IFIC/23-49

The majoron coupling to charged leptons

A. Herrero-Brocal${}^{\text{a}}$, Avelino Vicente${}^{\text{a,b}}$

${}^{(\text{a})}$ Instituto de Física Corpuscular, CSIC-Universitat de
València, 46980 Paterna, Spain

${}^{(\text{b})}$ Departament de Física Teòrica, Universitat de València,
46100 Burjassot, Spain

<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The particle spectrum of all Majorana neutrino mass models with spontaneous
violation of global lepton number include a Goldstone boson, the so-called
majoron. The presence of this massless pseudoscalar changes the phenomenology
dramatically. In this work we derive general analytical expressions for the
1-loop coupling of the majoron to charged leptons. These can be applied to any
model featuring a majoron that have a clear hierarchy of energy scales,
required for an expansion in powers of the low-energy scale to be valid. We
show how to use our general results by applying them to some example models,
finding full agreement with previous results in several popular scenarios and
deriving novel ones in other setups.

##  1 Introduction

Lepton number is an accidental symmetry in the Standard Model (SM) of particle
physics. Many extensions of this minimal model actually include sources of
lepton number violation. This is the case of all Majorana neutrino mass
models, in which the $\mathrm{U(1)}_{L}$ lepton number symmetry is broken.
When this symmetry is global and its breaking is spontaneous, a massless
Goldstone boson arises in the spectrum, the majoron ($J$) [1, 2, 3, 4, 5].
This massless pseudoscalar has dramatic phenomenological implications and can
be probed by many experiments, including the search for rare low-energy
processes [6] and invisible Higgs decays at high-energy colliders [7], as well
as due to its impact on many cosmological observables [8, 9, 10, 11].

The flavor structure of the majoron couplings to charged leptons is crucial
for the phenomenology [12, 13]. As explained below, flavor diagonal couplings
are strongly constrained by astrophysical observations, while the flavor off-
diagonal ones may induce exotic processes like
$\ell_{\alpha}\to\ell_{\beta}\,J$, with $\alpha\neq\beta$. Different models
also induce majoron couplings to charged leptons at different loop orders.
Tree-level diagonal majoron couplings to charged leptons are actually induced
in many scenarios beyond the SM. For instance, in models that generate them
via mixing between the Higgs and the singlet that breaks lepton number
spontaneously. However, in this case the couplings turn out to be diagonal in
the charged lepton mass basis. An alternative mechanism must be introduced in
order to generate tree-level off-diagonal couplings. For instance, these are
induced if the majoron couples directly to the charged leptons or to other
fermions that mix with them [14].

We study the 1-loop coupling of the majoron to a pair of charged leptons. In
contrast to previous works that focus on specific models, typically the type-I
seesaw with spontaneous lepton number violation [2, 15, 16, 17], we derive
general expressions valid for (virtually) any model. We consider all 1-loop
diagrams leading to a majoron coupling to charged leptons and expand the
resulting analytical expressions in powers of the light neutrino masses (or
other related low-energy scale). The main result of our work is a set of
formulae that can be readily applied to any majoron model of interest to
obtain the couplings to charged leptons. In order to demonstrate their use, we
illustrate the application of our analytical results to several example
models. This allows us to recover well-known results in some popular
scenarios, which constitutes a non-trivial cross-check of our calculation. We
also obtain novel results in other less studied models.

The rest of the manuscript is organized as follows. We present our setup,
discuss the current bounds on the majoron couplings to charged leptons and
derive general expressions for them in Sec. 2. These general results are
applied to specific models in Sec. 3. This allows us to show how to use our
analytical results. We summarize our work and discuss further directions in
Sec. 4. Finally, appendices A and B contain additional details about our
results.

##  2 The majoron coupling to charged leptons

The Lagrangian describing the interaction of a majoron with a pair of charged
leptons can be generally written as [6],

$\mathcal{L}_{\ell\ell
J}=J\,\bar{\ell}_{\beta}\left(S_{L}^{\beta\alpha}\,P_{L}+S_{R}^{\beta\alpha}\,P_{R}\right)\ell_{\alpha}+\text{h.c.}=J\,\bar{\ell}_{\beta}\left[S^{\beta\alpha}\,P_{L}+\left(S^{\alpha\beta}\right)^{*}\,P_{R}\right]\ell_{\alpha}\,.$
(1)

Here $P_{L,R}=\frac{1}{2}\left(1\mp\gamma_{5}\right)$ are the usual chiral
projectors while $\ell_{\alpha,\beta}$ are the charged leptons, with
$\alpha,\beta=1,2,3$ two flavor indices, and

$S^{\beta\alpha}=S_{L}^{\beta\alpha}+\left(S_{R}^{\alpha\beta}\right)^{*}\,.$
(2)

All flavor combinations are considered:
$\beta\alpha=\left\\{ee,\mu\mu,\tau\tau,e\mu,e\tau,\mu\tau\right\\}$. We note
that majorons are pseudoscalar states. This implies that the diagonal
$S^{\beta\beta}=S_{L}^{\beta\beta}+\left(S_{R}^{\beta\beta}\right)^{*}$
couplings are purely imaginary. We are interested in models that induce flavor
off-diagonal couplings at the 1-loop level. Therefore, while flavor diagonal
couplings will be allowed at tree-level, the off-diagonal ones will be absent
and only generated at 1-loop. In models that generate off-diagonal couplings
at tree-level, the 1-loop contributions considered in our work can be regarded
as just corrections, hence expected to be subdominant.

There are stringent constraints on both diagonal and off-diagonal majoron
couplings to charged leptons [6]. The former are constrained by astrophysical
observations. If produced, majorons would escape astrophysical scenarios
without interacting with the medium, thus leading to a very efficient energy
loss mechanism. This has been studied in several recent works [18, 19, 20, 21,
22, 23], which have derived the limits

$\text{Im}\,S^{ee}<2.1\times 10^{-13}$ (3)

and

$\text{Im}\,S^{\mu\mu}<3.1\times 10^{-9}\,.$ (4)

The off-diagonal majoron couplings to charged leptons also receive strong
constraints, in this case from searches of the flavor violating decays
$\ell_{\alpha}\to\ell_{\beta}\,J$. In order to specify the bound, it proves
convenient to define the combination

$\left|\widetilde{S}^{\beta\alpha}\right|=\left(\left|S^{\beta\alpha}_{L}\right|^{2}+\left|S^{\beta\alpha}_{R}\right|^{2}\right)^{1/2}\,,$
(5)

that enters the $\ell_{\alpha}\to\ell_{\beta}\,J$ decay width as

$\Gamma(\ell_{\alpha}\to\ell_{\beta}\,J)=\frac{m_{\alpha}}{32\,\pi}\,\left|\widetilde{S}^{\beta\alpha}\right|^{2}\,.$
(6)

The current limit on the branching ratio of $\mu^{+}\to e^{+}\,J$ was obtained
at TRIUMF [24]. Taking into account all possible chiral structures for the
majoron coupling, one can estimate the limit $\text{BR}\left(\mu\to
e\,J\right)\lesssim 10^{-5}$ [25], which in turn implies

$\left|\widetilde{S}^{e\mu}\right|<5.3\times 10^{-11}\,.$ (7)

Finally, the currently best experimental limits on $\tau$ decays including
majorons were set by the BelleII collaboration [26]. They can be used to
derive the bounds

$\begin{split}&\left|\widetilde{S}^{e\tau}\right|<3.5\times 10^{-7}\,,\\\
&\left|\widetilde{S}^{\mu\tau}\right|<2.7\times 10^{-7}\,.\end{split}$ (8)

###  2.1 General setup

Our goal is to obtain generic expressions for the majoron coupling to a pair
of charged leptons. We adopt a general setup with $3$ light charged leptons,
$\ell_{\alpha}=\left\\{e,\mu,\tau\right\\}$, $N$ neutral leptons,
$n_{i}=\left\\{n_{1},\dots,n_{N}\right\\}$, out of which at least $3$ must be
light. 111In the following, we denote the charged lepton flavor index with a
Greek character, $\alpha=1,2,3$, while the neutral fermion flavor index will
be denoted with a Latin character, $i=1,\dots,N$. We also introduce a massive
scalar $\rho$, a massive pseudoscalar $\sigma$ and a singly charged scalar
$\eta^{\pm}$. All these fields are mass eigenstates. In case more than one
massive scalar, pseudoscalar or charged scalar is present in the model, our
results can be easily adapted by summing over all mass eigenstates. In
addition, we note that at least one pseudoscalar will always be part of the
spectrum, namely the massless majoron. We will consider the following general
interaction Lagrangian:

$\mathcal{L}=\mathcal{L}_{J}+\mathcal{L}_{\rho}+\mathcal{L}_{\sigma}+\mathcal{L}_{\eta}+\mathcal{L}_{Z}+\mathcal{L}_{W}+\mathcal{L}_{S}\,.$
(9)

The first term describes the interaction of the majoron with a pair of
neutrinos or a pair of charged leptons and is given by

$\mathcal{L}_{J}=\bar{n}_{i}\left(A_{ij}P_{R}+A_{ji}^{*}P_{L}\right)\,n_{j}\,J+\bar{\ell}_{\alpha}\,K_{\alpha\alpha}\gamma_{5}\,\ell_{\alpha}\,J\,.$
(10)

As already explained, we have considered only flavor-diagonal couplings
between the majoron and charged leptons (here, $K$ is a purely imaginary
diagonal matrix). The Yukawa interactions of the scalar $\rho$ and the
pseudoscalar $\sigma$ can be written as

$\displaystyle\mathcal{L}_{\rho}$
$\displaystyle=\bar{\ell}_{\alpha}\left(G_{\alpha\beta}P_{L}+G_{\beta\alpha}^{*}P_{R}\right)\,\ell_{\beta}\,\rho\,,$
(11) $\displaystyle\mathcal{L}_{\sigma}$
$\displaystyle=\bar{n}_{i}\left(B_{ij}P_{R}+B_{ji}^{*}P_{L}\right)\,n_{j}\,\sigma+\bar{\ell}_{\alpha}\left(C_{\alpha\beta}P_{L}+C_{\beta\alpha}^{*}P_{R}\right)\,\ell_{\beta}\,\sigma\,.$
(12)

In principle, the scalar $\rho$ can also couple to a pair of neutrinos, but we
have decided to ignore these couplings, since they would not contribute (at
1-loop) to the majoron coupling to charged leptons. We also note that Eq. (12)
reduces to
$\bar{\ell}_{\alpha}\,C_{\alpha\alpha}^{*}\gamma_{5}\,\ell_{\alpha}\,\sigma$,
with $C_{\alpha\alpha}$ purely imaginary, when one considers the interaction
term between the pseudoscalar $\sigma$ and a pair of charged leptons with the
same flavor. Similarly, Eq. (11) reduces to
$\bar{\ell}_{\alpha}\,G_{\alpha\alpha}\,\ell_{\alpha}\,\sigma$, with
$G_{\alpha\alpha}$ purely real, in the flavor diagonal case. The charged
scalar $\eta$ can also have a Yukawa coupling to a charged lepton and a
neutrino, given by

$\mathcal{L}_{\eta}=\bar{\ell}_{\alpha}\left(D_{L}^{\alpha
i}P_{L}+D_{R}^{\alpha i}P_{R}\right)\,n_{i}\,\eta+\text{h.c.}\,.$ (13)

$\mathcal{L}_{Z}$ and $\mathcal{L}_{W}$ contain the usual gauge bosons
interactions with neutrinos and charged leptons, which can be written as [27,
28, 15, 16, 17]

$\displaystyle\mathcal{L}_{Z}$
$\displaystyle=\frac{g}{4\,\cos\theta_{W}}\bar{n}_{i}\,\not{Z}\left(E_{ij}\,P_{L}-E_{ji}\,P_{R}\right)n_{j}+\frac{g}{\,\cos\theta_{W}}\bar{\ell}_{\alpha}\,\not{Z}\left(v_{\ell}-a_{\ell}\gamma_{5}\right)\ell_{\alpha}\,,$
(14) $\displaystyle\mathcal{L}_{W}$
$\displaystyle=\frac{g}{\sqrt{2}}\left(\bar{\ell}_{\alpha}F_{\alpha
i}\not{W}^{-}P_{L}\,n_{i}+\text{h.c.}\right)\,,$ (15)

where $E$ is an Hermitian matrix and $v_{\ell}$ and $a_{\ell}$ are the usual
$Z$-boson couplings to charged leptons. Finally, $\mathcal{L}_{S}$ describes a
cubic scalar interaction of the majoron with one scalar and one pseudoscalar,

$\mathcal{L}_{S}=\omega\,J\,\sigma\,\rho\,,$ (16)

where $\omega$ is a real parameter with dimensions of mass. Sums over charged
lepton and neutrino flavor indices are implicitly assumed in Eqs. (10)-(15).
The generic couplings $A$, $B$, $C$, $D_{L,R}$, $E$, $F$ and $G_{L,R}$ have
model-dependent expressions in terms of the parameters of the specific model
under consideration. Nevertheless, they can generally be written as

$\displaystyle A_{ij}$ $\displaystyle=U_{ki}\,\bar{A}_{kr}\,U_{rj}\,,$ (17)
$\displaystyle B_{ij}$ $\displaystyle=U_{ki}\,\bar{B}_{kr}\,U_{rj}\,,$ (18)
$\displaystyle D_{L}^{\alpha i}$ $\displaystyle=\bar{D}_{L}^{\alpha
k}\,U_{ki}^{*}\,,$ (19) $\displaystyle D_{R}^{\alpha i}$
$\displaystyle=\bar{D}_{R}^{\alpha k}\,U_{ki}\,,$ (20) $\displaystyle E_{ij}$
$\displaystyle=U_{ki}\,\bar{E}_{kr}\,U_{rj}^{*}=\sum_{k=1}^{3}\,U_{ki}\,U_{kj}^{*}\,,$
(21) $\displaystyle F_{\alpha i}$ $\displaystyle=\bar{F}_{\alpha
k}\,U_{ki}^{*}=U_{\alpha i}^{*}\,,$ (22)

where $U$ is the unitary matrix implicitly defined by

$\displaystyle M_{ij}$ $\displaystyle=U_{ki}\,\bar{M}_{kr}\,U_{rj}\,,$ (23)

with $M\equiv\text{diag}\left(m_{n_{1}},\dots,m_{n_{N}}\right)$ and $\bar{M}$
the neutral lepton mass matrices in the mass and gauge bases, respectively. We
have assumed that we work in the charged lepton mass basis. Therefore,
$\bar{A}$, $\bar{B}$, $\bar{D}_{L,R}$, $\bar{E}$ and $\bar{F}$ are the
couplings in the gauge basis, while $C=\bar{C}$ and $G=\bar{G}$. Eqs. (21) and
(22) are completely general, since they just correspond to the transformation
of the usual neutral and charged currents, and the index $i=1,2,3$ in these
equations tags a gauge non-singlet.

###  2.2 1-loop coupling

(a) $\boldsymbol{W}$ boson contribution: $\mathcal{M}_{W}$

(b) $\boldsymbol{Z}$ boson contribution: $\mathcal{M}_{Z}$

(c) $\boldsymbol{\eta}$, $\boldsymbol{\rho}$ and $\boldsymbol{\sigma}$
contributions: $\mathcal{M}_{\eta}+\mathcal{M}_{\rho}+\mathcal{M}_{\sigma}$

(d) $\boldsymbol{\sigma n}$ and $\boldsymbol{\sigma\ell}$ contributions:
$\mathcal{M}_{\sigma n}+\mathcal{M}_{\sigma\ell}$

(e) $\boldsymbol{S}$ contribution: $\mathcal{M}_{S}$

Figure 1: Feynman diagrams leading to the 1-loop coupling of the majoron to a
pair of charged leptons.

