.

#  Towards deconstructing the simplest seesaw mechanism

Sanjoy Mandal<EMAIL_ADDRESS>Korea Institute for Advanced Study, Seoul
02455, Korea O. G. Miranda<EMAIL_ADDRESS>Departamento de Física, Centro
de Investigación y de Estudios Avanzados del IPN
Apartado Postal 14-740 07000 Ciudad de México, Mexico G. Sanchez Garcia
<EMAIL_ADDRESS>Departamento de Física, Centro de Investigación y de
Estudios Avanzados del IPN
Apartado Postal 14-740 07000 Ciudad de México, Mexico J. W. F. Valle
<EMAIL_ADDRESS>AHEP Group, Institut de Física Corpuscular –
C.S.I.C./Universitat de València, Parc Cientific de Paterna.
C/Catedratico José Beltrán, 2 E-46980 Paterna (València) - SPAIN Xun-Jie Xu
<EMAIL_ADDRESS>Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China

(August 28, 2024)

###### Abstract

The triplet or type-II seesaw mechanism is the simplest way to endow neutrinos
with mass in the Standard Model (SM). Here we review its associated theory and
phenomenology, including restrictions from $S$, $T$, $U$ parameters, neutrino
experiments, charged lepton flavour violation as well as collider searches. We
also examine restrictions coming from requiring consistency of electroweak
symmetry breaking, i.e. perturbative unitarity and stability of the vacuum.
Finally, we discuss novel effects associated to the scalar mediator of
neutrino mass generation namely, (i) rare processes, e.g. $l_{\alpha}\to
l_{\beta}\gamma$ decays, at the intensity frontier, and also (ii) four-lepton
signatures in colliders at the high-energy frontier. These can be used to
probe neutrino properties in an important way, providing a test of the
absolute neutrino mass and mass-ordering, as well as of the atmospheric
octant. They may also provide the first evidence for charged lepton flavour
violation in nature. In contrast, neutrino non-standard interaction strengths
are found to lie below current detectability.

##  I Introduction

The discovery of neutrino oscillations [1, 2] has opened a new era in particle
physics, implying the need for small neutrino mass and, as a result, providing
the first strong evidence for new physics beyond the Standard Model (SM) [3].
So far, we have not been able to underpin the ultimate mechanism responsible
for neutrino mass generation. An attractive possibility is the seesaw
mechanism, which leads to small active neutrino masses through the exchange of
heavy lepton or scalar mediators. Most generally the seesaw is formulated
within the SM picture, i.e., using the $\mathrm{SU(3)_{c}\otimes
SU(2)_{L}\otimes U(1)_{Y}}$ gauge group [4, 5].

Here we re-examine the simplest scalar-mediated $\mathrm{SU(3)_{c}\otimes
SU(2)_{L}\otimes U(1)_{Y}}$ seesaw mechanism, i.e., the type-II seesaw with
explicit breaking of lepton number. We discuss both its theoretical
consistency and the resulting restrictions, including the current experimental
constraints, and those from electroweak precision data. We show that future
high-energy colliders can play a key role in probing neutrino properties such
as the atmospheric octant, the neutrino mass-ordering, and the absolute-
neutrino-mass-scale, as recently sketched in [6].

Moreover, such high-energy collider experiments may also provide the discovery
site for processes with charged lepton flavor non-conservation in nature [7,
8, 9, 10, 11]. In contrast, current charged lepton flavour violation limits
severely restrict the strength of neutrino non-standard interactions (NSI),
posing a real challenge for the next generation of neutrino experiments.

After a brief review of the status of the type-II seesaw mechanism we show how
it provides a neat realization of the idea that neutrino properties can be
well-studied also within high energy experiments. The paper extends the
presentation given in [6], providing the full theoretical discussion and
relevant technical details. The organization is as follows. In Sec. II, we
briefly describe the type-II seesaw model with explicit violation of lepton
number. In Sec. III, we analyze the constraints, including consistency of the
scalar sector in III.1, electroweak precision data in III.2, constraints from
neutrino physics in III.3, from charged lepton flavour violation bounds in
III.4, and from collider experiments in III.5. We show how most of these
restrictions can be significant.

In Sec. IV, we describe how the type-II seesaw model can lead to promising
signatures at hadron colliders such as LHC and planned FCC, as well as future
$e^{+}e^{-}$ colliders, such as the ILC, CLIC, and the CEPC project in China.
Altogether, as discussed in Sec. V, we find that current and future
experiments can ultimately fully deconstruct the type-II seesaw mechanism. We
discuss the interplay between collider probes and charged lepton flavour
violation searches, illustrating the complementarity of high-energy and high-
intensity frontiers. As a result one will be able to probe important current
unknowns from neutrino experiments, such as the atmospheric octant, the mass
ordering, and the absolute neutrino mass.

In Sec. VI, we examine the possibility of having sizeable non-standard
neutrino interactions (NSI) given the constraints from searches for lepton
flavour violation described previously. Summary and outlook are given in Sec.
VII. Complementary material is presented in Appendices A, B and C.

##  II Type-II seesaw mechanism

Small neutrino masses required by neutrino oscillation experiments can be
generated by the non-renormalizable dimension-5 Weinberg operator:

${\cal L}\propto\frac{1}{\Lambda}{L}\Phi\Phi L.$ (1)

Out of the three possible UV-completions of the Weinberg operator, here we
focus on the simplest, the type-II seesaw mechanism [4, 5] 111The type-II is
the simplest seesaw mechanism, hence originally named type-I in the above
references.. It introduces one heavy $SU(2)_{L}$ triplet scalar
$\Delta=(\Delta^{++},\Delta^{+},\Delta^{0})^{T}$, with hypercharge
$Y_{\Delta}=2$. It is convenient to describe $\Delta$ in its matrix form as:

$\Delta=\left(\begin{array}[]{cc}\Delta^{+}/\sqrt{2}&\Delta^{++}\\\
\Delta^{0}&-\Delta^{+}/\sqrt{2}\end{array}\right),$ (2)

which is a traceless bidoublet under $SU(2)_{L}$, i.e., $\Delta\to U\Delta
U^{\dagger}$ for $U\in SU(2)_{L}$. This triplet couples only to the SM lepton
doublets, and to the scalar Higgs doublet $\Phi=(\Phi^{+}\leavevmode\nobreak\
\leavevmode\nobreak\ \Phi^{0})^{T}$. The relevant Yukawa Lagrangian reads:

${\cal L}_{{\rm type-
II}}=\left[iY_{\Delta\alpha\beta}L_{\alpha}^{T}C^{-1}\tau_{2}\Delta
L_{\beta}+\text{h.c.}\right]+\left(D_{\mu}\Phi\right)^{\dagger}\left(D^{\mu}\Phi\right)+\left(D_{\mu}\Delta\right)^{\dagger}\left(D^{\mu}\Delta\right)-V(\Phi,\Delta),$
(3)

where $Y_{\Delta\alpha\beta}$ is a complex symmetric matrix, $L_{\alpha}$ are
the lepton doublets, $C$ is the charge conjugation operator, and $D_{\mu}$
denotes the covariant derivative of the corresponding scalar field. We can
assume without loss of generality that the charged lepton Yukawa coupling is
in diagonal form. Note that after symmetry breaking, the first term in Eq. (3)
will generate the Majorana mass term for light neutrinos as

$\displaystyle{\cal L}_{\text{Majorana}}=\overline{\nu_{\alpha
L}^{c}}m_{\alpha\beta}^{\nu}\nu_{\beta L}+\text{h.c.}$ (4)

The scalar potential $V(\Phi,\Delta)$ is given as,

$\displaystyle V(\Phi,\Delta)=$ $\displaystyle-
m_{\Phi}^{2}\Phi^{\dagger}\Phi+\frac{\lambda}{4}(\Phi^{\dagger}\Phi)^{2}+\tilde{M}_{\Delta}^{2}{\rm
Tr}\left[\Delta^{\dagger}\Delta\right]+\lambda_{2}\left[{\rm
Tr}\Delta^{\dagger}\Delta\right]^{2}+\lambda_{3}{\rm
Tr}\left[\Delta^{\dagger}\Delta\right]^{2}$ (5)
$\displaystyle+\left[\mu\Phi^{T}i\sigma_{2}\Delta^{\dagger}\Phi+\text{h.c.}\right]+\lambda_{1}(\Phi^{\dagger}\Phi){\rm
Tr}\left[\Delta^{\dagger}\Delta\right]+\lambda_{4}\Phi^{\dagger}\Delta\Delta^{\dagger}\Phi.$

The Higgs triplet UV-completion of the Weinberg operator is characterized by a
small induced vacuum expectation value (VEV) for the triplet. Indeed, within
most of the parameter space, the minimization of the potential can flavour
nonzero VEVs $v_{\Phi}$ and $v_{\Delta}$ for the doublet and the triplet,
respectively 222For more details, see Ref. [12].. Minimization of the total
potential $V(\Phi,\Delta)$ leads to the relations:

$\displaystyle\tilde{M}_{\Delta}^{2}$ $\displaystyle=\displaystyle
M_{\Delta}^{2}-\frac{1}{2}\left[2v_{\Delta}^{2}(\lambda_{2}+\lambda_{3})+v_{\Phi}^{2}(\lambda_{1}+\lambda_{4})\right],\,\,\text{
with }M_{\Delta}^{2}\equiv\frac{v_{\Phi}^{2}\mu}{\sqrt{2}v_{\Delta}}.$ (6)
$\displaystyle m_{\Phi}^{2}$
$\displaystyle=\displaystyle\frac{1}{2}\left[\frac{v_{\Phi}^{2}\lambda}{2}+v_{\Delta}^{2}(\lambda_{1}+\lambda_{4})-2\sqrt{2}\mu
v_{\Delta}\right].$ (7)

In the limit $M_{\Delta}\gg v_{\Phi}$, which is consistent with all the
constraints (see below), we can solve Eq. (6) for $v_{\Delta}$. Keeping terms
of $\mathcal{O}(v_{\Phi}/M_{\Delta})$ we get the small induced triplet vacuum
expectation value :

$\boxed{v_{\Delta}\approx\frac{\mu
v_{\Phi}^{2}}{\sqrt{2}\tilde{M}_{\Delta}^{2}}}\leavevmode\nobreak\ .$ (8)

Eq. (8) explicitly shows that the smallness of the triplet VEV can be induced
either by a small $\mu$, or by a large value for $\tilde{M}_{\Delta}$. A small
$\mu$ is in agreement with t’Hooft’s naturalness argument, since it is the
only parameter that sources lepton number violation in Eq. (5) 333Of course,
within a more complete theory one can give a dynamical meaning to the
parameter $\mu$ [5].. As a result, in the limit $\mu\to 0$, lepton number
symmetry is recovered. After spontaneous symmetry breaking, we can write the
doublet and triplet neutral fields as:

$\Delta=\frac{1}{\sqrt{2}}\begin{pmatrix}\Delta^{+}&\sqrt{2}\Delta^{++}\\\
v_{\Delta}+h_{\Delta}+i\eta_{\Delta}&-\leavevmode\nobreak\
\Delta^{+}\end{pmatrix},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\Phi=\frac{1}{\sqrt{2}}\begin{pmatrix}\sqrt{2}{\Phi}^{+}\\\
v+h_{\Phi}+i\eta_{\Phi}\end{pmatrix}.$ (9)

Our scalar sector contains 10 degrees of freedom. After EW breaking, seven
remain as physical fields with definite masses. These scalar fields are the
charged Higgs bosons $H^{\pm\pm}$, $H^{\pm}$, and the neutral Higgs bosons
$h$, $H^{0}$ and $A^{0}$. The doubly-charged Higgs $H^{\pm\pm}$ is simply the
$\Delta^{\pm\pm}$ present in $\Delta$. The two singly-charged scalar fields
$\Phi^{\pm}$ and $\Delta^{\pm}$ from $\Phi$ and $\Delta$ mix, giving the
singly-charged Higgs boson $H^{\pm}$ and the unphysical charged Goldstone
$G^{\pm}$ 444Note that, the mixing within the singly-charged Higgs sector is
characterized by $\tan\beta_{\pm}=\sqrt{2}\frac{v_{\Delta}}{v_{\Phi}}$, hence
can be neglected for small $v_{\Delta}$.. Similarly, $\eta_{\Delta}$ and
$\eta_{\Phi}$ will mix and give rise to the CP-odd scalar $A^{0}$ and the
neutral Goldstone boson $G^{0}$ which becomes the longitudinal mode of $Z$.
Finally, two CP-even fields $h_{\Delta}$ and $h_{\Phi}$ mix and give rise to
the SM Higgs boson $h$ and a heavy Higgs boson $H^{0}$. For a detailed
discussion of charged and neutral scalar mass matrices and mixings, see [13,
14, 15].

The physical masses of doubly- and singly-charged Higgs bosons $H^{\pm\pm}$
and $H^{\pm}$ can be written as:

$\displaystyle
m_{H^{++}}^{2}=M_{\Delta}^{2}-v_{\Delta}^{2}\lambda_{3}-\frac{\lambda_{4}}{2}v_{\Phi}^{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
m_{H^{+}}^{2}=\left(M_{\Delta}^{2}-\frac{\lambda_{4}}{4}v_{\Phi}^{2}\right)\left(1+\frac{2v_{\Delta}^{2}}{v_{\Phi}^{2}}\right),$
(10)

whereas the CP-odd Higgs field $A^{0}$ has the following mass:

$\displaystyle m_{A^{0}}^{2}$ $\displaystyle=$ $\displaystyle
M_{\Delta}^{2}\left(1+\frac{4v_{\Delta}^{2}}{v_{\Phi}^{2}}\right).$ (11)

The common $M_{\Delta}^{2}$ is expected, since the scalars come from the same
triplet. Finally, the CP-even Higgs bosons $h$ and $H^{0}$ have the following
physical masses:

$\displaystyle m_{h}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[A+C-\sqrt{(A-C)^{2}+4B^{2}}],$ (12) $\displaystyle
m_{H^{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[A+C+\sqrt{(A-C)^{2}+4B^{2}}],$ (13)

where,

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{\lambda}{2}{v_{\Phi}^{2}},\;\;B=v_{\Phi}(-\sqrt{2}\mu+(\lambda_{1}+\lambda_{4})v_{\Delta}),\;\;C=M_{\Delta}^{2}+2(\lambda_{2}+\lambda_{3})v_{\Delta}^{2}.$
(14)

For $v_{\Delta}\ll v_{\Phi}$, the masses of the physical Higgs bosons can be
approximated as follows:

$\displaystyle\boxed{m_{H^{\pm\pm}}^{2}\simeq
M_{\Delta}^{2}-\frac{\lambda_{4}}{2}v_{\Phi}^{2},\,\,\,\,m_{H^{\pm}}^{2}\simeq
M_{\Delta}^{2}-\frac{\lambda_{4}}{4}v_{\Phi}^{2},\,\,\,\,m_{h}^{2}\simeq
2\lambda v_{\Phi}^{2}\,\,\,\,\text{and}\,\,\,\,m_{H^{0}}^{2}\approx
m_{A^{0}}^{2}\simeq M_{\Delta}^{2}}\leavevmode\nobreak\ ,$ (15)

so their mass-squared differences are given by:

$\displaystyle m_{H^{\pm\pm}}^{2}-m_{H^{\pm}}^{2}\approx
m_{H^{\pm}}^{2}-m_{H^{0}/A^{0}}^{2}\approx-\frac{\lambda_{4}}{4}v_{\Phi}^{2}.$
(16)

We further define the two mass-splittings as follows:

$\displaystyle\delta m_{1}=m_{H^{\pm}}-m_{H^{0}},\,\,\,\,\,\,\,\,\delta
m_{2}=m_{H^{\pm\pm}}-m_{H^{\pm}}.$ (17)

With the assumptions $v_{\Delta}\ll v_{\Phi}$ and
$M_{\Delta}^{2}\gg|\lambda_{4}|v_{\Phi}^{2}$, the two mass-splittings $\delta
m_{1,2}$ can be approximated as

$\displaystyle\delta
m_{1,2}\approx-\frac{\lambda_{4}}{8}\frac{v_{\Phi}^{2}}{M_{\Delta}}.$ (18)

Hence, the tree-level mass-splitting is dictated by the dimensionless quartic
coupling $\lambda_{4}$, so $\delta m_{1}\approx\delta m_{2}$ at tree-level.
Note also that a relatively large value of this quartic coupling at the
electroweak scale can become non-perturbative at high energies even below the
Planck scale. Hence $\lambda_{4}$ should be small, and as a result, the mass-
splitting $\delta m_{1,2}$ should also be small.

Note also that, three distinct mass spectra are expected depending on the
value (sign) of $\lambda_{4}$, as: $(1).\,\,\lambda_{4}=0:\,\,\delta m\approx
0\,(m_{H^{\pm\pm}}\simeq m_{H^{\pm}}\simeq m_{H^{0}/A})$,
$(2).\,\,\lambda_{4}<0:\,\,\delta
m>0\,(m_{H^{\pm\pm}}>m_{H^{\pm}}>m_{H^{0}/A})$ and
$(3).\,\,\lambda_{4}>0:\,\,\delta
m<0\,(m_{H^{\pm\pm}}<m_{H^{\pm}}<m_{H^{0}/A})$. These will be important when
discussing the collider constraints in the triplet seesaw, Sec. III.5.

Figure 1: Feynman diagram generating Majorana masses in type-II seesaw
mechanism.

On the other hand, the Yukawa interaction in the first term in Eq. (3) is
responsible for generating light Majorana neutrino masses. After spontaneous
symmetry breaking, we have:

$\boxed{m_{\alpha\beta}^{\nu}\equiv(Y_{\Delta})_{\alpha\beta}\frac{v_{\Delta}}{\sqrt{2}}}\leavevmode\nobreak\
.$ (19)

This mass matrix $m_{\alpha\beta}^{\nu}$ in the flavor basis must be
diagonalized in order to obtain the physical neutrino masses. The smallness of
the latter will follow from the small VEV, Eq. (8).

