of
being tested at colliders, see Fig. 6, our scoto-seesaw model with gauged
$B-L$ has sizeable charged lepton flavour violating phenomena. These include
also processes involving the emission of a Goldstone boson associated to an
accidental global symmetry present in the theory, see Fig. 2. Rate estimates
for muon number violating processes are given in Figs. 4 and 5. They indicate
that these processes lie within reach of present and upcoming searches.
Likewise, we also expect sizeable tau number violating processes.

###### Acknowledgements.

This work was supported by the Spanish grants PID2020-113775GB-I00
(AEI/10.13039/501100011033) and Prometeo CIPROM/2021/054 (Generalitat
Valenciana). S.S. thanks SERB, DST, Govt. of India for a SIRE grant with
number SIR/2022/000432 supporting an academic visit to the AHEP group at IFIC.
J. V. wishes to thank Yoshi Kuno for a discussion on COMET sensitivities.

##  Appendix A Scalar sector analytical expressions

In this appendix, we provide in full form some of the important, but longer,
analytical expressions obtained while deriving the scalar spectrum. The
$5\times 5$ squared mass matrix for the CP- and $M_{P}$-even fields
$(S_{H},S_{\Phi},S_{\varphi_{1}},S_{\varphi_{2}},S_{\varphi_{3}})$ is given by
the symmetric matrix

$\displaystyle M_{S}^{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccc}d_{1}&\star&\star&\star&\star\\\
\frac{v_{\varphi_{2}}v_{\varphi_{3}}\lambda_{2}}{2}+v_{H}v_{\Phi}(\lambda_{H\Phi}+\lambda_{H\Phi}^{\prime})&d_{2}&\star&\star&\star\\\
v_{H}v_{\varphi_{1}}\lambda_{H\varphi_{1}}&v_{\Phi}v_{\varphi_{1}}\lambda_{\Phi\varphi_{1}}&d_{3}&\star&\star\\\
\frac{v_{\Phi}v_{\varphi_{3}}\lambda_{2}}{2}+v_{H}v_{\varphi_{2}}\lambda_{H\varphi_{2}}&\frac{v_{H}v_{\varphi_{3}}\lambda_{2}}{2}+v_{\Phi}v_{\varphi_{2}}\lambda_{\Phi\varphi_{2}}&v_{\varphi_{1}}v_{\varphi_{2}}\lambda_{\varphi_{1}\varphi_{2}}+\frac{v_{\varphi_{3}}\mu_{2}}{2}&d_{4}&\star\\\
\frac{v_{\Phi}v_{\varphi_{2}}\lambda_{2}}{2}+v_{H}v_{\varphi_{3}}\lambda_{H\varphi_{3}}&\frac{v_{H}v_{\varphi_{2}}\lambda_{2}}{2}+v_{\Phi}v_{\varphi_{3}}\lambda_{\Phi\varphi_{3}}&v_{\varphi_{1}}v_{\varphi_{3}}\lambda_{\varphi_{1}\varphi_{3}}+\frac{v_{\varphi_{2}}\mu_{2}}{2}&\frac{v_{H}v_{\Phi}\lambda_{2}}{2}+\frac{v_{\varphi_{1}}\mu_{2}}{2}&d_{5}\\\
\end{array}\,\right)$ (46)

with

$\displaystyle d_{1}$ $\displaystyle=$ $\displaystyle
2v_{H}^{2}\lambda_{H}-\frac{v_{\Phi}v_{\varphi_{2}}v_{\varphi_{3}}\lambda_{2}}{2v_{H}},\quad
d_{2}=2v_{\Phi}^{2}\lambda_{\Phi}-\frac{v_{H}v_{\varphi_{2}}v_{\varphi_{3}}\lambda_{2}}{2v_{\Phi}},\quad
d_{3}=2v_{\varphi_{1}}^{2}\lambda_{\varphi_{1}}-\frac{v_{\varphi_{2}}v_{\varphi_{3}}\mu_{2}}{2v_{\varphi_{1}}},$
(47) $\displaystyle d_{4}$ $\displaystyle=$ $\displaystyle
2\lambda_{\varphi_{2}}v_{\varphi_{2}}^{2}-\frac{v_{\varphi_{3}}(v_{H}v_{\Phi}\lambda_{2}+v_{\varphi_{1}}\mu_{2})}{2v_{\varphi_{2}}},\quad
d_{5}=2\lambda_{\varphi_{3}}v_{\varphi_{3}}^{2}-\frac{v_{\varphi_{2}}(v_{H}v_{\Phi}\lambda_{2}+v_{\varphi_{1}}\mu_{2})}{2v_{\varphi_{3}}}.$

Similarly, for the $M_{P}$-even but CP-odd fields, in the basis
$(A_{H},A_{\Phi},A_{\varphi_{1}},A_{\varphi_{2}},A_{\varphi_{3}})$, we have
the symmetric matrix