Let us now move on to the computation of the 1-loop coupling of the majoron to
a pair of charged leptons. First of all, we consider all possible 1-loop
diagrams leading to this coupling. They are shown in Fig. 1. Other 1-loop
diagrams vanish due to the pseudoscalar nature of the majoron. For instance,
this would be the case for the self-energy diagram in Fig. 1(d) if we replaced
the pseudoscalar $\sigma$ by the scalar $\rho$. It is important to notice a
key aspect. In the scalar diagrams, the fermion in the loop can either be a
neutral or a charged lepton. We shall consider both scenarios. We could in
principle include charged leptons in the $Z$ boson diagram. However, given
that this diagram is always flavor diagonal, such an inclusion would
constitute a correction to the tree-level majoron coupling, thus rendering it
negligible. For this reason we will not consider this option.

The amplitudes of the Feynman diagrams in Fig. 1 will be denoted by
$\mathcal{M}_{A}$, with $A=\left\\{W,Z,\eta,\rho,\sigma,\sigma
n,\sigma\ell,S\right\\}$, and each amplitude corresponds to a different
diagram. They are given by

$\displaystyle\mathcal{M}_{W}=$ $\displaystyle
2\,\bar{v}_{\ell_{\beta}}\sum_{i,j}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{g}{\sqrt{2}}\,F_{\beta
j}\,\gamma_{\mu}P_{L}\frac{1}{k^{2}-M_{W}^{2}}\left(g^{\mu\nu}-\frac{q^{\mu}q^{\nu}}{M_{W}^{2}}\right)\frac{\xout{k}-\xout{r}+m_{j}}{(k-r)^{2}-m_{j}^{2}}$
$\displaystyle\left(A_{ij}P_{R}+A_{ji}^{*}P_{L}\right)\,\frac{\xout{k}+\xout{p}+m_{i}}{(k+p)^{2}-m_{i}^{2}}\frac{g}{\sqrt{2}}\,\gamma_{\nu}P_{L}\,F_{\alpha
i}^{*}\,u_{\ell_{\alpha}}\,,$ (24) $\displaystyle\mathcal{M}_{Z}=$
$\displaystyle-2\frac{g}{\cos\theta_{W}}\bar{v}_{\ell_{\beta}}\gamma_{\mu}\left(v_{\ell}-a_{\ell}\gamma_{5}\right)\,u_{\ell_{\alpha}}\frac{1}{-M_{Z}^{2}}\left(g^{\mu\nu}-\frac{q^{\mu}q^{\nu}}{M_{Z}^{2}}\right)$
$\displaystyle\sum_{i,j}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{g}{4\cos\theta_{W}}\,\text{Tr}\left[\gamma_{\nu}\left(E_{ij}\,P_{L}-E_{ji}\,P_{R}\right)\,\frac{\xout{k}+\xout{q}+m_{j}}{(k+q)^{2}-m_{j}^{2}}\left(A_{ij}P_{R}+A_{ji}^{*}P_{L}\right)\frac{\xout{k}+m_{i}}{k^{2}-m_{i}^{2}}\right]\,,$
(25) $\displaystyle\mathcal{M}_{\eta}=$
$\displaystyle\,2\sum_{i,j}\bar{v}_{\ell_{\beta}}\int\frac{d^{4}k}{(2\pi)^{4}}\,\left(D_{L}^{\beta
j}P_{L}+D_{R}^{\beta
j}P_{R}\right)\,\frac{\xout{k}-\xout{r}+m_{j}}{(k-r)^{2}-m_{j}^{2}}\left(A_{ij}P_{R}+A_{ji}^{*}P_{L}\right)$
$\displaystyle\frac{\xout{k}+\xout{p}+m_{i}}{(k+p)^{2}-m_{i}^{2}}\,\left[\left(D_{L}^{\alpha
i}\right)^{*}P_{R}+\left(D_{R}^{\alpha
i}\right)^{*}P_{L}\right]\,\frac{1}{k^{2}-m_{\eta}^{2}}\,u_{\ell_{\alpha}}\,,$
(26) $\displaystyle\mathcal{M}_{\rho}=$
$\displaystyle\,\sum_{i,j}\bar{v}_{\ell_{\beta}}\int\frac{d^{4}k}{(2\pi)^{4}}\,\left(G_{\beta\gamma}P_{L}+G_{\gamma\beta}^{*}P_{R}\right)\,\frac{\xout{k}-\xout{r}+m_{\ell_{\gamma}}}{(k-r)^{2}-m_{\ell_{\gamma}}^{2}}K_{\gamma\gamma}\gamma_{5}\frac{\xout{k}+\xout{p}+m_{\ell_{\gamma}}}{(k+p)^{2}-m_{\ell_{\gamma}}^{2}}$
$\displaystyle\,\left(G_{\gamma\alpha}P_{L}+G_{\alpha\gamma}^{*}P_{R}\right)\,\frac{1}{k^{2}-m_{\rho}^{2}}\,u_{\ell_{\alpha}}\,,$
(27) $\displaystyle\mathcal{M}_{\sigma}=$
$\displaystyle\mathcal{M}_{\rho}\left(G\leftrightarrow C\right)\,,$ (28)
$\displaystyle\mathcal{M}_{\sigma n}=$
$\displaystyle-2\sum_{i,j}\int\frac{d^{4}k}{(2\pi)^{4}}\,\text{Tr}\left[\left(B_{ij}P_{R}+B_{ji}^{*}P_{L}\right)\,\frac{\xout{k}+\xout{q}+m_{j}}{(k+q)^{2}-m_{j}^{2}}\,\left(A_{ij}P_{R}+A_{ji}^{*}P_{L}\right)\,\frac{\xout{k}+m_{i}}{k^{2}-m_{i}^{2}}\right]$
$\displaystyle\bar{v}_{\ell_{\beta}}\,\left(C_{\beta\alpha}P_{L}+C_{\alpha\beta}^{*}P_{R}\right)\,\frac{1}{-m_{\sigma}^{2}}u_{\ell_{\alpha}}\,,$
(29) $\displaystyle\mathcal{M}_{\sigma\ell}=$
$\displaystyle-\sum_{i,j}\int\frac{d^{4}k}{(2\pi)^{4}}\,\text{Tr}\left[K_{\gamma\gamma}\gamma_{5}\,\frac{\xout{k}+\xout{q}+m_{\ell_{\gamma}}}{(k+q)^{2}-m_{\ell_{\gamma}}^{2}}\,C_{\gamma\gamma}^{*}\gamma_{5}\,\frac{\xout{k}+m_{\ell_{\gamma}}}{k^{2}-m_{\ell_{\gamma}}^{2}}\right]$
$\displaystyle\bar{v}_{\ell_{\beta}}\,\left(C_{\beta\alpha}P_{L}+C_{\alpha\beta}^{*}P_{R}\right)\,\frac{1}{-m_{\sigma}^{2}}\,u_{\ell_{\alpha}}\,,$
(30) $\displaystyle\mathcal{M}_{S}=$
$\displaystyle\,\bar{v}_{\ell_{\beta}}\int\frac{d^{4}k}{(2\pi)^{4}}\,\left(G_{\beta\gamma}P_{L}+G_{\gamma\beta}^{*}P_{R}\right)\frac{\xout{k}+m_{\ell_{\gamma}}}{k^{2}-m_{\ell_{\gamma}}^{2}}\left(C_{\gamma\alpha}P_{L}+C_{\alpha\gamma}^{*}P_{R}\right)$
$\displaystyle\frac{1}{\left(k-r\right)^{2}-m_{\rho}^{2}}\frac{1}{\left(k+p\right)^{2}-m_{\sigma}^{2}}\,\omega\,u_{\ell_{\alpha}}\,.$
(31)

The amplitudes in the previous equations have been computed with the help of
Package X [29]. Since the resulting exact analytical expressions are very
involved, we have derived approximate expressions, valid in most scenarios of
interest. We now explain the main ingredients in our calculation:

  * •

A hierarchy of scales has been assumed, namely

$m_{\rm light}\ll m_{\rm EW}\ll M_{H}\,.$ (32)

Here $m_{\rm light}$ is a low-energy scale, to be identified with the scale of
neutrino masses or with other related low mass scales, $m_{\rm EW}\sim m_{W}$
is the usual electroweak scale and $M_{H}$ is a high-energy scale where the
heavy mediators responsible for neutrino masses lie. This hierarchy of scales
allows us to expand our results in powers of the small mass ratios $m_{\rm
light}/m_{\rm EW}$ and $m_{\rm EW}/M_{H}$ and obtain approximate (but also
simpler) analytical expressions.

  * •

As usual, all states at a given energy scale will be assumed to be degenerate
at order zero in $m_{\rm light}$. For instance, all the heavy mediators will
be assumed to have a mass $M_{H}$, with possible corrections of order $m_{\rm
light}$. In practice, this means that $m_{\rm light}/m_{\rm EW}$ and $m_{\rm
EW}/M_{H}$ will be single expansion parameters and our approximations may be
invalid if large mass splittings exist among the states that lie at a given
energy scale.

  * •

In order to properly apply the mass hierarchy in Eq. (32), we split the sum
over all neutral fermion mass eigenstates as

$\sum_{i,j}=\sum_{i,j\sim l}+\sum_{i,j\sim h}+\sum_{i\sim l}\sum_{j\sim
h}+\sum_{j\sim l}\sum_{i\sim h}\,~{},$ (33)

where $\sim l$ and $\sim h$ refer to the sum over light and heavy neutral
fermions, respectively.

  * •

The couplings in the interaction vertices may include the unitary matrix $U$,
see Eqs. (17)-(22). This must be taken into account to properly expand our
results, since the entries of the $U$ matrix involve the mass ratios that we
use as expansion parameters. In fact, we will make use of the identity

$\sum_{j\sim h}U_{rj}\,m_{j}\,U_{kj}=\bar{M}^{\dagger}_{rk}-\sum_{j\sim
l}U_{rj}\,m_{j}\,U_{kj}\,,$ (34)

which can be readily derived from the definition of $U$. This expression
allows us to identify up to which order in $m_{\rm light}$ we must expand our
analytical expressions in order to be fully consistent. The first term,
$\bar{M}^{\dagger}_{rk}$, is much larger than the neutrino mass scale, while
$\sum_{j\sim l}U_{rj}\,m_{j}\,U_{kj}\sim m_{\nu}$ is of order one in $m_{\rm
light}$. Therefore, if the model under consideration predicts
$\bar{M}_{kr}\neq 0$, then the part proportional to the light neutrino masses
does not contribute at leading order. In contrast, if $\bar{M}_{kr}=0$, the
dominant term will be proportional to $m_{\rm light}$, and we must expand up
to this order. In general, when we expand the integrals using Eq. (33), we
find, mostly, that we must compare expressions of the form $\sum_{j\sim
h}U_{rj}\,m_{j}\,U_{kj}$ with $\sum_{j\sim l}U_{rj}\,m_{j}\,U_{kj}$. A priori,
one can think that the first one is dominant because $m_{j}$ is much larger if
$j\sim h$, rather than when $j\sim l$. However, we do not know the shape of
the $U$ matrix. In some models, the suppression introduced by $U$ in the heavy
block compensates for the suppression of the light mass. Ultimately, we cannot
compare these expressions. For this reason, we use Eq. (34) intensively to
transform these expressions into comparable ones. 222The exception to this
rule takes place when the heavy mass appears in the denominator, $\sum_{j\sim
h}U_{rj}\,m_{j}^{-1}\,U_{kj}$. In this case, it does not make sense to perform
this transformation, as we would artificially introduce the dominant term into
the total sum, and this will cancel out with the sum over the light states.
Therefore, we will include it in the zeroth order, as it is not explicitly
proportional to any power of the light mass. The actual order will depend on
the model under consideration. Then, given a model, one can certainly derive
the form of the light neutrino mass matrix, and therefore, upon substitution,
determine the order of this term.

  * •

We have simplified the results and written them using matrix notation. In this
process, sums over repeated indices have been identified as matrix products
whenever possible.

  * •

The new scalar states will be assumed to be much heavier than $m_{\rm light}$
and all charged leptons.

Before we present our results it proves convenient to introduce some matrices
that will allow us to write them in a more compact way:

$\displaystyle\Gamma_{spj}^{n,m,t}$
$\displaystyle\equiv\sum_{k,r}\bar{A}_{kr}\left[\left(\bar{M}^{\dagger}\right)^{n}\left(\bar{M}\bar{M}^{\dagger}\right)^{m}\right]_{pk}U_{rj}m_{j}^{t}U_{sj}^{*}\,,$
(35) $\displaystyle\tilde{\Gamma}_{spj}^{n,m,t}$
$\displaystyle\equiv\sum_{k,r}\bar{A}_{kr}\left[\left(\bar{M}^{\dagger}\right)^{n}\left(\bar{M}\bar{M}^{\dagger}\right)^{m}\right]_{sr}U_{kj}m_{j}^{t}U_{pj}^{*}\,,$
(36) $\displaystyle\Delta_{spj}^{n,m,t}$
$\displaystyle\equiv\sum_{k,r}\bar{A}_{kr}\left[\left(\bar{M}^{\dagger}\right)^{n}\left(\bar{M}\bar{M}^{\dagger}\right)^{m}\right]_{pk}U_{rj}m_{j}^{t}U_{sj}\,,$
(37) $\displaystyle\tilde{\Delta}_{spj}^{n,m,t}$
$\displaystyle\equiv\sum_{k,r}\bar{A}_{kr}\left[\left(\bar{M}^{\dagger}\right)^{n}\left(\bar{M}\bar{M}^{\dagger}\right)^{m}\right]_{sr}U_{kj}m_{j}^{t}U_{pj}\,,$
(38) $\displaystyle\gamma_{sp}$ $\displaystyle\equiv\sum_{k,r}\sum_{i,j\sim
l}\bar{A}_{kr}^{*}U_{kj}^{*}U_{sj}U_{ri}^{*}U_{pi}\,.$ (39)

We are now in position to present our results. We will do so explicitly
splitting them into different orders of $m_{\rm light}$. The leading order
contributions, as well as the divergent pieces (when present), will be
provided here. However, one should keep in mind that this splitting in orders
of $m_{\rm light}$ is performed based on the explicit appearance of $m_{\rm
light}$ in the resulting analytical expressions. As explained above, a term
that is a priori of order zero in $m_{\rm light}$ might actually be of order
one if the specific shape of the $U$ matrix introduces one power of the low
scale. Therefore, the leading order contributions given below might be
polluted with higher order terms, which can in general be safely neglected.
This is no longer true if the order zero contributions vanish or turn out to
be of order one due to cancellations. In this case, a consistent calculation
requires the consideration of all order one contributions. For this reason,
the expressions for higher order terms are given in Appendix A.