##  III Constraints

We now examine the constraints that follow from electroweak precision tests,
theoretical consistency, as well as from various experiments. Theoretical
consistency requirements include stability of the electroweak vacuum as well
as perturbative unitarity.

###  III.1 Stability, Unitarity and Perturbativity

The stability of the vacuum requires the potential given in Eq. (5) to be
bounded from below when the scalar fields become large in any field space
direction. At large field values, the potential Eq. (5) is generically
dominated by the part containing the quartic terms, i.e.

$V^{(4)}(\Phi,\Delta)=\frac{\lambda}{4}(\Phi^{\dagger}{\Phi})^{2}+\lambda_{1}(\Phi^{\dagger}{\Phi})\text{Tr}(\Delta^{\dagger}{\Delta})+\lambda_{2}(\text{Tr}\Delta^{\dagger}{\Delta})^{2}+\lambda_{3}\text{Tr}(\Delta^{\dagger}{\Delta})^{2}+\lambda_{4}{\Phi^{\dagger}\Delta\Delta^{\dagger}\Phi}.$
(20)

The quartic piece of the potential suffices to obtain the conditions for tree-
level stability. Taking into account all field directions, the necessary and
sufficient conditions for the potential to be bounded from below (BFB) have
been studied in Ref. [14, 13]. They can be written as follows,

$\displaystyle\lambda>0\;\;{\rm\&}\;\;\lambda_{2}+\lambda_{3}>0\;\;{\rm\&}\;\;\lambda_{2}+\frac{\lambda_{3}}{2}>0$
(21)
$\displaystyle{\rm\&}\;\;\lambda_{1}+\sqrt{\lambda(\lambda_{2}+\lambda_{3})}>0\;\;{\rm\&}\;\;\lambda_{1}+\sqrt{\lambda(\lambda_{2}+\frac{\lambda_{3}}{2})}>0$
(22)
$\displaystyle{\rm\&}\;\;\lambda_{1}+\lambda_{4}+\sqrt{\lambda(\lambda_{2}+\lambda_{3})}>0\;\;{\rm\&}\;\;\lambda_{1}+\lambda_{4}+\sqrt{\lambda(\lambda_{2}+\frac{\lambda_{3}}{2})}>0.$
(23)

The above conditions ensure a BFB potential for all possible directions in
field space and provide the most general necessary and sufficient BFB
conditions at tree-level. As quartic couplings change with energy due to
renormalisation group evolution, the BFB conditions must be satisfied at all
energy scales for a consistent vacuum.

In addition, quartic couplings can also be constrained by demanding tree-level
unitarity, which should be preserved in a variety of scattering processes such
as scalar-scalar scattering, gauge-boson-gauge-boson scattering, and scalar-
gauge-boson scattering. These have been studied in great detail for the type-
II seesaw model in Ref. [13]. Demanding tree-level unitarity to be preserved
for different elastic scattering processes, one obtains the following
constraints:

$\displaystyle|\lambda_{1}+\lambda_{4}|\leq\kappa\pi,\,\,|\lambda_{1}|\leq\kappa\pi,\,\,|2\lambda_{1}+3\lambda_{4}|\leq
2\kappa\pi,\,\,|\lambda|\leq
2{\kappa}\pi,\,\,|\lambda_{2}|\leq\frac{\kappa}{2}\pi,\,\,|\lambda_{2}+\lambda_{3}|\leq\frac{\kappa}{2}\pi,$
(24)
$\displaystyle|\lambda+4\lambda_{2}+8\lambda_{3}\pm\sqrt{(\lambda-4\lambda_{2}-8\lambda_{3})^{2}+16\lambda_{4}^{2}}\;|\leq
4\kappa\pi,$ (25)
$\displaystyle|3\lambda+16\lambda_{2}+12\lambda_{3}\pm\sqrt{(3\lambda-16\lambda_{2}-12\lambda_{3})^{2}+24(2\lambda_{1}+\lambda_{4})^{2}}\;|\leq
4\kappa\pi,$ (26) $\displaystyle|2\lambda_{1}-\lambda_{4}|\leq
2\kappa\pi,\,\,|2\lambda_{2}-\lambda_{3}|\leq\kappa\pi.$ (27)

where $\kappa=8$. Moreover, we also impose the perturbativity condition, that
is, all quartic couplings should be less than $4\pi$ up to the Planck scale.

Let us now examine the stability of the electroweak vacuum in more detail. To
see how the couplings evolve with energy we use the full two-loop
renormalisation group equations (RGEs) governing the evolution of the Higgs
quartic coupling.

Effective Theory

Below the scale $\mu_{r}=M_{\Delta}$ one needs to integrate out the triplet
field, so that the resulting theory is the Standard Model plus an effective
dimension-five Weinberg operator,

$\displaystyle\mathcal{L}_{\nu}^{d=5}=\kappa_{0}L\Phi\Phi
L,\,\,\text{with}\,\,\kappa_{0}=\frac{Y_{\Delta}\mu}{M_{\Delta}^{2}}.$ (28)

Hence, below the scale $\mu_{r}=M_{\Delta}$, only the Standard Model couplings
as well as the coefficient $\kappa_{0}$ will evolve.

As seen in Fig. 1, the same operator which generates the neutrino mass below
the scale $\mu_{r}=M_{\Delta}$, also provides a correction to the running of
the Higgs quartic self-coupling $\lambda$ below that scale, see Fig. 2. This
generic contribution is of order $v_{\Delta}^{2}\kappa_{0}^{2}$ and hence
negligible [16], as $\kappa_{0}$ is very small to account for the small
neutrino masses. However, in the type-II seesaw, there are tree-level
contributions to the effective potential arising from Fig. 3. These are
obtained by integrating out the triplet and introducing threshold corrections
to the Standard Model Higgs quartic coupling $\lambda$ at the scale
$\mu_{r}=M_{\Delta}$.

Figure 2: Effect of Weinberg’s effective operator on the Higgs quartic
interaction in the effective theory [17].

Figure 3: Contributions to the quartic coupling $\lambda$ in effective theory
obtained by integrating out the triplet.

The effective potential below the scale $\mu_{r}=M_{\Delta}$ is then given as:

$\displaystyle
V=-m_{\Phi}^{2}\Phi^{\dagger}\Phi+\frac{\lambda^{\prime}}{4}(\Phi^{\dagger}\Phi)^{2},$
(29)

where $\lambda^{\prime}$ is given as:

$\displaystyle\lambda^{\prime}=\lambda-\frac{4\mu^{2}}{M_{\Delta}^{2}}.$ (30)

Eq. (30) shows how the matching condition at the scale $\mu_{r}=M_{\Delta}$
induces a positive shift in the Higgs quartic coupling,
$\delta\lambda=\frac{4\mu^{2}}{M_{\Delta}^{2}}$. This corresponds to a larger
Higgs quartic coupling above the scale $\mu_{r}=M_{\Delta}$ and improves the
chances of keeping $\lambda$ positive all the way at higher scales.

Figure 4: The evolution of the SM quartic coupling $\lambda_{\text{SM}}$ is
shown by the red dashed curves. This nearly coincides with the evolution of
the scalar quartic coupling $\lambda$ below $\mu_{r}=M_{\Delta}$. The solid
green curve illustrates the positivity of the quartic coupling throughout its
evolution in both cases.

Full Theory

Building upon the discussion of the previous section, we now turn to the
region above the scale $\mu_{r}=M_{\Delta}$. As discussed earlier, below the
$\mu_{r}=M_{\Delta}$ scale, the theory is an effective Standard Model
supplemented by the dimension-five Weinberg operator. However, above the
$\mu_{r}=M_{\Delta}$ scale, the theory is UV-complete. Therefore, all the new
couplings, such as the Yukawa coupling $Y_{\Delta}$ and the quartic couplings
$\lambda_{1,2,3,4}$ will take part in the system of RGEs, and will
consequently affect the evolution of the Standard Model couplings, especially
$\lambda$. In Appendix A, we present the one-loop RGEs for the full theory,
and in Fig. 4 we give illustrative examples of the evolution of $\lambda$.

Notice that below the scale $M_{\Delta}$, the RGEs are the ones with
${\lambda_{1,2,3,4}}$ removed and $\lambda$ replaced by $\lambda^{\prime}$.
Above $M_{\Delta}$ one needs to include $\beta_{\lambda_{1,2,3,4}}$ and find
$\lambda$ using the full RGEs with the matching condition as in Eq. (30) at
$\mu_{r}=M_{\Delta}$. As far as the new Yukawa coupling $Y_{\Delta}$ is
concerned, it can be obtained by substituting
$Y_{\Delta}=\theta(\mu_{r}-M_{\Delta})Y_{\Delta}$ on the right side of the
RGEs of the full theory.

In Fig. 4, we show the running of the quartic coupling $\lambda$ for two
benchmarks. In the left panel we fix $M_{\Delta}=1$ TeV,
$\lambda_{1,2,3,4}=0.1$ and
$\text{Tr}\left(Y_{\Delta}^{\dagger}Y_{\Delta}\right)=0.5$. In the right panel
we fix $M_{\Delta}=10$ TeV, $\lambda_{1,2,3,4}=0.05$, and also 555Note that
this large Yukawa coupling is chosen just for illustration, and may not be
consistent with neutrino masses. However, if stability holds even for such
large Yukawa values, then it will be certainly consistent with smaller Yukawas
adequate for neutrino masses.
$\text{Tr}\left(Y_{\Delta}^{\dagger}Y_{\Delta}\right)=0.5$. We choose the
parameter $\mu$ in such a way that the positive shift $\delta\lambda=0.05$.
One sees from the left panel that, just with the effect of running, the
coupling $\lambda$ can be kept positive all the way up to Planck scale even
for relatively large Yukawa coupling
$\text{Tr}(Y_{\Delta}^{\dagger}Y_{\Delta})$ as long as we choose adequately
large values for the quartic couplings $\lambda_{1,2,3,4}$. In the right
panel, we show that if the positive shift $\delta\lambda$ coming from the
matching condition Eq. (30) is relatively large, this can also help achieving
vacuum stability. The evolution of other quartic couplings and stability
conditions given in Eq. (21)-(23) are not shown to avoid cluttering the plot.
In conclusion, one sees that even for relatively large Yukawa coupling
$Y_{\Delta}$, one can have a wide range of parameters with stable vacuum all
the way up to the Planck scale.

###  III.2 Electroweak precision $S$, $T$, $U$ parameters

The presence of the Higgs triplet in the type-II seesaw will modify the
prediction for different radiative corrections, especially for the so-called
$\rho$ parameter [4]. This will imply restrictions on the vacuum expectation
value, $v_{\Delta}$, that will translate into theoretical restrictions for the
mass of the triplet scalars.

The triplet VEV $v_{\Delta}$ leads to a tree-level contribution to the $\rho$
parameter which is defined by $\rho\equiv m_{W}^{2}/(m_{Z}^{2}c_{W}^{2})$. The
Standard Model predicts $\rho=1$ at tree level. At loop level, it is redefined
so as to absorb loop corrections so that the SM value remains $1$. The global
fit of electroweak precision data currently gives [18],

$\rho_{\text{global fit}}=1.00038\pm 0.00020.$ (31)

In the type-II seesaw, $m_{W}^{2}$ and $m_{Z}^{2}$ are modified to
$m_{W}^{2}=g^{2}(v_{\Phi}^{2}+2v_{\Delta}^{2})/4$ and
$m_{Z}^{2}=g^{2}(v_{\Phi}^{2}+4v_{\Delta}^{2})/(4c_{W}^{2})$. This leads to
the theoretical relation

$\rho_{\text{th.}}=\frac{v_{\Phi}^{2}+2v_{\Delta}^{2}}{v_{\Phi}^{2}+4v_{\Delta}^{2}}\approx
1-2\frac{v_{\Delta}^{2}}{v^{2}},$ (32)

where the electroweak VEV $v\equiv\sqrt{v_{\Phi}^{2}+2v_{\Delta}^{2}}\approx
246$ GeV. Comparing Eq. (32) with Eq. (31), we obtain

$v_{\Delta}\leq 2.6\ {\rm GeV}\ {\rm at}\ 3\sigma\text{ C.L.}$ (33)

Since the triplet couples to the $W$ and $Z$ bosons, it affects the oblique
parameters (often referred to as the $S$, $T$, $U$ parameters) [19]. The
general form of these are given in Appendix B. Assuming that the triplet is
heavy and the mass splittings amongst its components are small, the result
reads as:

$\displaystyle S$
$\displaystyle\approx-\frac{(2-4s_{W}^{2}+5s_{W}^{4})m_{Z}^{2}}{30\pi
m_{H^{\pm}}^{2}}-\frac{2(\delta m_{1}+\delta m_{2})}{3\pi
m_{H^{\pm}}}\thinspace,$ (34) $\displaystyle T$
$\displaystyle\approx\frac{\delta m_{1}^{2}+\delta m_{2}^{2}}{6\pi
s_{W}^{2}m_{W}^{2}}\thinspace,$ (35) $\displaystyle U$
$\displaystyle\approx\frac{\delta m_{1}-\delta m_{2}}{3\pi
m_{H^{\pm}}}\thinspace,$ (36)

where $\delta m_{1}$ and $\delta m_{2}$ are given in Eq. (17). The $U$
parameter is highly suppressed, since in type-II seesaw we have $\delta
m_{1}\approx\delta m_{2}$. The $T$ parameter is mainly sensitive to the mass
splitting and it is amplified by the $s_{W}^{2}$ in the denominator. The $S$
parameter, in principle, could place a constraint on the value of
$m_{H^{\pm}}$.

When $U$ is fixed at zero, the current global fit of electroweak precision
data (EWPD) gives [18]:

$\displaystyle S$ $\displaystyle=0.00\pm 0.07\thinspace,$ (37) $\displaystyle
T$ $\displaystyle=0.05\pm 0.06\thinspace.$ (38)

Comparing Eq. (38) with Eq. (35), we obtain

$|m_{H^{\pm}}-m_{H^{0}}|\approx|m_{H^{\pm\pm}}-m_{H^{\pm}}|<45.5\ {\rm GeV}\ \
{\rm at}\ 90\%\text{ C.L.}$ (39)

As for the $S$ parameter, the first term in Eq. (34) is negligible ($\lesssim
0.01$ for $m_{H^{\pm}}\gtrsim 100$ GeV) and the second term gives
$m_{H^{\pm}}>169\ {\rm GeV}$ (90 %C.L.) for
$|m_{H^{\pm\pm}}-m_{H^{\pm}}|=45.5\ {\rm GeV}$. This is less stringent than
the bound from direct searches at the LHC to be discussed below.

###  III.3 Constraints from neutrino physics

We now examine the observational restrictions on the type-II seesaw model
parameters that follow from neutrino oscillation experiments, as well as the
absolute neutrino mass scale probes.

Neutrino oscillations

The discovery of neutrino oscillations made in solar and atmospheric neutrino
studies was quickly confirmed by reactor and accelerator-based experiments
which provided a better determination of the oscillation parameters. Current
experimental data converge towards a consistent global picture called the
“three-neutrino paradigm” which provides tight restrictions on the allowed
neutrino mass and mixing parameters [20, 21] (note, however, that the octant
of the atmospheric angle, the mass-ordering, and CP determination still
require further input from the next generation of oscillation searches). These
constraints apply to any model of neutrino mass generation, such as the type-
II seesaw.

If the type-II seesaw mechanism is responsible for the small neutrino masses,
then the Yukawa couplings from Eq. (19), must reproduce the corresponding
flavor neutrino mass matrix. This matrix is related to the mass basis matrix
through

$m_{\alpha\beta}=[U\textup{diag}\left\\{m_{1},m_{2},m_{3}\right\\}U^{T}]_{\alpha\beta},$
(40)

with $U$ denoting the neutrino mixing matrix, and $m_{i}$ the eigenvalues of
the massive neutrino states. Note that the matrix $U$ depends on three mixing
angles ($\theta_{12},\theta_{13},\theta_{23}$), and three CP phases. The Dirac
phase affects oscillations, while the two Majorana phases are crucial in the
description of neutrinoless double beta decay as discussed below. Note that we
express the mixing matrix in a symmetrical way, where each phase is associated
to each of the mixing angles [4]. The best fit values for the mixing angles as
well as for the squared mass differences are determined using a global
analysis of the neutrino oscillation data [20].

Neutrinoless double-beta decay ($0\nu\beta\beta$ for short) is the prime
lepton number violating process [22, 23, 24] that is sensitive not only to the
absolute scale of neutrino mass but also to the Majorana phases inaccessible
to oscillation experiments. The Majorana-type mass term of the type-II seesaw
model implies that the exchange of light neutrinos yields a $0\nu\beta\beta$
decay amplitude given as

$\left\langle
m_{\beta\beta}\right\rangle=\left|\sum_{j=1}^{3}U_{ej}^{2}m_{j}\right|=\left|c^{2}_{12}c^{2}_{13}m_{1}+s^{2}_{12}c^{2}_{13}m_{2}e^{2i\phi_{12}}+s^{2}_{13}m_{3}e^{2i\phi_{13}}\right|\leavevmode\nobreak\
.$ (41)

Here the neutrino mixing matrix is expressed in terms of the symmetrical
parametrization [4]. The latter provides a most transparent description of
$0\nu\beta\beta$ [25], in contrast with the convention adopted by the PDG.

