aainstitutetext: Instituto Superior de Engenharia de Lisboa, Instituto
Politécnico de Lisboa 1959-007 Lisboa, Portugalbbinstitutetext: Centro de
Física Teórica e Computacional, Faculdade de Ciências, Universidade de Lisboa,
Campo Grande, Edifício C8 1749-016 Lisboa, Portugalccinstitutetext:
Departamento de Física and CFTP, Instituto Superior Técnico, Universidade de
Lisboa, Lisboa, Portugal

# The hidden side of scalar-triplet models with spontaneous CP violation

P.M. Ferreira c,b B.L. Gonçalves c and F.R. Joaquim<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Scalar triplet extensions of the Standard Model provide an interesting
playground for the explanation of neutrino mass suppression through the type-
II seesaw mechanism. Propelled by the possible connections with leptonic CP
violation, we explore under which conditions spontaneous CP violation can
arise in models with extra scalar triplets. The minimal model satisfying such
conditions requires adding two such triplets to the SM field content. For this
model, the scalar mass spectrum in both the CP-conserving and CP-violating
scenarios is studied. In the former case, a decoupling limit for the new
scalars can be achieved, while this is not the case when CP is spontaneously
broken. In particular, we show that the existence of two light neutral scalars
with masses below a few tenths of GeVs is unavoidable in the CP-violating
case. Using matrix theory theorems, we derive upper bounds for the masses of
those light scalars and briefly examine whether they can still be
experimentally viable. Other interesting features of the scalar mass spectrum
are discussed as, e.g., the existence of relations among the charged and
neutral scalar masses.

##  1 Introduction

With the discovery of the Higgs boson by the ATLAS and CMS Collaborations
Aad:2012tfa ; Chatrchyan:2012xdj  at the Large Hadron Collider (LHC), the
particle content of the Standard Model (SM) has been confirmed. Though the SM
is in remarkable agreement with most experimental data delivered so far, there
are still important questions which remain unanswered within the standard
theory: among others, the origin of fermion masses; the observed
matter–antimatter asymmetry; the existence of dark matter; and even the recent
confirmation of a 4.2$\sigma$ discrepancy on the muon anomalous magnetic
moment. There is a plethora of SM extensions which provide answers for one or
several of these questions, some of them relying on enlarging the SM field
content with extra spin-zero fields. Popular models include scalar gauge
singlets McDonald:1993ex ; Barger:2008jx , or a second scalar doublet,
yielding the two Higgs Doublet Model (2HDM), first introduced by T. D. Lee in
1973 Lee:1973iz . In both cases, spontaneous CP violation (SCPV) may be
realized, suitable dark-matter candidates are provided, and the dynamics of
the first-order electroweak phase transition may be altered. Another scenario
much studied in the literature is the scalar-triplet model where the usual
hypercharge $Y=1$ SM Higgs doublet is complemented with a scalar SU(2) triplet
with hypercharge $Y=2$ Konetschny:1977bn . The main motivation for considering
such possibility is that small neutrino masses may be generated via the type-
II seesaw mechanism Mohapatra:1979ia ; Magg:1980ut ; Cheng:1980qt ;
Schechter:1980gr ; Lazarides:1980nt . Although some versions of the model can
also accommodate dark matter candidates Kanemura:2012rj , SCPV is not possible
in the case with only one triplet. The possibility of dynamical CP violation
triggered by complex vacuum expectation values (VEVs) of neutral scalar-
triplet components is interesting in the sense that such CP-violating effects
can, in principle, be communicated to the lepton sector via Yukawa
interactions. This would, for instance, lead to a connection between Dirac and
Majorana leptonic CP violation (for a review see Ref. Branco:2011zb ) and the
fundamental properties of the vacuum Branco:2012vs . In more elaborated
scenarios, relations can also be established with the problem of the matter-
antimatter asymmetry in the Universe in the context of leptogenesis induced by
heavy triplet decays into leptons Ma:1998dx ; Felipe:2013kk .

SM extensions with scalar triplets have a much richer spectrum, including new
neutral, charged and doubly-charged scalars, opening up the possibility for
new experimental searches. On the other hand, any multiscalar model must
account for the fact that the LHC has established that the 125 GeV Higgs boson
has properties similar, within current experimental uncertainties, to those
expected for the SM Higgs particle (see for instance Aad:2015zhl ;
Khachatryan:2016vau ), and also for the non-observation, thus far, of other
scalars. These requirements can be met in several ways – for instance, in the
inert 2HDM there exists a scalar with tree-level couplings identical to those
of the SM Higgs boson, the extra scalar particles being inert, i.e. they do
not couple to fermions or gauge bosons at tree level. This feature stems from
imposing a discrete $Z_{2}$ symmetry which is unbroken by the vacuum,
guaranteeing the existence of a suitable dark matter candidate. Another
possibility is to choose (though without fine tuning) the parameters of a
certain model such that the extra scalars are very heavy. For instance, within
the 2HDM, where sum rules for the scalar-gauge boson couplings exist, such
regime could explain in a fairly natural way why the heavier CP-even state
would have escaped detection at the LHC in searches for resonances in $Z$
boson pairs. This decoupling limit is achieved due to one of the quadratic
coefficients of the scalar potential being unconstrained by the minimisation
conditions which determine the CP-conserving vacuum of the theory. However, it
has been shown that a 2HDM with a CP-violating minimum does not exhibit a
decoupling limit Nebot:2020niz . In that case, there is an upper bound for the
extra scalar masses resulting from unitarity and perturbativity constraints
which limit the magnitude of the quartic couplings.

In this work, we investigate whether SCPV violation may occur in models with
extra scalar triplets. One possible candidate would be the Georgi-Machacek
model Georgi:1985nv , which includes a real and a complex triplet with
hypercharges $Y=1$ and $Y=2$, respectively. Although much work has been done
in this framework Chanowitz:1985ug ; Gunion:1989ci ; Gunion:1990dt ;
Cheung:1994rp ; Aoki:2007ah ; Logan:2010en ; Chiang:2012cn ; Englert:2013wga ;
Chiang:2013rua ; Kanemura:2014bqa ; Hartling:2014zca ; Chiang:2014bia ;
Degrande:2015xnm ; Chang:2017niy ; Keeshan:2018ypw ; Ghosh:2019qie ;
Chiang:2018cgb ; Ismail:2020zoz ; Hartling:2014xma , SCPV violation is not
possible in it. Motivated by the possible connection with leptonic CP
violation, in this work we consider instead the SM extended with two $Y=2$
complex triplets which we dub as the two-scalar-triplet model (2STM). It turns
out that this is the minimal triplet model which meets the conditions for SCPV
to occur and lead to interesting leptonic CP-violating effects which are being
probed, for instance, at neutrino oscillation experiments. Moreover, we will
see that the 2STM reveals itself as an instructive example of how requiring
that the vacuum breaks CP can restrain the decoupling of extra scalar
particles by introducing novel features, namely the presence of very light
scalars. It also shows how decisive the mass spectrum analysis can be for
model-building scenarios with extended scalar sectors.

This paper is organized as follows: in Section 2, after recapping the main
features of the SM extended with one scalar triplet, we introduce the 2STM.
The possibility of SCPV is considered and we show that it cannot occur with a
single triplet field, but is possible when two triplets are added. For the
sake of simplicity, we consider the 2STM endowed with a (softly-broken)
continuous global symmetry, which turns out to be the minimal model with two
triplets where SCPV can be realised. In Section 3 we study in detail the
scalar mass spectrum of the 2STM for both CP-conserving and CP-violating
minima of the scalar potential. By using matrix-theory theorems we obtain some
upper bounds on the scalar masses and investigate whether decoupling is
possible for both types of minima. We conclude that the model is non-
decoupling when SCPV occurs. These results are then confirmed by a numerical
analysis presented in Section 4, where several experimental constraints are
considered. Finally, in Section 5, we draw our final conclusions.