$\displaystyle
M^{2}_{A}=\frac{1}{2}\left(\begin{array}[]{ccccc}\lambda_{2}\frac{v_{\Phi}v_{\varphi_{2}}v_{\varphi_{3}}}{v_{H}}&\star&\star&\star&\star\\\
-\lambda_{2}v_{\varphi_{2}}v_{\varphi_{3}}&\lambda_{2}\frac{v_{H}v_{\varphi_{2}}v_{\varphi_{3}}}{v_{\Phi}}&\star&\star&\star\\\
0&0&\mu_{2}\frac{v_{\varphi_{2}}v_{\varphi_{3}}}{v_{\varphi_{1}}}&\star&\star\\\
\lambda_{2}v_{\Phi}v_{\varphi_{3}}&-\lambda_{2}v_{H}v_{\varphi_{3}}&-\mu_{2}v_{\varphi_{3}}&\frac{\lambda_{2}v_{H}v_{\Phi}+\mu_{2}v_{\varphi_{1}}}{v_{\varphi_{2}}}v_{\varphi_{3}}&\star\\\
-\lambda_{2}v_{\Phi}v_{\varphi_{2}}&\lambda_{2}v_{H}v_{\varphi_{2}}&-\mu_{2}v_{\varphi_{2}}&-\lambda_{2}v_{H}v_{\Phi}+\mu_{2}v_{\varphi_{1}}&\frac{v_{H}v_{\Phi}\lambda_{2}+v_{\varphi_{1}}\mu_{2}}{v_{\varphi_{3}}}v_{\varphi_{2}}\\\
\end{array}\right).$ (53)

Out of the five eigenvalues of $M_{A}^{2}$, only the two below are non-
vanishing

$m_{A_{1,2}}^{2}=\frac{1}{4v_{H}v_{\Phi}v_{\varphi_{1}}v_{\varphi_{2}}v_{\varphi_{3}}}\left[C_{1}+C_{2}\mp\sqrt{(C_{1}-C_{2})^{2}+4C_{3}^{2}}\right],$
(54)

where

$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle\lambda_{2}v_{\varphi_{1}}\left(v_{H}^{2}v_{\Phi}^{2}v_{\varphi_{2}}^{2}+v_{H}^{2}v_{\Phi}^{2}v_{\varphi_{3}}^{2}+v_{H}^{2}v_{\varphi_{2}}^{2}v_{\varphi_{3}}^{2}+v_{\Phi}^{2}v_{\varphi_{2}}^{2}v_{\varphi_{3}}^{2}\right),$
(55) $\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle\mu_{2}v_{H}v_{\Phi}\left(v_{\varphi_{1}}^{2}v_{\varphi_{2}}^{2}+v_{\varphi_{1}}^{2}v_{\varphi_{3}}^{2}+v_{\varphi_{2}}^{2}v_{\varphi_{3}}^{2}\right)\quad\mbox{and}\quad
C_{3}^{2}=\lambda_{2}\mu_{2}v_{H}^{3}v_{\Phi}^{3}v_{\varphi_{1}}^{3}(v_{\varphi_{2}}^{2}-v_{\varphi_{3}}^{2})^{2}.$

Finally, the exact expression for the physical Nambu-Goldstone, $G$, is

$\displaystyle G$ $\displaystyle=$ $\displaystyle
N_{G}^{-1/2}\left\\{v_{H}v_{\Phi}(10v_{\varphi_{1}}^{2}+4v_{\varphi_{2}}^{2}+v_{\varphi_{3}}^{2})(v_{\Phi}A_{H}-v_{H}A_{\Phi})+v_{\varphi_{1}}\left[6v_{H}^{2}v_{\Phi}^{2}+(v_{H}^{2}+v_{\Phi}^{2})(4v_{\varphi_{2}}^{2}-v_{\varphi_{3}}^{2})\right]A_{\varphi_{1}}\right.$
$\displaystyle\left.+v_{\varphi_{2}}\left[3v_{H}^{2}v_{\Phi}^{2}-(v_{H}^{2}+v_{\Phi}^{2})(5v_{\varphi_{1}}^{2}+v_{\varphi_{3}}^{2})\right]A_{\varphi_{2}}+v_{\varphi_{3}}\left[3v_{H}^{2}v_{\Phi}^{2}+(v_{H}^{2}+v_{\Phi}^{2})(5v_{\varphi_{1}}^{2}+4v_{\varphi_{2}}^{2})\right]A_{\varphi_{3}}\right\\},$
with $\displaystyle\quad N_{G}$ $\displaystyle=$
$\displaystyle\left[9v_{H}^{2}v_{\Phi}^{2}+(v_{H}^{2}+v_{\Phi}^{2})(25v_{\varphi_{1}}^{2}+16v_{\varphi_{2}}^{2}+v_{\varphi_{3}}^{2})\right]\left[v_{H}^{2}v_{\Phi}^{2}(4v_{\varphi_{1}}^{2}+v_{\varphi_{2}}^{2}+v_{\varphi_{3}}^{2})+(v_{H}^{2}+v_{\Phi}^{2})(v_{\varphi_{1}}^{2}v_{\varphi_{2}}^{2}+v_{\varphi_{1}}^{2}v_{\varphi_{3}}^{2}+v_{\varphi_{2}}^{2}v_{\varphi_{3}}^{2})\right].$

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