Notice that the amplitude $\left\langle m_{\beta\beta}\right\rangle$ coincides
precisely with the $|m_{ee}|$ entry in Eq. (40). Given the current oscillation
results one can determine the allowed ranges for the expected $0\nu\beta\beta$
amplitude. As depicted in the upper-left panel in Fig. 5 (the other panels
will be explained below) the result depends on whether the ordering of the
neutrino mass spectrum is normal (NO) or inverted (IO). Thanks to the effect
of Majorana phases in Eq. (41) the amplitude can vanish due to possible
destructive interference among the three neutrinos. This is a key feature of
the normal neutrino mass ordering, currently preferred by the global
oscillation fit [20], One sees how oscillations leave an important imprint
upon neutrinoless double beta decay studies. Limits on $\left\langle
m_{\beta\beta}\right\rangle$ from $0\nu\beta\beta$ searches are also
indicated, taking into account the uncertainty in the nuclear matrix elements
relevant for the computation of the decay rates. This leads to the K1 and K2
lines, indicating the current 95%CL limits from the KamLAND-Zen 400
experiment, for extreme $0\nu\beta\beta$ nuclear matrix element choices [26].
Other bounds come from GERDA [27], CUORE [28] and EXO-200 [29]. There is a
reasonable chance that, perhaps, $0\nu\beta\beta$ decay could be seen in the
next round of experiments such as SNO+Phase-II [30], LEGEND-1000 [31] and
nEXO-10 yr [32]. According to the black-box theorem [33], this would be a
major discovery, as it would imply that at least one of the neutrinos is a
Majorana particle.

Probing the absolute scale of neutrino masses is also the goal of single beta
decay experiments such as KATRIN which leads to an upper limit $m_{\beta}<0.8$
eV (90% CL) [34, 35]. Absolute neutrino masses can also be measured through
cosmological observations, like those of the cosmic microwave background
anisotropy spectrum or of the clustering of cosmic structures. The vertical
shaded band in Fig. 5 indicates the current sensitivity of cosmological data
[36] on $m_{\text{lightest}}$.

We now summarize the above neutrino mass constraints. To do so we perform a
scan over the parameter space to calculate the allowed modulus for each of the
elements of the mass matrix in the flavor basis as a function of $m_{1}$. For
this analysis we consider the range $m_{1}<0.8$ eV, in agreement with KATRIN’s
upper limit for $m_{\beta}$ [34]. We generate different combinations of random
values for the $U$ parameters within their 3$\sigma$ limits, taking into
account the most recent neutrino oscillation analysis [20]. We varied the
Majorana phases randomly in the range between 0 and 2$\pi$. The resulting
values for $|m_{ee}|$ obtained after this scan, are depicted in the upper-left
panel in Fig. 5 as a shaded blue region. The interval found in this scan, for
each of the neutrino mass entries, in units of eV, is given by:

$\left|Y_{\Delta\alpha\beta}\right|\frac{v_{\Delta}}{\sqrt{2}}=|m_{\alpha\beta}|\in\begin{pmatrix}(6.77\times
10^{-6},0.80)&(6.79\times 10^{-6},0.66)&(4.07\times 10^{-6},0.68)\\\
(6.79\times 10^{-6},0.66)&(1.05\times 10^{-4},0.80)&(5.53\times
10^{-5},0.80)\\\ (4.07\times 10^{-6},0.68)&(5.53\times
10^{-5},0.79)&(1.64\times 10^{-5},0.80)\end{pmatrix}.$ (42)

Note that, as a result of the limits of the sampling region, all absolute
values of the neutrino neutrino mass entries are bounded from above. Moreover,
the $ee$-entry is bounded from below. These are just artifacts of the sampling
procedure.

Figure 5:  The allowed neutrino mass matrix element regions in the flavor
basis, as a function of $m_{1}$, are shown in blue, for normal neutrino mass
ordering, NO. They are arranged as follows: $|m_{ee}|$ (top-left),
$|m_{e\mu}|$ (top-right), $|m_{\mu\mu}|$ (center-left), $|m_{e\tau}|$ (center-
right), $|m_{\tau\tau}|$ (bottom-left), and $|m_{\mu\tau}|$ (bottom-right).
Mixing angles, squared mass differences, and $\delta$ are taken within their
3$\sigma$ ranges [20]. Vertical shaded bands are the KATRIN [34], CMB [36],
and CMB+BAO [37] limits for $m_{1}$. The top-left panel coincides with the
effective $0\nu\beta\beta$ mass parameter $|m_{\beta\beta}|$ and the shaded
horizontal bands marked K1 and K2 are excluded by KAMLAND-Zen [26], see text.

###  III.4 Charged lepton flavor violaton

Processes with charged lepton flavor violation (cLFV) are those that do not
conserve lepton family number in transitions between $e$, $\mu$ and $\tau$
families. Their existence is expected, at some level, from the observation of
neutrino oscillations. Within the SM, leptonic flavor is conserved at any
order of perturbation theory. Flavor can be violated in the absence of
neutrino mass within specific type-I seesaw models [7, 38, 8].

Within the type-II seesaw mechanism, the same triplet Yukawa coupling
responsible for neutrino masses also induce cLFV effects. In particular,
processes $l_{\alpha}\to l_{\beta}\gamma$ are predicted at one-loop level,
with contributions from both singly- and doubly-charged scalars, while the
processes $l_{\alpha}\to\bar{l_{\beta}}l_{\beta}l_{\beta}$ proceed at the tree
level, as seen in Figs. 6 and 7.

Figure 6:  Feynman diagrams for the doubly-charged contribution to the cLFV
process $\ell_{\alpha}\to\gamma\ell_{\beta}$.

Figure 7:  Left: Feynman diagram for the singly-charged contribution to the
cLFV process $\ell_{\alpha}\to\gamma\ell_{\beta}$. Right: Feynman diagram for
the doubly-charged contribution to the cLFV process
$\ell_{\alpha}\to\ell_{\beta}\ell_{\rho}\ell_{\sigma}$.

It is straightforward to determine the corresponding expressions for the
branching ratios. For example, for $\mu\to e\gamma$ and $\mu\to eee$ the
corresponding branching ratios are given as [39, 40]:

$BR(\mu\to
e\gamma)\approx\frac{\alpha}{192\pi}\frac{\left|(Y_{\Delta}^{\dagger}Y_{\Delta})_{e\mu}\right|^{2}}{G_{F}^{2}}\left(\frac{1}{m_{H^{+}}^{2}}+\frac{8}{m_{H^{++}}^{2}}\right)^{2},$
(43)

$BR(\mu\to
eee)=\frac{1}{G_{F}^{2}}\frac{\left|(Y_{\Delta}^{\dagger})_{ee}(Y_{\Delta})_{\mu
e}\right|^{2}}{m_{H^{++}}^{4}}.$ (44)

In a way similar to section III.3 we can determine the allowed ranges for the
other entries of $(Y_{\Delta})_{\alpha\beta}$. One can obtain simple
expressions that constrain the model parameters such as the Yukawa couplings
and charged scalar masses using the negative searches for charged lepton
flavour violating transitions, such as $\mu\to e\gamma$ decay,
$\mu\to\bar{e}ee$ decay and similar $\tau$ decays which were searched, e.g.,
in BaBar. We summarize the current experimental status of radiative charged
lepton flavour violation in the second column of Table 1. These are the
experimental restrictions on the decay branching ratios [18]. To obtain the
third column we assumed that the difference between $m_{H^{++}}$ and
$m_{H^{+}}$ is relatively small, see Sec.III.2. The branching ratios are
inversely proportional to $M^{2}_{\Delta}$ at first order. The resulting
constraints from each channel are listed in the third column of Table 1.
Notice that from processes that have three leptons in the final state we can
constrain the product of two Yukawa couplings, while the radiative processes
allow us to constrain the sum of three of such products. This will be
important for the study of Non-Standard neutrino interaction parameters, as we
will see at the end. Moreover, one must also include the restrictions from
negative searches at collider experiments, such the LHC.

Processes | Experimental limits on BR |  Bounds on $G_{F}^{-1}M_{\Delta}^{-2}Y_{\Delta}^{2}$xxxxx
---|---|---
$\mu\to e\gamma$ | ${\rm BR}<4.2\times 10^{-13}$ | $G_{F}^{-1}M_{\Delta}^{-2}|Y_{\Delta}^{\dagger}Y_{\Delta}|_{e\mu}<2.1\times 10^{-5}=L_{1}$
$\mu\to 3e$ | ${\rm BR}<1.0\times 10^{-12}$ | $G_{F}^{-1}M_{\Delta}^{-2}|Y_{\Delta}^{\dagger}|_{\mu e}|Y_{\Delta}|_{ee}<2.0\times 10^{-6}=L_{2}$
$\tau\to 3e$ | ${\rm BR}<2.7\times 10^{-8}$ | $G_{F}^{-1}M_{\Delta}^{-2}|Y_{\Delta}^{\dagger}|_{\tau e}|Y_{\Delta}|_{ee}<7.9\times 10^{-4}=L_{3}$
$\tau\to e^{+}e^{-}\mu^{-}$ | ${\rm BR}<1.8\times 10^{-8}$ | $G_{F}^{-1}M_{\Delta}^{-2}|Y_{\Delta}^{\dagger}|_{\tau e}|Y_{\Delta}|_{e\mu}<4.5\times 10^{-4}=L_{4}$
$\tau\to e\gamma$ | ${\rm BR}<3.3\times 10^{-8}$ | $G_{F}^{-1}M_{\Delta}^{-2}|Y_{\Delta}^{\dagger}Y_{\Delta}|_{e\tau}<1.4\times 10^{-2}=L_{5}$
$\tau\to\mu\gamma$ | ${\rm BR}<4.4\times 10^{-8}$ | $G_{F}^{-1}M_{\Delta}^{-2}|Y_{\Delta}^{\dagger}Y_{\Delta}|_{\mu\tau}<1.6\times 10^{-2}=L_{6}$
$\tau\to\mu^{+}\mu^{-}e^{-}$ | ${\rm BR}<2.7\times 10^{-8}$ | $G_{F}^{-1}M_{\Delta}^{-2}|Y_{\Delta}^{\dagger}|_{\tau\mu}|Y_{\Delta}|_{\mu e}<5.6\times 10^{-4}=L_{7}$
$\tau\to e^{+}\mu^{-}\mu^{-}$ | ${\rm BR}<1.7\times 10^{-8}$ | $G_{F}^{-1}M_{\Delta}^{-2}|Y_{\Delta}^{\dagger}|_{\tau e}|Y_{\Delta}|_{\mu\mu}<6.3\times 10^{-4}=L_{8}$
$\tau\to\mu^{+}e^{-}e^{-}$ | ${\rm BR}<1.5\times 10^{-8}$ | $G_{F}^{-1}M_{\Delta}^{-2}|Y_{\Delta}^{\dagger}|_{\tau\mu}|Y_{\Delta}|_{ee}<5.9\times 10^{-4}=L_{9}$
$\tau\to 3\mu$ | ${\rm BR}<2.1\times 10^{-8}$ | $G_{F}^{-1}M_{\Delta}^{-2}|Y_{\Delta}^{\dagger}|_{\tau\mu}|Y_{\Delta}|_{\mu\mu}<6.9\times 10^{-4}=L_{10}$
Table 1:  Constraints from cLFV processes on the Yukawa coupling matrix
$Y_{\Delta}$. Experimental limits on the branching ratios (BR) are taken from
Ref. [18], and the bounds on $G_{F}^{-1}M_{\Delta}^{-2}Y_{\Delta}^{2}$ are
adapted from Ref. [41].

###  III.5 Collider constraints

The Higgs triplet interacts with gauge bosons, the lepton doublets, as well as
the scalar Higgs doublet. Its decay modes have been explored at the LEP and
LHC experiments to some extent. Unfortunately, most of the experimental
searches for doubly-charged Higgs bosons make specific model assumptions
concerning their decay modes, whose features do not fully cover the type-II
seesaw expectations. On the other hand, searches for the singly-charged and
neutral scalars do not apply to the type-II seesaw, since the relevant
couplings are neutrino-mass-suppressed ($v_{\Delta}$-suppression, see Eq.
(19)) in the type-II seesaw. This applies, for example, to their couplings to
quarks, which arise only due to mixing, as the only triplet scalar Yukawa
coupling involves leptons, Eq. (3). These inadequacies undermine the
robustness of the constraints obtained. In this subsection we present a
dedicated analysis of the decay modes of the type-II seesaw scalars which
should serve as basis for deriving truly robust constraints on their masses.

The doubly charged scalar

We start with the case of small mass splitting $\delta m\approx 0$. Depending
on the magnitude of the triplet VEV $v_{\Delta}$, the doubly-charged Higgs
$H^{\pm\pm}$ may mainly decay to same-sign dileptons or gauge bosons. Clearly,
for small $v_{\Delta}$, the $H^{\pm\pm}$ predominantly decays into same-sign
leptonic states $H^{\pm\pm}\to\ell^{\pm}\ell^{\pm}$, whereas for larger
$v_{\Delta}$, the gauge boson decay mode $H^{\pm\pm}\to W^{\pm}W^{\pm}$
becomes dominant [42, 43, 15]. The relevant decay widths are given as:

$\Gamma(H^{\pm\pm}\to
l^{\pm}_{i}l^{\pm}_{j})=\Gamma_{l_{i}l_{j}}=\frac{m_{H}^{\pm\pm}}{(1+\delta_{ij})8\pi}\left|\frac{m_{ij}^{\nu}}{v_{\Delta}}\right|^{2},\,\,m^{\nu}=Y_{\Delta}v_{\Delta}/\sqrt{2},$
(45) $\Gamma(H^{\pm\pm}\to
W^{\pm}W^{\pm})=\Gamma_{W^{\pm}W^{\pm}}=\frac{g^{2}v^{2}_{\Delta}}{8\pi
m_{H^{\pm\pm}}}\sqrt{1-\frac{4}{r^{2}_{W}}}\left[\left(2+(r_{W}/2-1)^{2}\right)\right].$
(46)

where $r_{W}=\frac{m_{H^{\pm\pm}}}{M_{W}}$. Here $m^{\nu}$ denotes the
neutrino mass matrix, $i,j$ are the generation indices, $\Gamma_{l_{i}l_{j}}$
and $\Gamma_{W^{\pm}W^{\pm}}$ are the partial decay widths for the
$H^{\pm\pm}\to l^{\pm}_{i}l^{\pm}_{j}$, and $H^{\pm\pm}\to W^{\pm}W^{\pm}$
channels, respectively.

For the case of positive mass splitting $(\delta m>0)$, one must consider
additonal decay channels

$\displaystyle\Gamma(H^{\pm\pm}\to
H^{\pm}W^{\pm*})=\frac{9g^{4}m_{H^{\pm\pm}}\cos^{2}\beta_{\pm}}{128\pi^{3}}G\left(\frac{m_{H^{\pm}}^{2}}{m_{H^{\pm\pm}}^{2}},\frac{m_{W}^{2}}{m_{H^{\pm\pm}}^{2}}\right),$
(47)

where $\tan\beta_{\pm}=\frac{\sqrt{2}v_{\Delta}}{v_{\Phi}}$ and $G(x,y)$ is
defined in the Appendix. C.

Figure 8:  $H^{\pm\pm}$ branching ratios for $m_{H^{\pm\pm}}=500$ GeV. The
left and right panels are for $\delta m\approx 0$ and $\delta m=20$ GeV,
respectively. Red, green and cyan lines correspond to $H^{\pm\pm}\to
l^{\pm}l^{\pm}$, $H^{\pm\pm}\to W^{\pm}W^{\pm}$ and $H^{\pm\pm}\to
H^{\pm}W^{\pm*}$, respectively.

In Fig. 8 we display the $H^{\pm\pm}$ branching ratio to the inclusive
dilepton channel $l^{\pm}l^{\pm}$, diboson channel $W^{\pm}W^{\pm}$, and the
cascade channel $H^{+}W^{+*}$ for $m_{H^{\pm\pm}}=500$ GeV, with $\delta
m\approx 0$ (left panel) or $\delta m=20$ GeV (right panel). One clearly sees
that the collider search strategy for the charged Higgs boson crucially
depends on the value of the triplet VEV $v_{\Delta}$ and mass splitting
$\delta m$. The same qualitative behaviour of the $H^{\pm\pm}$ branching
ratios holds for different $H^{\pm\pm}$ masses. The direct limit on the
$H^{\pm\pm}$ has been derived from collider searches of multi-lepton final
states. Below we discuss the existing constraints on $m_{H^{\pm\pm}}$ from LEP
and LHC searches for the case $\delta m\approx 0$.

  * •

Constraint from LEP-II: LEP has searched for $H^{++}H^{--}$ pair production
through s-channel $\gamma/Z$ exchange, with subsequent decay of $H^{\pm\pm}$
into charged lepton pairs, which is the dominant decay mode for
$v_{\Delta}\leq 0.1$ MeV. This constrains the mass $m_{H^{\pm\pm}}>97.3$ GeV
[44] at $95\%$ C.L.

  * •

Constraints from doubly-charged Higgs boson production in three and four
lepton final states at 13 TeV LHC: These searches analysed Drell-Yan
production $pp\to H^{++}H^{--}$ and subsequent decay in the
$H^{\pm\pm}\to\ell^{\pm}\ell^{\pm}$ channel. The CMS collaboration looked for
various leptonic final states such as $ee,\mu\mu,\tau\tau,e\mu,e\tau$ and
$\mu\tau$. They also studied the associated production of $H^{\pm\pm}H^{\mp}$
through s-channel $W^{\pm}$ exchange, followed by $H^{\pm\pm}$ decay to same-
sign di-lepton and $H^{\pm}\to\ell^{\pm}\nu$. This combined channel of Drell-
Yan production and associated production gives the constarint
$m_{H^{\pm\pm}}>820$ GeV [45] at $95\%$ C.L for $e,\mu$ flavor. ATLAS searches
only include the Drell-Yan production and the bound is $m_{H^{\pm\pm}}>870$
GeV at $95\%$ C.L [46]. Note that these limits hold only for small triplet
VEV, $v_{\Delta}<0.1$ MeV. For larger triplet $v_{\Delta}>0.1$ MeV, the gauge
boson decay mode is dominant. Hence for this region, a search via pair-
production $H^{++}H^{--}$, with subsequent decay into same-sign gauge bosons
and further to leptonic final states is required. The ATLAS collaboration has
looked for this channel and constrained the doubly-charged Higgs mass
$m_{H^{\pm\pm}}$ to lie above 220 GeV [47]. Hence, we can conclude that low
masses, $m_{H^{\pm\pm}}>220$ GeV, are still allowed, as long as the triplet
VEV is “large”, $v_{\Delta}>10^{-4}$ GeV. For lower triplet VEVs
$v_{\Delta}<10^{-4}$ GeV, the doubly-charged Higgs mass constraint is much
more stringent, $m_{H^{\pm\pm}}>870$ GeV.