##  2 SCPV in the two-scalar-triplet model

Before introducing the 2STM, which will be the main focus of this work, it is
instructive to revisit the most relevant aspects of the one-triplet model,
sometimes referred to as the scalar-triplet model (STM). Its scalar sector
contains the hypercharge one ($Y=1$) SM Higgs doublet $\Phi$, and a $Y=2$
scalar triplet $\Delta$. The field components of the scalar multiplets $\Phi$
and $\Delta$ are denoted with the usual notation,

$\Phi=\begin{pmatrix}\phi^{+}\\\ \phi^{0}\end{pmatrix}\,\ ,\,\
\Delta=\begin{pmatrix}\delta^{+}/\sqrt{2}&\delta^{++}\\\
\delta^{0}&-\delta^{+}/\sqrt{2}\end{pmatrix}\,,$ (1)

where the $\text{SU}(2)$ matrix representation for $\Delta$ has been
considered. With this scalar field content, the most general renormalizable
SU(2)${}_{L}\times$U(1)Y gauge-invariant scalar potential is given by

$\begin{split}V_{S}=&\,m^{2}\Phi^{\dagger}\Phi+M^{2}\text{Tr}(\Delta^{\dagger}\Delta)+(\mu\Phi^{T}i\tau_{2}\Delta^{\dagger}\Phi+\text{H.c.})\\\
+&\,\lambda_{1}(\Phi^{\dagger}\Phi)^{2}+\lambda[\text{Tr}(\Delta^{\dagger}\Delta)]^{2}+\tilde{\lambda}\text{Tr}[(\Delta^{\dagger}\Delta)^{2}]+\lambda^{\prime}(\Phi^{\dagger}\Phi)\text{Tr}(\Delta^{\dagger}\Delta)+\hat{\lambda}\Phi^{\dagger}\Delta\Delta^{\dagger}\Phi\,.\end{split}$
(2)

Note that the trilinear $\mu$ term softly breaks a global $\text{U}(1)$
transformation of the form $\Phi\rightarrow e^{i\alpha}\Phi$ and
$\Delta\rightarrow e^{i\alpha^{\prime}}\Delta$. It is straightforward to see
that, without loss of generality, the $\mu$ coefficient can always be made
real via rephasings of the doublet and triplet scalars. Electroweak symmetry
breaking (EWSB) occurs if the parameters of the potential are such that the
neutral components of $\Phi$ and $\Delta$ acquire vacuum expectation values
(VEVs), which we represent as

$\displaystyle\langle\phi^{0}\rangle=\frac{v}{\sqrt{2}}\,,\,\langle\delta^{0}\rangle=\frac{ue^{i\theta}}{\sqrt{2}}\,,\,v^{2}+2u^{2}\simeq(246\,\text{GeV})^{2}\,,$
(3)

where we have used the freedom to redefine $\Phi$ such that
$\langle\phi^{0}\rangle$ is real. The last relation above ensures consistent
EWSB, i.e. this vacuum yields correct values for the masses of the electroweak
gauge bosons. Notice, however, that unlike models with an arbitrary number of
Higgs doublets, in the triplet model the electroweak precision parameter
$\rho$ deviates from unity already at tree-level. As such, compatibility with
electroweak precision data forces the triplet VEV $u$ to be, at most,
$8\,\text{GeV}$ Arhrib:2011uy ; Arhrib:2011vc ; Kanemura:2012rj ; Aoki:2012yt
; Aoki:2012jj . We mention in passing that the STM can have charge-breaking
vacua, some of which can coexist with minima of the type of Eq. (3). An
analytical and numerical analysis of the STM vacuum structure was performed in
Ferreira:2019hfk .

Regarding the scalar-fermion couplings, the doublet $\Phi$ couples to fermions
in exactly the same way as in the SM, while the triplet $\Delta$ can couple to
the SM lepton doublets $L_{\alpha}$ as

$\mathcal{L}_{\rm
Yuk}=\mathbf{Y}_{\alpha\beta}^{\Delta}\,L_{\alpha}^{T}\mathcal{C}^{\dagger}i\tau_{2}\Delta\,L_{\beta}+{\rm
H.c.}\,,\,L_{\alpha}=\begin{pmatrix}\nu_{L}\\\
\ell_{L}\end{pmatrix}_{\alpha}\,,\,(\alpha,\beta)=e,\mu,\tau\,.$ (4)

Here, $\mathcal{C}$ is the charge-conjugation matrix, $\tau_{2}$ is the
complex Pauli matrix and $\mathbf{Y}^{\Delta}$ is a general $3\times 3$
complex (symmetric) matrix. The triplet VEV $u$, which is induced upon EWSB,
generates a neutrino Majorana mass matrix $m_{\nu}=\mathbf{Y}_{\Delta}u/2$.
Small neutrino masses may result from either suppressed Yukawa couplings
and/or a tiny $u$ VEV. In the latter case, and considering $\mathcal{O}(1)$
Yukawa couplings, one may get $m_{\nu}\sim u\sim\mu v^{2}/M^{2}\sim{\rm
0.1~{}eV}$ by considering $\mu v/M^{2}\sim 10^{-13}$ which, for $\mu\sim v$,
implies $M\sim 10^{9}$ GeV. However, if $\mu\ll v$ (which would be natural in
the t’Hooft sense in the context of a broken softly-broken lepton number
symmetry), the scale $M$ can be lowered to e.g. the electroweak scale.

Let us now investigate the possibility of SCPV in the STM. As already
mentioned, it is always possible to render the $\mu$ parameter real through
field redefinitions, making $V_{S}$ real and the Lagrangian CP conserving.
Then, the non-trivial extremisation conditions are:

$\displaystyle
v\left[2m^{2}+2\lambda_{1}v^{2}+(\lambda^{\prime}+\hat{\lambda})v^{2}-2\sqrt{2}\mu
u\cos{\theta}\right]$ $\displaystyle=0\,,$ (5) $\displaystyle-\sqrt{2}\mu
v^{2}+u\left[2M^{2}+(\lambda^{\prime}+\hat{\lambda})v^{2}+2(\lambda+\tilde{\lambda})u^{2}\right]\cos{\theta}$
$\displaystyle=0\,,$ $\displaystyle\mu\,vu\sin{\theta}$ $\displaystyle=0\,.$

It is straightforward to see that, for the cases of interest with $\mu,v,u\neq
0$, the only possible solution for the last equation above is $\theta=n\pi$.
Thus, CP cannot be spontaneously broken in a model with a single $\Delta$ and,
therefore, leptonic CP violation can only be accounted for with complex Yukawa
couplings. With these minimization conditions, we find that the STM has a
richer scalar spectrum than the SM – it includes two CP-even scalars, $h$ (the
125 GeV SM-like Higgs boson) and $H$, a CP-odd one $A$, a charged scalar
$H^{\pm}$ and a doubly-charged, $H^{\pm\pm}$. The $h$ and $H$ masses are the
eigenvalues of the matrix

$\mathcal{M}\,=\,\left(\begin{array}[]{cc}2\lambda_{1}v^{2}&-\dfrac{2u}{v}\,M_{\Delta}^{2}+(\lambda^{\prime}+\hat{\lambda})vu\\\
-\dfrac{2u}{v}\,M_{\Delta}^{2}+(\lambda^{\prime}+\hat{\lambda})vu&M_{\Delta}^{2}+2(\lambda+\tilde{\lambda})u^{2}\end{array}\right)\,,$
(6)

while for $A$, $H^{\pm}$ and $H^{\pm\pm}$ we have

$\displaystyle m^{2}_{A}$
$\displaystyle=M_{\Delta}^{2}\left(1+\dfrac{4u^{2}}{v^{2}}\right)\;\;,\;\;m^{2}_{+}=\left(M_{\Delta}^{2}-\dfrac{\hat{\lambda}}{4}v^{2}\right)\left(1+\dfrac{2u^{2}}{v^{2}}\right)\;\;,\;\;m^{2}_{++}=M_{\Delta}^{2}-u^{2}\tilde{\lambda}-\dfrac{\hat{\lambda}}{2}v^{2}\,,$
(7)

where we have defined $M_{\Delta}^{2}$

$M_{\Delta}^{2}\,\equiv\,\frac{v^{2}\,\mu}{\sqrt{2}\,u}\,.$ (8)

It is then clear that, except for $h$, all scalars may have masses much larger
than $m_{h}$ if $M_{\Delta}$ is very large. It is also possible to choose a
region of parameter space such that the matrix of Eq. (6) is close to diagonal
– this can be achieved by taking $|\mu|\ll v$ and remembering that $u\ll v$,
so that one obtains $uM_{\Delta}^{2}/v\ll v^{2}$. Thus, in this situation,
there is almost no mixing between the doublet and the triplet, and the $h$
scalar has couplings to gauge bosons and fermions almost identical to those of
the SM Higgs. At the same time, since the extra scalars stem from the triplet,
their interactions with gauge bosons and fermions are heavily suppressed
Arhrib:2011uy ; Kanemura:2012rj  111This is obvious for the quark Yukawa
interactions since only the doublet couples to quarks. For the leptons, the
triplet Yukawa interactions of Eq. (4) induce interactions between the extra
scalars and leptons, but in this limit those too are suppressed.. Thus the STM
is found to have a decoupling limit for this region of parameter space.