First of all, the $Z$ boson contribution can be written as

$\mathcal{M}_{Z}=\frac{1}{8\pi^{2}}\,\bar{v}_{\ell_{\alpha}}\gamma_{5}\,\Gamma_{Z}^{\alpha}\,u_{\ell_{\alpha}}\,,$
(40)

where

$\displaystyle\Gamma_{Z}^{\alpha}$
$\displaystyle=\,i\,\frac{m_{\ell_{\alpha}}}{v^{2}}\,\textup{Im}\Big{\\{}\left(\Gamma_{Z}\right)^{\rm(div)}+\left(\Gamma_{Z}\right)^{(0)}+\mathcal{O}(m_{\rm
light})\Big{\\}}\,,$ (41)

with $\langle H^{0}\rangle=\frac{v}{\sqrt{2}}$ the usual Higgs vacuum
expectation value (VEV) that breaks the electroweak symmetry and

$\displaystyle\left(\Gamma_{Z}\right)^{\rm(div)}$
$\displaystyle=\sum_{s=1}^{3}\delta_{ks}\,\bar{A}_{kr}\left[\bar{M}^{\dagger}_{rk}\left(\frac{1}{\epsilon}+\log\left(\frac{\mu^{2}}{M_{H}^{2}}\right)-\frac{5}{3}\right)-M_{H}\sum_{j\sim
h}U_{kj}U_{rj}\right]\,,$ (42) $\displaystyle\left(\Gamma_{Z}\right)^{(0)}$
$\displaystyle=\sum_{s=1}^{3}\left(\sum_{j\sim
l}\frac{\tilde{\Gamma}_{ssj}^{1,0,0}+\Gamma_{ssj}^{1,0,0}}{6}-\sum_{j\sim
h}\frac{\tilde{\Delta}_{ssj}^{0,1,-1}+\Delta_{ssj}^{0,1,-1}}{3}\right)\,.$
(43)

The term $\left(\Gamma_{Z}\right)^{\rm(div)}$ contains the dimensional
regularization divergence $\displaystyle\frac{1}{\epsilon}$. It will only be
relevant in models leading to $\delta_{ks}\,\bar{A}_{kr}\neq 0$. Since the
index $s=1,2,3$ runs over gauge non-singlet states, this will happen in models
in which the majoron couples to non-singlet fermion representations and in
this case one expects a majoron coupling to charged leptons already at tree-
level. We have also included a finite piece in
$\left(\Gamma_{Z}\right)^{\rm(div)}$, the last term in Eq. (42), simply
because it is also proportional to $\delta_{ks}\,\bar{A}_{kr}$ and vanishes
whenever the divergent piece does. Furthermore, the rest of the contributions
can be generally written as

$\mathcal{M}_{B}=\frac{1}{8\pi^{2}}\,\bar{v}_{\ell_{\beta}}\left(L_{B}^{\beta\alpha}\,P_{L}+R_{B}^{\beta\alpha}\,P_{R}\right)u_{\ell_{\alpha}}\,,$
(44)

with $B=\left\\{W,\eta,\rho,\sigma,\sigma n,\sigma\ell,S\right\\}$. The
$W$-boson contributions are given by

$\displaystyle L_{W}^{\beta\alpha}$
$\displaystyle=\frac{2\,m_{\ell_{\beta}}}{v^{2}}\,\left\\{\left(L_{W}^{\beta\alpha}\right)^{(0)}+\mathcal{O}(m_{\rm
light})\right\\}\,,$ (45)

with

$\displaystyle\left(L_{W}^{\beta\alpha}\right)^{(0)}$
$\displaystyle=\sum_{j\sim l}\left(\frac{\Gamma_{\alpha\beta
j}^{1,0,0\,*}}{12}+\frac{2}{3}\Gamma_{\beta\alpha
j}^{1,0,0}\right)-\sum_{j\sim h}\left(\frac{\tilde{\Delta}_{\alpha\beta
j}^{0,1,-1\,*}}{6}+\frac{7}{12}\tilde{\Delta}_{\beta\alpha
j}^{0,1,-1}\right)\,.$ (46)

The charged scalar contribution is given by

$\displaystyle L_{\eta}^{\beta\alpha}=$
$\displaystyle\,m_{\ell_{\alpha}}\left[\bar{D}_{L}^{\beta
p}\left(\bar{D}_{L}^{\alpha
s}\right)^{*}\left(L_{\eta}^{LL}\right)^{*}_{sp}-\left(\bar{D}_{L}^{\alpha
p}\right)^{*}\bar{D}_{L}^{\beta
s}\left(\widetilde{L}_{\eta}^{LL}\right)_{sp}\right]$
$\displaystyle+m_{\ell_{\beta}}\left[\bar{D}_{R}^{\beta
p}\left(\bar{D}_{R}^{\alpha
s}\right)^{*}\left(L_{\eta}^{RR}\right)^{*}_{sp}-\left(\bar{D}_{R}^{\alpha
p}\right)^{*}\bar{D}_{R}^{\beta
s}\left(\widetilde{L}_{\eta}^{RR}\right)_{sp}\right]$ (47)
$\displaystyle+\bar{D}_{L}^{\beta p}\left(\bar{D}_{R}^{\alpha
s}\right)^{*}\left(L_{\eta}^{RL}\right)^{*}_{sp}\,,$

where

$\displaystyle L_{\eta}^{LL}=$
$\displaystyle\left(L_{\eta}^{LL}\right)^{(0)}+\mathcal{O}(m_{\rm light})\,,$
(48) $\displaystyle L_{\eta}^{RR}=$
$\displaystyle\left(L_{\eta}^{RR}\right)^{(0)}+\mathcal{O}(m_{\rm light})\,,$
(49) $\displaystyle L_{\eta}^{RL}=$
$\displaystyle\left(L_{\eta}^{RL}\right)^{(0)}+\mathcal{O}(m_{\rm light})\,,$
(50) $\displaystyle\widetilde{L}_{\eta}^{LL}=$ $\displaystyle
L_{\eta}^{LL}\,\left(f_{\left(1,\,2,\,5,\,6,\,7,\,8\right)}\leftrightarrow
f_{\left(3,\,4,\,9,\,10,\,15,\,16\right)},\>f_{13}\leftrightarrow
F_{1,-3},\>f_{14}\leftrightarrow F_{2,-4}\right)\,,$ (51)
$\displaystyle\widetilde{L}_{\eta}^{RR}=$ $\displaystyle
L_{\eta}^{RR}\,\left(f_{\left(1,\,2,\,5,\,6,\,7,\,8\right)}\leftrightarrow
f_{\left(3,\,4,\,9,\,10,\,15,\,16\right)},\>f_{13}\leftrightarrow
F_{1,-3},\>f_{14}\leftrightarrow F_{2,-4}\right)\,.$ (52)

Here we have introduced the $f$ and $F$ loop functions. Their explicit
expressions can be found in Appendix B. We note that in the $Z$ and $W$ bosons
contributions given above, no loop functions appeared. This is because
$m_{Z}\sim m_{W}\sim m_{\rm EW}$ and we already expanded our results in powers
of $m_{\rm EW}/M_{H}$. However, in contributions involving $\eta$, $\rho$ and
$\sigma$ a new mass scale appears, namely the mass of the new heavy scalar.
Then, a complete expansion cannot be made without introducing further
assumptions on their mass scale. As already explained above, our calculation
assumes $m_{\eta},m_{\rho},m_{\sigma}\gg m_{\rm light}$ but, for the sake of
generality, we prefer to stay agnostic about how these scalar masses compare
to $M_{H}$. Furthermore, the notation
$f_{\left(i_{1}\,,...,\,i_{n}\right)}\leftrightarrow
f_{\left(j_{1}\,,...,\,j_{n}\right)}$ means that the replacement
$f_{i_{1}}\leftrightarrow f_{j_{1}}$, …, $f_{i_{n}}\leftrightarrow f_{j_{n}}$
is to be applied. We found

$\displaystyle\left(L_{\eta}^{LL}\right)^{(0)}=$
$\displaystyle\left(L_{\eta}^{RR}\right)^{(0)}\left(\Gamma\leftrightarrow\tilde{\Gamma},\Delta\leftrightarrow\tilde{\Delta}\right)\,,$
(53) $\displaystyle\left(L_{\eta}^{RR}\right)^{(0)}_{sp}=$ $\displaystyle
f_{7}\sum_{j\sim
h}\Delta^{0,1,-1}_{spj}+\sum_{j}\left\\{f_{8}\,\Delta^{0,1,1}_{spj}-F_{5,7}\,\tilde{\Gamma}^{1,0,0}_{spj}-F_{6,8}\,\tilde{\Gamma}^{1,1,0}_{spj}\right\\}$
$\displaystyle+$ $\displaystyle\sum_{j\sim
l}\Big{\\{}F_{5,7,-1}\,\tilde{\Gamma}^{1,0,0}_{spj}+F_{6,8,-2}\,\tilde{\Gamma}^{1,1,0}_{spj}\Big{\\}}\,,$
(54) $\displaystyle\left(L_{\eta}^{RL}\right)^{(0)}_{sp}=$
$\displaystyle\gamma_{sp}^{*}\frac{m_{\ell_{\beta}}^{2}+m_{\ell_{\alpha}}^{2}}{2m_{\eta}^{2}}-\gamma_{sp}\frac{m_{\ell_{\beta}}m_{\ell_{\alpha}}}{2m_{\eta}^{2}}+f_{11}\sum_{j\sim
h}\Delta_{spj}^{1,1,-1}$ $\displaystyle+$
$\displaystyle\sum_{j}\left[-f_{10}\left(\tilde{\Gamma}_{psj}^{0,2,0}\right)^{*}-f_{9}\left(\tilde{\Gamma}_{psj}^{0,1,0}\right)^{*}-F_{3,11}\,\tilde{\Delta}_{spj}^{1,0,1}+f_{12}\,\Delta_{spj}^{1,1,1}-F_{4,12}\,\tilde{\Delta}_{spj}^{1,1,1}\right]$
$\displaystyle+$ $\displaystyle\sum_{j\sim
l}\left[F_{9,-1}\left(\tilde{\Gamma}_{psj}^{0,1,0}\right)^{*}+F_{10,-2}\left(\tilde{\Gamma}_{psj}^{0,2,0}\right)^{*}+F_{3,-9}\left(\Gamma_{psj}^{0,1,0}\right)^{*}+F_{4,-10}\left(\Gamma_{psj}^{0,2,0}\right)^{*}\right]\,.$
(55)

Let us note that we have included a term proportional not to the mass of
neutrinos but to the square of the mass of charged leptons in the zeroth order
expressions. Although this term is not explicitly proportional to $m_{\rm
light}$, we expect it to be subdominant, simply because $m_{\ell}\ll M_{H}$.
However, in the event that both remaining terms of order zero and order one
were to cancel out, then this term would dominate over a possible order two,
as $m_{\rm light}\ll m_{\ell}$. We move on to the contribution induced by the
neutral scalar $\rho$, which can be written as

$\displaystyle L_{\rho}^{\beta\alpha}$
$\displaystyle=\frac{1}{2m_{\rho}^{2}}\left[\left(L_{\rho}^{\beta\alpha}\right)^{\rm(div)}+\left(L_{\rho}^{\beta\alpha}\right)^{(0)}\right]\,,$
(56)

with

$\displaystyle\left(L_{\rho}^{\beta\alpha}\right)^{\rm(div)}$
$\displaystyle=-\frac{1}{2}\left[\left(m_{\ell_{\beta}}^{2}+m_{\ell_{\alpha}}^{2}\right)+2m_{\rho}^{2}\left(1+\frac{1}{\epsilon}+\log\frac{\mu^{2}}{m_{\rho}^{2}}\right)\right]\left(GKG\right)_{\beta\alpha}\,,$
(57) $\displaystyle\left(L_{\rho}^{\beta\alpha}\right)^{(0)}$
$\displaystyle=m_{\ell_{\beta}}\left[\left(G^{\dagger}\left(\mathbb{I}_{3}+\log\frac{M_{\ell}^{2}}{m_{\rho}^{2}}\right)M_{\ell}KG\right)_{\beta\alpha}-\text{h.c.}\right]-\left(G\log\frac{M_{\ell}^{2}}{m_{\rho}^{2}}M_{\ell}^{2}KG\right)_{\beta\alpha}$
$\displaystyle+\frac{m_{\ell_{\alpha}}m_{\ell_{\beta}}}{2}\left(G^{\dagger}KG^{\dagger}\right)_{\beta\alpha}\,.$
(58)

where $\mathbb{I}_{n}$ is the $n\times n$ identity matrix. Again, we have
included a finite piece in $\left(L_{\rho}^{\beta\alpha}\right)^{\rm(div)}$
because it vanishes whenever the divergent piece does. A similar triangle
diagram contribution is induced by the pseudoscalar $\sigma$, with

$\displaystyle L_{\sigma}^{\beta\alpha}$
$\displaystyle=L_{\rho}^{\beta\alpha}\left(G\leftrightarrow
C,m_{\rho}\leftrightarrow m_{\sigma}\right)\,.$ (59)

Note that the expressions in Eqs. (56) and (59) are exactly of order zero (no
expansion has been made). This is easy to understand since there are no
neutrinos in the associated loops. Lastly, the $\sigma n$ and $\sigma\ell$
contributions are given by

$\displaystyle L_{\sigma n}^{\beta\alpha}$
$\displaystyle=\frac{1}{m_{\sigma}^{2}}\,C_{\beta\alpha}\left[\left(L_{\sigma
n}\right)^{(\text{div})}+\left(L_{\sigma n}\right)^{(0)}+\mathcal{O}(m_{\rm
light})\right]\,,$ (60)

with

$\displaystyle\left(L_{\sigma n}\right)^{(\text{div})}$
$\displaystyle=\text{Re}\left\\{\sum_{j}\left[\bar{B}_{sp}\Delta_{psj}^{1,0,1}+\bar{B}_{sp}^{*}\left(\Gamma_{spj}^{0,1,0}+\tilde{\Gamma}_{spj}^{0,1,0}\right)\right]\left(3+2\,\frac{1}{\epsilon}+2\,\log\frac{\mu^{2}}{M_{H}^{2}}\right)\right\\}\,,$
(61) $\displaystyle\left(L_{\sigma n}\right)^{(0)}$
$\displaystyle=\text{Re}\Bigg{\\{}\sum_{j}2\bar{B}_{sp}^{*}\left[\tilde{\Gamma}_{spj}^{0,1,0}-\Gamma_{spj}^{0,1,0}-\frac{1}{M_{H}^{2}}\left(\tilde{\Gamma}_{spj}^{0,2,0}+\tilde{\Gamma}_{spj}^{0,1,2}\right)\right]-\frac{2}{M_{H}^{2}}\bar{B}_{sp}\tilde{\Delta}_{psj}^{1,1,1}$
$\displaystyle-\sum_{j\sim
l}[\bar{B}_{sp}^{*}\left(\frac{2}{M_{H}^{2}}\Gamma_{spj}^{0,2,0}-3\Gamma_{spj}^{0,1,0}+\tilde{\Gamma}_{spj}^{0,1,0}\right)-\sum_{j\sim
h}\bar{B}_{sp}\Delta_{psj}^{1,1,-1}\Bigg{\\}}\,,$ (62)

and

$\displaystyle L_{\sigma\ell}^{\beta\alpha}$
$\displaystyle=-\frac{1}{m_{\sigma}^{2}}\,C_{\beta\alpha}\,\text{Tr}\left[\left(L_{\sigma\ell}\right)^{\rm(div)}+\left(L_{\sigma\ell}\right)^{(0)}\right]\,,$
(63)

with

$\displaystyle\left(L_{\sigma\ell}\right)^{\rm(div)}$
$\displaystyle=KM_{\ell}^{2}C^{\dagger}\left(I+\frac{1}{\epsilon}\right)\,$
(64) $\displaystyle\left(L_{\sigma\ell}\right)^{(0)}$
$\displaystyle=KM_{\ell}^{2}\log\frac{\mu^{2}}{M_{\ell}^{2}}C^{\dagger}\,.$
(65)

The cubic scalar interaction contribution leads to

$\displaystyle
L_{S}^{\beta\alpha}=\frac{\omega}{4\left(m_{\rho}^{2}-m_{\sigma}^{2}\right)^{2}}\left(L_{\sigma\rho}^{\beta\alpha}\right)^{(0)}\,,$
(66)

with

$\displaystyle\left(L_{S}^{\beta\alpha}\right)^{(0)}$
$\displaystyle=\left(G\,C^{\dagger}\,M_{\ell}\right)_{\beta\alpha}\left(-m_{\rho}^{2}+m_{\sigma}^{2}+m_{\rho}^{2}\log\frac{m_{\rho}^{2}}{m_{\sigma}^{2}}\right)-\left(M_{\ell}G^{\dagger}\,C\right)_{\beta\alpha}\left(-m_{\rho}^{2}+m_{\sigma}^{2}+m_{\sigma}^{2}\log\frac{m_{\rho}^{2}}{m_{\sigma}^{2}}\right)$
$\displaystyle-2\left(G\,M_{\ell}\,C\right)\left(m_{\rho}^{2}-m_{\sigma}^{2}\right)\log\frac{m_{\rho}^{2}}{m_{\sigma}^{2}}\,.$
(67)

Again, the last three expressions are exactly of order zero in $m_{\rm light}$
since there are no neutrinos in the loop. We also note that only the real
parts contribute in Eqs. (61) and (62). To conclude, the right chirality
couplings are given by

$R_{B}^{\beta\alpha}=L_{B}^{\alpha\beta\,\ast}\,.$ (68)

Finally, the majoron coupling to a pair of charged leptons, defined in Eq.
(1), can be written in terms of the results presented in this Section as

$S^{\beta\alpha}=\frac{1}{8\pi^{2}}\left(\delta^{\beta\alpha}\,\Gamma_{Z}^{\alpha}+\sum_{B}L_{B}^{\beta\alpha}\right)\,.$
(69)

Some comments are in order. First of all, we emphasize that these expressions
are very long because they are meant to be completely general. In specific
models, most of the terms will simply vanish or take very simple forms, as we
will show explicitly in Sec. 3. Secondly, while most contributions are
perfectly finite, some include divergent pieces. As already explained, these
only appear in contributions that would necessarily lead to the existence of a
majoron coupling to a pair of charged leptons already at tree-level. In this
case they are expected to induce just small (finite) corrections, but we
included them for the sake of completeness.