The above-mentioned constraints do not apply to the full parameter space, but
rather to a specific range. For example, the search in Refs. [45, 46] only
holds for $\delta m=0$ and $v_{\Delta}<10^{-4}$ GeV, while the the search in
Ref. [47] is only valid for $\delta m=0$ and $v_{\Delta}>10^{-4}$ GeV.

Notice that in a realistic type-II seesaw scenario the branching fractions of
the triplet-like scalars into different lepton flavours are determined by the
neutrino oscillation parameters. However, most of the aforementioned limits
are derived in the context of simplified scenarios that do not take into
account the footprints of the low-energy neutrino parameters.

Moreover, the triplet scalars in this model are in general non-degenerate
($|\delta m|\neq 0$). For moderate values of triplet VEV $v_{\Delta}$ and
relatively large mass splitting $|\delta m|\sim 10$ GeV, cascade decays
quickly dominate over the leptonic and diboson decay modes, see the right
panel of Fig. 8. Thus, for nonzero mass splitting, the cascade decays are
entitled to play a significant role in the phenomenology, resulting in
collider signatures that may differ significantly from those of the degenerate
scenario. In order to probe the full parameter space, one should take into
account all the complexity of tripet scalar decays, see Sec. IV below.

The singly charged scalar

Let us now turn our attention to the singly-charged scalar, $H^{\pm}$,
starting with all the relevant partial decay widths, for the case $\delta
m=0$. They are given as [48, 49, 50, 51, 52, 53]:

$\displaystyle\Gamma(H^{\pm}\to
q\bar{q}^{\prime})=\frac{3m_{H^{\pm}}^{3}\sin^{2}\beta_{\pm}}{8\pi
v_{\Phi}^{2}}\left[\left(\frac{m_{q}^{2}}{m_{H^{\pm}}^{2}}+\frac{m_{q^{\prime}}^{2}}{m_{H^{\pm}}^{2}}\right)\left(1-\frac{m_{q}^{2}}{m_{H^{\pm}}^{2}}-\frac{m_{q^{\prime}}^{2}}{m_{H^{\pm}}^{2}}\right)-4\frac{m_{q}^{2}}{m_{H^{\pm}}^{2}}\frac{m_{q^{\prime}}^{2}}{m_{H^{\pm}}^{2}}\right]\lambda^{\frac{1}{2}}\left(\frac{m_{q}^{2}}{m_{H^{\pm}}^{2}},\frac{m_{q^{\prime}}^{2}}{m_{H^{\pm}}^{2}}\right),$
(48)
$\displaystyle\Gamma(H^{\pm}\to\ell_{i}^{\pm}\nu_{j})=\frac{m_{H^{\pm}}}{8\pi
v_{\Phi}^{2}}\left(\delta_{ij}m_{i}^{2}\sin^{2}\beta_{\pm}+|Y^{ij}_{\Delta}|^{2}v_{\Phi}^{2}\cos^{2}\beta_{\pm}\right)\left(1-\frac{m_{i}^{2}}{m_{H^{\pm}}^{2}}\right)^{2},$
(49) $\displaystyle\Gamma(H^{\pm}\to
W^{\pm}Z)=\frac{g^{4}v_{\Delta}^{2}\cos^{2}\beta_{\pm}}{32\pi\cos^{2}\theta_{w}m_{H^{\pm}}}\left[\lambda\left(\frac{m_{W}^{2}}{m_{H^{\pm}}^{2}},\frac{m_{Z}^{2}}{m_{H^{\pm}}^{2}}\right)\right]^{1/2}\left[2+\frac{m_{H^{\pm}}^{4}}{4m_{W}^{2}m_{Z}^{2}}\left(1-\frac{m_{W}^{2}}{m_{H^{\pm}}^{2}}-\frac{m_{Z}^{2}}{m_{H^{\pm}}^{2}}\right)^{2}\right],$
(50)

where the relevant functions we have listed in Appendix. C.

For the case of non-zero mass splitting $|\delta m|\neq 0$, one also has the
following decay channels for $H^{\pm}$:

$\displaystyle\Gamma(H^{\pm}\to
H^{0}/AW^{\pm*})=\frac{9g^{4}m_{H^{\pm}}}{512\pi^{3}}\xi_{H^{\pm}W^{\mp}H^{0}/A}^{2}G\left(\frac{m_{H^{0}/A}^{2}}{m_{H^{\pm}}^{2}},\frac{m_{W}^{2}}{m_{H^{\pm}}^{2}}\right);\text{
for }\delta m>0,$ (51) $\displaystyle\Gamma(H^{\pm}\to
H^{\pm\pm}W^{\mp*})=\frac{9g^{4}m_{H^{\pm}}\cos^{2}\beta_{\pm}}{128\pi^{3}}G\left(\frac{m_{H^{\pm\pm}}^{2}}{m_{H^{\pm}}^{2}},\frac{m_{W}^{2}}{m_{H^{\pm}}^{2}}\right);\text{
for }\delta m<0.$ (52)

Figure 9: Branching ratio of $H^{\pm}$ for $m_{H^{\pm}}=250$ GeV (top panel)
and $m_{H^{\pm}}=1000$ GeV (bottom panel) as a function of the triplet VEV
$v_{\Delta}$. The left and right panels are for $|\delta m|=1$ GeV and
$|\delta m|=20$ GeV, respectively.

In Fig. 9, we have shown the branching ratio of the singly-charged Higgs
$H^{\pm}$ into various channels. The singly-charged Higgs boson $H^{\pm}$ has
four decay modes: (i) leptonic decay, i.e. $\ell^{\pm}\nu$, (ii) hadronic
decay, i.e. $q\bar{q^{\prime}}$, (iii) diboson decay, i.e. $W^{\pm}Z$,
$W^{\pm}h$ and (iv) cascade decay, i.e. $H^{0}/AW^{\pm*}$ ($\delta m>0$) or
$H^{\pm\pm}W^{\mp*}$ ($\delta m<0$). Comparing left and right panels in Fig.
9, we see that for relatively large mass splitting the cascade decays dominate
in the intermediate triplet VEV region. One sees that, as for the doubly-
charged scalar, the branching ratio pattern for singly-charged scalar decays
is quite sensitive to mass splitting and the triplet VEV. In order to probe
the singly-charged scalar, one should take into account all the complexity of
its decays which we sketch in Sec. IV.

For singly-charged Higgs bosons $H^{\pm}$ in the type-II seesaw model, the
coupling to a pair of quarks is suppressed by $\frac{v_{\Delta}}{v_{\Phi}}$.
Both the CMS and the ATLAS official searches [54, 55, 56] employ the $gg\to
tbH^{\pm}$ and $gb\to tH^{\pm}$ production channels for the singly-charged
Higgs. The $H^{\pm}$ has also been looked for in the vector boson fusion
production channel [57, 58]. However, these production channels depend on the
$H^{\pm}-W^{\pm}-Z$ coupling, which is again suppressed by $v_{\Delta}$, hence
not directly applicable to the type-II seesaw model. In summary, all the
limits derived from the LHC searches discussed above do not apply in the case
of type-II seesaw model. Only the combined trully model-independent LEP limit
of around 80 GeV on $m_{H^{\pm}}$ applies [59].

The neutral scalars

We now discuss the existing constraints on neutral scalars $H^{0}/A$. In order
to do that let us first discuss all the relevant $H^{0}/A$ partial decay
widths. Assuming $\delta m=0$ we have that the $H^{0}$ decay widths are given
as [48, 49, 50, 51, 52, 53]:

$\displaystyle\Gamma(H^{0}\to
f\bar{f})=\frac{N_{c}^{f}m_{f}^{2}m_{H^{0}}}{8\pi
v_{\Phi}^{2}}\sin^{2}\alpha\left[\beta\left(\frac{m_{f}^{2}}{m_{H^{0}}^{2}}\right)\right]^{3},$
(53) $\displaystyle\Gamma(H^{0}\to\nu\nu)=\frac{\kappa
m_{H^{0}}\cos^{2}\alpha}{8\pi}\sum_{i,j=1}^{3}|Y^{ij}_{\Delta}|^{2},$ (54)
$\displaystyle\Gamma(H^{0}\to W^{+}W^{-})=\frac{g^{4}m_{H^{0}}^{3}}{256\pi
m_{W}^{4}}\left(v_{\Phi}\sin\alpha-2v_{\Delta}\cos\alpha\right)^{2}\left(1-4\frac{m_{W}^{2}}{m_{H^{0}}^{2}}+12\frac{m_{W}^{4}}{m_{H^{0}}^{4}}\right)\beta\left(\frac{m_{W}^{2}}{m_{H^{0}}^{2}}\right),$
(55) $\displaystyle\Gamma(H^{0}\to ZZ)=\frac{g^{4}m_{H^{0}}^{3}}{512\pi
m_{W}^{4}}\left(v_{\Phi}\sin\alpha-4v_{\Delta}\cos\alpha\right)^{2}\left(1-4\frac{m_{Z}^{2}}{m_{H^{0}}^{2}}+12\frac{m_{Z}^{4}}{m_{H^{0}}^{4}}\right)\beta\left(\frac{m_{Z}^{2}}{m_{H^{0}}^{2}}\right)$
(56) $\displaystyle\Gamma(H^{0}\to
hh)=\frac{\lambda_{H^{0}hh}^{2}v_{\Phi}^{2}}{8\pi
m_{H^{0}}}\beta\left(\frac{m_{h}^{2}}{m_{H^{0}}^{2}}\right)$ (57)

where $\tan(2\alpha)=\frac{2B}{A-C}$, $N_{c}^{f}$ is the color factor with
$N_{c}^{q}=3$ and $N_{c}^{\ell}=1$, $\kappa=2(1)$ for $i\neq j(i=j)$.

For nonzero mass splitting, in this case one must have $\delta m<0$, and the
relevant CP–even decay width is

$\displaystyle\Gamma(H^{0}\to
H^{\pm}W^{\mp*})=\frac{9g^{4}m_{H^{0}}}{512\pi^{3}}\xi_{H^{\pm}W^{\mp}H^{0}}^{2}G\left(\frac{m_{H^{\pm}}^{2}}{m_{H^{0}}^{2}},\frac{m_{W}^{2}}{m_{H^{0}}^{2}}\right).$
(58)

On the other hand the relevant decay widths of the CP–odd neutral scalar $A$
with $\delta m=0$ are given as:

$\displaystyle\Gamma(A\to f\bar{f})=\frac{N_{c}^{f}m_{f}^{2}m_{A}}{8\pi
v_{\Phi}^{2}}\sin^{2}\beta_{0}\leavevmode\nobreak\
\beta\left(\frac{m_{f}^{2}}{m_{A}^{2}}\right),$ (59)
$\displaystyle\Gamma(A\to\nu\nu)=\frac{m_{A}\cos^{2}\beta_{0}}{8\pi}\sum_{i,j=1}^{3}|Y^{ij}_{\Delta}|^{2},$
(60) $\displaystyle\Gamma(A\to hZ)=\frac{g^{2}m_{A}^{3}}{64\pi
m_{W}^{2}}(\cos\alpha\sin\beta_{0}-2\sin\alpha\cos\beta_{0})^{2}\left[\lambda\left(\frac{m_{h}^{2}}{m_{A}^{2}},\frac{m_{Z}^{2}}{m_{A}^{2}}\right)\right]^{3/2},$
(61)

where $\tan\beta_{0}=\frac{2v_{\Delta}}{v_{\Phi}}$.

Similarly for the case of nonzero mass splitting one should also consider the
following decay width

$\displaystyle\Gamma(A\to
H^{\pm}W^{\mp*})=\frac{9g^{4}m_{A}}{512\pi^{3}}\xi_{H^{\pm}W^{\mp}A}^{2}G\left(\frac{m_{H^{\pm}}^{2}}{m_{A}^{2}},\frac{m_{W}^{2}}{m_{A}^{2}}\right).$
(62)

Again we refer to the Appendix. C for all the relevant functions. The
qualitative features of the branching ratio plots for neutral scalar bosons
$H^{0}/A$ are very similar to those of the singly-charged Higgs case, and will
not be presented. Similarly to the singly-charged Higgs boson $H^{\pm}$, the
neutral scalar bosons $H^{0}/A$ have four types of decay modes: (i) leptonic
decays, i.e. $H^{0}/A\to\ell^{\pm}\ell^{\mp}+\nu\nu$, (ii) hadronic decays,
i.e. $H^{0}/A\to q\bar{q}$, (iii) diboson decays, i.e. $H^{0}\to
W^{\pm}W^{\mp}+ZZ+hh$ or $A\to hZ$ and (iv) cascade decays, i.e. $H^{0}/A\to
H^{\pm}W^{\mp*}$ ($\delta m<0$). In the nearly degenerate scenario the cascade
decay is very small and depending on the triplet VEV, either the leptonic
decays ($v_{\Delta}<10^{-4}$ GeV) or bosonic decays ($v_{\Delta}>10^{-4}$ GeV)
dominate. On the other hand for relatively large mass splitting, the bosonic
decays dominate in the intermediate triplet VEV region. Therefore, in order to
probe neutral type-II seesaw scalars in a robust manner one must take into
account these decay channels.

The official ATLAS and CMS searches for neutral scalars $H^{0}/A$ focus on
gluon fusion production [60, 61, 62], associated with b-jet productions [60,
61, 63], and associated production with a vector boson [62]. Most of these
searches are relevant for two Higgs doublet models or supersymmetric
extensions. However, these production channels are again suppressed by
$v_{\Delta}$ in the type-II seesaw, since they involve the $H^{0}/A-q-q$
coupling, rendering them again irrelevant for constraining the type-II seesaw
scalar bosons.

In summary, in contrast to the doubly-charged scalar boson discussed above (at
least for $\delta m\approx 0$), there has been no dedicated LHC searches for
neutral scalar bosons ($H^{\pm},H^{0}/A$) within the context of the triplet
seesaw. For example, all the above mentioned singly-charged and neutral Higgs
boson production channels at the LHC are suppressed in the type-II seesaw
model. Instead the triplet-like scalar bosons are produced a la Drell-Yan,
Fig. 10. Indeed, Drell-Yan quark-antiquark annihilation through s-channel
$\gamma/Z$ and $W$ exchanges at the LHC produces triplet scalar bosons and
hence a number of signatures which we discuss in detail in the next section.
We find that all of these Drell-Yan production cross-sections can be sizeable.
In particular, the production of the singly-charged scalars in association
with the neutral ones, often overlooked by both CMS and ATLAS searches, has
the largest cross-sections. Hence the need for including these production
channels when probing the type-II seesaw parameter space.

##  IV Collider signatures from type-II seesaw scalars

Neutrino mass generation, especially when mediated by scalars, as in our
simple type-II seesaw scheme, provides interesting signatures at high-energy
colliders. Although this is the first and simplest SM-based seesaw scheme [4,
5], these signatures are still far under-studied in a comprehensive manner. By
that we mean one in which the proper connection with neutrino mass generation
and the associated theoretical and phenomenological constraints is duly taken
into account. The very interesting results recently obtained in [6] further
motivate us to present here a comprehensive theoretical study.

We first sketch the main channels that can produce the type-II seesaw neutrino
mass mediators and the relevant cross sections. We start by discussing the
triplet scalar signatures at the LHC, and then move to future hadron as well
as lepton colliders. Our study illustrates their potential for probing
fundamental neutrino properties within the type-II seesaw setup. Throughout
this section we will see that, depending on the value of the triplet VEV
$v_{\Delta}$ there are three important regimes. Two of them are either far
below or far above $10^{-4}$ GeV or so. There is however an intermediate
region around $10^{-4}$ GeV where additional decay channels appear depending
on the sign and magnitude of the mass-splitting $\delta m$ between the triplet
scalars. This region can play an important role and should be taken into
account.

Large Hadron Collider constraints

As discussed earlier, one must take into account all possible production
channels and decay modes in order to fully constrain the simplest triplet
seesaw [4, 5]. In this section we focus on possible signatures at $pp$ and
$e^{+}e^{-}$ colliders to either discover or constrain the parameter space.

The relevant pair and associated triplet scalar production $\Phi\Phi^{\prime}$
(with $\Phi,\Phi^{\prime}\in\\{H^{0},A,H^{\pm},H^{\pm\pm}\\}$) proceeds via
the neutral- or charged-current Drell-Yan mechanism, respectively, involving
s-channel $\gamma/Z$ and $W^{\pm}$ exchange, Fig. 10:

$\displaystyle q\bar{q^{\prime}}\to W^{*}\to
H^{\pm\pm}H^{\mp},H^{\pm}H^{0},H^{\pm}A,$ (63) $\displaystyle
q\bar{q}\to\gamma^{*}/Z^{*}\to H^{\pm\pm}H^{\mp\mp},H^{\pm}H^{\mp},H^{0}A.$
(64)

Figure 10: Feynman diagrams for Drell-Yan production of the triplet Higgs
bosons $\Phi,\Phi^{\prime}\in\\{H^{0},A,H^{\pm},H^{\pm\pm}\\}$ in type-II
seesaw.