We will now investigate whether the aforementioned properties of the STM still
hold if one introduces a second triplet field. In particular, we wish to
investigate the possibility of SCPV. In the 2STM the scalar sector includes
the usual doublet $\Phi$, as well as two $Y=2$ triplets $\Delta_{1,2}$ which,
following the conventions in (1), we represent by

$\Delta_{1,2}=\begin{pmatrix}\delta^{+}_{1,2}/\sqrt{2}&\delta^{++}_{1,2}\\\
\delta^{0}_{1,2}&-\delta^{+}_{1,2}/\sqrt{2}\end{pmatrix}\,.$ (9)

The CP-conserving scalar potential invariant under the
SU(2)${}_{L}\times$U(1)L gauge symmetry and the global U(1) symmetry
$\Phi\rightarrow e^{i\alpha}\Phi$ and $\Delta_{1,2}\rightarrow
e^{i\alpha_{1,2}}\Delta_{1,2}$ is

$\begin{split}V_{\rm
U(1)}=&\,m^{2}\Phi^{\dagger}\Phi+M_{11}^{2}\text{Tr}(\Delta_{1}^{\dagger}\Delta_{1})+M_{22}^{2}\text{Tr}(\Delta_{2}^{\dagger}\Delta_{2})+\lambda_{0}(\Phi^{\dagger}\Phi)^{2}\\\
&+{\lambda}_{1}[\text{Tr}(\Delta_{1}^{\dagger}\Delta_{1})]^{2}+{\lambda}_{2}[\text{Tr}(\Delta_{2}^{\dagger}\Delta_{2})]^{2}+{\lambda}_{21}\text{Tr}(\Delta_{2}^{\dagger}\Delta_{2})\text{Tr}(\Delta_{1}^{\dagger}\Delta_{1})+{\lambda}_{12}\text{Tr}(\Delta_{1}^{\dagger}\Delta_{2})\text{Tr}(\Delta_{2}^{\dagger}\Delta_{1})\\\
&+{\tilde{\lambda}}_{1}\text{Tr}[(\Delta_{1}^{\dagger}\Delta_{1})^{2}]+{\tilde{\lambda}}_{2}\text{Tr}[(\Delta_{2}^{\dagger}\Delta_{2})^{2}]+{\tilde{\lambda}}_{21}\text{Tr}(\Delta_{2}^{\dagger}\Delta_{2}\Delta_{1}^{\dagger}\Delta_{1})+{\tilde{\lambda}}_{12}\text{Tr}(\Delta_{1}^{\dagger}\Delta_{2}\Delta_{2}^{\dagger}\Delta_{1})\\\
&+{\lambda}_{1}^{\prime}\text{Tr}(\Delta_{1}^{\dagger}\Delta_{1})\Phi^{\dagger}\Phi+{\lambda}_{2}^{\prime}\text{Tr}(\Delta_{2}^{\dagger}\Delta_{2})\Phi^{\dagger}\Phi+{\hat{\lambda}}_{1}\Phi^{\dagger}\Delta_{1}\Delta_{1}^{\dagger}\Phi+{\hat{\lambda}}_{2}\Phi^{\dagger}\Delta_{2}\Delta_{2}^{\dagger}\Phi\end{split}$
(10)

where all parameters are real. To avoid unwanted massless Goldstone bosons
arising upon spontaneous breaking of the exact global U(1) symmetry, we
introduce the most general soft symmetry-breaking potential

$\displaystyle V_{\rm
SB}=M_{12}^{2}[\text{Tr}(\Delta_{1}^{\dagger}\Delta_{2})+\text{Tr}(\Delta_{2}^{\dagger}\Delta_{1})]+(\mu_{1}\Phi^{T}\mathrm{i}\tau_{2}\Delta_{1}^{\dagger}\Phi+\mu_{2}\Phi^{T}\mathrm{i}\tau_{2}\Delta_{2}^{\dagger}\Phi+\text{H.c.})\,,$
(11)

with real $M_{1,2}^{2}$ and $\mu_{1,2}$, to ensure CP invariance. Bear in mind
that it would be possible to write a fully symmetric potential with only part
of those soft-breaking terms: for instance, one could set $M_{12}=0$ and
$\mu_{2}=0$ by considering the symmetry $\Phi\rightarrow i\Phi$ and
$\Delta_{1}\rightarrow-\Delta_{1}$.

As discussed above, in the STM it is not possible to obtain VEVs with a
relative complex phase. What about in the 2STM? Given that a complex VEV for
$\Phi$ can always be made real through a phase redefinition of $\Delta_{1,2}$,
we take

$\langle\Phi\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ v\end{pmatrix}\,\
,\,\ \langle\Delta_{1}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0\\\
u_{1}e^{i\theta_{1}}&0\end{pmatrix}\,\ ,\,\
\langle\Delta_{2}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0\\\
u_{2}e^{i\theta_{2}}&0\end{pmatrix}\,,$ (12)

being $v$, $u_{1}$ and $u_{2}$ real. We must still guarantee that
$v^{2}+2u^{2}\simeq(246\,\text{GeV})^{2}$, with $u^{2}=u_{1}^{2}+u_{2}^{2}$,
to ensure viable EWSB and, as before, $u$ must be below about 8 GeV to comply
with the $\rho$ parameter constraints. Let us now define
$\tan\beta=u_{2}/u_{1}$, so that $u_{1}=u\,c_{\beta}$ and
$u_{2}=u\,s_{\beta}$, where we use the simplified notation
$c_{x}\equiv\cos{x}$, $s_{x}\equiv\sin{x}$. Eq. (12) is the most general VEV
configuration for a neutral vacuum, and the minimization of the scalar
potential leads to five non-trivial conditions:

$\displaystyle
v\left[2m^{2}+2\lambda_{0}v^{2}+(\Lambda_{1}c_{\beta}^{2}+\Lambda_{2}s_{\beta}^{2})u^{2}-2\sqrt{2}u(\mu_{1}c_{\beta}c_{\theta_{1}}+\mu_{2}s_{\beta}c_{\theta_{2}})\right]$
$\displaystyle=0\,,$ (13) $\displaystyle
uc_{\beta}c_{\theta_{1}}\left[2M_{11}^{2}+\Lambda_{1}v^{2}+(2\Lambda_{3}c_{\beta}^{2}+\Lambda_{4}s_{\beta}^{2})u^{2}\right]+2M_{12}^{2}us_{\beta}c_{\theta_{2}}-\sqrt{2}\mu_{1}v^{2}$
$\displaystyle=0\,,$ $\displaystyle
us_{\beta}c_{\theta_{2}}\left[2M_{22}^{2}+\Lambda_{2}v^{2}+(\Lambda_{4}\,c_{\beta}^{2}+2\Lambda_{5}s_{\beta}^{2})u^{2}\right]+2M_{12}^{2}uc_{\beta}c_{\theta_{1}}-\sqrt{2}\mu_{2}v^{2}$
$\displaystyle=0\,,$ $\displaystyle
2M_{12}^{2}us_{\beta}s_{\theta_{2}}+uc_{\beta}s_{\theta_{1}}\left[2M_{11}^{2}+\Lambda_{1}v^{2}+(2\Lambda_{3}c_{\beta}^{2}+\Lambda_{4}s_{\beta}^{2})u^{2}\right]$
$\displaystyle=0\,,$ $\displaystyle
uv\left(\mu_{1}c_{\beta}s_{\theta_{1}}+\mu_{2}s_{\beta}s_{\theta_{2}}\right)$
$\displaystyle=0\,,$

where the $\Lambda_{i}$ coefficients are defined in terms of the dimensionless
parameters in $V_{\rm U(1)}$ as,

$\Lambda_{1}=\lambda^{\prime}_{1}+\hat{\lambda}_{1}\;\;,\;\;\Lambda_{2}=\lambda^{\prime}_{2}+\hat{\lambda}_{2}\;\;,\;\;\Lambda_{3}=\lambda_{1}+\tilde{\lambda}_{1}\;\;,\;\;\Lambda_{4}=\lambda_{21}+\lambda_{12}+\tilde{\lambda}_{21}+\tilde{\lambda}_{12}\;\;,\;\;\Lambda_{5}=\lambda_{2}+\tilde{\lambda}_{2}\,.$
(14)

Consider now the last of Eqs. (13) which, for non-zero values of the VEVs $v$
and $u$, allows for non-zero independent values for the phases $\theta_{1}$
and $\theta_{2}$. For this to happen, notice how crucial is the presence of
both soft-breaking terms $\mu_{1}$ and $\mu_{2}$ in the scalar potential. If
either of these parameters vanishes, we fall into a situation analogous to
that of the STM where SCPV cannot occur. It is also important to stress that,
from what we have just shown, SCPV can be realised in the 2STM with a softly-
broken U(1) symmetry – this is not possible in the 2HDM Ferreira:2010hy , but
it can already be achieved in a 3HDM for which only one of the doublets is
charged under the global U(1).