##  3 Application to specific models

In this Section we show several example models and illustrate how our general
results for the majoron coupling to charged leptons apply to these specific
cases. We consider several neutrino mass models, some of them very well known.
Instead of breaking lepton number explicitly, as usually done, we promote it
to a global conserved $\mathrm{U(1)}_{L}$ symmetry that gets spontaneously
broken by the vacuum expectation value (VEV) of a scalar $\chi$. This implies
the presence of a majoron in the particle spectrum of the theory.

###  3.1 Type-I seesaw

field | spin | generations | $\mathrm{SU(3)}_{c}$ | $\mathrm{SU(2)}_{L}$ | $\mathrm{U(1)}_{Y}$ | $\mathrm{U(1)}_{L}$
---|---|---|---|---|---|---
$N$ | $\frac{1}{2}$ | 3 | $\mathbf{1}$ | $\mathbf{1}$ | 0 | 1
$\chi$ | 0 | 1 | $\mathbf{1}$ | $\mathbf{1}$ | 0 | -2
Table 1: New particles in the type-I seesaw with spontaneous lepton number
violation.

The type-I seesaw with spontaneous lepton number violation [2] extends the SM
particle content with 3 singlet fermions $N$ and a scalar singlet $\chi$, all
charged under the global $\mathrm{U(1)}_{L}$ as shown in Tab. 1. The Yukawa
terms relevant for our discussion are

$-\mathcal{L}=y\,\bar{L}\tilde{H}N+\frac{1}{2}\,\lambda\,\chi\,\bar{N}^{c}N+\text{h.c.}\,.$
(70)

Here $H$ is the SM Higgs doublet, with $\tilde{H}=i\sigma_{2}H^{*}$, $y$ a
general $3\times 3$ matrix and $\lambda$ a symmetric $3\times 3$ matrix. We
will work in the basis in which $\lambda$ is a diagonal matrix with real
entries. The $\chi$ singlet can be decomposed as

$\chi=\frac{1}{\sqrt{2}}\left(v_{\chi}+\rho+iJ\right)\,,$ (71)

where $\left\langle\chi\right\rangle=\frac{v_{\chi}}{\sqrt{2}}$ is the $\chi$
VEV responsible for the breaking of $\mathrm{U(1)}_{L}$, $\rho$ is a massive
CP-even scalar and $J$ is the massless majoron, the Goldstone boson associated
to the spontaneous violation of lepton number. After symmetry breaking, a
Dirac mass term, $M_{D}$, that mixes the left-handed neutrinos in the lepton
doublets with the singlet neutrinos, as well as Majorana mass term for the
singlet neutrinos, $M_{R}$, are induced. They are given by

$M_{D}=y\,\frac{v}{\sqrt{2}}\,,\quad
M_{R}=\lambda\,\frac{v_{\chi}}{\sqrt{2}}\,.$ (72)

In the basis $\left\\{\nu_{L}^{c},N\right\\}$, the resulting $6\times 6$
Majorana mass matrix for the neutral fermions can be written as

$\bar{M}=\begin{pmatrix}0&M_{D}\\\ M_{D}^{T}&M_{R}\end{pmatrix}\,.$ (73)

If one assumes the hierarchy $M_{D}\ll M_{R}$ (equivalent to $m_{\rm EW}\ll
M_{H}$ in Eq. (32)), the light neutrinos mass matrix is given by
$m_{\nu}=-M_{D}\,M_{R}^{-1}\,M_{D}^{T}$, hence leading to naturally suppressed
masses. This is the usual type-I seesaw mechanism [30, 31, 32, 33, 34].

Since $\rho$ does not couple to the charged leptons at tree-level, the only
diagrams that induce a majoron coupling to charged leptons are those with
gauge bosons, shown in Fig. 1(a) and 1(b). Therefore, we only need to adapt
the general results for $\Gamma_{Z}$ and $L_{W}$ to the type-I seesaw. We
first identify the general couplings of Eqs. (17)-(22) that are involved in
these contributions. In this model, the majoron coupling to a pair a neutral
fermions is given in the gauge basis by

$\bar{A}_{kr}=\left\\{\begin{array}[]{cl}\displaystyle\frac{i}{2v_{\chi}}\,\bar{M}_{kr}\,,&\text{if}\,\,k,r=4,5,6\\\
&\\\ \displaystyle 0\,,&\text{otherwise}\end{array}\right.$ (74)

The $A$, $E$ and $F$ couplings in the mass basis are then given by Eqs. (17),
(21) and (22), although they are not necessary for our computation. First, we
note that $\left(\Gamma_{Z}\right)^{\rm(div)}$ vanishes exactly since
$\delta_{ks}\bar{A}_{kr}=0$ for $s=1,2,3$. Then, we just need to compute
$\sum_{j\sim l}\tilde{\Gamma}_{\beta\alpha j}^{1,0,0}$, $\sum_{j\sim
l}\Gamma_{\beta\alpha j}^{1,0,0}$, $\sum_{j\sim h}\Delta_{\beta\alpha
j}^{0,1,-1}$ and $\sum_{j\sim h}\tilde{\Delta}_{\beta\alpha j}^{0,1,-1}$. In
order to do that we need to know the form of the $U$ matrix. In the type-I
seesaw this unitary matrix can be written as [35]

$U=\begin{pmatrix}U_{l}&0\\\
0&U_{h}\end{pmatrix}\,\begin{pmatrix}\sqrt{\mathbb{I}_{3}-PP^{\dagger}}&P\\\
-P^{\dagger}&\sqrt{\mathbb{I}_{3}-P^{\dagger}P}\end{pmatrix}\equiv
U_{2}\,U_{1}\,.$ (75)

Here $U_{1}$ brings the neutral fermions mass matrix into a block-diagonal
form, while $U_{2}$ finally diagonalizes, independently, the light and heavy
sectors of the matrix. $U_{l}$, $U_{h}$ and $P$ are $3\times 3$ matrices. The
matrix $P$ depends on the matrices $M_{D}$ and $M_{R}$ in a non-trivial way.
However, one can expand $P$ in inverse powers of the large $M_{R}$ scale as

$P=\sum_{i=1}^{\infty}P_{i}\,,$ (76)

with $P_{i}\propto M_{R}^{-i}$. At leading order in $M_{R}^{-1}$, one finds

$\displaystyle\sqrt{\mathbb{I}_{3}-PP^{\dagger}}=\sqrt{\mathbb{I}_{3}-P^{\dagger}P}=\mathbb{I}_{3}+\mathcal{O}(M_{R}^{-2})\,,$
(77)

and

$\displaystyle
P=P_{1}+\mathcal{O}(M_{R}^{-3})=M_{D}^{*}\,M_{R}^{-1}+\mathcal{O}(M_{R}^{-3})\,.$
(78)

With these expressions at hand it is straightforward to expand the sums in
Eqs. (35)-(38) and find

$\displaystyle\sum_{j\sim l}\Gamma_{\beta\alpha j}^{1,0,0}$
$\displaystyle=\frac{i}{2v_{\chi}}\sum_{k,r=4}^{6}\sum_{j=1}^{3}\bar{M}_{kr}\bar{M}^{\dagger}_{\alpha
k}U_{rj}U_{\beta j}^{*}$
$\displaystyle\simeq\frac{i}{2v_{\chi}}\left(-\mathbb{I}_{3}\,M_{D}M_{R}^{-1}M_{R}M_{D}^{\dagger}\right)_{\beta\alpha}=-\frac{i}{2v_{\chi}}\left(M_{D}M_{D}^{\dagger}\right)_{\beta\alpha}\,,$
(79) $\displaystyle\sum_{j\sim l}\tilde{\Gamma}_{\beta\alpha j}^{1,0,0}$
$\displaystyle=\frac{i}{2v_{\chi}}\sum_{k,r=4}^{6}\sum_{j=1}^{3}\bar{M}_{kr}\bar{M}^{\dagger}_{\beta
r}U_{kj}U_{\alpha j}^{*}$
$\displaystyle\simeq\frac{i}{2v_{\chi}}\left(-M_{D}^{*}M_{R}M_{R}^{-1}M_{D}^{T}\,\mathbb{I}_{3}\right)_{\beta\alpha}=-\frac{i}{2v_{\chi}}\left(M_{D}M_{D}^{\dagger}\right)_{\alpha\beta}\,,$
(80) $\displaystyle\sum_{j\sim h}\Delta_{\beta\alpha j}^{0,1,-1}$
$\displaystyle=\frac{i}{2v_{\chi}}\sum_{k,r=4}^{6}\sum_{j=4}^{6}\bar{M}_{kr}\left(\bar{M}\bar{M}^{\dagger}\right)_{\alpha
k}U_{rj}m_{j}^{-1}U_{\beta j}$
$\displaystyle\simeq\frac{i}{2v_{\chi}}\left(M_{D}^{*}M_{R}^{-1}M_{R}^{-1}M_{R}M_{R}M_{D}^{T}\right)_{\beta\alpha}=\frac{i}{2v_{\chi}}\left(M_{D}M_{D}^{\dagger}\right)_{\alpha\beta}\,,$
(81) $\displaystyle\sum_{j\sim h}\tilde{\Delta}_{\beta\alpha j}^{0,1,-1}$
$\displaystyle=\frac{i}{2v_{\chi}}\sum_{k,r=4}^{6}\sum_{j=4}^{6}\bar{M}_{kr}\left(\bar{M}\bar{M}^{\dagger}\right)_{\beta
r}U_{kj}m_{j}^{-1}U_{\alpha j}$
$\displaystyle\simeq\frac{i}{2v_{\chi}}\left(M_{D}M_{R}M_{R}M_{R}^{-1}M_{R}^{-1}M_{D}^{\dagger}\right)_{\beta\alpha}=\frac{i}{2v_{\chi}}\left(M_{D}M_{D}^{\dagger}\right)_{\beta\alpha}\,.$
(82)

At this point we just need to combine all these results into a final
expression for the majoron coupling to charged leptons. For the $Z$ boson
contribution we make use of Eq. (43) to obtain

$\left(\Gamma_{Z}\right)^{(0)}=-\frac{i}{2v_{\chi}}\,\textup{Tr}(M_{D}M_{D}^{\dagger})\,,$
(83)

whereas for the $W$ boson contribution, using Eq. (46), we find

$\left(L_{W}^{\beta\alpha}\right)^{(0)}=-\frac{i}{2v_{\chi}}\,\left(M_{D}M_{D}^{\dagger}\right)_{\beta\alpha}\,.$
(84)

Finally, replacing these contributions into Eqs. (41) and (45), and using the
generic Lagrangian in Eqs. (1) and the expression for the coupling $S$ in Eq.
(69), we recover the known formula for the majoron coupling to charged leptons
in the type-I seesaw [15, 16, 17],

$\mathcal{L}_{\ell\ell
J}=\frac{iJ}{16\pi^{2}v^{2}v_{\chi}}\bar{\ell}\left[M_{\ell}~{}\textup{Tr}(M_{D}M_{D}^{\dagger})\gamma_{5}+2M_{\ell}M_{D}M_{D}^{\dagger}P_{L}-2M_{D}M_{D}^{\dagger}M_{\ell}P_{R}\right]\ell\,.$
(85)

The natural size of these couplings can be readily estimated from this
expression. The coupling $S$ in Eq. (69) is estimated as

$S_{\rm
Type-I}\sim\frac{M_{\ell}\,M_{D}^{2}}{16\pi^{2}\,v^{2}\,v_{\chi}}\sim\frac{M_{\ell}\,m_{\nu}}{16\pi^{2}\,v^{2}}\,,$
(86)

where we have used that $m_{\nu}\sim M_{D}^{2}/M_{R}$ and $M_{R}\sim v_{\chi}$
in this model. The resulting coupling is strongly suppressed, not only by the
loop factor, but also by charged lepton and neutrino masses. For instance,
with $M_{\ell}\sim m_{\mu}$ and $m_{\nu}\sim 0.1$ eV, one finds $S_{\rm
Type-I}\sim 10^{-18}$. This generic result clearly respects the bounds in Sec.
2 (Eqs. (3), (4), (7) and (8)). In fact, using Eq. (6) one finds branching
ratios as low as BR$(\mu\to e\,J)\sim 10^{-20}$, well below any foreseen
experimental search.

In summary, in the type-I seesaw one generally expects tiny majoron couplings
to the charged leptons. The only way out of this conclusion is to take
advantage of the fact that neutrino masses and the majoron coupling to charged
leptons actually depend on a different product of matrices
($M_{D}M_{R}^{-1}M_{D}^{T}$ versus $M_{D}M_{D}^{\dagger}/v_{\chi}$).
Therefore, one can in principle evade the generic expectation of Eq. (86) by
adopting specific textures for the $y$ Yukawa matrix.

###  3.2 Inverse seesaw

field | spin | generations | $\mathrm{SU(3)}_{c}$ | $\mathrm{SU(2)}_{L}$ | $\mathrm{U(1)}_{Y}$ | $\mathrm{U(1)}_{L}$
---|---|---|---|---|---|---
$N$ | $\frac{1}{2}$ | 3 | $\mathbf{1}$ | $\mathbf{1}$ | 0 | -1
$S$ | $\frac{1}{2}$ | 3 | $\mathbf{1}$ | $\mathbf{1}$ | 0 | 1
$\chi$ | 0 | 1 | $\mathbf{1}$ | $\mathbf{1}$ | 0 | -2
Table 2: New particles in the inverse seesaw with spontaneous lepton number
violation.