Figure 11: Cross section for pair and associated production of triplet seesaw
scalars vs the doubly-charged scalar mass $m_{H^{\pm\pm}}$ at a $pp$ collider.
The center of mass energy is $\sqrt{s}=14$ TeV. The left panel is for $\delta
m>0$ ($m_{H^{\pm\pm}}>m_{H^{\pm}}>m_{H^{0}/A}$) while the right panel is for
$\delta m<0$ ($m_{H^{\pm\pm}}<m_{H^{\pm}}<m_{H^{0}/A}$).

In Fig. 11 we display these cross sections versus the mass of $H^{\pm\pm}$,
for $\sqrt{s}=14$ TeV. Depending on the value and sign of this mass splitting
$\delta m$, the cross section also varies. The left and right panels of Fig.
11 correspond to cross sections for positive ($\delta m=20$ GeV) and negative
($\delta m=-20$ GeV) mass splitting, respectively. We see that most of these
production channels have sizable cross sections, suggesting that significant
event numbers can be produced. This also stresses the need for including all
of the possible channels in both the CMS and ATLAS analyses. For example, we
see from Fig. 11 that the production cross section for
$H^{\pm}H^{0}/H^{\pm}A$, neglected in current ATLAS and CMS analyses, are
substantial, even larger than $H^{\pm\pm}H^{\mp}$ and $H^{\pm\pm}H^{\mp\mp}$,
especially in the $\delta m>0$ region.

Future hadron colliders

In Fig. 12, we also show the production cross section for a future $pp$
collider with center-of-mass energy $\sqrt{s}=100$ TeV, such as FCC-hh. As
shown in the Fig. 12, the production cross section at such center of mass
energy is large enough that multi-TeV Higgs masses can be explored.

Figure 12: Cross section for pair and associated production of triplet seesaw
scalars vs the doubly-charged scalar mass at the FCC-hh with $\sqrt{s}=100$
TeV. Same conventions as Fig. 11.

The LHC signatures depend also on the decay channels of each scalar boson
being produced. Apart from fully invisible processes, all relevant branching
ratios are given in Eq. (45) to Eq. (62). We have already shown the branching
ratio of $H^{\pm\pm}$ and $H^{\pm}$ in Figs. 8 and 9, respectively. The
behaviour of the $H^{0}/A$ branching ratios is similar to the singly-charged
Higgs. The doubly-charged Higgs decays to dilepton, diboson or into cascade
mode depending on the triplet VEV $v_{\Delta}$ and mass splitting $\delta m$.
The singly-charged Higgs boson $H^{\pm}$ or neutral boson $H^{0}/A$ has four
decay modes: leptonic decay, hadronic decay, diboson decay and cascade decay,
i.e. $H^{0}/AW^{*}$ ($\delta m>0$) or $H^{\pm\pm}W^{\mp*}$ ($\delta m<0$)
depending on the triplet VEV and mass splitting $\delta m$.

In what follows we discuss possible signatures at colliders depending on what
we assume for the mass splitting $\delta m$ [64, 48, 65, 66]. There are three
regimes to consider: (A) $\lambda_{4}=0:\,\,\delta m\approx
0\,(m_{H^{\pm\pm}}\simeq m_{H^{\pm}}\simeq m_{H^{0}/A})$, (B)
$\lambda_{4}>0:\,\,\delta m<0\,(m_{H^{\pm\pm}}<m_{H^{\pm}}<m_{H^{0}/A})$ and
(C) $\lambda_{4}<0:\,\,\delta m>0\,(m_{H^{\pm\pm}}>m_{H^{\pm}}>m_{H^{0}/A})$.
As the official ATLAS and CMS searches already considered the production
channel $H^{\pm\pm}H^{\mp\mp}$, we do not repeat here.

###  IV.1 $\lambda_{4}=0:\,\,\delta m\approx 0\,(m_{H^{\pm\pm}}\simeq
m_{H^{\pm}}\simeq m_{H^{0}/A})$

In this case, all the pair- and associated pair-production, except for
$H^{\pm}H^{\mp}$, have a sizeable cross section. For small triplet VEV
$v_{\Delta}<10^{-4}$ GeV, $H^{\pm\pm}$, $H^{\pm}$ and $H^{0}/A$ predominantly
decay to $\ell^{\pm}\ell^{\pm}$, $\ell^{\pm}\nu$ and $\nu\nu$, respectively.
Hence, to probe this $|\delta m|\approx 0$ and $v_{\Delta}<10^{-4}$ GeV
region, we must include the following three possible multilepton final states:

$\displaystyle pp\to
H^{\pm\pm}H^{\mp}\to\ell^{\pm}\ell^{\pm}\ell^{\mp}\nu,\,\,\,\,pp\to
H^{\pm}(H^{0}/A)\to\ell^{\pm}\nu\nu\nu,\,\,\,\,\text{and}\,\,\,\,\,\,pp\to
H^{0}A\to\nu\nu\nu\nu.$ (65)

From the above, we see that despite having sizeable cross-sections,
$H^{\pm}H^{0}/A$ and $H^{0}A$ can not compete with the three-lepton final
state coming from $H^{\pm\pm}H^{\mp}$ due to their invisible decays.

On the other hand, for larger triplet VEV $v_{\Delta}>10^{-4}$ GeV,
$H^{\pm\pm}$, $H^{\pm}$, $H^{0}$ and $A$ decay to $W^{\pm}W^{\pm}$,
$W^{\pm}Z/h$, $ZZ/WW/hh$ and $hZ$, respectively. Therefore, to probe the
region $|\delta m|\approx 0$ and $v_{\Delta}>10^{-4}$ GeV we must use the
following multi-lepton final states

$\displaystyle pp\to H^{\pm\pm}H^{\mp}\to
W^{\pm}W^{\pm}W^{\mp}(Z/h)\to\ell^{\pm}\ell^{\pm}\ell^{\mp}\ell^{+}\ell^{-}\nu\nu\nu,$
(66) $\displaystyle pp\to
H^{\pm}H^{0}\to(W^{\pm}Z/h)(ZZ/hh/W^{\pm}W^{\mp})\to(\ell^{\pm}\ell^{+}\ell^{-}\nu)(\ell^{+}\ell^{-}\ell^{+}\ell^{-}/\ell^{\pm}\ell^{\mp}\nu\nu),$
(67) $\displaystyle pp\to
H^{0}A\to(ZZ/hh/W^{\pm}W^{\mp})Zh\to(\ell^{+}\ell^{-}\ell^{+}\ell^{-}/\ell^{\pm}\ell^{\mp}\nu\nu)(\ell^{+}\ell^{-}\ell^{+}\ell^{-}).$
(68)

###  IV.2 $\lambda_{4}>0:\,\,\delta
m<0\,(m_{H^{\pm\pm}}<m_{H^{\pm}}<m_{H^{0}/A})$

For small mass splitting such as $\delta m\lesssim\mathcal{O}(1\,\text{GeV})$,
the signatures resemble the degenerate scenario $\delta m\approx 0$. On the
other hand, for triplet VEV in between
$10^{-5}\,\text{GeV}<v_{\Delta}<10^{-3}\,\text{GeV}$ and relatively large mass
splitting $\delta m$, the cascade decay of $H^{\pm}$ into $H^{\pm\pm}W^{\mp*}$
is dominant. This effectively enhances the production cross section for
$H^{\pm\pm}$. In the same triplet VEV region, $H^{0}/A\to H^{\pm}W^{\mp*}$
decay is also significantly large. Moreover, for $v_{\Delta}>10^{-4}$ GeV,
$H^{\pm\pm}\to W^{\pm}W^{\pm}$ has large branching ratio. Therefore the
following processes are important in order to probe the region $\delta m<0$
and moderate $v_{\Delta}\sim 10^{-4}$ GeV:

$\displaystyle pp\to H^{\pm}H^{0}/A\to H^{\pm\pm}W^{\mp*}H^{\pm}W^{\mp*}\to
H^{\pm\pm}W^{\mp*}H^{\pm\pm}W^{\mp*}W^{\mp*}\to 4W^{\pm}+X,$ (69)
$\displaystyle pp\to H^{0}A\to H^{\pm}W^{\mp*}H^{\pm}W^{\mp*}\to
H^{\pm\pm}W^{\mp*}W^{\mp*}H^{\pm\pm}W^{\mp*}W^{\mp*}\to 4W^{\pm}+Y,$ (70)
$\displaystyle pp\to H^{\pm\pm}H^{\mp}\to H^{\pm\pm}H^{\mp\mp}W^{\pm*}\to
2W^{\pm}+2W^{\mp}+Z.$ (71)

Taking into account the subsequent leptonic decay of the produced on-shell
$W^{\pm}$, these processes will lead to same-sign and opposite-sign tetra-
lepton signatures [15]. Again for small $v_{\Delta}$,
$H^{\pm\pm},H^{\pm},H^{0}/A$ dominantly decay leptonically giving rise to
multilepton final states. However these can not compete with multilepton final
states coming from pair production of $H^{\pm\pm}H^{\mp\mp}$, hence we have
not discussed them.

###  IV.3 $\lambda_{4}<0:\,\,\delta
m>0\,(m_{H^{\pm\pm}}>m_{H^{\pm}}>m_{H^{0}/A})$

For small $\delta m$, this scenario will be again the same as the case of
$\delta m\approx 0$. If the mass splitting $\delta m$ is large, above $1$ GeV,
and the triplet VEV is moderate, then $H^{\pm\pm}\to H^{\pm}W^{\pm*}$ and
$H^{\pm}\to H^{0}/AW^{\pm*}$ decay modes dominate over others. This
effectively enhances the $H^{0}$ and $A$ production cross section. Further for
moderate triplet VEV $v_{\Delta}\sim\mathcal{O}(10^{-4}\,\text{GeV})$, $H^{0}$
and $A$ dominantly decay to $ZZ/W^{\pm}W^{\mp}/hh$ and $hZ$, respectively.
Hence, the following are the possible signatures to probe the region of
moderate $v_{\Delta}$ and large enough $\delta m$:

$\displaystyle pp\to
H^{\pm}(H^{0}/A)\to(H^{0}/A)W^{\pm*}(H^{0}/A)\to(ZZ/W^{\pm}W^{\mp}/hh/hZ)W^{\pm*}(ZZ/W^{\pm}W^{\mp}/hh/hZ)$
$\displaystyle\to(\ell^{+}\ell^{-}\ell^{+}\ell^{-}/\ell^{\pm}\nu\ell^{\mp}\nu)(\ell^{+}\ell^{-}\ell^{+}\ell^{-}/\ell^{\pm}\nu\ell^{\mp}\nu)+X,$
(72) $\displaystyle pp\to
H^{0}A\to(ZZ/W^{\pm}W^{\mp}/hh)hZ\to\ell^{+}\ell^{-}\ell^{+}\ell^{-}\ell^{\pm}\nu\ell^{\mp}\nu,$
(73) $\displaystyle pp\to H^{\pm\pm}H^{\mp}\to
W^{\pm}\,\,W^{\pm}H^{0}/AW^{\mp*}\to
W^{\pm}W^{\pm}(ZZ/W^{\pm}W^{\mp}/hh/hZ)W^{\mp*}$
$\displaystyle\to\ell^{\pm}\nu\ell^{\pm}\nu(\ell^{+}\ell^{-}\ell^{+}\ell^{-}/\ell^{\pm}\nu\ell^{\mp}\nu)+X.$
(74)

From the discussion in IV.2 and IV.3, we conclude that the cascade decays
($H^{\pm}\to H^{\pm\pm}W^{\mp*},H^{0}/A\to H^{\pm}W^{\mp*}$ for $\delta m<0$
and $H^{\pm\pm}\to H^{\pm}W^{\pm*}$, $H^{\pm}\to H^{0}/AW^{\pm*}$ for $\delta
m>0$) should play a significant role in restricting the model parameter space
in the non-degenerate situation ($|\delta m|\neq 0$) for intermediate region
$v_{\Delta}\sim 10^{-4}$ GeV.

###  IV.4 Displaced vertices

In certain parameter regions, the doubly-charged Higgs $H^{\pm\pm}$ can be
long-lived, leading to the possibility of displaced vertices that could be
searched at HL-LHC ($\sqrt{s}=14$ TeV) or FCC-hh ($\sqrt{s}=100$ TeV). We
already saw that for large triplet VEV $v_{\Delta}$, $H^{\pm\pm}$ dominantly
decay to on-shell $W^{\pm}W^{\pm}$, but is kinematically forbidden if
$m_{H^{\pm\pm}}\lesssim 160$ GeV. In that case, $H^{\pm\pm}$ decays mainly via
$H^{\pm\pm}\to W^{\pm}W^{\pm*}\to W^{\pm}f\bar{f^{\prime}}$ and this rate is
proportional to triplet VEV $v_{\Delta}$. Depending upon the parameter
choices, the decay width can be small, i.e. the decay length can be
macroscopic. For example, with $v_{\Delta}\sim 10^{-4}$ GeV and
$m_{H^{\pm\pm}}\sim 150$ GeV, proper decay length can be order of 1 mm. In
such parameter region one has very good prospects for displaced vertex
searches. For a detailed discussion, see Ref. [65].

Future lepton colliders

As seen in Figs. 11 and 12, the pair production ($H^{\pm\pm}H^{\mp\mp}$) cross
section at hadron colliders becomes small for large doubly-charged Higgs
masses. In addition to this, the existence of multiple backgrounds reduces
their physics reach for doubly-charged Higgs boson discovery. A lepton
$e^{+}e^{-}$ collider, with a considerably cleaner environment, is more
suitable for searching $H^{\pm\pm}$. In this section, we discuss the possible
collider signatures associated to the doubly-charged Higgs boson triplet of
the type-II seesaw mechanism. We examine its production cross section at
$e^{+}e^{-}$ colliders, such as CLIC [67], FCC-ee [68], ILC [69] and CEPC
[70]. First note that the doubly-charged Higgs boson $H^{\pm\pm}$ can be
produced at $e^{+}e^{-}$ colliders through $\gamma/Z$-mediated processes.

In Fig. 13, we show the $H^{\pm\pm}H^{\mp\mp}$ production cross section at an
$e^{+}e^{-}$ collider for center of mass energies $\sqrt{s}=1$ TeV and 3 TeV,
respectively. For small triplet VEV $v_{\Delta}<10^{-4}$ GeV, the
$H^{\pm\pm}\to\ell^{\pm}\ell^{\pm}$ decay mode dominates and one can have
tetra-lepton signatures.

Figure 13: The production cross section for $H^{\pm\pm}H^{\mp\mp}$ at
$e^{+}e^{-}$ collider. The center of mass energies are $\sqrt{s}=1$ TeV (red-
solid line) and 3 TeV (blue-dashed line).

Figure 14: The production cross section for $H^{\pm}H^{\mp}$ (left panel) and
$H^{0}/A$ (right panel) at $e^{+}e^{-}$ collider. The center of mass energies
are $\sqrt{s}=1$ TeV (red-solid line) and 3 TeV (blue-dashed line).

For large triplet VEV $v_{\Delta}>10^{-4}$ GeV, $H^{\pm\pm}$ will decay
dominantly into $W^{\pm}W^{\pm}$ gauge bosons. One can consider either the
leptonic or hadronic decay modes of $W^{\pm}$. Hence to probe the large
triplet VEV $v_{\Delta}>10^{-4}$ GeV region, one can look for the following
multilepton or multijet final states:

$\displaystyle e^{+}e^{-}\to H^{\pm\pm}H^{\mp\mp}\to
W^{\pm}W^{\pm}W^{\mp}W^{\mp}\to\ell^{\pm}\ell^{\pm}\ell^{\mp}\ell^{\mp}\nu\nu\nu\nu/8j.$
(75)

If the $H^{\pm\pm}$ mass $m_{H^{\pm\pm}}\sim\mathcal{O}(1\,\text{TeV})$, the
gauge bosons produced from $H^{\pm\pm}$ decay will be highly boosted, leading
to a fat-jet [71]:

$\displaystyle e^{+}e^{-}\to H^{\pm\pm}H^{\mp\mp}\to
W^{\pm}W^{\pm}W^{\mp}W^{\mp}\to 4\,\text{fat-jets}.$ (76)

Note that for such a heavy Higgs boson ($m_{H^{\pm\pm}}\sim 1$ TeV), the
produced $W^{\pm}$ or fat-jet will have large transverse momenta ($p_{T}\sim
1$ TeV), which can be used effectively to reduce the SM backgrounds.

For completeness, in Fig. 14, we show the $H^{\pm}H^{\mp}$ (left panel) and
$H^{0}/A$ (right panel) production cross section at an $e^{+}e^{-}$ collider
for center of mass energies $\sqrt{s}=1$ TeV and 3 TeV, respectively. Although
smaller than $H^{\pm\pm}H^{\mp\mp}$, the production cross sections are still
sizeable.

##  V Deconstructing the type-II seesaw at colliders

As already mentioned, charged lepton flavor violaton is mediated by the same
triplet scalar responsible for neutrino mass generation. A single universal
symmetrical Yukawa coupling $Y_{\Delta}$ is involved, see Eqs. (19) and (43),
(44). The relevant Yukawa coupling matrix can be parameterised in terms of
neutrino oscillation parameters as,

$\displaystyle
Y_{\Delta}=\frac{\sqrt{2}}{v_{\Delta}}U\textup{diag}\left\\{m_{1},m_{2},m_{3}\right\\}U^{T}.$
(77)

The structure of the restrictions on $m_{\nu}=Y_{\Delta}v_{\Delta}$ that
follow from oscillations and $0\nu\beta\beta$ decay have been given in Fig. 5.
One sees from Eqs. (43) and (44) that the cLFV branching fractions are
detemined by the same matrix $Y_{\Delta}$. In Fig. 15, we show contours of
$\text{BR}(\mu\to e\gamma)$ in the $m_{H^{\pm\pm}}-v_{\Delta}$ plane, obtained
for normal ordered light neutrino spectrum and best-fit values for the
neutrino oscillation parameters [20, 21]. One sees how $\text{BR}(\mu\to
e\gamma)$ may easily lie within reach of experiment, such as the current
measurements of MEG [72] that lead to the limit $\text{BR}(\mu\to
e\gamma)<4.2\times 10^{-13}$. One sees that the $\mu\to e\gamma$ branching
ratio can easily exceed current senstivities for small triplet VEV
$v_{\Delta}$. The same happens for other cLFV processes.