In the following, we will study the properties of the 2STM, especially in what
concerns its scalar mass spectrum. To do so, we need to solve the minimization
conditions for both CP-conserving and CP-violating vacua. For convenience, we
take as input parameters the $\Phi$ and $\Delta_{1,2}$ VEVs and solve Eqs.
(13) with respect to mass parameters of the scalar potential. In particular,
in the case of CP-conserving vacua ($\theta_{1}=\theta_{2}=0$), we have

$\displaystyle m^{2}$
$\displaystyle=-\lambda_{0}v^{2}-\dfrac{1}{2}(\Lambda_{1}c_{\beta}^{2}+\Lambda_{2}s_{\beta}^{2})u^{2}+\sqrt{2}u\left(\mu_{1}c_{\beta}+\mu_{2}s_{\beta}\right)\,,$
(15) $\displaystyle M_{11}^{2}$
$\displaystyle=-\dfrac{1}{2}\Lambda_{1}v^{2}-\dfrac{1}{2}(2\Lambda_{3}c_{\beta}^{2}+\Lambda_{4}s_{\beta}^{2})u^{2}+\dfrac{\sqrt{2}}{2}\dfrac{\mu_{1}v^{2}}{u}\dfrac{1}{c_{\beta}}-M_{12}^{2}t_{\beta}\,,$
$\displaystyle M_{22}^{2}$
$\displaystyle=-\dfrac{1}{2}\Lambda_{2}v^{2}-\dfrac{1}{2}(\Lambda_{4}c_{\beta}^{2}+2\Lambda_{5}s_{\beta}^{2})u^{2}+\dfrac{\sqrt{2}}{2}\dfrac{\mu_{2}v^{2}}{u}\dfrac{1}{s_{\beta}}-M_{12}^{2}\dfrac{1}{t_{\beta}}\,,$

where the compact notation $t_{\beta}\equiv\tan\beta$ has been used. Notice
that the soft-breaking parameters $\mu_{1,2}$ and $M_{12}^{2}$ will be later
considered to vary together with $u$, $\tan\beta$ and the quartic couplings of
$V_{\rm U(1)}$. Instead, for the CP-violating case with ($\theta_{1,2}\neq 0$)
we get

$\displaystyle m^{2}$
$\displaystyle=-\lambda_{0}v^{2}-\dfrac{1}{2}(\Lambda_{1}c_{\beta}^{2}+\Lambda_{2}s_{\beta}^{2})u^{2}-\sqrt{2}u\mu_{1}c_{\beta}\dfrac{s_{\theta_{1}-\theta_{2}}}{s_{\theta_{2}}}\,,$
(16) $\displaystyle M_{11}^{2}$
$\displaystyle=-\dfrac{1}{2}\Lambda_{1}v^{2}-\dfrac{1}{2}(2\Lambda_{3}c_{\beta}^{2}+\Lambda_{4}s_{\beta}^{2})u^{2}-\dfrac{\sqrt{2}}{2}\dfrac{\mu_{1}v^{2}}{u}\dfrac{1}{c_{\beta}}\dfrac{s_{\theta_{2}}}{s_{\theta_{1}-\theta_{2}}}\,,$
$\displaystyle M_{22}^{2}$
$\displaystyle=-\dfrac{1}{2}\Lambda_{2}v^{2}-\dfrac{1}{2}(\Lambda_{4}c_{\beta}^{2}+2\Lambda_{5}s_{\beta}^{2})u^{2}-\dfrac{\sqrt{2}}{2}\dfrac{\mu_{1}v^{2}}{u}\dfrac{1}{t_{\beta}s_{\beta}}\dfrac{s_{\theta_{1}}^{2}}{s_{\theta_{1}-\theta_{2}}s_{\theta_{2}}}\,,$
$\displaystyle M_{12}^{2}$
$\displaystyle=\dfrac{\mu_{1}v^{2}}{\sqrt{2}u}\dfrac{1}{s_{\beta}}\dfrac{s_{\theta_{1}}}{s_{\theta_{1}-\theta_{2}}}\;\;,\;\;\mu_{2}=-\mu_{1}\dfrac{1}{t_{\beta}}\dfrac{s_{\theta_{1}}}{s_{\theta_{2}}}\,.$

The last two equations, which stem from the $\theta_{1,2}$ extrema conditions,
determine $M_{12}^{2}$ and $\mu_{2}$ as functions of the VEVs, the soft-
breaking parameter $\mu_{1}$ and the $\theta_{1,2}$ phases. It is worth
commenting on the cases with $\theta_{1}=0$ or $\theta_{2}=0$ or
$\theta_{1}=\theta_{2}$ which, according to the above equations, seem to lead
to inconsistent results. These unphysical outcomes originate from the fact
that the minimisation conditions (16) are no longer valid in these specific
cases, as they have been deduced assuming all phases generic and non-zero. For
instance, considering the $\theta_{1}=\theta_{2}=\theta$ limit from the
beginning, Eqs. (16) are replaced by Eqs. (15) together with
$\mu_{1}=\mu_{2}=0$, where $M_{12}^{2}$ becomes now a free parameter. In other
words, the $\theta_{1}=\theta_{2}$ case corresponds to a potential with a
different symmetry, $\Phi\rightarrow e^{i\alpha}\Phi$,
$\Delta_{1,2}\rightarrow e^{i\beta}\Delta_{1,2}$. Without the $\mu_{1,2}$
terms, the VEVs for the triplets will break this symmetry and yield a massless
scalar (other than the desired gauge Goldstone boson).

Let us now look at the fermion sector of the model. As in the STM case, the
quark fields only couple to the doublet $\Phi$ in an identical manner to the
SM. The leptons also couple to $\Phi$ as in the SM, and with the two triplets
$\Delta_{1,2}$ similarly to Eq. (4). Namely, the Yukawa interactions are now

$\mathcal{L}_{\rm
Yuk}=\mathbf{Y}_{\alpha\beta}^{\Delta_{1}}\,L_{\alpha}^{T}\mathcal{C}^{\dagger}i\tau_{2}\Delta_{1}\,L_{\beta}+\mathbf{Y}_{\alpha\beta}^{\Delta_{2}}\,L_{\alpha}^{T}\mathcal{C}^{\dagger}i\tau_{2}\Delta_{2}\,L_{\beta}{\rm
H.c.}\,,\;(\alpha,\beta)=e,\mu,\tau\,.$ (17)

Notice that now the existence of two Yukawa matrices,
$\mathbf{Y}^{\Delta_{1}}$ and $\mathbf{Y}^{\Delta_{2}}$, opens up the
possibility of inducing CP violation in the leptonic sector from SCPV with
$\theta_{1}\neq\theta_{2}\neq 0$, in which case the effective neutrino mass
matrix becomes

$\displaystyle
m_{\nu}=\frac{1}{2}\left(\mathbf{Y}^{\Delta_{1}}u_{1}e^{i\theta_{1}}+\mathbf{Y}^{\Delta_{2}}u_{2}e^{i\theta_{2}}\right)\,.$
(18)

Demanding CP conservation at the Lagrangian level, implies the Yukawa matrices
to be real and leptonic CP violation may only arise from the VEV complex
phases $\theta_{1,2}$ – it is clear that only the phase difference
$\theta_{1}-\theta_{2}$ is relevant since the lepton doublets can always be
rephased as $L_{\alpha}\rightarrow e^{-i\theta_{2}/2}L_{\alpha}$. Thus, not
only both $\theta_{1,2}$ must be non-zero, but also $\theta_{1}-\theta_{2}\neq
n\pi$, if leptonic CP violation is to be originated from the vacuum. Further,
the two $\mathbf{Y}^{\Delta}$ matrices must be linearly independent, otherwise
the complex phase would be eliminated when one rotates to the mass basis. With
generic $\mathbf{Y}^{\Delta}$ matrices, though, the complex phase
$\theta_{1}-\theta_{2}$ will irreducibly appear in the lepton mixing matrix.
The situation is analogous to obtaining a complex CKM matrix in the 2HDM from
SCPV – it cannot occur if the model is endowed with $Z_{2}$ or global U(1)
symmetries, which prevent tree-level flavour changing neutral currents by
forcing each doublet to couple exclusively to fermions with the same electric
charge. It is only when such symmetries are removed, and there are two
distinct real Yukawa matrices for the interactions between each doublet and up
and down quarks, that CP violation in the CKM matrix is obtained from SCPV.
Finally, concerning the neutrino mass matrix, without any further assumptions
or symmetry principle constraining the flavour structure of
$\mathbf{Y}^{\Delta_{1,2}}$, the relation between the Dirac and Majorana CP
phases appearing in the lepton mixing matrix and the VEV phases cannot be
directly established. This may not be the case when flavour symmetries are
considered to shape the Yukawa couplings among lepton doublets and scalar
triplets.