The inverse seesaw with spontaneous lepton number violation [36] introduces
the new fields in Tab. 2. In this case, in addition to the usual 3 $N$ fermion
singlets, 3 $S$ fermion singlets are included too, with opposite lepton
number. The relevant Yukawa Lagrangian terms are given by

$-\mathcal{L}=y\,\bar{L}\tilde{H}N+M_{R}\,\bar{N}^{c}S+\frac{1}{2}\,\lambda^{\prime}\,\chi^{*}\bar{N}^{c}N+\frac{1}{2}\,\lambda\,\chi\bar{S}^{c}S+\text{h.c.}\,.$
(87)

We note the presence of two Yukawa terms involving $\chi$, with couplings
$\lambda^{\prime}$ and $\lambda$, two symmetric $3\times 3$ matrices.
Moreover, $M_{R}$ is a general $3\times 3$ matrix with dimensions of mass.
However, we will work in the basis in which $M_{R}$ is a diagonal matrix with
real entries. This choice is particularly convenient for our computation,
since $M_{R}$ will be taken below as expansion parameter. Again, the singlet
$\chi$ can be decomposed as in Eq. (71), including a massless majoron, $J$.
Furthermore, symmetry breaking leads to a Dirac mass term, $M_{D}$, as in the
type-I seesaw, as well as to Majorana mass terms for the singlet neutrinoss,
$\mu^{\prime}$ and $\mu$, given by

$M_{D}=y\,\frac{v}{\sqrt{2}}\,,\quad\mu^{\prime}=\lambda^{\prime}\,\frac{v_{\chi}}{\sqrt{2}}\,,\quad\mu=\lambda\,\frac{v_{\chi}}{\sqrt{2}}\,.$
(88)

In the basis $\left\\{\nu_{L}^{c},N,S^{c}\right\\}$, the $9\times 9$ Majorana
mass matrix for the neutral fermions can be written as

$\bar{M}=\begin{pmatrix}0&M_{D}&0\\\ M_{D}^{T}&\mu^{\prime}&M_{R}\\\
0&M_{R}&\mu\end{pmatrix}\equiv\begin{pmatrix}0&m_{D}\\\
m_{D}^{T}&Q\end{pmatrix}\,,$ (89)

with

$Q=\begin{pmatrix}\mu^{\prime}&M_{R}\\\ M_{R}&\mu\end{pmatrix}\,,$ (90)

and

$m_{D}=\begin{pmatrix}M_{D}&0\\\ \end{pmatrix}\,.$ (91)

If one assumes the hierarchy $\mu,\mu^{\prime}\ll M_{D}\ll M_{R}$, the light
neutrinos mass matrix is given by $m_{\nu}=M_{D}M_{R}^{-1}\mu
M_{R}^{-1}M_{D}^{T}$, hence leading to naturally small neutrino masses. We
note that the $\mu^{\prime}$ parameter does not contribute to neutrino masses
at leading order in the expansion.

In this model, just as in the type-I seesaw, the only diagrams contributing to
the majoron coupling to charged leptons are those with gauge bosons, shown in
Fig. 1(a) and 1(b). Then, we only need to compute $\Gamma_{Z}$ and $L_{W}$ at
leading order. In order to do that we must identify the general couplings of
Eqs. (17)-(22) that participate in these contributions. In the gauge basis,
the majoron coupling to a pair a neutral fermions is given by

$\bar{A}_{kr}=\left\\{\begin{array}[]{cl}\displaystyle-\frac{i}{2v_{\chi}}\,\bar{M}_{kr}\,,&\text{if}\,\,k,r=4,5,6\\\
&\\\
\displaystyle\frac{i}{2v_{\chi}}\,\bar{M}_{kr}\,,&\text{if}\,\,k,r=7,8,9\\\
&\\\ \displaystyle 0\,,&\text{otherwise}\end{array}\right.$ (92)

The $A$, $E$ and $F$ couplings in the mass basis can be readily obtained with
the help of Eqs. (17), (21) and (22). Again, the divergent piece
$\left(\Gamma_{Z}\right)^{\rm(div)}$ vanishes exactly due to
$\delta_{ks}\bar{A}_{kr}=0$ for $s=1,2,3$. Therefore, we must compute
$\sum_{j\sim l}\tilde{\Gamma}_{\beta\alpha j}^{1,0,0}$, $\sum_{j\sim
l}\Gamma_{\beta\alpha j}^{1,0,0}$, $\sum_{j\sim h}\Delta_{\beta\alpha
j}^{0,1,-1}$ and $\sum_{j\sim h}\tilde{\Delta}_{\beta\alpha j}^{0,1,-1}$. In
order to do that we need the form of the $U$ matrix, which in this model is a
$9\times 9$ unitary matrix. When the mass matrix $\bar{M}$ in Eq. (89) is
written in terms of the matrices $Q$ and $m_{D}$ in Eqs. (90) and (91), one
obtains an expression that is formally equivalent to that in the type-I
seesaw, see Eq. (73). Therefore, $U$ takes the same form, namely

$U=\begin{pmatrix}U_{l}&0\\\
0&U_{h}\end{pmatrix}\,\begin{pmatrix}\sqrt{\mathbb{I}_{3}-PP^{\dagger}}&P\\\
-P^{\dagger}&\sqrt{\mathbb{I}_{6}-P^{\dagger}P}\end{pmatrix}\equiv
U_{2}\,U_{1}\,,$ (93)

with $U_{l}$ and $U_{h}$ two $3\times 3$ and $6\times 6$ matrices,
respectively. $P$ is a $3\times 6$ matrix that can be expanded in inverse
powers of the large $M_{R}$ scale as in Eq. (90). At leading order one finds

$\displaystyle\sqrt{\mathbb{I}_{3}-PP^{\dagger}}$
$\displaystyle=\mathbb{I}_{3}+\mathcal{O}(M_{R}^{-2})\,,$ (94)
$\displaystyle\sqrt{\mathbb{I}_{6}-P^{\dagger}P}$
$\displaystyle=\mathbb{I}_{6}+\mathcal{O}(M_{R}^{-2})\,,$ (95)

and

$\displaystyle
P=P_{1}+\mathcal{O}(M_{R}^{-3})=m_{D}^{*}\,\left(Q^{-1}\right)^{\dagger}+\mathcal{O}(M_{R}^{-3})\,.$
(96)

With these expressions, one can readily find

$\displaystyle\sum_{j\sim l}\Gamma_{\beta\alpha j}^{1,0,0}$
$\displaystyle=\frac{i}{2v_{\chi}}\sum_{j=1}^{3}\left(\sum_{k,r=7}^{9}-\sum_{k,r=4}^{6}\right)\bar{M}_{kr}\left(\bar{M}^{\dagger}\right)_{\alpha
k}U_{rj}U_{\beta j}^{*}$
$\displaystyle\simeq-\frac{i}{2v_{\chi}}\left(M_{D}M_{R}^{-1}\mu\,M_{R}^{-1}\mu^{\prime}M_{D}^{\dagger}\right)_{\beta\alpha}\,,$
(97) $\displaystyle\sum_{j\sim l}\tilde{\Gamma}_{\beta\alpha j}^{1,0,0}$
$\displaystyle=\frac{i}{2v_{\chi}}\sum_{j=1}^{3}\left(\sum_{k,r=7}^{9}-\sum_{k,r=4}^{6}\right)\bar{M}_{kr}\left(\bar{M}^{\dagger}\right)_{\beta
r}U_{kj}U_{\alpha j}^{*}$
$\displaystyle\simeq-\frac{i}{2v_{\chi}}\left(M_{D}^{*}\mu^{\prime}M_{R}^{-1}\mu\,M_{R}^{-1}M_{D}^{T}\right)_{\beta\alpha}\,,$
(98) $\displaystyle\sum_{j\sim h}\Delta_{\beta\alpha j}^{0,1,-1}$
$\displaystyle=\frac{i}{2v_{\chi}}\sum_{j=4}^{9}\left(\sum_{k,r=7}^{9}-\sum_{k,r=4}^{6}\right)\left(\bar{M}\bar{M}^{\dagger}\right)_{\alpha
k}U_{rj}m_{j}^{-1}U_{\beta j}$
$\displaystyle\simeq-\frac{i}{2v_{\chi}}\left[\left(M_{D}^{*}M_{R}^{-1}\mu^{\dagger}M_{R}^{-2}\mu\,M_{R}M_{D}^{T}\right)_{\beta\alpha}+\left(M_{D}^{*}M_{R}^{-2}\mu^{\prime}\mu^{{}^{\prime}\dagger}M_{D}^{T}\right)_{\beta\alpha}\right]\,,$
(99) $\displaystyle\sum_{j\sim h}\tilde{\Delta}_{\beta\alpha j}^{0,1,-1}$
$\displaystyle=\frac{i}{2v_{\chi}}\sum_{j=4}^{9}\left(\sum_{k,r=7}^{9}-\sum_{k,r=4}^{6}\right)\bar{M}_{kr}\left(\bar{M}\bar{M}^{\dagger}\right)_{\beta
r}U_{kj}m_{j}^{-1}U_{\alpha j}$
$\displaystyle\simeq-\frac{i}{2v_{\chi}}\left[\left(M_{D}M_{R}\,\mu\,M_{R}^{-2}\mu^{\dagger}M_{R}^{-1}M_{D}^{\dagger}\right)_{\beta\alpha}+\left(M_{D}\mu^{{}^{\prime}\dagger}\mu^{\prime}M_{R}^{-2}M_{D}^{\dagger}\right)_{\beta\alpha}\right]\,.$
(100)

Now we just need to introduce these results into the expressions for the
leading order $Z$ and $W$ boson contributions. For the $Z$ boson contribution
we use Eq. (43) to obtain

$\left(\Gamma_{Z}\right)^{(0)}=-\frac{i}{6v_{\chi}}\,\textup{Tr}\left(M_{D}\left[\mu\mu^{\prime}-2\mu\mu^{\dagger}-2\mu^{{}^{\prime}\dagger}\mu^{\prime}\right]M_{R}^{-2}M_{D}^{\dagger}\right)\,,$
(101)

whereas for the $W$ boson contribution, using Eq. (46), we find

$\left(L_{W}^{\beta\alpha}\right)^{(0)}=-\frac{i}{24v_{\chi}}\,\left(M_{D}\left[8\mu\mu^{\prime}-\mu^{{}^{\prime}\dagger}\mu^{\dagger}+5\mu\mu^{\dagger}+5\mu^{\prime}\mu^{{}^{\prime}\dagger}\right]
M_{R}^{-2}M_{D}^{\dagger}\right)_{\beta\alpha}\,\,.$ (102)

Finally, replacing these contributions into Eqs. (41) and (45), and using the
generic expressions in Eqs. (1) and (69) we can write the coupling of the
majoron to a pair of charged leptons in the inverse seesaw as

$\displaystyle\mathcal{L}_{J\ell\ell}=\frac{iJ}{96\pi^{2}v^{2}v_{\chi}}\bar{\ell}\,\Bigg{\\{}$
$\displaystyle
2M_{\ell}\,\textup{Tr}\left(M_{D}\left[\mu\mu^{\prime}-2\mu\mu^{\dagger}-2\mu^{{}^{\prime}\dagger}\mu^{\prime}\right]M_{R}^{-2}M_{D}^{\dagger}\right)\gamma_{5}$
$\displaystyle+M_{\ell}\,M_{D}\left(8\mu\mu^{\prime}-\mu^{{}^{\prime}\dagger}\mu^{\dagger}+5\mu\mu^{\dagger}+5\mu^{\prime}\mu^{{}^{\prime}\dagger}\right)M_{R}^{-2}M_{D}^{\dagger}\,P_{L}$
$\displaystyle-
M_{D}\left(8\mu^{{}^{\prime}\dagger}\mu^{\dagger}-\mu\mu^{\prime}+5\mu\mu^{\dagger}+5\mu^{\prime}\mu^{{}^{\prime}\dagger}\right)
M_{R}^{-2}M_{D}^{\dagger}\,M_{\ell}\,P_{R}\Bigg{\\}}\,\ell\,.$ (103)

In the case of $\mu^{\prime}=0$ (or, equivalently, $\lambda^{\prime}=0$), an
assumption that is often made in the literature, this expression reduces to

$\displaystyle\mathcal{L}_{J\ell\ell}=$
$\displaystyle\frac{iJ}{96\pi^{2}v^{2}v_{\chi}}\bar{\ell}\,\Bigg{\\{}-4M_{\ell}\,\textup{Tr}\left(M_{D}\,\mu\mu^{\dagger}M_{R}^{-2}M_{D}^{\dagger}\right)\gamma_{5}$
$\displaystyle+5M_{\ell}\,M_{D}\,\mu\mu^{\dagger}M_{R}^{-2}M_{D}^{\dagger}\,P_{L}-5M_{D}\,\mu\mu^{\dagger}M_{R}^{-2}M_{D}^{\dagger}\,M_{\ell}\,P_{R}\Bigg{\\}}\,\ell\,.$
(104)

To the best of our knowledge, this is the first time that this coupling is
computed in the context of the inverse seesaw. We can now proceed in the same
way as in the type-I seesaw and estimate its size in the inverse seesaw. In
this case, and assuming $\mu^{\prime}=0$ for simplicity, the coupling $S$ can
be estimated as

$S_{\rm
ISS}\sim\frac{M_{\ell}\,M_{D}^{2}\,\mu^{2}}{96\pi^{2}\,v^{2}\,v_{\chi}\,M_{R}^{2}}\sim\frac{M_{\ell}\,\mu\,m_{\nu}}{96\pi^{2}\,v^{2}\,v_{\chi}}\sim\frac{M_{\ell}\,m_{\nu}}{96\pi^{2}\,v^{2}}\,,$
(105)

where we have used that $m_{\nu}\sim M_{D}^{2}\mu/M_{R}^{2}$ and $\mu\sim
v_{\chi}$ in this model. The resulting coupling is again very strongly
suppressed. In fact, one finds the same suppression factors as in the type-I
seesaw (see Eq. (86)) and thus expects $S_{\rm ISS}\sim S_{\rm Type-I}$. One
could in principle increase $S_{\rm ISS}$ by introducing a large non-vanishing
$\mu^{\prime}$ (or $\lambda^{\prime}$), since this coupling does not affect
neutrino masses at leading order. However, it does enter subleadingly, see for
instance [37]. Therefore, $\mu^{\prime}$ cannot be increased indefinitely.

We conclude that in the version of the inverse seesaw with spontaneous lepton
number violation considered here one also expects tiny majoron couplings to
the charged leptons. Again, as in the type-I seesaw, one can evade this
conclusion by properly tuning the Yukawa matrices. This may be used to break
the generic expectation in Eq. (151). Furthermore, this conclusion may not
hold in alternative versions of the inverse seesaw with spontaneous lepton
number violation [38, 39].

###  3.3 Scotogenic model

field | spin | generations | $\mathrm{SU(3)}_{c}$ | $\mathrm{SU(2)}_{L}$ | $\mathrm{U(1)}_{Y}$ | $\mathrm{U(1)}_{L}$ | $\mathbb{Z}_{2}$
---|---|---|---|---|---|---|---
$N$ | $\frac{1}{2}$ | 3 | $\mathbf{1}$ | $\mathbf{1}$ | 0 | 1 | -
$\eta$ | 0 | 1 | $\mathbf{1}$ | $\mathbf{2}$ | $\frac{1}{2}$ | 0 | -
$\chi$ | 0 | 1 | $\mathbf{1}$ | $\mathbf{1}$ | 0 | -2 | +
Table 3: New particles in the Scotogenic model with spontaneous lepton number
violation.