Figure 15: Contour of $\text{BR}(\mu\to e\gamma)$ in the
$m_{H^{\pm\pm}}-v_{\Delta}$ plane. The oscillation parameters are fixed to
their best fit values.

Hence, given the neutrino mass ordering, the magnitude of the Yukawa coupling
is completely determined by $v_{\Delta}$. We checked that for the triplet VEV
range $10^{-10}\,\text{GeV}\leq v_{\Delta}\leq 10^{-3}\,\text{GeV}$ relevant
for Fig. 15, the Yukawa coupling always remains within the perturbative
regime.

Distinguising the neutrino mass ordering at colliders

We have already discussed that the decays of the doubly-charged Higgs boson to
leptonic final states are determined by the Yukawa coupling $Y_{\Delta}$. The
decay width is given by

$\Gamma(H^{\pm\pm}\to
l^{\pm}_{i}l^{\pm}_{j})=\Gamma_{l_{i}l_{j}}=\frac{m_{H}^{\pm\pm}}{(1+\delta_{ij})16\pi}|Y_{\Delta}^{ij}|^{2}.$
(78)

where the Yukawa coupling is determined by Eq. (77). Current measurements of
neutrino oscillation parameters [20, 21] restrict the pattern of diagonal and
off-diagonal entries of Yukawa coupling $Y_{\Delta}$. Note that for
$v_{\Delta}<10^{-4}$ GeV, $H^{++}$ decays predominantly to leptons and as
$\Gamma_{\ell_{i}\ell_{j}}\propto|Y_{\Delta}^{ij}|^{2}$, the patterns of
various leptonic channels will exactly follow the pattern of
$Y_{\Delta}^{ij}$, partially determined by oscillation experiments.

In Fig. 16, we show the branching fractions for the decays into same-flavour
leptonic final states. To obtain this we randomly varied the mixing angles
within their $3\sigma$ ranges of allowed values [20, 21], varying
$\delta_{\text{CP}}$ in the range $[-\pi:\pi]$, and choosing the triplet VEV
as $v_{\Delta}<10^{-4}$ GeV. Since there is no infomation on Majorana phases
we set them to zero.

Figure 16: $H^{++}$ branching ratio to same-flavour like-sign dilepton final
states versus the lightest neutrino mass for NO (left panel), and IO (right
panel). Oscillation parameters are varied within their $3\sigma$ ranges. We
have fixed triplet VEV, $v_{\Delta}<10^{-4}$ GeV and $m_{H^{\pm\pm}}=1$ TeV.
Note that for NO, di-muon and di-tau final states are always larger than di-
electron final state, whereas for IO, di-electron final states are always
larger than di-muon and di-tau final sates.

Figure 17: Same as Fig. 16 but now for $H^{\pm\pm}$ decay to different flavour
like-sign dilepton final states. Except for “large” values of the lightest
neutrino mass above few$\times 10^{-2}$eV or so, in both NO and IO cases, the
$\mu\tau$ like-sign final state is dominant.

Depending on the ordering of the light-neutrino mass spectrum we obtain the
following decay branching ratio patterns [73]:

$\displaystyle\text{BR}(H^{++}\to\mu\mu),\text{BR}(H^{++}\to\tau\tau)\gg\text{BR}(H^{++}\to
ee)\,\,\,\,\text{for {\bf NO}},$ (79) $\displaystyle\text{BR}(H^{++}\to
ee)\gg\text{BR}(H^{++}\to\mu\mu),\text{BR}(H^{++}\to\tau\tau)\,\,\,\,\text{for
{\bf IO}}.$ (80)

The above two relations suggest that, depending on the ordering of the light
neutrino masses, $H^{++}$ can decay to either $\mu\mu$, $\tau\tau$ (for NO) or
$ee$ (IO) with relatively large strength. Hence, it might be possible to probe
the ordering (NO or IO) by looking into the decay patterns of $H^{++}$ to
same-flavour leptonic final states.

In Fig. 17, we show the $H^{++}$ decay branching fraction into different-
flavour like-sign dilepton final states. One finds that for both NO and IO
cases the $H^{++}$ charged-lepton-flavour-violating branching ratios are of
the same order of magnitude as the lepton-flavour-conserving ones, and
especially the $\mu\tau$ channel can be quite large for both mass spectra.

Figure 18:  $H^{\pm\pm}$ decay branching ratios versus the atmospheric angle
within its allowed $3\sigma$ range, for both NO (left panel) and IO (right
panel), repectively. The lightest neutrino mass is fixed to zero, with triplet
VEV $v_{\Delta}<10^{-4}$ GeV, so that the dilepton decay channel is dominant.
The Dirac phase $\delta_{\text{CP}}$ is varied in the range $[-\pi:\pi]$ and
other oscillation parameters are fixed to their best fit values [20, 21]. One
sees that like-sign di-muon and di-tau branching fractions correlate with the
octant in different ways depending on the mass ordering.

Likewise, in Fig. 18, we display the leptonic branching ratios (only those
which are appreciable) in terms of the allowed $3\sigma$ range of the
“atmospheric” mixing angle $\theta_{23}$, both for NO (left panel) and IO
(right panel). As before, the triplet VEV is chosen as $v_{\Delta}<10^{-4}$
GeV, $\delta_{\text{CP}}$ is varied in the range $[-\pi:\pi]$, and other
oscillation parameters are fixed to their best fit values [20, 21]. The
vertical line in each panel denotes $\theta_{23}=45^{\circ}$. One sees that
for the NO case, electron final-states are penalized with respect to those
into muons and taus and one can have sizeable branching ratios for $\mu\mu$
and $\mu\tau$ final states. On the other hand, for the case of IO, the $ee$
final state is dominant. One sees that like-sign di-muon and di-tau branching
fractions correlate with the atmospheric octant in different ways depending on
the neutrino mass ordering.

Note that, although $H^{\pm\pm}$ pair production at $pp$ or $e^{+}e^{-}$
colliders is gauge-mediated, its decays to like-sign dileptons are determined
by the Yukawa coupling $Y_{\Delta}$. This coupling matrix is also responsible
for inducing lepton flavour violation processes at low energies. As a result
one expects that collider observables such as the cross sections
$\sigma(pp/e^{+}e^{-}\to
H^{\pm\pm}H^{\mp\mp}\to\ell_{i}^{\pm}\ell_{j}^{\pm}\ell_{k}^{\mp}\ell_{m}^{\mp})$,
where at least one of the final-state leptons differs from the others, may
strongly correlate with low-energy flavour violating phenomena such as
neutrino oscillations or rare $\ell_{i}\to\ell_{j}\gamma$ decays.

Figure 19:  Probing neutrino mass and mass ordering in neutrino oscillations
through the 4-lepton cross sections such as $e^{\pm}e^{\pm}e^{\mp}e^{\mp}$,
$\mu^{\pm}\mu^{\pm}\mu^{\mp}\mu^{\mp}$, $e^{\pm}e^{\pm}\mu^{\mp}\mu^{\mp}$,
$e^{\pm}e^{\pm}\tau^{\mp}\tau^{\mp}$ and
$\mu^{\pm}\mu^{\pm}\tau^{\mp}\tau^{\mp}$. Red points correspond to inverted
neutrino mass-ordering IO, while blue points correspond to NO. Here we chose
the doubly-charged Higgs mass as $m_{H^{\pm\pm}}=1$ TeV, suitable for a
generic X-collider that can be either FCC-hh [74] or a lepton collider (CLIC
[67]/FCC-ee [68]/ILC [69], CEPC [70]). Oscillation parameters are varied
within their $3\sigma$ ranges [20, 21]. Vertical shaded bands indicate the
KATRIN [34], CMB [36], and CMB+BAO [37] limits on the lightest neutrino mass.
We see from these results how one can probe the mass ordering through these
multilepton final sates at future colliders.

In Fig. 19 we display $\sigma(pp,e^{+}e^{-}\to H^{\pm\pm}H^{\mp\mp}\to
eeee/\mu\mu\mu\mu/ee\mu\mu/ee\tau\tau/\mu\mu\tau\tau)$ versus the lightest
neutrino mass, both for NO and IO mass spectra. We fix the doubly-charged
Higgs mass at $m_{H^{\pm\pm}}=1$ TeV, so that both the 3 TeV $e^{+}e^{-}$ as
well as 100 TeV $pp$ colliders have similar cross section (5 fb), as seen in
Figs. 12 and 13. We denote by “X” such future “generic” collider. As before,
rather than fixing reference values for the oscillation parameters, we varied
them within their allowed $3\sigma$ ranges [20, 21], keeping the triplet VEV
small, $v_{\Delta}<10^{-4}$ GeV. In this case the Yukawa coupling $Y_{\Delta}$
is large, so $H^{\pm\pm}$ decays predominantly to like-sign dileptons, hence
the four-lepton cross sections can be rather large. Indeed, the cross sections
into final states such as $e^{\pm}e^{\pm}e^{\mp}e^{\mp}$,
$e^{\pm}e^{\pm}\mu^{\mp}\mu^{\mp}$ and $e^{\pm}e^{\pm}\tau^{\mp}\tau^{\mp}$
($\mu^{\pm}\mu^{\pm}\mu^{\mp}\mu^{\mp}$ and
$\mu^{\pm}\mu^{\pm}\tau^{\mp}\tau^{\mp}$) are always larger for IO (NO) and
also that the difference increases with a smaller mass of the lightest
neutrino. Moreover, comparing blue and red points in all of the panels, one
notes that the magnitudes of this multilepton signal differences for normal
and inverted neutrino mass ordering strongly correlate with the magnitude of
the lightest neutrino mass. This provides a way to probe the neutrino mass
ordering through these four-lepton final states in a robust manner, throughout
the parameter region allowed by neutrino oscillations.

Lepton flavour violation first at colliders?

It was suggested in the LEP days that charged lepton flavour [7] and leptonic
CP symmetry [8] non-conservation could be first seen at high-energies. This
proposal was scrutinized in the context of the LEP-1 experiments and
subsequently revived in [9, 10, 11] for the case of the LHC [75] (see also
[76, 77, 78, 68, 79] for recent references). It has recently been noted that
the type-II seesaw mechanism provides a neater seesaw realization of the same
idea [6]. We now provide a complete discussion including also the relevant
theoretical aspects. First of all, comparing Figs. 16, 17 and Fig. 18 one sees
that same-flavour ($\mu\mu$) and different-flavour ($\mu\tau$) doubly-charged
Higgs decay branching fractions have very similar values. Hence one can
conclude that within the type-II seesaw mechanism, lepton flavour violating
4-lepton cross sections need not be suppressed when compared to those for
same-flavour leptonic final states.

Figure 20: Charged lepton flavour violating multilepton cross sections versus
$\text{BR}(\mu\to e\gamma)$, $\text{BR}(\tau\to e\gamma)$ or
$\text{BR}(\tau\to\mu\gamma)$. Red points correspond to IO, while blue points
correspond to NO. We have varied the oscillation parameters within their
$3\sigma$ ranges [20, 21]. The gray band is excluded by the MEG limit [72] or
the BaBar limit [80]. We see specially that for final states such as
$e^{\pm}e^{\pm}e^{\mp}\mu^{\mp}$, $e^{\pm}e^{\pm}e^{\mp}\tau^{\mp}$ and
$e^{\pm}e^{\pm}\mu^{\mp}\tau^{\mp}$ one can clearly distinguish NO from IO for
all “interesting” values of $\text{BR}(\mu\to e\gamma)$, $\text{BR}(\tau\to
e\gamma)$ and $\text{BR}(\tau\to\mu\gamma)$, respectively.

Figure 21:  Charged lepton flavour violating multilepton cross sections versus
$\text{BR}(\mu\to e\gamma)$ or $\text{BR}(\tau\to e\gamma)$. Although these
Charged lepton flavour violating cross sections are sizeable even for very
tiny $\text{BR}(\mu\to e\gamma)$ or $\text{BR}(\tau\to e\gamma)$ values, here
it is difficult to distinguish the mass orderings, as they overlap.

The fact that, as mentioned, charged lepton flavour violation may be first
observed as a high-energy phenomenon is seen neatly from Fig. 20, where we
show the correlation between $\sigma(e^{+}e^{-}\to H^{\pm\pm}H^{\mp\mp}\to
eee\mu/\mu\mu\mu\tau/\tau\tau\mu\tau/eee\tau/ee\mu\tau)$ and $\text{BR}(\mu\to
e\gamma)$, $\text{BR}(\tau\to\mu\gamma)$ or $\text{BR}(\tau\to e\gamma)$ both
for NO and IO. For simplicity we took the lightest neutrino mass
$m_{\nu_{1(3)}}=0$. Indeed, one sees that the various flavour-violating multi-
lepton cross-sections can be sizeable, even when the low-energy
$\text{BR}(\mu\to e\gamma)$, $\text{BR}(\tau\to\mu\gamma)$ or
$\text{BR}(\tau\to e\gamma)$ decay branching ratios lie well below
detectability.

In Fig. 20, comparing red and blue points, we see that the cross section into
$e^{\pm}e^{\pm}e^{\mp}\mu^{\mp}$, $e^{\pm}e^{\pm}e^{\mp}\tau^{\mp}$ and
$e^{\pm}e^{\pm}\mu^{\mp}\tau^{\mp}$ final states (see left panels and lower
one) is always larger in the case of IO compared to NO, whereas for final
state such as $\mu^{\pm}\mu^{\pm}\mu^{\mp}\tau^{\mp}$ and
$\tau^{\pm}\tau^{\pm}\mu^{\mp}\tau^{\mp}$ the behaviour is the opposite.
Moreover, although the cross sections can be large they are somewhat less
sensitive to mass ordering.

Similarly, in Fig. 21 we show how the 4-lepton cross sections
$\sigma(e^{+}e^{-}\to
H^{\pm\pm}H^{\mp\mp}\to\mu^{\pm}\mu^{\pm}e^{\mp}\mu^{\mp})$ (left panel) and
$\sigma(e^{+}e^{-}\to
H^{\pm\pm}H^{\mp\mp}\to\mu^{\pm}\mu^{\pm}e^{\mp}\tau^{\mp})$ (right panel)
correlate with $\text{BR}(\mu\to e\gamma)$ and $\text{BR}(\tau\to e\gamma)$,
respectively. One sees that although the cross-sections for these flavour-
violating four-lepton final states can be sizeable even for tiny values of
$\text{BR}(\mu\to e\gamma)$ or $\text{BR}(\tau\to e\gamma)$, these can not
probe the inverted mass ordering, since its prediction for the cross section
can also occur for the case of normal ordering. However, predicted lepton
flavour violating multi-lepton cross section values within the blue region and
outside the red one would be a signature for normal mass ordering. This can be
understood from the doubly-charged Higgs decay branching ratios into flavor
conserving and flavor violating channels, depicted in Figs. 16 and 17. We also
note that there are other multilepton final states, such as
$\tau^{\pm}\tau^{\pm}e^{\mp}\mu^{\mp}$ and
$\tau^{\pm}\tau^{\pm}e^{\mp}\tau^{\mp}$, whose cross sections exhibit a
similar behaviour as the one discussed above.

In conclusion, we have shown that multilepton signatures can be used to probe
the neutrino mass ordering. Even when not useful as a probe of the neutrino
mass ordering, our four-lepton cross sections to different-flavour leptonic
final states can probe the fundamental flavour structure of the neutrino
sector, and perhaps provide a way of reconstructing the neutrino mixing matrix
at collider experiments 666For example, this happens in supersymmetric models
with bilinear violation of R-parity [81, 82, 83, 84], where the atmospheric
mixing angle can be independently probed at collider experiments [85, 86,
87]..

##  VI non-standard neutrino interactions

A feature of low-scale neutrino mass generation models is the existence of
potentially sizeable non-standard neutrino interactions (NSI). These can be
induced either at the tree level or through radiative corrections [88], and
can manisfest phenomenologically in a variety of ways [89, 90, 91, 92, 93, 94,
95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110].
Neutral-current NSI (denoted as NC-NSI) can be expressed as follows:

${\cal L}_{{\rm NC-
NSI}}=-2\sqrt{2}G_{F}\overline{\nu}_{\alpha}\gamma^{\mu}{\cal
P}_{L}\nu_{\beta}\overline{f}\gamma_{\mu}{\cal
P}_{X}\epsilon_{\alpha\beta}^{fX}f,$ (81)

factorizing $G_{F}$, for convenience. They lead to flavour changing (FC)
neutrino-matter interactions, as well as to non-universal (NU) interactions.
Note that in the above equation ${\cal P}_{L}=\left(1-\gamma^{5}\right)/2$,
${\cal P}_{R}=\left(1+\gamma^{5}\right)/2$, $X=L$ or $R$, $\nu_{\alpha}$ are
neutrinos of flavor $\alpha$ and $\beta$ ($\alpha,\ \beta=e$, $\mu$, $\tau$)
respectively. Here $f$ denote charged SM fermions such as electrons ($e^{-}$)
and quarks ($u$, $d$). The NC-NSIs in Eq. (81) are particularly responsible
for the scattering processes $\nu_{\alpha}+f\to\nu_{\beta}+f$. Therefore they
induce matter effects on neutrino oscillations, modifying their propagation
properties [89], as well as the elastic neutrino-electron scattering [97,
98]777Neutrino-nucleus scattering, e.g. CEvNS [100, 99, 101, 102, 103, 104,
105, 106] is not affected in our case, as the triplet couplings are purely
leptonic..