##  3 Scalar mass spectrum of the 2STM

Extensions of the SM with scalar triplets boast a rich phenomenology due to
the existence of new neutral, charged and doubly-charged spin-0 particles.
Actually, the latter are not present when only scalar-doublet fields are added
to the SM and thus the presence of scalar triplets opens up the window to
interesting collider signatures Aaboud:2018qcu ; Sirunyan:2017ret . As already
mentioned, viable EWSB requires $\sqrt{v^{2}+2u^{2}}\simeq 246\,\text{GeV}$.
Furthermore, constraints on the value of the $\rho$ parameter impose
$u\lesssim 8\,\text{GeV}$. Since $u\ll v$, we find it convenient to introduce
a small parameter $\varepsilon$ defined as:

$\varepsilon\equiv\frac{u}{v}\lesssim 3.3\times 10^{-2}\,.$ (19)

In the 2STM there are six neutral, three charged and two doubly-charged scalar
mass eigenstates, of which one neutral and one charged are massless,
corresponding to the SM Goldstone bosons. One of the physical neutral scalars
is expected to be the 125 GeV Higgs boson discovered at LHC. Taking as
examples the 2HDM and the one-triplet model, one expects that some of the new
scalars (or even all) can be made to decouple from the SM electroweak sector.
In this section, we aim at studying the general properties of the neutral and
charged scalar mass spectra for both CP-conserving and CP-violating vacuum
configurations of the 2STM. We are especially interested in verifying if the
model exhibits decoupling features or whether, on the other hand, there are
inevitably light scalars which might have escaped experimental detection.

###  3.1 CP-conserving minima

Let us first address the neutral scalar spectrum for the case with real VEVs,
i.e. with $\theta_{1,2}=0$ in Eq. (12). Using the notation from Eqs. (1) and
(9), we consider that the neutral complex fields are decomposed as

$\phi^{0}=\dfrac{v+\rho_{0}+i\eta_{0}}{\sqrt{2}}\;,\;\delta^{0}_{1,2}=\dfrac{u_{1,2}+\rho_{1,2}+i\eta_{1,2}}{\sqrt{2}}\,,$
(20)

with $\rho_{i}$ and $\eta_{i}$ real. As usual, the neutral scalar mass matrix,
$\mathcal{M}_{0}^{2}$, is obtained by computing the second derivatives of the
scalar potential with respect to the $\rho_{i}$ and $\eta_{i}$ degrees of
freedom. The resulting $6\times 6$ matrix $\mathcal{M}_{0}^{2}$ in the basis
$S=(\rho_{0},\rho_{1},\rho_{2},\eta_{0},\eta_{1},\eta_{2})^{T}$ is such that
the corresponding mass term in the Lagrangian is in the form
$S^{T}\mathcal{M}_{0}^{2}S$. The six mass eigenstates will be denoted by
$h_{i}^{0}$ with $i=1,...,6$. In terms of the parameter $\varepsilon$ defined
in Eq. (19), $\mathcal{M}_{0}^{2}$ can be decomposed as:

$\mathcal{M}_{0}^{2}=(\mathcal{H}_{0}+\mathcal{P})\,v^{2}\;,\;\mathcal{P}=\varepsilon\mathcal{V}_{1}+\varepsilon^{2}\mathcal{V}_{2}\;,\;\mathcal{H}_{0}=\left(\begin{matrix}\mathcal{A}&0\\\
0&\mathcal{B}\end{matrix}\right)\,,$ (21)

where $\mathcal{H}_{0}$, $\mathcal{P}$ and $\mathcal{V}_{1,2}$ ($\mathcal{A}$
and $\mathcal{B}$) are dimensionless $6\times 6$ ($3\times 3$) real symmetric
matrices. It is worth stressing that the above expressions are exact in the
sense that no perturbative expansion in the small parameter $\varepsilon$ has
been done. Taking into account the minimization conditions for a CP-conserving
minimum in Eqs. (15), we find

$\displaystyle\mathcal{B}=\begin{pmatrix}\dfrac{2\sqrt{2}u(\mu_{1}c_{\beta}+\mu_{2}s_{\beta})}{v^{2}}&\\!\\!\\!\dfrac{-\sqrt{2}\mu_{1}}{v}&\dfrac{-\sqrt{2}\mu_{2}}{v}\\\\[5.0pt]
\dfrac{-\sqrt{2}\mu_{1}}{v}&\\!\\!\\!\quad\dfrac{\mu_{1}}{\sqrt{2}uc_{\beta}}-\dfrac{M_{12}^{2}t_{\beta}}{v^{2}}&\dfrac{M_{12}^{2}}{v^{2}}\\\\[5.0pt]
\dfrac{-\sqrt{2}\mu_{2}}{v}&\dfrac{M_{12}^{2}}{v^{2}}&\dfrac{\mu_{2}}{\sqrt{2}us_{\beta}}-\dfrac{M_{12}^{2}}{v^{2}t_{\beta}}\end{pmatrix}\,,$
(22)

with $\mathcal{A}$ being the same as $\mathcal{B}$, except for
$\mathcal{A}_{11}=2\lambda_{0}$. As for $\mathcal{V}_{1,2}$, they are given by

$\mathcal{V}_{1,2}=\left(\begin{matrix}\mathcal{C}_{1,2}&0\\\
0&0\end{matrix}\right)\;,\;\mathcal{C}_{1}=\begin{pmatrix}0&\Lambda_{1}c_{\beta}&\Lambda_{2}s_{\beta}\\\\[6.0pt]
\Lambda_{1}c_{\beta}&0&0\\\\[6.0pt]
\Lambda_{2}s_{\beta}&0&0\end{pmatrix}\;,\;\mathcal{C}_{2}=\begin{pmatrix}0&0&0\\\\[6.0pt]
0&2\Lambda_{3}c^{2}_{\beta}&\Lambda_{4}\,c_{\beta}s_{\beta}\\\\[6.0pt]
0&\Lambda_{4}\,c_{\beta}s_{\beta}&2\Lambda_{5}s^{2}_{\beta}\end{pmatrix}\,,$
(23)

where the dimensionless parameters $\Lambda_{i}$ are those defined in Eqs.
(14). As expected, $\mathcal{M}_{0}^{2}$ is block-diagonal in the chosen basis
due to absence of CP violation – the neutral imaginary components of the
fields do not mix with the real ones, a hallmark of CP conservation when one
is in a basis where all parameters in the potential are real, as is our case.
Direct computation shows that the full matrix $\mathcal{M}_{0}^{2}$ has one
vanishing eigenvalue corresponding to the SM massless neutral Goldstone boson.
This zero eigenvalue is also the only one of $\mathcal{H}_{0}$, hinting at the
fact that, since $\varepsilon$ is a small parameter, the non-zero masses
should also mainly come from the eigenvalues of $\mathcal{H}_{0}$. As we shall
see in the next section, this turns out to not be the case when the triplet
VEVs are complex. Full analytical expressions for the five non vanishing
masses $m^{2}_{h^{0}_{2-6}}$ cannot be easily obtained in the most general
case. However, one can adopt a simple, yet insightful, approach based on
matrix-theory results and theorems which will allow us to infer certain
features of the mass spectrum. Moreover, we will resort to benchmark examples
to anticipate results which will be confirmed by a full general numerical
treatment in Section 4.

From now on, the eigenvalues of a generic $n\times n$ matrix $\mathcal{M}$ are
dubbed $\lambda_{k}(\mathcal{M})$, with $k$ running from $1$ to $n$. For the
neutral mass matrix $\mathcal{M}_{0}^{2}$ the lowest eigenvalue is always
identified by $k=1$ (it corresponds to the neutral Goldstone boson) – thus we
have
$\lambda_{1}(\mathcal{H}_{0}+\varepsilon\mathcal{V}_{1}+\varepsilon^{2}\mathcal{V}_{2})=0$,
and the remaining eigenvalues are all positive, since we are assuming to be at
a minimum. Therefore the full matrix spectrum follows:

$0<\lambda_{2}(\mathcal{H}_{0}+\varepsilon\mathcal{V}_{1}+\varepsilon^{2}\mathcal{V}_{2})\leq...\leq\lambda_{6}(\mathcal{H}_{0}+\varepsilon\mathcal{V}_{1}+\varepsilon^{2}\mathcal{V}_{2})\,.$
(24)

Let us then enunciate Weyl’s inequalities (see e.g. Ref. matrix:analysis ), a
theorem of linear algebra about changes in the eigenvalues of Hermitian
matrices which are “perturbed” by addition of a matrix with typical entries
much smaller than the original one.
Weyl’s inequalities: Let $A,B\in\mathbb{R}^{n\times n}$ be Hermitian matrices.
Each matrix can be diagonalised using a sequence
$\lambda_{1}(A,B)\leq...\leq\lambda_{n}(A,B)$ of $n$ real eigenvalues,
together with an orthonormal basis of complex eigenvectors. Then, Weyl’s
inequalities state that

$\lambda_{n-i-j+2}(A+B)\leq\lambda_{n-i+1}(A)+\lambda_{n-j+1}(B)\,.$ (25)

valid whenever $i,j\geq 1$ and $i+j-1\leq n$.