In the Scotogenic model [40, 41] with spontaneous lepton number violation, the
particle spectrum is extended with 3 fermion singlets, $N$, a scalar doublet,
$\eta$, and a scalar singlet, $\chi$. A new $\mathbb{Z}_{2}$ parity is
introduced as well, under which $N$ and $\eta$ are odd while the rest of the
fields in the model are even. The fields and their charges under the
symmetries of the model are given in Tab. 3. The terms of the Yukawa
Lagrangian that are relevant for our discussion are given by

$-\mathcal{L}=y\,\bar{L}\tilde{\eta}N+\frac{1}{2}\,\lambda\,\chi\bar{N}^{c}N+\text{h.c.}\,.$
(106)

with $\tilde{\eta}=i\sigma_{2}\eta^{*}$. The doublet $\eta$ is assumed to have
a vanishing VEV, $\left\langle\eta\right\rangle=0$ in order to preserve the
$\mathbb{Z}_{2}$ symmetry. This ensures that the lightest $\mathbb{Z}_{2}$-odd
state is completely stable. The singlet $\chi$ can be decomposed as in Eq.
(71), giving rise to a massless majoron, $J$. The breaking of
$\mathrm{U(1)}_{L}$ induces a Majorana mass term for the singlet fermions,
$M_{N}$, defined as

$M_{N}=\lambda\,\frac{v_{\chi}}{\sqrt{2}}\,.$ (107)

Note, however, that the $\mathbb{Z}_{2}$ symmetry precludes any Dirac mass
term mixing the SM neutrinos with the fermion singlets. This in turn implies
that neutrinos remain massless at tree-level. 333In the Scotogenic model,
neutrinos get non-zero masses at the 1-loop level. This is due to the breaking
of lepton number in two units by the $\chi$ VEV. In this work we are
interested in the 1-loop coupling of the majoron to charged leptons. Any
effect on this coupling associated to the 1-loop neutrino mass would be of
higher order in pertubation theory, hence not relevant for our discussion. In
fact, in the basis $\left\\{\nu_{L}^{c},N\right\\}$, the tree-level $6\times
6$ Majorana mass matrix for the neutral fermions can be written as

$\bar{M}=\begin{pmatrix}0&0\\\ 0&M_{N}\end{pmatrix}\,,$ (108)

which trivially leads to $U=\mathbb{I}_{6}$.

In the Scotogenic model, the majoron interacts at tree-level only with the $N$
fermion singlets, which are mass eigenstates since the $\nu_{L}-N$ mixing is
exactly zero. For this reason, the gauge bosons contributions to the majoron
coupling to charged leptons are absent in this model. The only 1-loop
contribution is given by the $\eta$ charged scalar, as shown in Fig. 1(e). We
just need to compute $L_{\eta}$, given in Eq. (47). The relevant couplings in
the gauge basis are

$\bar{A}_{kr}=\left\\{\begin{array}[]{cl}\displaystyle
i\frac{M_{N}}{2v_{\chi}}\,\delta_{kr}\,,&\text{if}\,\,k,r=4,5,6\\\ &\\\
\displaystyle 0\,,&\text{otherwise}\end{array}\right.$ (109)

and

$\displaystyle D_{L}^{\alpha i}$ $\displaystyle=0\,,$ (110) $\displaystyle
D_{R}^{\alpha i}$ $\displaystyle=-\sum_{k=4}^{6}y^{\alpha k}\,\delta_{ki}\,.$
(111)

The couplings in the mass basis can be computed using Eqs. (17), (19) and
(20). Since $D_{L}^{\alpha i}=0$, we only need to compute
$\left(L_{\eta}^{RR}\right)^{(0)}$. This contribution involves the sums over
light states $\sum_{j\sim l}\tilde{\Gamma}^{1,0,0}_{spj}$ and $\sum_{j\sim
l}\tilde{\Gamma}^{1,1,0}_{spj}$, but these are trivially zero, since they both
involve the product $\bar{A}_{kr}U_{kj}=\bar{A}_{kr}\delta_{kj}=0$ for
$j=1,2,3$. Therefore, we are left with $\sum_{j\sim h}\Delta_{spj}^{0,1,-1}$,
$\sum_{j}\Delta_{spj}^{0,1,1}$, $\sum_{j}\tilde{\Gamma}_{spj}^{1,0,0}$ and
$\sum_{j}\tilde{\Gamma}_{spj}^{1,1,0}$. They are found to be

$\displaystyle\sum_{j\sim h}\Delta_{spj}^{0,1,-1}$
$\displaystyle=i\frac{M_{N}}{2v_{\chi}}\sum_{k,r=4}^{6}\delta_{kr}\left(\bar{M}\bar{M}^{\dagger}\right)_{pk}\delta_{rj}\,m_{j}^{-1}\,\delta_{sj}=i\frac{M_{N}^{2}}{2v_{\chi}}\,\delta_{sp}\,,$
(112) $\displaystyle\sum_{j}\Delta_{spj}^{0,1,1}$
$\displaystyle=i\frac{M_{N}}{2v_{\chi}}\sum_{k,r=4}^{6}\delta_{kr}\left(\bar{M}\bar{M}^{\dagger}\right)_{pk}\delta_{rj}\,m_{j}^{1}\,\delta_{sj}=i\frac{M_{N}^{4}}{2v_{\chi}}\,\delta_{sp}\,,$
(113) $\displaystyle\sum_{j}\tilde{\Gamma}_{spj}^{1,0,0}$
$\displaystyle=i\frac{M_{N}}{2v_{\chi}}\sum_{k,r=4}^{6}\left(\bar{M}^{\dagger}\right)_{sr}\delta_{kj}\,\delta_{pj}=i\frac{M_{N}^{2}}{2v_{\chi}}\,\delta_{sp}\,,$
(114) $\displaystyle\sum_{j}\tilde{\Gamma}_{spj}^{1,1,0}$
$\displaystyle=i\frac{M_{N}}{2v_{\chi}}\sum_{k,r=4}^{6}\left(\bar{M}^{\dagger}\bar{M}\bar{M}^{\dagger}\right)_{sr}\delta_{kj}\,\delta_{pj}=i\frac{M_{N}^{4}}{2v_{\chi}}\,\delta_{sp}\,.$
(115)

With these results one can directly obtain from (54) and (52)

$\displaystyle\left(L_{\eta}^{RR}\right)_{sp}^{(0)}=-\frac{i}{2v_{\chi}}\left(f_{5}\,M_{N}^{2}+f_{6}\,M_{N}^{4}\right)\delta_{sp}\,,\quad\left(\tilde{L}_{\eta}^{RR}\right)_{sp}^{(0)}=-\frac{i}{2v_{\chi}}\left(f_{9}\,M_{N}^{2}+f_{10}\,M_{N}^{4}\right)\delta_{sp}\,.$
(116)

Using the $f_{i}$ and $F_{i,j,...,n}$ loop functions defined in Appendix B, we
find

$\displaystyle\left(L_{\eta}^{RR}\right)^{(0)}=i\,\frac{M_{N}^{2}}{2v_{\chi}\left(M_{N}^{2}-m_{\eta}^{2}\right)^{2}}\left(M_{N}^{2}-m_{\eta}^{2}+m_{\eta}^{2}\log\frac{m_{\eta}^{2}}{M_{N}^{2}}\right)\delta_{sp}$
(117)

and we recover the known result of [42, 43] 444The definition of the $y$
Yukawa matrix in [43] differs from ours by a complex conjugation.

$\mathcal{L}_{J\ell\ell}=\frac{iJ}{16\pi^{2}v_{\chi}}\bar{\ell}\left(M_{\ell}\,y\,\Theta\,y^{\dagger}\,P_{L}-y\,\Theta\,y^{\dagger}\,M_{\ell}\,P_{R}\right)\ell\,,$
(118)

with

$\Theta_{sp}\equiv\frac{M_{N}^{2}}{\left(M_{N}^{2}-m_{\eta}^{2}\right)^{2}}\left(M_{N}^{2}-m_{\eta}^{2}+m_{\eta}^{2}\log\frac{m_{\eta}^{2}}{M_{N}^{2}}\right)\delta_{sp}\,.$
(119)

It is important to emphasize here that calculating this coupling in this
general form in this model is an overkill. In this model, as explained above,
the majoron only interacts with the fermion singlets and the coupling can be
taken to be diagonal without loss of generality. This fact implies that the
resulting loop integral is straightforward, since the light neutrinos do not
enter the loop. Therefore, if we were to perform the explicit calculation, we
would obtain a simple integral that would directly yield the reproduced
result. In other words, we would not need to carry out any expansion. Hence,
the fact that our calculation, after considering the most general possible
model and carrying out all expansions while accounting for all possibilities,
returns the correct result is a very strong validation of our outcome.

Finally, one can estimate the size of the majoron couplings to charged leptons
in the Scotogenic model. In this model, the coupling $S$ can be estimated as

$S_{\rm
Scot}\sim\frac{M_{\ell}\,y^{2}}{16\pi^{2}\,v_{\chi}}\sim\frac{2\,M_{\ell}\,m_{\nu}}{\lambda_{5}\,v^{2}}\sim\frac{32\pi^{2}}{\lambda_{5}}\,S_{\rm
Type-I}\,,$ (120)

where we have used that $m_{\nu}\sim\lambda_{5}\,v^{2}y^{2}/(32\pi^{2}M_{N})$,
with $\lambda_{5}$ the coupling of the quartic scalar term
$\left(H^{\dagger}\eta\right)^{2}$, and $M_{N}\sim v_{\chi}$ in this model. We
have also assumed that all loop functions are of $\mathcal{O}(1)$ and have
compared to the result for the type-I seesaw in Eq. (86). We find a large
enhancement compared to the tiny value obtained in the type-I seesaw. In fact,
a relatively small $\lambda_{5}$ coupling would naturally lead to the
violation of the bounds in Sec. 2. For instance, the current limit on
BR$(\mu\to e\,J)\sim 10^{-17}/\lambda_{5}^{2}$ would be violated for
$\lambda_{5}\lesssim 10^{-6}$. This result confirms previous findings in the
literature, which have already shown a very rich majoron phenomenology in the
Scotogenic model. We refer to [43] for a detailed exploration of the
phenomenology of the $\ell_{\alpha}\to\ell_{\beta}\,J$ decays in this model.

###  3.4 Type-I seesaw with two Higgs doublets

field | spin | generations | $\mathrm{SU(3)}_{c}$ | $\mathrm{SU(2)}_{L}$ | $\mathrm{U(1)}_{Y}$ | $\mathrm{U(1)}_{L}$
---|---|---|---|---|---|---
$N$ | $\frac{1}{2}$ | 3 | $\mathbf{1}$ | $\mathbf{1}$ | 0 | 1
$\chi$ | 0 | 1 | $\mathbf{1}$ | $\mathbf{1}$ | 0 | -2
$H_{2}$ | 0 | 1 | $\mathbf{1}$ | $\mathbf{2}$ | $\frac{1}{2}$ | 0
Table 4: New particles in the type-I seesaw with two Higgs doublets and
spontaneous lepton number violation.

In order to better illustrate our results, we now consider a non-minimal
model: the type-I seesaw with two Higgs doublets and spontaneous lepton number
violation. This model extends the Two Higgs Doublet Model (2HDM) particle
content with 3 singlet fermions $N$ and a scalar singlet $\chi$, all charged
under the global $\mathrm{U(1)}_{L}$ as shown in Tab. 4, just as the type-I
seesaw discussed in Sec. 3.1. In addition, this model adds a second Higgs
doublet, $H_{2}$. In the following we assume that we are working in the Higgs
basis, in which $H_{2}$ does not acquire a VEV. The Yukawa terms relevant for
our discussion are

$-\mathcal{L}=Y\,\bar{L}He_{R}+Y_{2}\,\bar{L}H_{2}e_{R}+y\,\bar{L}\tilde{H}N+y_{2}\,\bar{L}\tilde{H_{2}}N+\frac{1}{2}\,\lambda\,\chi\,\bar{N}^{c}N+\text{h.c.}\,.$
(121)

Since we work in the Higgs basis, $M_{D}$ and $M_{R}$ are given by the same
expressions as in Eq. (72). Similarly, the charged leptons mass matrix is

$M_{\ell}=Y\,\frac{v}{\sqrt{2}}\,.$ (122)

Furthermore, $Y_{2}$ and $y_{2}$ are two general $3\times 3$ matrices.
Neutrino mass generation takes place in exactly the same way as in the minimal
type-I seesaw and the same holds true for the majoron. Consequently, the gauge
diagrams remain identical to those we computed previously in Sec. 3.1.
However, as in any 2HDM, the presence of a second Higgs doublet implies the
existence of physical charged and CP-odd scalars. Following the same notation
as in Sec. 2, they will be denoted as $\eta$ and $\sigma$. These states induce
the Feynman diagrams with amplitudes $\mathcal{M}_{\eta}$ and
$\mathcal{M}_{\sigma n}$ shown in Figs. 1(e) and 1(d). In order to compute
them, we first identify the new couplings, not present in Sec. 3.1, defined in
Eqs. (17)-(22). These are

$\displaystyle\bar{B}_{kr}$
$\displaystyle=\left\\{\begin{array}[]{cl}\displaystyle
i\left(y_{2}\right)_{kr}\,,&\text{if}\,\,k=1,2,3\,\,\text{and}\,\,r=4,5,6\\\
&\\\ \displaystyle 0\,,&\text{otherwise}\end{array}\right.$ (126)
$\displaystyle C^{\alpha\beta}$
$\displaystyle=-\left(Y_{2}\right)_{\beta\alpha}^{*}\,,$ (127)
$\displaystyle\bar{D}_{L}^{\alpha k}$
$\displaystyle=\left\\{\begin{array}[]{cl}\displaystyle\left(Y_{2}\right)_{\alpha
k}\,,&\text{if}\,\,k=1,2,3\\\ &\\\ \displaystyle
0\,,&\text{otherwise}\end{array}\right.$ (131)
$\displaystyle\bar{D}_{R}^{\alpha k}$
$\displaystyle=\left\\{\begin{array}[]{cl}\displaystyle\left(y_{2}\right)_{\alpha
k}\,,&\text{if}\,\,k=4,5,6\\\ &\\\ \displaystyle
0\,,&\text{otherwise}\end{array}\right.$ (135)

In our computation we will assume that $m_{\eta},m_{\sigma}\ll M_{H}$. Let us
start with the computation of the charged scalar contribution. We have to
obtain expressions for $L_{\eta}^{RR}$, $\tilde{L}_{\eta}^{RR}$,
$L_{\eta}^{LL}$, $\tilde{L}_{\eta}^{LL}$ and $L_{\eta}^{RL}$. We will start by
computing the first order of $L_{\eta}^{RR}$ and $\tilde{L}_{\eta}^{RR}$. One
can easily find

$\displaystyle\sum_{j}\Delta_{spj}^{0,1,1}=\sum_{j}\tilde{\Gamma}_{spj}^{1,1,0}=\frac{i}{2v_{\chi}}M_{R}^{4}\,\delta_{sp}+\mathcal{O}(M_{R}^{2})\,,$
(136) $\displaystyle\sum_{j\sim
h}\Delta_{spj}^{1,1,-1}=\sum_{j}\tilde{\Gamma}_{spj}^{1,0,0}=\frac{i}{2v_{\chi}}M_{R}^{2}\,\delta_{sp}+\mathcal{O}(M_{R}^{0})\,,$
(137)

while other terms are of lower order in $M_{R}$ and hence subdominant. With
these results one can directly obtain

$\displaystyle\left(L_{\eta}^{RR}\right)_{sp}^{(0)}=-\frac{i}{2v_{\chi}}\left(f_{5}\,M_{R}^{2}+f_{6}\,M_{R}^{4}\right)\,\delta_{sp}\,,\quad\left(\tilde{L}_{\eta}^{RR}\right)_{sp}^{(0)}=-\frac{i}{2v_{\chi}}\left(f_{9}\,M_{R}^{2}+f_{10}\,M_{R}^{4}\right)\,\delta_{sp}\,.$
(138)