Within the simplest type-II seesaw, the same triplet that mediates neutrino
mass generation also induces NSI as illustrated in Fig. 22. To see this,
notice that the singly-charged Higgs of the triplet $\Delta$ has Yukawa
interactions with neutrinos and charged leptons described by the Lagrangian
shown previously in Eq. (3). More specifically

${\cal
L}_{\Delta^{\pm}}=\sqrt{2}Y_{\Delta\alpha\beta}\nu_{L\alpha}^{T}C\ell_{L\beta}\Delta^{+}+{\rm
h.c.},$ (82)

Figure 22:  Effective non-standard neutrino interactions arising in generic
neutrino mass generation schemes.

Integrating out the physical field $H^{\pm}\approx\Delta^{\pm}$, and applying
Fierz transformations888In the Dirac notation, we have the Fierz
transformation $\overline{f}_{1}{\cal P}_{L}f_{2}\overline{f}_{3}{\cal
P}_{R}f_{4}=-\frac{1}{2}\overline{f}_{1}\gamma_{\mu}{\cal
P}_{R}f_{4}\overline{f}_{3}\gamma^{\mu}{\cal P}_{L}f_{2}$. to this charged
current interaction, we get the effective operator 999There is a rotation
between doublet and triplet Higgs scalars, but the rotation angle is neutrino-
mass suppressed.:

${\cal
L}\supset-\frac{{\color[rgb]{1,0,0}i}}{m_{H^{\pm}}^{2}}Y_{\Delta\alpha^{\prime}\beta^{\prime}}^{*}Y_{\Delta\alpha\beta}\overline{\nu_{L\alpha^{\prime}}}\gamma^{\mu}\nu_{L\alpha}\overline{\ell_{L\beta^{\prime}}}\gamma_{\mu}\ell_{L\beta}.$
(83)

Taking $\beta=\beta^{\prime}$ in Eq. (83) we can identify the NC-NSI of
neutrinos with charged leptons as given in Ref. [111]. Comparing Eq. (83) and
Eq. (81), and again assuming $m_{H^{\pm\pm}}\approx m_{H^{\pm}}\approx
M_{\Delta}^{2}$, one can extract the corresponding NC-NSI parameters:

$\epsilon_{\alpha\beta}^{\ell
L}=-\frac{Y_{\Delta\alpha\ell}^{*}Y_{\Delta\beta\ell}}{2\sqrt{2}G_{F}M_{\Delta}^{2}}.$
(84)

Constraining NSI from cLFV

Figure 23:  The blue regions are the allowed neutrino mass matrix element
magnitudes $|m_{ee}|$ (left) and $|m_{e\mu}|$ (right) in the flavor basis, as
a function of $m_{1}$ for normal neutrino mass ordering (NO). The mixing
angles, squared mass differences, and $\delta$ are taken within their
3$\sigma$ limits as reported in [20]. The vertical shaded regions illustrate
KATRIN’s limit for $m_{1}$ as well as the cosmological limits. The left panel
coincides with the effective mass from neutrinoless double beta decay
$|m_{ee}|$ and the shaded horizontal bands are excluded by KAMLAND-Zen [26].
Magenta dots represent the regions where the corresponding matrix elements
satisfy the cLFV limits with the NSI parameter
$|\varepsilon_{ee}^{eL}|>1\times 10^{-4}$. Brown dots represent the analogue
for $|\varepsilon_{e\tau}^{eL}|$.

As in Fig. 5 we can now determine the allowed regions for the magnitudes of
the neutrino mass matrix elements obeying the constraints discussed in the
sections above. In order to determine the allowed modulus for each of the
elements of the mass matrix in the flavor basis we have performed a scan over
the parameter space. For this analysis we consider the range $m_{1}<0.8$ eV
given by KATRIN’s upper limit for $m_{\beta}$ [34]. We generate random values
for the neutrino mixing $U$ parameters within their 3$\sigma$ limits, taking
into account the most recent global neutrino oscillation analysis [20]. We
vary the Majorana phases randomly in the range between 0 and 2$\pi$. The
results obtained by this scan for $|m_{ee}|$ and $|m_{e\mu}|$ are shown as
blue regions in Fig. 23. They are given in the flavor basis as a function of
$m_{1}$, for the case normal neutrino mass ordering. Notice that the left
panel in the figure coincides wih the absolute value $|m_{ee}|$ of the
effective Majorana neutrino mass in neutrinoless double beta decay. Notice
that for the determination of the allowed NC-NSI strength we have taken into
account consistency with the triplet VEV and neutrino mass matrix
restrictions, Eqs. (8), (19), and (84). Moreover, we have also included
consistency with LFV constraints in Table 1, where we show the explicit Yukawa
coupling combinations relevant for each of the decay modes, and the
corresponding experimental branching ratio limit denoted by $L_{i}$.

One can determine the conditions that maximize the value of any of the six
different NSI parameters, for example the diagonal NSI parameter
$|\varepsilon_{ee}^{eL}|$. From the previous definitions for
$\varepsilon_{ee}^{eL}$ (Eq. 84) and mass matrix element
($|m_{ee}|=v_{\Delta}|Y_{\Delta ee}|/\sqrt{2}$) we have:

$\frac{|m_{ee}|^{2}}{\sqrt{2}G_{F}}=|\varepsilon_{ee}^{eL}|\,M_{\Delta}^{2}v_{\Delta}^{2}.$
(85)

NSI | Explicit Form | Estimated Limit (NO) | Estimated Limit (IO)
---|---|---|---
$|\epsilon_{ee}^{eL}|$ | $(2\sqrt{2}G_{F})^{-1}M_{\Delta}^{-2}|Y_{\Delta ee}^{*}Y_{\Delta ee}|$ | $<8.0\times 10^{-4}$ | $<8.0\times 10^{-4}$
$|\epsilon_{e\mu}^{eL}|$ | $(2\sqrt{2}G_{F})^{-1}M_{\Delta}^{-2}|Y_{\Delta ee}^{*}Y_{\Delta\mu e}|$ | $<7.0\times 10^{-7}$ | $<7.0\times 10^{-7}$
$|\epsilon_{e\tau}^{eL}|$ | $(2\sqrt{2}G_{F})^{-1}M_{\Delta}^{-2}|Y_{\Delta ee}^{*}Y_{\Delta\tau e}|$ | $<2.0\times 10^{-4}$ | $<2.1\times 10^{-4}$
$|\epsilon_{\mu\mu}^{eL}|$ | $(2\sqrt{2}G_{F})^{-1}M_{\Delta}^{-2}|Y_{\Delta\mu e}^{*}Y_{\Delta\mu e}|$ | $<6.8\times 10^{-6}$ | $<2.5\times 10^{-6}$
$|\epsilon_{\mu\tau}^{eL}|$ | $(2\sqrt{2}G_{F})^{-1}M_{\Delta}^{-2}|Y_{\Delta\mu e}^{*}Y_{\Delta\tau e}|$ | $<4.8\times 10^{-6}$ | $<2.5\times 10^{-6}$
$|\epsilon_{\tau\tau}^{eL}|$ | $(2\sqrt{2}G_{F})^{-1}M_{\Delta}^{-2}|Y_{\Delta\tau e}^{*}Y_{\Delta\tau e}|$ | $<9.5\times 10^{-5}$ | $<9.9\times 10^{-5}$
Table 2:  Constraints on NSI parameters from cLFV processes. See text for
details.

Besides NSI parameters, we also express cLFV branching ratios in terms of the
neutrino mass matrix elements. For example, from Table 1 we see that from
$\mu\to e\gamma$ we have the restriction:

$\frac{|m^{\dagger}m|_{e\mu}}{G_{F}}<L_{1}\,M_{\Delta}^{2}v_{\Delta}^{2},$
(86)

where $L_{1}$ is the corresponding limit quoted in Table 1. Using the previous
equations to eliminate the product $M_{\Delta}v_{\Delta}$, we find:

$\frac{|m^{\dagger}m|_{\alpha\beta}}{|m_{ee}|^{2}}<\frac{L_{i}}{2\sqrt{2}\leavevmode\nobreak\
|\varepsilon_{ee}^{eL}|},\hskip 56.9055pti=1,3,4$ (87)
$\frac{|m^{\dagger}|_{\alpha\beta}|m|_{\sigma\rho}}{|m_{ee}|^{2}}<\frac{L_{i}}{2\sqrt{2}\leavevmode\nobreak\
|\varepsilon_{ee}^{eL}|},\hskip 42.67912pti\neq 1,3,4$ (88)

where $L_{i}$ are the limits found from cLFV in Table 1 and $\alpha$, $\beta$,
$\sigma$, and $\rho$ represent the relevant flavors for each $L_{i}$, also
given in Table 1. In order to ensure an efficient parameter scan, we will use
these ratios as a guidance to search for potentially large NSI parameters that
satisfy all the cLFV experimental conditions, as well as those from the
neutrino data. We have performed the scan both for NO as well as for the IO
case. The latter is also viable with current oscillation data [20], and is
characterized by having the lightest neutrino corresponding to $m_{3}$.

To perform the scan, we start by studying the restrictions on NU-NSI
parameters in the NO case. In order to determine each of the
$|m_{\alpha\beta}|$ we generate random values for the neutrino parameters in
the mixing matrix $U$, within their 3$\sigma$ ranges from Ref. [20]. Then we
give different values of $|\varepsilon_{ee}^{eL}|$ until we find the maximum
for which the ten conditions in Eqs. (87) and (88) are satisfied. This gives
us the strength of the NSI that is consistent with oscillation data as well as
cLFV constraints. No combination of parameters was found with an NSI strength
of order $10^{-3}$ or larger. However, we found several combinations in
parameter space that induced an NSI of order $10^{-4}$, which are indicated in
Fig. 23 as magenta dots. The panels in this figure give an idea of the
$|m_{\alpha\beta}|$ elements required to have an NSI of order $10^{-4}$. For
simplicity we only show $|m_{ee}|$ (left) and $|m_{e\mu}|$ (right), but
similar plots can be obtained for each $|m_{\alpha\beta}|$. In the left panel
we also indicate as shaded horizontal bands, marked as K1 and K2, those
excluded by the KAMLAND-Zen collaboration [26]. Notice that a diagonal NSI of
order $10^{-4}$ is reachable for masses just below KAMLAND-Zen limits and the
current cosmological limits from the CMB [36] (see figure). We followed the
same procedure to maximize the possible value for each of the other two
diagonal NSIs, for which we found upper limits below $10^{-4}$. The best
result for each case is shown in Table 2.

Figure 24:  Combinations of $M_{\Delta}$ and $v_{\Delta}$ where one can have
$|\varepsilon_{ee}^{eL}|>1\times 10^{-4}$ consistent with cLFV limits and
normal neutrino mass ordering, NO. The gray region is excluded by collider
data (see Section III), and the dashed line indicates $v_{\Delta}=0.1$ MeV.
The brown points correspond to $|\varepsilon_{e\tau}^{eL}|>1\times 10^{-4}$.
For an explanation of the different excluded regions in grey, see text.

For the case of non-diagonal NSIs, we follow a similar method. First, by
comparing the definition of the non-diagonal NSIs with the second, third, and
forth rows in Table 1, one sees that we expect these NSIs to lie below
8$\times 10^{-4}$. The most severely constrained is
$|\varepsilon_{e\mu}^{eL}|<L_{2}/(2\sqrt{2})=7.1\times 10^{-7}$, because of
the stringent limits on $\mu\to 3e$. Similarly, for the case of
$|\varepsilon_{e\tau}^{eL}|$ we again generate random values for $U$ in order
to produce the neutrino mass matrix elements $|m_{\alpha\beta}|$. Then we look
for the maximum value of $|\varepsilon_{e\tau}^{eL}|$ so that Eqs. (87) and
(88), with $|m_{ee}|^{2}\to|m_{e\tau}^{*}||m_{ee}|$, are satisfied. Brown dots
in Fig. 23 allow for $|\varepsilon_{e\tau}^{eL}|>1\times 10^{-4}$. A similar
analysis was done for $|\varepsilon_{\mu\tau}^{eL}|$. The maximum limits found
with this method for all the NSI are given in Table 2. All in all, there are
only three NSIs that can exceed $\approx 10^{-4}$, still far from the current
sensitivity of reactor and solar neutrino experiments.

Figure 25:  Same as Fig. 23 but for the case of inverted neutrino mass
ordering, IO. Magenta dots denote the regions where the corresponding matrix
elements obey cLFV limits with the NSI parameter
$|\varepsilon_{ee}^{eL}|>1\times 10^{-4}$. Brown dots correspond to
$|\varepsilon_{e\tau}^{eL}|$.

From the previous analysis, we can also get information about the triplet mass
and VEV that could induce an NSI of a desired order. From Eq. (85), notice
that for a given NSI, there is an inverse relation between $M_{\Delta}$ and
$v_{\Delta}$. This means that there are different combinations of the product
of these parameters that reproduce the same NSI of a desired order. For
instance, the magenta dots in Fig. 23 correspond to combinations for which
$|\varepsilon_{ee}^{eL}|>1\times 10^{-4}$. For each of these dots we can
generate a plot $M_{\Delta}$ vs $v_{\Delta}$ with the NSI value fixed. Fig. 24
shows the resulting curves, which combined produce the magenta region of
allowed values for the mass and the VEV that can induce an NSI of order
$10^{-4}$. The shaded grey regions in this figure correspond to the excluded
regions for $m_{H^{++}}$ from collider constraints discussed in sec. III.5.
For lower triplet VEV $v_{\Delta}<10^{5}$ eV, the doubly-charged Higgs mass
constraint is $m_{H^{++}}>870$ GeV, while for $v_{\Delta}>10^{5}$ eV, we have
a weaker bound $m_{H^{++}}>220$ GeV.

Our analysis has been focused on the assumption of normal ordering of the
neutrino masses. The alternative case of inverse ordering leads to the blue
regions in Fig. 25. One sees the $|m_{ee}|$ (left) and $|m_{e\mu}|$ (right)
values obtained from oscillation data, when varying the neutrino mixing
parameters within their 3$\sigma$ ranges. Following a similar analysis we
conclude that in the IO case we can only have NSI above $10^{-4}$ and
consistent with cLFV limits for $|\varepsilon_{ee}^{eL}|$ and
$|\varepsilon_{e\tau}^{eL}|$. The magenta dots show the regions for
$|\varepsilon_{ee}^{eL}|$, and the brown dots those for
$|\varepsilon_{e\tau}^{eL}|$. Table 2 summarizes the largest NSI strengths
consistent with current experimental restrictions.

Finally, we comment on the consistency of our NSI results with current limits
on $\mu\to e$ conversion in nuclei [112, 113]. Within the type-II seesaw
mechanism, both the singly- and doubly-charged scalars contribute to $\mu\to
e$ conversion. For relatively light nuclei the conversion rate is given by

$\mathrm{BR}(\mu\to
e)\approx\frac{\alpha^{5}}{36\pi^{4}}\frac{m_{\mu}^{5}}{\Gamma_{\mathrm{capt}}}Z_{\rm
eff}^{4}ZF^{2}(-m_{\mu}^{2})\left|(Y_{\Delta}^{\dagger}Y_{\Delta})_{e\mu}\left[\frac{5}{24m_{H^{+}}^{2}}+\frac{1}{m_{H^{++}}^{2}}\right]+\frac{1}{m_{H^{++}}^{2}}\sum_{\alpha}Y^{\dagger}_{\Delta
e\alpha}Y_{\Delta\alpha\mu}f(m_{\alpha},m_{H^{++}})\right|^{2},$ (89)

where $F(q^{2})$ is the nuclear form factor, $m_{\mu}$ is the muon mass,
$Z_{\rm eff}$ is an effective atomic charge, and the function $f$ is given by
[114, 115]:

$f(m_{\alpha},m_{H^{++}})=4\frac{m_{\alpha}^{2}}{m_{\mu}^{2}}+\mathrm{ln}\left(\frac{m_{\alpha}^{2}}{m_{H^{++}}^{2}}\right)+\left(1-2\frac{m_{\alpha}^{2}}{m_{\mu}^{2}}\right)\sqrt{1+4\frac{m_{\alpha}^{2}}{m_{\mu}^{2}}}\mathrm{ln}\left(\frac{\sqrt{m_{\mu}^{2}m_{H^{++}}^{-2}}+\sqrt{m_{\mu}^{2}m_{H^{++}}^{-2}+4m_{\alpha}^{2}m_{H^{++}}^{-2}}}{\sqrt{m_{\mu}^{2}m_{H^{++}}^{-2}}-\sqrt{m_{\mu}^{2}m_{H^{++}}^{-2}+4m_{\alpha}^{2}m_{H^{++}}^{-2}}}\right).$
(90)

For the case of Ti, the most stringent experimental limit comes from SINDRUM
II, i.e. $\mathrm{BR}(\mu\to e)<4.2\times 10^{-12}$ [116]. To test the
consistency of this experimental bound with NSIs of order $10^{-4}$, we
considered the NO oscillation parameters for which
$|\varepsilon_{ee}^{eL}|=8\times 10^{-4}$ is the largest allowed value from
Table II. We found that the predicted branching ratio in Eq. (89),
$\mathrm{BR}(\mu\to e)=1.6\times 10^{-13}$, is one order of magnitude smaller
than the experimental limit for this choice, assuming $m_{{H}^{++}}^{2}\approx
m_{{H}^{+}}^{2}$ again, which is consistent with the current limit of SINDRUM
II. Hence the results in Table 2 obtained for the $\mu\to e\gamma$ case seem
unaffected by the inclusion of nuclear $\mu\to e$ conversion bounds. A visible
modification of the NSI sensitivity would require an improvement of the
current experimental bound on the latter by over an order of magnitude.