The above result implies that if $B$ is small compared to $A$ (in the sense
that all its eigenvalues are small compared to $A$’s), then the eigenvalues of
$A+B$ are close to those of $A$. This is actually the case we are dealing with
since the entries of $\mathcal{P}$ are, at most, of order
$\Lambda_{i}\varepsilon\ll 1$, while the potentially large ratios $\mu_{i}/u$,
$\mu_{i}/v$ and $M_{1,2}^{2}/v^{2}$ will render the eigenvalues of
$\mathcal{H}_{0}$ large. Considering the case with $n-i-j+2=n-i+1\equiv k$,
Weyl’s inequalities imply $\lambda_{k}(A+B)-\lambda_{k}(A)\leq\lambda_{n}(B)$,
meaning that the difference between the $k$-th eigenvalue of $A+B$ and the
$k$-th eigenvalue of $A$ is bounded by the spectral radius of $B$ (the largest
absolute value of its eigenvalues).

Identifying $A$ and $B$ with $\mathcal{H}_{0}$ and $\mathcal{P}$ given in Eqs.
(21), respectively, we get

$\lambda_{k}(\mathcal{M}_{0}^{2}/v^{2})-\lambda_{k}(\mathcal{H}_{0})\leq\lambda_{6}(\mathcal{P})\,,$
(26)

where we have used the fact that the largest eigenvalue of $\mathcal{P}$ is
$\lambda_{6}(\mathcal{P})$. As for $\mathcal{M}_{0}/v^{2}$, the matrix
$\mathcal{H}_{0}$ has a single vanishing eigenvalue
$\lambda_{1}(\mathcal{H}_{0})=0$, which, as we already mentioned, corresponds
to the expected neutral Goldstone boson. If, in general, the magnitude of
$\lambda_{k}\left(\mathcal{H}_{0}\right)$ for $k>1$ is much larger than that
of $\lambda_{6}(\mathcal{P})$, then, according to Eq. (26), the difference
between the eigenvalues of $\mathcal{M}_{0}/v^{2}$ and the ones of
$\mathcal{H}_{0}$ will be relatively small. Thus, we expect the non-vanishing
scalar masses to stem mainly from the five non-vanishing eigenvalues of
$\mathcal{H}_{0}$, as anticipated above.

Under this premise, the neutral scalar masses are approximately given by

$\begin{split}m_{h_{2}^{0}}^{2}\simeq&\,2\lambda_{0}v^{2}-2\sqrt{2}u\bar{\mu}\,,\quad\bar{\mu}=\mu_{1}c_{\beta}+\mu_{2}s_{\beta}\\\
m_{h_{3,4}^{0}}^{2}\simeq&\,-\frac{M_{12}^{2}}{s_{2\beta}}+\frac{\sqrt{2}}{2}\frac{v^{2}\bar{\mu}}{us_{2\beta}}+\,\sqrt{\frac{v^{2}}{u^{2}s_{2\beta}}\left(-\mu_{1}\mu_{2}v^{2}+\sqrt{2}M_{12}^{2}u\bar{\mu}\right)+\left(-\frac{M_{12}^{2}}{s_{2\beta}}+\frac{\sqrt{2}}{2}\frac{v^{2}\bar{\mu}}{us_{2\beta}}\right)^{2}}\\\
m_{h_{5,6}^{0}}^{2}\simeq&\,-\frac{M_{12}^{2}}{s_{2\beta}}+\frac{\sqrt{2}}{2}\frac{v^{2}\bar{\mu}}{us_{2\beta}}-\,\sqrt{\frac{v^{2}}{u^{2}s_{2\beta}}\left(-\mu_{1}\mu_{2}v^{2}+\sqrt{2}M_{12}^{2}u\bar{\mu}\right)+\left(-\frac{M_{12}^{2}}{s_{2\beta}}+\frac{\sqrt{2}}{2}\frac{v^{2}\bar{\mu}}{us_{2\beta}}\right)^{2}}\,.\end{split}$
(27)

The above equations show one can choose a region of parameter space that
yields a decoupling limit. To wit, since $|u|\ll v$, taking $u\bar{\mu}$ to
be, at most, of order $\mathcal{O}(v^{2})$ and $|M_{12}^{2}|\gg v^{2}$, we see
that we obtain one neutral scalar with mass $m_{h_{2}^{0}}\sim v$, and two
pairs of quasi-degenerate states whose masses can be arbitrarily large, since
both $M^{2}_{12}$ and $\bar{\mu}/u$ can have arbitrarily large values.
Aditionally, we can show that the states $h^{0}_{3}$ and $h^{0}_{5}$ are CP-
even, whereas $h^{0}_{4}$ and $h^{0}_{6}$ are CP-odd and, thus, pseudoscalars.

We now turn our attention to the three singly- and two doubly-charged scalars
of the 2STM, labelled as $H_{i}^{+}$ $(i=1,2,3)$ and $H_{i}^{++}$ $(j=1,2)$,
respectively. The $H_{i}^{+}$ masses are determined by the $3\times 3$ matrix
$\mathcal{M}_{+}^{2}$ which, similarly to $\mathcal{M}_{0}^{2}$, can be
decomposed as

$\mathcal{M}_{+}^{2}=\left(\mathcal{H}_{0}^{+}+\mathcal{H^{+}_{\lambda}}+\varepsilon\mathcal{V}_{1}^{+}+\varepsilon^{2}\mathcal{V}_{2}^{+}\right)v^{2}\,,$
(28)

where $\varepsilon$ is, once more, the small parameter defined in Eq. (19).
The zeroth order matrices in $\varepsilon$ are given by

$\mathcal{H}_{0}^{+}=\begin{pmatrix}\dfrac{\sqrt{2}u(\mu_{1}c_{\beta}+\mu_{2}s_{\beta})}{v^{2}}&-\dfrac{\mu_{1}}{v}&-\dfrac{\mu_{2}}{v}\\\\[10.0pt]
-\dfrac{\mu_{1}}{v}&\dfrac{\mu_{1}}{\sqrt{2}uc_{\beta}}-\dfrac{M_{12}^{2}t_{\beta}}{v^{2}}&\dfrac{M_{12}^{2}}{v^{2}}\\\\[10.0pt]
-\dfrac{\mu_{2}}{v}&\dfrac{M_{12}^{2}}{v^{2}}&\dfrac{\mu_{2}}{\sqrt{2}us_{\beta}}-\dfrac{M_{12}^{2}}{v^{2}t_{\beta}}\end{pmatrix}\,,\,\mathcal{H}_{\lambda}^{+}={\rm
diag}\left(0,-\dfrac{\hat{\lambda}_{1}}{4},-\dfrac{\hat{\lambda}_{2}}{4}\right)$
(29)

while $\mathcal{V}_{1,2}^{+}$ are written as

$\mathcal{V}_{1}^{+}=\begin{pmatrix}0&\dfrac{\hat{\lambda}_{1}c_{\beta}}{2\sqrt{2}}&\dfrac{\hat{\lambda}_{2}s_{\beta}}{2\sqrt{2}}\\\\[7.0pt]
\dfrac{\hat{\lambda}_{1}c_{\beta}}{2\sqrt{2}}&0&0\\\\[7.0pt]
\dfrac{\hat{\lambda}_{2}s_{\beta}}{2\sqrt{2}}&0&0\end{pmatrix}\,,\,\mathcal{V}_{2}^{+}=\begin{pmatrix}-\dfrac{\hat{\lambda}_{1}c_{\beta}^{2}+\hat{\lambda}_{2}s_{\beta}^{2}}{2}&0&0\\\\[7.0pt]
0&-\dfrac{\Lambda_{6}s_{\beta}^{2}}{4}&\dfrac{\Lambda_{6}s_{2\beta}}{8}\\\\[7.0pt]
0&\dfrac{\Lambda_{6}s_{2\beta}}{8}&-\dfrac{\Lambda_{6}c_{\beta}^{2}}{4}\end{pmatrix}\,,$
(30)

with $\Lambda_{6}=2\lambda_{12}+\tilde{\lambda}_{21}+\tilde{\lambda}_{12}$.