It is now useful to notice that $f_{i}\sim M_{H}^{-2}$ for $i$ odd and
$f_{i}\sim M_{H}^{-4}$ for $i$ even. Therefore, the two terms in the previous
expressions are of the same order in $M_{R}$. In fact, making use of the
expressions for the $f_{i}$ and $F_{i,j,...,n}$ loop functions in Appendix B,
and using $m_{\eta}\ll M_{H}$, we find

$\displaystyle\left(L_{\eta}^{RR}\right)_{sp}^{(0)}=\frac{-3i}{8v_{\chi}}\,\delta_{sp}\,,\quad\left(\tilde{L}_{\eta}^{RR}\right)_{sp}^{(0)}=\frac{-i}{8v_{\chi}}\,\delta_{sp}\,,$
(139)

where we have used that $\frac{\left(M_{R}\right)_{sp}}{M_{H}}=\delta_{sp}$ at
first order. Our result is easy to understand: since both the majoron and the
charged scalar couple directly to the singlets $N$, which are essentially the
heavy neutrinos, there is no mixing in the loop (no mass suppression) at
leading order. For this reason, one expects $L_{\eta}^{LL}$ and
$\tilde{L}_{\eta}^{LL}$ to be subdominant, as two mixings are required in the
loop. This fact is reflected in our results and an explicit calculation
confirms this reasoning. Moreover, one could be tempted to make a similar
argument for $L_{\eta}^{RL}$, which requires one mixing. However, we must be
careful with this term because it is not proportional to the charged lepton
masses in Eq. (2.2). In fact, the introduction of $M_{D}$ is equivalent to the
charged lepton mass, since $M_{D}\sim M_{\ell}$. Let us see it explicitly. One
can easily find

$\displaystyle-\sum_{j}\left(\tilde{\Gamma}_{psj}^{0,2,0}\right)^{*}=\sum_{j}\tilde{\Delta}_{spj}^{1,1,1}=\sum_{j}\Delta_{spj}^{1,1,1}=\sum_{j\sim
l}\left(\Gamma_{psj}^{0,2,0}\right)^{*}=\frac{i}{2v_{\chi}}\left(M_{R}^{4}M_{D}^{\dagger}\right)_{s-3,\,p}+\mathcal{O}(M_{R}^{2})\,,$
(140) $\displaystyle\sum_{j\sim
h}\Delta_{spj}^{0,1,-1}=\sum_{j}\tilde{\Delta}_{spj}^{1,0,1}=-\sum_{j}\left(\tilde{\Gamma}_{psj}^{0,1,0}\right)^{*}=\sum_{j\sim
l}\left(\Gamma_{psj}^{0,1,0}\right)^{*}=\frac{i}{2v_{\chi}}\left(M_{R}^{2}M_{D}^{\dagger}\right)_{s-3,\,p}+\mathcal{O}(M_{R}^{0})\,,$
(141)

while other terms are of lower order in $M_{R}$ and can be neglected. With
these results,

$\displaystyle\left(L_{\eta}^{RL}\right)_{sp}^{(0)}=-2\frac{i}{2v_{\chi}}\left[\left(f_{3}\,M_{R}^{2}+f_{4}\,M_{R}^{4}\right)M_{D}^{\dagger}\right]_{s-3,\,p}\,,$
(142)

and, using the expressions for $f_{i}$ and $F_{i,j,...,n}$, we obtain

$\displaystyle\left(L_{\eta}^{RL}\right)_{sp}^{(0)}=-\frac{i}{2v_{\chi}}\left(M_{D}^{\dagger}\right)_{s-3,\,p}\,.$
(143)

And, with both results, finally,

$\displaystyle\left(L_{\eta}\right)^{\beta\alpha}=\frac{i}{2v_{\chi}}\left(M_{\ell}\,y_{2}\,y_{2}^{\dagger}+Y_{2}\,M_{D}\,y_{2}^{\dagger}\right)_{\beta\alpha}\,.$
(144)

Let us continue with the CP-odd scalar contribution, $\mathcal{M}_{\sigma n}$.
First, we note that the divergent piece vanishes, as expected. This is because
only the real part contributes, see Eq. (61), and $\left(L_{\sigma
n}\right)^{\textup{(div)}}$ turns out to be purely imaginary. To show it we
compute $\sum_{j}\Delta_{psj}^{1,0,1}$, $\sum_{j}\Gamma_{spj}^{0,1,0}$ and
$\sum_{j}\tilde{\Gamma}_{spj}^{0,1,0}$,

$\displaystyle\sum_{j}\Delta_{psj}^{1,0,1}$
$\displaystyle=\frac{i}{2v_{\chi}}\left(M_{R}^{2}M_{D}^{\dagger}\right)_{p-3,\,s}\,,$
(145) $\displaystyle\sum_{j}\tilde{\Gamma}_{spj}^{0,1,0}$
$\displaystyle=\frac{i}{2v_{\chi}}\left(M_{D}M_{R}^{2}\right)_{p-3,\,s}\,,$
(146) $\displaystyle\sum_{j}\Gamma_{spj}^{0,1,0}$ $\displaystyle=0\,.$ (147)

Therefore,

$\displaystyle\left(L_{\sigma n}\right)^{\textup{(div)}}$
$\displaystyle=\text{Re}\,\Bigg{\\{}-\frac{1}{2v_{\chi}}\left[
\textup{Tr}\left(y_{2}M_{R}^{2}M_{D}^{\dagger}\right)-\textup{Tr}\left(y_{2}^{*}M_{R}^{2}M_{D}^{T}\right)\right]\left(3+2\,\frac{1}{\epsilon}+2\,\log\frac{\mu^{2}}{M_{H}^{2}}\right)\Bigg{\\}}=0\,.$
(148)

We then consider the finite piece. We have already computed several terms
required for the evaluation of $L_{\eta}^{RL}$. With these results (and a few
more), one can show that $L_{\sigma n}^{(0)}$ vanishes for the same reason as
for the divergent piece. This means that the leading order contribution from
the $\sigma$ scalar is, at most, of order one. This includes the explicit
order one contribution given in Appendix A as well as the hidden order one
contribution in $L_{\sigma n}^{(0)}$. However, one can show that this order
also cancels out, so $L_{\sigma n}=\mathcal{O}(M_{R}^{-2})$, and in this way,
we can safely neglect this contribution. In summary, the majoron coupling to a
pair charged leptons in the type-I seesaw with two Higgs doublets is given by

$\displaystyle\mathcal{L}_{\ell\ell
J}=\frac{iJ}{16\pi^{2}v_{\chi}}\bar{\ell}\Bigg{\\{}$ $\displaystyle
M_{\ell}~{}\textup{Tr}\left(\frac{M_{D}M_{D}^{\dagger}}{v^{2}}\right)\gamma_{5}+2\left(M_{\ell}\frac{M_{D}M_{D}^{\dagger}}{v^{2}}-M_{\ell}\,y_{2}\,y_{2}^{\dagger}-Y_{2}\,M_{D}\,y_{2}^{\dagger}\right)P_{L}$
$\displaystyle-2\left(\frac{M_{D}M_{D}^{\dagger}}{v^{2}}M_{\ell}-y_{2}\,y_{2}^{\dagger}\,M_{\ell}-y_{2}\,M_{D}^{\dagger}\,Y_{2}^{\dagger}\right)P_{R}\Bigg{\\}}\ell\,.$
(149)

Let us now study the size of the coupling $S$ in this model. It can be split
as

$S_{\rm Type-I+2HDM}=S_{\rm Type-I}+S_{\rm 2HDM}\,,$ (150)

where $S_{\rm Type-I}$ is given in Eq. (86) and $S_{\rm 2HDM}$ contains the
new terms associated to our 2HDM version of the model, not present in the
type-I seesaw (with one Higgs doublet). The latter can be estimated as

$S_{\rm
2HDM}\sim\frac{1}{16\pi^{2}\,v_{\chi}}\left(M_{\ell}\,y_{2}^{2}+y_{2}\,Y_{2}\,M_{D}\right)\sim\frac{1}{16\pi^{2}\,v^{2}\,v_{\chi}}\left(M_{\ell}\,M_{D}^{{}^{\prime}2}+M_{\ell}^{\prime}\,M_{D}\,M_{D}^{\prime}\right)\,,$
(151)

where we have defined $M_{\ell}^{\prime}$ and $M_{D}^{\prime}$ analogously to
$M_{\ell}$ and $M_{D}$, simply replacing $Y$ and $y$ by $Y_{2}$ and $y_{2}$.
This allows us to obtain the ratio

$\frac{S_{\rm 2HDM}}{S_{\rm
Type-I}}=\left(\frac{M_{D}^{\prime}}{M_{D}}\right)^{2}+\frac{M_{\ell}^{\prime}\,M_{D}^{\prime}}{M_{\ell}\,M_{D}}\,.$
(152)

This result hints at an enhacement of the majoron coupling to charged leptons
compared to the type-I seesaw if $M_{D}^{\prime}\gg M_{D}$ and/or
$M_{\ell}^{\prime}\gg M_{\ell}$. Let us suppose $M_{\ell}^{\prime}=0$ (or,
equivalently, $Y_{2}=0$) and consider $M_{D}^{\prime}\gg M_{D}$ (or,
equivalently, $y_{2}\gg y$). One should note that $y_{2}$ does not participate
in the type-I seesaw contribution to neutrino masses, but it does induce a
Scotogenic-like 1-loop contribution, $m_{\nu}^{\rm
1-loop}\sim\lambda_{5}\,v^{2}y_{2}^{2}/(32\pi^{2}M_{R})$. 555In the absence of
mixing between the CP-even scalars, $\lambda_{5}$ is the coupling of the
quartic scalar term $\left(H^{\dagger}H_{2}\right)^{2}$. Otherwise, if the CP-
even components of $H^{0}$ and $H_{2}^{0}$ mix, $\lambda_{5}$ is a combination
of scalar potential parameters associated to this mixing. Imposing
$m_{\nu}^{\rm 1-loop}\lesssim m_{\nu}\equiv m_{\nu}^{\rm tree}$ then leads to

$\frac{\lambda_{5}}{32\pi^{2}}\left(\frac{y_{2}}{y}\right)^{2}\lesssim 1\,.$
(153)

Assuming now $y\sim 10^{-6}$, the typical size of the Yukawa couplings for a
type-I seesaw at the TeV scale, Eq. (153) implies
$\lambda_{5}\,y_{2}^{2}\lesssim 10^{-10}$. This is fulfilled for $y_{2}\sim 1$
if $\lambda_{5}\lesssim 10^{-10}$. In this case, $M_{D}^{\prime}/M_{D}\sim
10^{6}$ and one expects an enhancement of the majoron couplings of about $12$
orders of magnitude with respect to the type-I seesaw. Such a huge increase
would indeed lead to conflict with the bounds in Eqs. (3), (4), (7) and (8).
However, less pronounced hierarchies lead to observable effects, for instance
in $\ell_{\alpha}\to\ell_{\beta}\,J$ decays, in agreement with the current
constraints. This is shown in Fig. 2, which displays contours of BR($\mu\to
e\,J$) and $\lambda_{5}^{\rm max}$ in the $y_{2}^{11}$-$y_{2}^{21}$ plane.
Here $\lambda_{5}^{\rm max}$ is the maximal value of $\lambda_{5}$ compatible
with all the entries in $m_{\nu}^{\rm 1-loop}$ being smaller than those in
$m_{\nu}^{\rm tree}$. The rest of $y_{2}$ Yukawa couplings are set to zero for
simplicity, while $\lambda=1$ and $v_{\chi}=5$ TeV. This plot clearly shows
that the model can achieve large $\mu\to e\,J$ braching ratios, almost as
large as the current bound, in the region of parameter space that has been
considered. In conclusion, the extension of the type-I seesaw with a second
Higgs doublet induces novel contributions to the majoron couplings to charged
leptons that may increase them notably and lead to observable consequences.

Figure 2: Contours of BR($\mu\to e\,J$) (black) and $\lambda_{5}^{\rm max}$
(blue, dashed) in the $y_{2}^{11}$-$y_{2}^{21}$ plane. The rest of $y_{2}$
Yukawa couplings are set to zero for simplicity, while $\lambda=1$ and
$v_{\chi}=1$ TeV. Here $\lambda_{5}^{\rm max}$ is the maximal value of
$\lambda_{5}$ compatible with all the entries in $m_{\nu}^{\rm 1-loop}$ being
smaller than those in $m_{\nu}^{\rm tree}$. The shaded region is excluded by
$\mu\to e\,J$ searches while the vertical and horizontal red dotted lines
determine the upper limits on $y_{2}^{11}$ and $y_{2}^{21}$, respectively,
derived from Eqs. (3) and (4).

##  4 Final discussion

The majoron is present in any model that breaks a global lepton number
symmetry spontaneously. In the absence of explicit sources of
$\mathrm{U(1)}_{L}$ breaking, its mass is exactly zero. This has a strong
impact on the phenomenology of the model, which changes dramatically with
respect to the variant with explicit breaking. We have derived general
analytical expressions for the 1-loop coupling of the majoron to charged
leptons. These couplings are known to be crucial for the phenomenology of the
model and, in fact, they often lead to the most stringent constraints. Our
analytical results can be used in virtually all models featuring a majoron. We
have provided several example models that illustrate how to use them. Due to
our lack of imagination, we could not find examples for all possible
contributions, but they can be present in some (less popular) models. In any
case, we found perfect agreement with the results obtained in all models for
which the majoron coupling to charged leptons was known. This is a clearly
non-trivial cross-check of our analytical expressions.

The main limitation of our approach is the assumption of a hierarchy of mass
scales, which is nevertheless common to most Majorana neutrino mass models,
such as those based on the seesaw mechanism. We have also focused on scenarios
that actually require the computation of the 1-loop majoron couplings to
charged leptons. For instance, since we did not consider additional fermions
in the particle spectrum, one may wonder about the applicability of our
results in models with heavy leptons. While our analytical expressions cannot
be directly applied in this case, we note that (i) the analogous expressions
with heavy neutral leptons can be easily adapted, and, more importantly, (ii)
models with heavy charged leptons induce majoron couplings with the light
charged leptons already at tree-level, thus making our 1-loop results mere
corrections. In conclusion, our analytical expressions are applicable to any
model in which the 1-loop majoron couplings to charged leptons must be
computed.

Our results go beyond the majoron. One can use them to obtain the 1-loop
couplings of any massless (or very light) pseudoscalar to a pair of charged
leptons. In fact, the majoron can be regarded as a particular and
theoretically well-motivated type of axion-like particle (ALP). Whenever the
assumptions of our approach are fulfilled, our results can also be applied to
compute the 1-loop couplings of an ALP to a pair of charged leptons.
Therefore, they may be of interest in scenarios not necessarily related to
neutrino mass generation and lepton number violation.

The study of majorons and lepton number violation represents a fascinating
frontier in particle physics, pushing the boundaries of our understanding of
the fundamental building blocks of the Universe. Our contribution constitutes
just another step in the ongoing quest to detect and characterize majorons. As
we delve deeper into the mechanisms behind neutrino masses and lepton number
violation, we not only expand our knowledge of the SM but also pave the way
for potential breakthroughs that could revolutionize our understanding of
particle interactions and the origins of mass.