##  VII Summary and outlook

We have examined the possibility of vindicating and “deconstructing” the
simplest type-II seesaw mechanism, with explicit violation of lepton number,
in which a Higgs triplet mediates neutrino mass generation in the Standard
Model [4, 5], see Fig. 1. We have first discussed the existing restrictions
from electroweak precision data, i.e. from the $S$, $T$, $U$ parameters, and
electroweak symmetry breaking, including perturbative unitarity and stability
of the vacuum, see Figs. 2, 3 and 4. Constraints from neutrino oscillations
were examined in Fig. 5 where one sees also present limits and future
sensitivities from cosmology and neutrinoless double beta decay searches.

Charged lepton flavour violation resulting from diagrams in Figs. 6 and 7 lead
to stringent restrictions, especially from $\mu\to e\gamma$. Note that, as
seen in Fig. 8, in the limit of small triplet vacuum expectation value, the
doubly-charged Higgs boson has dominant lepton number and charged lepton
flavour violating decays. The scalars mediating neutrino mass generation may
also be produced at present and future hadron colliders, see Figs. 10, 11 and
12. Indeed, we discussed current restrictions coming from collider searches.
In addition, we have discussed some of the novel experimental opportunities
associated to the scalar mediator of neutrino mass generation, extending the
discussion recently given in [6]. The decay pattern of the doubly- and singly-
charged Higgs bosons is shown in Figs. 8 and 9. Such decays could lead to new
impressive experimental signatures at high-energy hadron colliders, e.g. those
given in Eqs. (66) and (68), or in Eqs. (69) and (70). Likewise, these scalar
mediators would also be produced at lepton colliders, the cross section for
scalar production is given in Fig. 13 and Fig. 14. Especially interesting
signatures in this case are those given in Eqs. (75) and (76). For comparison,
Fig. 15 illustrates the cLFV sensitvities in terms of the scalar mass, and the
triplet seesaw VEV.

A remarkable feature of the triplet seesaw mechanism is the possibility of
full “deconstruction” free from the ambiguities that characterize the type-I
seesaw variant and which are seen neatly in the Casas-Ibarra description
[117]. The latter simply reflect the large number of parameters characterizing
the type-I seesaw mixing matrix, given in [4, 5].

As recently summarized in [6], one can probe neutrino oscillation physics at
collider energies through the pattern of the type-II scalar boson mediator
decays. This is seen, for example, in Figs. 16, 17 and 18, where one can see
that the doubly-charged Higgs boson has lepton-flavour-violating decay rates,
comparable to the lepton-flavour-conserving ones.

Indeed, Figs. 20 and 21 clearly show that charged lepton flavour violation
could be observed first as a high-energy phenomenon, as the corresponding
signal cross section can be sizeable even when low-energy charged lepton
flavour violating processes, such as $\mu\to e\gamma$, have very small rates.
High-energy probes clearly complement the efforts towards charged lepton
flavour violation searches in other high-intensity facilities such as BaBar.

The simplest seesaw mechanism leads to promising signatures not only at hadron
colliders such as LHC and FCC, but also at future $e^{+}e^{-}$ colliders such
as ILC, CLIC or CEPC in China. Indeed, the type-II seesaw illustrates how
high-energy signatures can be complementary to neutrino oscillation studies.
For example, Fig. 19 shows how the rates for multilepton final-state events
coming from pair-production of the doubly-charged Higgs boson may be used as a
probe of the neutrino mass ordering, currently not yet robustly determined by
oscillation experiments.

The results found here illustrate the complementarity and interplay of the
high-energy and high-intensity frontiers in particle physics, providing
encouragement for dedicated simulation studies in order to evaluate the
potential of these proposed facilities in probing the neutrino sector.

We have also examined the possibility of having sizeable non-standard neutrino
interactions (see Fig. 22) given the current constraints from lepton flavour
violation searches. We saw in Figs. 23, 24 and 25 that effects in neutrino
propagation associated to non-standard interactions are below the current
detectability, though they could lie above the $10^{-4}$ level, e.g.
$|\varepsilon_{ee}^{eL}|>1\times 10^{-4}$ providing a real challenge for the
future.

###### Acknowledgements.

Work of J.V. is supported by the Spanish grants PID2020-113775GB-I00
(AEI/10.13039/501100011033) and PROMETEO/2018/165 (Generalitat Valenciana) and
by CONACYT-Mexico under grant A1-S-23238. O. G. M. has been supported by SNI
(Sistema Nacional de Investigadores). S.M. has been supported by KIAS
Individual Grants (PG086001) at Korea Institute for Advanced Study. X.J.X has
been supported in part by the National Natural Science Foundation of China
under grant No. 12141501.

##  Appendix A Renormalisation group equations

In our work we have used the package SARAH [120] to perform the
renormalization group analysis of the type-II seesaw model. The $\beta$
function of a given parameter $c$ is given by,

$\displaystyle\mu_{r}\frac{dc}{d\mu_{r}}\equiv\beta_{c}=\frac{1}{16\pi^{2}}\beta_{c}^{(1)}+\frac{1}{(16\pi^{2})^{2}}\beta_{c}^{(2)}\,.$

where $\mu_{r}$ is the running scale and $\beta_{c}^{(1)}$, $\beta_{c}^{(2)}$
are the one-loop and two-loop renormalization group terms. Below we list only
one-loop RGEs for various quartic couplings and Yukawa couplings $Y_{\Delta}$,
$y_{t}$.

Quartic couplings:

$\displaystyle\beta_{\lambda}^{(1)}$
$\displaystyle=\frac{27}{50}g_{1}^{4}+\frac{9}{5}g_{1}^{2}g_{2}^{2}+\frac{9}{2}g_{2}^{4}-\left(\frac{18}{5}g_{1}^{2}+18g_{2}^{2}\right)\lambda+6\lambda^{2}+12\lambda_{1}^{2}+12\lambda_{1}\lambda_{4}+5{\lambda_{4}}^{2}$
$\displaystyle+12\lambda y_{t}^{2}-24y_{t}^{4}\,,$ (91)
$\displaystyle\beta_{\lambda_{1}}^{(1)}$
$\displaystyle=\frac{27}{25}g_{1}^{4}-\frac{18}{5}g_{1}^{2}g_{2}^{2}+6g_{2}^{4}-\left(\frac{9}{2}g_{1}^{2}+\frac{33}{2}g_{2}^{2}\right)\lambda_{1}+3\lambda\lambda_{1}+\lambda\lambda_{4}+4\lambda_{1}^{2}$
$\displaystyle+16\lambda_{2}\lambda_{1}+12\lambda_{3}\lambda_{1}+{\lambda_{4}}^{2}+6\lambda_{2}\lambda_{4}+2\lambda_{3}\lambda_{4}+6\lambda_{1}y_{t}^{2}+4\lambda_{1}\text{Tr}\Big{(}{Y_{\Delta}^{\dagger}Y_{\Delta}}\Big{)}\,,$
(92) $\displaystyle\beta_{\lambda_{4}}^{(1)}$
$\displaystyle=\frac{36}{5}g_{1}^{2}g_{2}^{2}-\left(\frac{9}{2}g_{1}^{2}+\frac{33}{2}g_{2}^{2}\right)\lambda_{4}+\lambda\lambda_{4}+8\lambda_{1}\lambda_{4}+4{\lambda_{4}}^{2}+4\lambda_{2}\lambda_{4}$
$\displaystyle+8\lambda_{3}\lambda_{4}+6\lambda_{4}y_{t}^{2}+4\lambda_{4}\mbox{Tr}\Big{(}{Y_{\Delta}^{\dagger}Y_{\Delta}}\Big{)}\,,$
(93) $\displaystyle\beta_{\lambda_{2}}^{(1)}$
$\displaystyle=\frac{108}{50}g_{1}^{4}-\frac{72}{10}g_{1}^{2}g_{2}^{2}+15g_{2}^{4}-\left(\frac{36}{5}g_{1}^{2}+24g_{2}^{2}\right)\lambda_{2}+2\lambda_{1}^{2}+2\lambda_{1}\lambda_{4}$
$\displaystyle+28\lambda_{2}^{2}+24\lambda_{2}\lambda_{3}+6{\lambda_{3}}^{2}-4\mbox{Tr}\Big{(}{Y_{\Delta}^{\dagger}Y_{\Delta}Y_{\Delta}^{\dagger}Y_{\Delta}}\Big{)}+8\lambda_{2}\mbox{Tr}\Big{(}{Y_{\Delta}^{\dagger}Y_{\Delta}}\Big{)}\,,$
(94) $\displaystyle\beta_{\lambda_{3}}^{(1)}$
$\displaystyle=\frac{144}{10}g_{1}^{2}g_{2}^{2}-6g_{2}^{4}+{\lambda_{4}}^{2}-\left(\frac{36}{5}g_{1}^{2}+24g_{2}^{2}\right)\lambda_{3}+24\lambda_{2}\lambda_{3}+18{\lambda_{3}}^{2}$
(95)
$\displaystyle+8\lambda_{3}\mbox{Tr}\Big{(}{Y_{\Delta}^{\dagger}Y_{\Delta}}\Big{)}+4\mbox{Tr}\Big{(}{Y_{\Delta}^{\dagger}Y_{\Delta}Y_{\Delta}^{\dagger}Y_{\Delta}}\Big{)}\,.$
(96)

Yukawa couplings:

$\displaystyle\beta_{Y_{\Delta}}^{(1)}$
$\displaystyle=\frac{1}{10}\Big{(}60\,{Y_{\Delta}Y_{\Delta}^{\dagger}Y_{\Delta}}+Y_{\Delta}\Big{(}20\mbox{Tr}\Big{(}{Y_{\Delta}^{\dagger}Y_{\Delta}}\Big{)}-9\Big{(}5g_{2}^{2}+g_{1}^{2}\Big{)}\Big{)}\Big{)},$
(97) $\displaystyle\beta_{y_{t}}^{(1)}$
$\displaystyle=\frac{3}{2}y_{t}^{3}+y_{t}\Big{(}3y_{t}^{2}-8g_{3}^{2}-\frac{17}{20}g_{1}^{2}-\frac{9}{4}g_{2}^{2}\Big{)},$
(98)
$\displaystyle\beta_{\kappa}^{(1)}=-\frac{27}{10}g_{1}^{2}\kappa-\frac{21}{2}g_{2}^{2}\kappa+\kappa\lambda+4\kappa\lambda_{1}+6\kappa\lambda_{4}+6\kappa
y_{t}^{2}+2\kappa\mbox{Tr}\Big{(}{Y_{\Delta}^{\dagger}Y_{\Delta}}\Big{)}.$
(99)

##  Appendix B Calculation of the oblique parameters $S$, $T$ and $U$
parameters

In this appendix, we present the explicit expressions used to evaluate the
oblique parameters ($S$, $T$, $U$) for type-II seesaw. Similar expressions
have already been presented in Ref. [118]. Since there was a typo in the $U$
parameter in Ref. [118], we re-drive the expressions using the result in Ref.
[119] which computed the oblique parameters for generic scalar multiplets
assuming their VEVs are negligibly small. Applying their results to the Higgs
triplet, we obtain:

$\displaystyle S$ $\displaystyle=-\frac{1}{3\pi}\sum_{k=-1}^{1}\left[k\ln
m_{k}^{2}+6(k-s_{W}^{2}Q_{k})^{2}\xi\left(\frac{m_{k}^{2}}{m_{Z}^{2}},\frac{m_{k}^{2}}{m_{Z}^{2}}\right)\right],$
(100) $\displaystyle T$ $\displaystyle=\frac{1}{16\pi
s_{W}^{2}}\sum_{k=-1}^{1}\left[(2+k-k^{2})\eta\left(\frac{m_{k}^{2}}{m_{W}^{2}},\frac{m_{k-1}^{2}}{m_{W}^{2}}\right)\right],$
(101) $\displaystyle U$
$\displaystyle=\frac{1}{6\pi}\ln\left(\frac{m_{H^{\pm}}^{4}}{m_{H^{\pm\pm}}^{2}m_{H^{0}}^{2}}\right)$
$\displaystyle+\frac{1}{\pi}\sum_{k=-1}^{1}\left[(-2-k+k^{2})\xi\left(\frac{m_{k}^{2}}{m_{W}^{2}},\frac{m_{k-1}^{2}}{m_{W}^{2}}\right)+2(k-s_{W}^{2}Q_{k})^{2}\xi\left(\frac{m_{k}^{2}}{m_{Z}^{2}},\frac{m_{k}^{2}}{m_{Z}^{2}}\right)\right],$
(102)

where

$m_{k}=\\{m_{H^{0}},\ m_{H^{\pm}},\ m_{H^{\pm\pm}}\\},\ Q_{k}=\\{0,\ 1,\ 2\\}\
\ \text{for}\ k=\\{-1,\ 0,\ 1\\}.$ (103)

Note that in Eqs. (101) and (102) when $k=-1$, although $m_{k-1}$ is
undefined, terms containing $m_{k-1}$ always vanish. The $\eta$ function is
defined as

$\eta(x,y)=x+y-\frac{2xy}{x-y}\log\left(\frac{x}{y}\right).$ (104)

For small mass splitting, it approximately reduces to

$\eta(x,y)\approx\frac{(y-x)^{2}}{3x},\ \ {\rm for}\ \frac{|y-x|}{x}\ll
1\thinspace.$ (105)

The full form of the $\xi$ function can be found in Refs. [118] and [119].
Since we are only interested in cases where the new particles are heavy and
the mass splitting is small, we take the following approximate form:

$\xi(x,y)\approx\frac{1}{15}\frac{1}{4x-1}-\left(\frac{2(y-x)}{15}+\frac{1}{21}\right)\left(\frac{1}{4x-1}\right)^{2},\
\ {\rm for}\ \frac{|y-x|}{x}\ll 1\ll 4x-1\thinspace.$ (106)

Using the above approximate forms and taking only the dominant contributions,
we obtain the results in Eqs. (34)-(36).

##  Appendix C Relevant definitions for decay-width calculations

Here we present the analytical forms of the quantities introduced when
computing the triplet scalar decay branching ratios in Sec. III.5. The
relevant formulae are given below.

$\displaystyle\xi_{H^{\pm}W^{\mp}\hat{\varphi}}=\cos\alpha\sin\beta_{\pm}-\sqrt{2}\sin\alpha\cos\beta_{\pm},\leavevmode\nobreak\
\sin\alpha\sin\beta_{\pm}+\sqrt{2}\cos\alpha\cos\beta_{\pm},\leavevmode\nobreak\
\sin\beta_{0}\sin\beta_{\pm}+\sqrt{2}\cos\beta_{0}\cos\beta_{\pm},$
$\displaystyle\text{ for }\hat{\varphi}=h^{0},H^{0}\text{ and }A$ (107)

$\displaystyle\lambda_{H^{0}hh}=\frac{1}{4v_{\Phi}^{3}}\Big{[}2v_{\Delta}\left\\{-2M_{\Delta}^{2}+v_{\Phi}^{2}(\lambda_{1}+\lambda_{4})\right\\}\cos^{3}\alpha+v_{\Phi}^{3}\left\\{-3\lambda+4(\lambda_{1}+\lambda_{4})\right\\}\cos^{2}\alpha\sin\alpha$
$\displaystyle+4v_{\Delta}\big{\\{}2M_{\Delta}^{2}+v_{\Phi}^{2}(3\lambda_{2}+3\lambda_{3}-\lambda_{1}-\lambda_{4})\big{\\}}\cos\alpha\sin^{2}\alpha-2v_{\Phi}^{3}(\lambda_{1}+\lambda_{4})\sin^{3}\alpha\Big{]}.$
(108)

$\displaystyle\lambda(x,y)=(1-x-y)^{2}-4xy\leavevmode\nobreak\ ,$ (109)
$\displaystyle\beta(x)=\sqrt{\lambda(x,x)}=\sqrt{1-4x}\leavevmode\nobreak\ ,$
(110) $\displaystyle
F(x)=\frac{3(1-8x+20x^{2})}{\sqrt{4x-1}}\cos^{-1}\left(\frac{3x-1}{2x^{3/2}}\right)-\frac{(1-x)(2-13x+47x^{2})}{2x}-\frac{3}{2}(1-6x+4x^{2})\log
x\leavevmode\nobreak\ ,$ (111) $\displaystyle
G(x,y)=\frac{1}{12y}\Bigg{[}2\left(-1+x\right)^{3}-9\left(-1+x^{2}\right)y+6\left(-1+x\right)y^{2}-6\left(1+x-y\right)y\sqrt{-\lambda(x,y)}\Bigg{\\{}\tan^{-1}\left(\frac{1-x+y}{\sqrt{-\lambda(x,y)}}\right)$
$\displaystyle+\tan^{-1}\left(\frac{1-x-y}{\sqrt{-\lambda(x,y)}}\right)\Bigg{\\}}-3\left(1+\left(x-y\right)^{2}-2y\right)y\log
x\Bigg{]}\leavevmode\nobreak\ ,$ (112) $\displaystyle
H(x,y)=\frac{\tan^{-1}\left(\frac{1-x+y}{\sqrt{-\lambda(x,y)}}\right)+\tan^{-1}\left(\frac{1-x-y}{\sqrt{-\lambda(x,y)}}\right)}{4x\sqrt{-\lambda(x,y)}}\Big{\\{}-3x^{3}+(9y+7)x^{2}-5(1-y)^{2}x+(1-y)^{3}\Big{\\}}$
$\displaystyle+\frac{1}{24xy}\Big{\\{}(-1+x)(2+2x^{2}+6y^{2}-4x-9y+39xy)-3y(1-3x^{2}+y^{2}-4x-2y+6xy)\log
x\Big{\\}}\leavevmode\nobreak\ .$ (113)

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