The full matrix $\mathcal{M}_{+}^{2}$ has a vanishing eigenvalue, which is
identified with the charged SM Goldstone boson, and $\mathcal{H}_{0}^{+}$ also
has a single vanishing eigenvalue. Thus, using the same arguments as before,
the singly-charged spectrum can be approximated by the spectrum of the matrix
$\mathcal{H}_{0}^{+}+\mathcal{H}_{\lambda}^{+}$, since the contributions from
$\mathcal{V}_{1}^{+}$ and $\mathcal{V}_{2}^{+}$ are affected by powers of
$\varepsilon$. In fact, we can further argue that the singly-charged masses
are, to very good approximation, given by the eigenvalues of
$\mathcal{H}_{0}^{+}$, since the non-zero elements of
$\mathcal{H}_{\lambda}^{+}$ are much smaller in magnitude than the $(2,2)$ and
$(3,3)$ entries of $\mathcal{H}_{0}^{+}$ when $|M^{2}_{12}|\gg v^{2}$, which
is in the decoupling region we found for the neutral scalars 222This can also
be understood as another application of Weyl’s inequalities, which we leave as
an exercise for the reader.. Thus, we obtain the following very good
approximations for the charged masses,

$m_{H_{1}^{+}}^{2}=0\quad,\quad m_{H_{2}^{+}}^{2}\approx
m_{h_{3}^{0}}^{2}\quad,\quad m_{H_{3}^{+}}^{2}\approx m_{h_{5}^{0}}^{2}\,,$
(31)

where the vanishing eigenvalue corresponds, as we already mentioned, to the
expected Goldstone boson. Notice the degeneracy between neutral and charged
particles, where $m_{h_{3,5}^{0}}^{2}$ are approximately given as shown in
Eqs. (27). A full numerical calculation shows that this degeneracy is not
exact – it is lifted by terms of order $\mathcal{O}(\varepsilon)$ – but it is
an excellent approximation nonetheless.

To conclude the CP-conserving mass spectrum analysis, we now address the
doubly-charged spectrum. There are two doubly-charged mass states,
$H_{i}^{++}$, with $i=1,2$, which stem from the triplet fields’ doubly-charged
fields, $\delta_{i}^{++}$ as in Eq. (9). Their squared mass matrix can be
written as

$\frac{\mathcal{M}_{++}^{2}}{v^{2}}=\mathcal{H}_{0}^{++}+\mathcal{H}_{\lambda}^{++}+\varepsilon^{2}\mathcal{V}_{2}^{++}\,,$
(32)

and these submatrices are given by

$\mathcal{H}_{0}^{++}=\begin{pmatrix}\dfrac{\mu_{1}}{\sqrt{2}uc_{\beta}}-\dfrac{M_{12}^{2}t_{\beta}}{v^{2}}&\dfrac{M_{12}^{2}}{v^{2}}\\\\[10.0pt]
\dfrac{M_{12}^{2}}{v^{2}}&\dfrac{\mu_{2}}{\sqrt{2}us_{\beta}}-\dfrac{M_{12}^{2}}{v^{2}t_{\beta}}\end{pmatrix}\,,$
(33)
$\mathcal{H}_{\lambda}^{++}=\begin{pmatrix}-\dfrac{\hat{\lambda}_{1}}{2}&0\\\\[10.0pt]
0&-\dfrac{\hat{\lambda}_{2}}{2}\end{pmatrix}\,,\,\mathcal{V}_{2}^{++}=\begin{pmatrix}-\tilde{\lambda}_{1}c_{\beta}^{2}-\dfrac{\Lambda_{7}}{2}s_{\beta}^{2}&\dfrac{\lambda_{12}}{2}c_{\beta}s_{\beta}\\\\[10.0pt]
\dfrac{\lambda_{12}}{2}c_{\beta}s_{\beta}&-\tilde{\lambda}_{2}s_{\beta}^{2}-\dfrac{\Lambda_{7}}{2}c_{\beta}^{2}\end{pmatrix}\,,$
(34)

with $\Lambda_{7}=\lambda_{12}+\tilde{\lambda}_{21}+\tilde{\lambda}_{12}$.

The matrix $\mathcal{H}_{0}^{++}$ does not have a vanishing eigenvalue as was
to be expected. Following the analysis of the singly-charged mass spectrum, we
can approximate the eigenvalues of the full matrix $\mathcal{M}_{++}^{2}$ by
the ones stemming from just $\mathcal{H}_{0}^{++}$. This way, we are able to
obtain approximate (albeit very accurate) expressions for the doubly-charged
squared masses, from which the following quasi degeneracies stand out:

$m_{h^{0}_{3,4}}^{2}\approx m_{H_{2}^{+}}^{2}\approx
m_{H_{1}^{++}}^{2}\quad,\quad m_{h^{0}_{5,6}}^{2}\approx
m_{H_{3}^{+}}^{2}\approx m_{H_{2}^{++}}^{2}\,.$ (35)

Therefore, in the region where $|M^{2}_{12}|\gg v^{2}$, the CP-conserving mass
spectrum has one scalar with mass of $\mathcal{O}(v)$, while the remaining
states, both neutral or charged, can be made much heavier and, thus, decouple
from the theory. We observe that we obtain two sets of quasi-degenerate
particles, each set containing a CP-even scalar ($h^{0}_{3}$ or $h^{0}_{5}$),
a pseudoscalar ($h^{0}_{4}$ or $h^{0}_{6}$), a singly-charged scalar
($H_{2}^{+}$ or $H_{3}^{+}$) and a doubly charged one ($H_{1}^{++}$ or
$H_{2}^{++}$). As further confirmation, we numerically checked that the
approximated expressions for the masses stemming from $\mathcal{H}_{0}$-like
matrices hold in the region of parameter space considered and hence the
spectrum exhibits the features described above.

###  3.2 CP-violating minima

We now draw our attention to the case of complex VEVs defined by Eqs. (12), to
which correspond the five independent minimisation conditions of Eqs. (13). It
is worth reminding that, in contrast with the CP-conserving case, the soft-
breaking parameters $M_{12}^{2}$ and $\mu_{2}$ can now be expressed as
function of the remaining parameters – see Eqs. (16). Once more, we write the
neutral scalar mass matrix in the form of Eq. (21) where now:

$\mathcal{A},\mathcal{B}=\begin{pmatrix}2\lambda_{0}\,,\,-\dfrac{2\sqrt{2}\mu_{1}uc_{\beta}s_{\theta_{1}-\theta_{2}}}{v^{2}s_{\theta_{2}}}&\dfrac{-\sqrt{2}\mu_{1}}{v}&\dfrac{\sqrt{2}\mu_{1}s_{\theta_{1}}}{vt_{\beta}s_{\theta_{2}}}\\\\[10.0pt]
\dfrac{-\sqrt{2}\mu_{1}}{v}&-\dfrac{\mu_{1}s_{\theta_{2}}}{\sqrt{2}us_{\theta_{1}-\theta_{2}}c_{\beta}}&\dfrac{\mu_{1}s_{\theta_{1}}}{\sqrt{2}us_{\beta}s_{\theta_{1}-\theta_{2}}}\\\\[10.0pt]
\dfrac{\sqrt{2}\mu_{1}s_{\theta_{1}}}{vt_{\beta}s_{\theta_{2}}}&\dfrac{\mu_{1}s_{\theta_{1}}}{\sqrt{2}us_{\beta}s_{\theta_{1}-\theta_{2}}}&-\dfrac{\mu_{1}s^{2}_{\theta_{1}}}{\sqrt{2}ut_{\beta}s_{\beta}s_{\theta_{1}-\theta_{2}}s_{\theta_{2}}}\end{pmatrix}\,.$
(36)

Due to the fact that $\theta_{1,2}\neq 0$, the matrices $\mathcal{V}_{1,2}$
are no longer block diagonal as for the CP-conserving case – this is a clear
indication of CP violation, since the real and imaginary neutral components of
the doublet and triplets have opposite CP quantum numbers. A simple
calculation shows that $\mathcal{V}_{1}$ is such that

$(\mathcal{V}_{1})_{1j}=(\mathcal{V}_{1})_{j1}\equiv
V_{j}=(0,\Lambda_{1}c_{\beta}c_{\theta_{1}},\Lambda_{2}s_{\beta}c_{\theta_{2}},0,\Lambda_{1}c_{\beta}s_{\theta_{1}},\Lambda_{2}s_{\beta}s_{\theta_{2}})_{j}\,,$
(37)

and all remaining entries are zero (notice the vanishing elements of the
fourth line and column of $\mathcal{V}_{1}$). As for $\mathcal{V}_{2}$ we
have:

$\mathcal{V}_{2}=\begin{pmatrix}0&0&0&0&0&0\\\\[6.0pt]
0&2\Lambda_{3}c^{2}_{\beta}c^{2}_{\theta_{1}}&\Lambda_{4}\,c_{\beta}c_{\theta_{1}}c_{\theta_{2}}s_{\beta}&0&\Lambda_{3}c^{2}_{\beta}s_{2\theta_{1}}&\Lambda_{4}\,c_{\beta}c_{\theta_{1}}s_{\theta_{2}}s_{\beta}\\\\[6.0pt]
0&\Lambda_{4}\,c_{\beta}c_{\theta_{1}}c_{\theta_{2}}s_{\beta}&2\Lambda_{5}s^{2}_{\beta}c^{2}_{\theta_{2}}&0&\Lambda_{4}\,c_{\beta}c_{\theta_{2}}s_{\theta_{1}}s_{\beta}&\Lambda_{5}s^{2}_{\beta}s_{2\theta_{2}}\\\\[6.0pt]
0&0&0&0&0&0\\\\[6.0pt]
0&\Lambda_{3}\Lambda_{3}c^{2}_{\beta}s_{2\theta_{1}}&\Lambda_{4}\,c_{\beta}c_{\theta_{2}}s_{\theta_{1}}s_{\beta}&0&2\Lambda_{3}c^{2}_{\beta}s^{2}_{\theta_{1}}&\Lambda_{5}\,c_{\beta}s_{\theta_{1}}s_{\theta_{2}}s_{\beta}\\\\[6.0pt]
0&\Lambda_{4}\,c_{\beta}c_{\theta_{1}}s_{\theta_{2}}s_{\beta}&\Lambda_{5}s^{2}_{\beta}s_{2\theta_{2}}&0&\Lambda_{5}\,c_{\beta}s_{\theta_{1}}s_{\theta_{2}}s_{\beta}&2\Lambda_{5}s^{2}_{\beta}s^{2}_{\theta_{2}}\end{pmatrix}\,.$
(38)