## Acknowledgements

The authors are grateful to Salvador Centelles Chuliá for discussions. Work
supported by the Spanish grants PID2020-113775GB-I00
(AEI/10.13039/501100011033) and CIPROM/2021/054 (Generalitat Valenciana). The
work of AHB is supported by the grant No. CIACIF/2021/100 (also funded by
Generalitat Valenciana). AV acknowledges financial support from MINECO through
the Ramón y Cajal contract RYC2018-025795-I.

##  Appendix A Higher order terms

We present here our results for the contributions of higher order in $m_{\rm
light}$ not included in Sec. 2, using exactly the same notation and
conventions:

$\displaystyle\left(\Gamma_{Z}\right)^{(1)}$
$\displaystyle=-\sum_{s=1}^{3}\sum_{i,j\sim
l}\bar{A}_{kr}U_{ki}U_{rj}\log\left(\frac{m_{j}^{2}}{M_{H}^{2}}\right)\left(U_{si}m_{i}U_{sj}^{*}+U_{sj}m_{j}U_{si}^{*}\right)\,,$
(154) $\displaystyle\left(L_{W}^{\beta\alpha}\right)^{(1)}$
$\displaystyle=\sum_{i,j\sim l}\bar{A}_{kr}^{*}U_{\beta
i}^{*}m_{i}U_{ki}^{*}U_{\alpha
j}U_{rj}^{*}\left[-\frac{5}{6}+3\log\left(\frac{M_{H}}{M_{W}}\right)+2\log\left(\frac{m_{i}}{M_{H}}\right)\right]$
$\displaystyle-\sum_{i,j\sim l}\bar{A}_{kr}U_{\alpha i}m_{i}U_{ki}U_{\beta
j}^{*}U_{rj}\left[\frac{7}{6}+2\log\left(\frac{m_{i}}{M_{H}}\right)\right]\,,$
(155) $\displaystyle\left(L_{\eta}^{LL}\right)^{(1)}=$
$\displaystyle\left(L_{\eta}^{RR}\right)^{(1)}\left(m_{i}\leftrightarrow
m_{j},k\leftrightarrow r,s\leftrightarrow p\right)\,,$ (156)
$\displaystyle\left(L_{\eta}^{RR}\right)^{(1)}_{sp}=$
$\displaystyle\sum_{j\sim
l}\bar{A}_{kr}U_{rj}m_{j}U_{sj}\Bigg{[}\left(F_{5,7,-1}-f_{13}\right)\delta_{kp}-\left(F_{8,2}+f_{14}\right)\left(\bar{M}^{\dagger}\bar{M}\right)_{kp}$
$\displaystyle+$ $\displaystyle\sum_{i\sim
l}U_{ki}U_{pi}^{*}\frac{\log\frac{m_{j}}{m_{\eta}}}{m_{\eta}^{2}}\Bigg{]}\,,$
(157) $\displaystyle\left(L_{\eta}^{RL}\right)^{(1)}_{sp}=$
$\displaystyle\sum_{j\sim
l}\left[F_{3,11,-1}\left(\tilde{\Delta}_{sp}^{1,0,1}+\Delta_{sp}^{1,0,1}\right)+F_{4,12,-2}\,\tilde{\Delta}_{sp}^{1,1,1}+F_{12,-2}\,\Delta_{sp}^{1,1,1}\right]\,,$
(158) $\displaystyle\left(L_{\sigma n}\right)^{(1)}$
$\displaystyle=\text{Re}\,\Bigg{\\{}\sum_{j\sim
l}\left[\bar{B}_{sp}\left(-\frac{2}{M_{H}^{2}}\Delta_{ps}^{1,1,1}+\Delta^{1,0,1}_{ps}+\tilde{\Delta}_{ps}^{1,0,1}\right)\right]\Bigg{\\}}\,,$
(159) $\displaystyle\left(L_{\sigma n}\right)^{(2)}$
$\displaystyle=\text{Re}\,\Bigg{\\{}\sum_{i,j\sim
l}\bar{A}_{kr}U_{rj}U_{ki}\left[\bar{B}_{sp}U_{pj}U_{si}m_{j}m_{i}+\bar{B}_{sp}^{*}U_{sj}^{*}U_{pi}^{*}\left(m_{i}^{2}+m_{j}^{2}\right)\right]\log\frac{M_{H}^{2}}{m_{j}^{2}}\Bigg{\\}}\,.$
(160)

##  Appendix B Loop functions

We define the following loop functions, which turn out to be relevant for the
charged $\eta$ scalar contributions:

$\displaystyle f_{1}$
$\displaystyle=\frac{-M_{H}^{2}+m_{\eta}^{2}+(2M_{H}^{2}-m_{\eta}^{2})\log\left(\frac{M_{H}^{2}}{m_{\eta}^{2}}\right)}{2(M_{H}^{2}-m_{\eta}^{2})^{2}}\,,$
(161) $\displaystyle f_{2}$
$\displaystyle=-\frac{-M_{H}^{2}+m_{\eta}^{2}+M_{H}^{2}\log\left(\frac{M_{H}^{2}}{m_{\eta}^{2}}\right)}{2M_{H}^{2}(M_{H}^{2}-m_{\eta}^{2})^{2}}\,,$
(162) $\displaystyle f_{3}$
$\displaystyle=\frac{2M_{H}^{2}(M_{H}^{2}-m_{\eta}^{2})+(-3M_{H}^{2}m_{\eta}^{2}+m_{\eta}^{4})\log\left(\frac{M_{H}^{2}}{m_{\eta}^{2}}\right)}{2(M_{H}^{2}-m_{\eta}^{2})^{3}}\,,$
(163) $\displaystyle f_{4}$
$\displaystyle=\frac{-M_{H}^{4}+m_{\eta}^{4}+2M_{H}^{2}m_{\eta}^{2}\log\left(\frac{M_{H}^{2}}{m_{\eta}^{2}}\right)}{2M_{H}^{2}(M_{H}^{2}-m_{\eta}^{2})^{3}}\,,$
(164) $\displaystyle f_{5}$
$\displaystyle=\frac{6M_{H}^{6}-5M_{H}^{4}m_{\eta}^{2}-2M_{H}^{2}m_{\eta}^{4}+m_{\eta}^{6}-2(6M_{H}^{4}m_{\eta}^{2}-4M_{H}^{2}m_{\eta}^{4}+m_{\eta}^{6})\log\left(\frac{M_{H}^{2}}{m_{\eta}^{2}}\right)}{4(M_{H}^{2}-m_{\eta}^{2})^{4}}\,,$
(165) $\displaystyle f_{6}$
$\displaystyle=\frac{-3M_{H}^{6}-2M_{H}^{4}m_{\eta}^{2}+7M_{H}^{2}m_{\eta}^{4}-2m_{\eta}^{6}+(8M_{H}^{4}m_{\eta}^{2}-2M_{H}^{2}m_{\eta}^{4})\log\left(\frac{M_{H}^{2}}{m_{\eta}^{2}}\right)}{4M_{H}^{2}(M_{H}^{2}-m_{\eta}^{2})^{4}}\,,$
(166) $\displaystyle f_{7}$
$\displaystyle=-\frac{-4M_{H}^{8}-19M_{H}^{6}m_{\eta}^{2}+27M_{H}^{4}m_{\eta}^{4}-5M_{H}^{2}m_{\eta}^{6}+m_{\eta}^{8}+6M_{H}^{4}m_{\eta}^{2}(3M_{H}^{2}+m_{\eta}^{2})\log\left(\frac{M_{H}^{2}}{m_{\eta}^{2}}\right)}{6(M_{H}^{2}-m_{\eta}^{2})^{5}}\,,$
(167) $\displaystyle f_{8}$
$\displaystyle=\frac{-M_{H}^{6}-9M_{H}^{4}m_{\eta}^{2}+9M_{H}^{2}m_{\eta}^{4}+m_{\eta}^{6}+6M_{H}^{2}m_{\eta}^{2}(M_{H}^{2}+m_{\eta}^{2})\log\left(\frac{M_{H}^{2}}{m_{\eta}^{2}}\right)}{3(M_{H}^{2}-m_{\eta}^{2})^{5}}\,,$
(168) $\displaystyle f_{9}$
$\displaystyle=\frac{2M_{H}^{6}-11M_{H}^{4}m_{\eta}^{2}+10M_{H}^{2}m_{\eta}^{4}-m_{\eta}^{6}+(8M_{H}^{2}m_{\eta}^{4}-2m_{\eta}^{6})\log\left(\frac{M_{H}^{2}}{m_{\eta}^{2}}\right)}{4(M_{H}^{2}-m_{\eta}^{2})^{4}}\,,$
(169) $\displaystyle f_{10}$
$\displaystyle=-\frac{M_{H}^{6}-6M_{H}^{4}m_{\eta}^{2}+3M_{H}^{2}m_{\eta}^{4}+2m_{\eta}^{6}+6M_{H}^{2}m_{\eta}^{4}\log\left(\frac{M_{H}^{2}}{m_{\eta}^{2}}\right)}{4M_{H}^{2}(M_{H}^{2}-m_{\eta}^{2})^{4}}\,,$
(170) $\displaystyle f_{11}$
$\displaystyle=\frac{2M_{H}^{6}+3M_{H}^{4}m_{\eta}^{2}-6M_{H}^{2}m_{\eta}^{4}+m_{\eta}^{6}-6M_{H}^{4}m_{\eta}^{2}\log\frac{M_{H}^{2}}{m_{\eta}^{2}}}{4\left(M_{H}^{2}-m_{\eta}^{2}\right)^{4}}\,,$
(171) $\displaystyle f_{12}$
$\displaystyle=-\frac{M_{H}^{4}+4M_{H}^{2}m_{\eta}^{2}-5m_{\eta}^{4}-4M_{H}^{2}m_{\eta}^{2}\log\frac{M_{H}^{2}}{m_{\eta}^{2}}-2m_{\eta}^{4}\log\frac{M_{H}^{2}}{m_{\eta}^{2}}}{4\left(M_{H}^{2}-m_{\eta}^{2}\right)^{4}}\,$
(172) $\displaystyle f_{13}$
$\displaystyle=-\frac{3M_{H}^{4}-4M_{H}^{2}m_{\eta}^{2}+m_{\eta}^{4}+2M_{H}^{4}\log\left(\frac{m_{\eta}^{2}}{M_{H}^{2}}\right)}{2(M_{H}^{2}-m_{\eta}^{2})^{3}}\,,$
(173) $\displaystyle f_{14}$
$\displaystyle=\frac{2(M_{H}^{2}-m_{\eta}^{2})+(M_{H}^{2}+m_{\eta}^{2})\log\left(\frac{m_{\eta}^{2}}{M_{H}^{2}}\right)}{2(M_{H}^{2}-m_{\eta}^{2})^{3}}\,,$
(174) $\displaystyle f_{15}$
$\displaystyle=\frac{M_{H}^{8}-8M_{H}^{6}m_{\eta}^{2}+8M_{H}^{2}m_{\eta}^{6}-m_{\eta}^{8}-12M_{H}^{4}m_{\eta}^{4}\log\left(\frac{m_{\eta}^{2}}{M_{H}^{2}}\right)}{6\left(M_{H}^{2}-m_{\eta}^{2}\right)^{5}}\,,$
(175) $\displaystyle f_{16}$
$\displaystyle=\frac{-M_{H}^{6}+9M_{H}^{4}m_{\eta}^{2}+9M_{H}^{2}m_{\eta}^{4}-17m_{\eta}^{6}+6(3M_{H}^{2}m_{\eta}^{4}+m_{\eta}^{6})\log\left(\frac{m_{\eta}^{2}}{M_{H}^{2}}\right)}{12(M_{H}^{2}-m_{\eta}^{2})^{5}}\,.$
(176)

In addition, we define

$\displaystyle
F_{i,j,k}\equiv\textup{sign}(i)f_{i}+\textup{sign}(j)f_{j}+\textup{sign}(k)f_{k}\,.$
(177)

The functions in this Appendix take simplified expressions if $m_{\eta}\ll
M_{H}$ and $m_{\eta}\gg M_{H}$. In the light $\eta$ limit ($m_{\eta}\ll
M_{H}$) they take the approximate expressions

$\displaystyle f_{1}$
$\displaystyle=\frac{1}{2M_{H}^{2}}\left[-1+2\log\left(\frac{M_{H}^{2}}{m_{\eta}^{2}}\right)\right]\,,$
(178) $\displaystyle f_{2}$
$\displaystyle=-\frac{1}{2M_{H}^{4}}\left[-1+\log\left(\frac{M_{H}^{2}}{m_{\eta}^{2}}\right)\right]\,,$
(179) $\displaystyle f_{13}$
$\displaystyle=-\frac{1}{2M_{H}^{2}}\left[3-2\log\left(\frac{M_{H}^{2}}{m_{\eta}^{2}}\right)\right]\,,$
(180) $\displaystyle f_{14}$
$\displaystyle=\frac{1}{2M_{H}^{4}}\left[2-\log\left(\frac{M_{H}^{2}}{m_{\eta}^{2}}\right)\right]\,,$
(181) $\displaystyle f_{3}$
$\displaystyle=\frac{2}{3}f_{5}=\frac{3}{2}f_{7}=2f_{9}=2f_{11}=6f_{15}=\frac{1}{M_{H}^{2}}\,,$
(182) $\displaystyle f_{4}$
$\displaystyle=\frac{2}{3}f_{6}=\frac{3}{2}f_{8}=2f_{10}=2f_{12}=6f_{16}=-\frac{1}{2M_{H}^{4}}\,,$
(183)

whereas in the heavy $\eta$ limit ($m_{\eta}\gg M_{H}$) they can be
approximately written as

$\displaystyle f_{1}$
$\displaystyle=-\frac{1}{2m_{\eta}^{2}}\left[-1+\log\left(\frac{M_{H}^{2}}{m_{\eta}^{2}}\right)\right]\,,$
(184) $\displaystyle f_{3}$
$\displaystyle=-\frac{1}{2m_{\eta}^{2}}\log\left(\frac{M_{H}^{2}}{m_{\eta}^{2}}\right)\,,$
(185) $\displaystyle f_{5}$
$\displaystyle=\frac{1}{4m_{\eta}^{2}}\left[1-2\log\left(\frac{M_{H}^{2}}{m_{\eta}^{2}}\right)\right]\,,$
(186) $\displaystyle f_{9}$
$\displaystyle=-\frac{1}{4m_{\eta}^{2}}\left[1+2\log\left(\frac{M_{H}^{2}}{m_{\eta}^{2}}\right)\right]\,,$
(187) $\displaystyle f_{12}$
$\displaystyle=\frac{1}{4m_{\eta}^{2}M_{H}^{2}}\left[5+2\log\left(\frac{M_{H}^{2}}{m_{\eta}^{2}}\right)\right]\,,$
(188) $\displaystyle f_{14}$
$\displaystyle=-\frac{1}{2m_{\eta}^{4}}\left[-2+\log\left(\frac{m_{\eta}^{2}}{M_{H}^{2}}\right)\right]\,,$
(189) $\displaystyle f_{16}$
$\displaystyle=\frac{1}{12m_{\eta}^{4}}\left[17-6\log\left(\frac{m_{\eta}^{2}}{M_{H}^{2}}\right)\right]\,,$
(190) $\displaystyle f_{2}$
$\displaystyle=f_{4}=f_{6}=f_{10}=-\frac{1}{2m_{\eta}^{2}M_{H}^{2}}\,,$ (191)
$\displaystyle 3f_{7}$
$\displaystyle=2f_{11}=f_{13}=3f_{15}=\frac{1}{2m_{\eta}^{2}}\,,$ (192)
$\displaystyle f_{8}$ $\displaystyle=-\frac{1}{3m_{\eta}^{4}}\,.$ (193)

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