In the present scenario $\mathcal{H}_{0}$ has three vanishing eigenvalues,
$\lambda_{1,2,3}(\mathcal{H}_{0})=0$, instead of just one as in the real-vacua
case. Resorting once more to Weyl’s inequalities in the form of Eq. (26), and
taking into account that $\lambda_{1}(\mathcal{M}_{0}^{2})=0$ (corresponding
to the SM neutral Goldstone boson), we can write

$\lambda_{k}(\mathcal{M}_{0}^{2}/v^{2})\leq\lambda_{6}(\mathcal{P})\,,k=2,3.$
(39)

Thus, we can already anticipate that
$\mathcal{P}=\varepsilon\mathcal{V}_{1}+\varepsilon^{2}\mathcal{V}_{2}$ shall
be responsible for, somehow, giving rise to two light neutral scalars and,
since the matrix $\mathcal{P}$ is of order $\sim\varepsilon$, the three
lightest eigenvalues of the full mass matrix $\mathcal{M}_{0}^{2}$ will be
bounded from above by masses which we expect to be small when compared with
the electroweak scale. We therefore see the possibility of a non-decoupling
regime emerging. It turns out that, as we will now show, the full neutral
spectrum includes two light scalars expected to have a mass upper bound coming
just from the $\varepsilon^{2}\mathcal{V}_{2}$ matrix. Following the previous
reasoning, the remaining three mass states (in which the 125 GeV Higgs boson
is included) stem mainly from the non-vanishing eigenvalues of
$\mathcal{H}_{0}$, expected to be much larger than those of $\mathcal{P}$. To
prove this, we start by considering the full mass matrix decomposed as in the
CP conserving case:

$\mathcal{M}_{0}^{2}=(\mathcal{H}_{0}+\varepsilon\mathcal{V}_{1}+\varepsilon^{2}\mathcal{V}_{2})\,v^{2}\;,$
(40)

and make use of another result from matrix theory: Cauchy’s Eigenvalue
Interlacing Theorem (EIT) matrix:analysis .

Eigenvalue Interlacing Theorem (EIT): Let $A\in\mathbb{R}^{n\times n}$ be a
symmetric matrix, and $B_{i}\in\mathbb{R}^{m\times m}$ (with $m<n$) a
principal submatrix of $A$ obtained by removing its $i$-th row and the $i$-th
column. Denoting the eigenvalues of $A$ and $B_{i}$ by
$\lambda_{1}(A)\leq...\leq\lambda_{n}(A)$ and
$\lambda_{1}(B_{i})\leq...\leq\lambda_{m}(B_{i})$, respectively, then

$\displaystyle\lambda_{k}(A)\leq\lambda_{k}(B_{i})\leq\lambda_{k+n-m}(A)\,,\quad
k=1,...,m\,.$ (41)

Let us apply this theorem to the $6\times 6$ normalised mass matrix
$\mathcal{M}_{0}^{2}/v^{2}$, which we identify with $A$ in the EIT. Removing
the first line and column of $\mathcal{M}_{0}^{2}/v^{2}$ (and denoting with a
tilde any matrix resulting from this operation), we define the principal
$5\times 5$ submatrix
$\tilde{\mathcal{M}}_{0}^{2}/v^{2}=\tilde{\mathcal{H}}_{0}+\tilde{\mathcal{P}}$.
Given that $\tilde{\mathcal{V}}_{1}=0$ [see Eq. (37)] we get
$\tilde{\mathcal{M}}_{0}^{2}/v^{2}=\tilde{\mathcal{H}}_{0}+\varepsilon^{2}\tilde{\mathcal{V}}_{2}$.
Considering the EIT with $n=6$ and $m=5$, we have
$\lambda_{k}(\mathcal{M}_{0}^{2}/v^{2})\leq\lambda_{k}(\tilde{\mathcal{H}}_{0}+\varepsilon^{2}\tilde{\mathcal{V}}_{2})\leq\lambda_{k+1}(\mathcal{M}_{0}^{2}/v^{2})$
which, for $k=2,3$, implies
$\lambda_{2,3}(\mathcal{M}_{0}^{2}/v^{2})\leq\lambda_{2,3}(\tilde{\mathcal{H}}_{0}+\varepsilon^{2}\tilde{\mathcal{V}}_{2})\leq\lambda_{3,4}(\mathcal{M}_{0}^{2}/v^{2})$.
Since $\lambda_{k}(\mathcal{M}_{0}^{2}/v^{2})>0$ for $k>1$ (all squared masses
are positive, since we are assuming that these VEVs correspond to a minimum),
we conclude that

$0<\lambda_{2,3}(\mathcal{M}_{0}^{2}/v^{2})\leq\lambda_{2,3}(\tilde{\mathcal{H}}_{0}+\varepsilon^{2}\tilde{\mathcal{V}}_{2})\,,$
(42)

i.e., the second and third largest eigenvalues of the full matrix are bounded
from above by
$\lambda_{2}(\tilde{\mathcal{H}}_{0}+\varepsilon^{2}\tilde{\mathcal{V}}_{2})$
and
$\lambda_{3}(\tilde{\mathcal{H}}_{0}+\varepsilon^{2}\tilde{\mathcal{V}}_{2})$,
respectively. A simple analytical expression for such upper bounds is hard to
obtain in the most general case. However, recalling Weyl’s Inequalities, we
can write:

$\lambda_{2,3}(\tilde{\mathcal{H}}_{0}+\varepsilon^{2}\tilde{\mathcal{V}}_{2})\leq\lambda_{2,3}(\tilde{\mathcal{H}}_{0})+\varepsilon^{2}\lambda_{4,5}(\tilde{\mathcal{V}}_{2})\,,$
(43)

where $\lambda_{4,5}(\tilde{\mathcal{V}}_{2})$ stand for the largest two
eigenvalues of the $5\times 5$ matrix $\tilde{\mathcal{V}}_{2}$. Since
$\tilde{\mathcal{H}}_{0}$ has three vanishing and two non-vanishing
eigenvalues it is straightforward to conclude that
$\lambda_{3}(\tilde{\mathcal{H}}_{0})=0$, regardless of the sign of both non-
vanishing eigenvalues. Although one cannot guarantee that
$\lambda_{2}(\tilde{\mathcal{H}}_{0})=0$ (since there can be negative
eigenvalues), one can show that this is actually the case when requiring all
masses to be positive. Thus, in the case of physical interest,
$\lambda_{2}(\tilde{\mathcal{H}}_{0})=\lambda_{3}(\tilde{\mathcal{H}}_{0})=0$
and Eq. (43) imply an upper bound on the lightest-scalar masses
$m^{2}_{h_{2,3}^{0}}$:

$m^{2}_{h_{2,3}^{0}}=\lambda_{2,3}(\mathcal{M}_{0}^{2})\leq
u^{2}\lambda_{4,5}(\tilde{\mathcal{V}}_{2})\,,$ (44)

which, after extracting $\tilde{\mathcal{V}}_{2}$ from $\mathcal{V}_{2}$ given
in Eq. (38), leads to

$m^{2}_{h_{2,3}^{0}}\leq\frac{u^{2}}{2}\left[\Lambda_{3}+\Lambda_{5}+(\Lambda_{3}-\Lambda_{5})c_{2\beta}\pm\sqrt{\left[\Lambda_{3}+\Lambda_{5}+(\Lambda_{3}-\Lambda_{5})c_{2\beta}\right]^{2}+\left(\Lambda_{4}^{2}-4\Lambda_{3}\Lambda_{5}\right)s^{2}_{2\beta}}\right]\,.$
(45)

This result implies that there is an upper bound on the two lightest neutral
scalar masses proportional to $f(\Lambda)u$, where $f(\Lambda)$ is a function
of $\Lambda_{i}$ and $\beta$ defined in Eqs. (14) and (12), respectively.
Considering all quartic parameters in the scalar potential to be at most 10,
we arrive at a rough estimate of $m^{2}_{h_{2}^{0}}\lesssim 3u$ and
$m^{2}_{h_{3}^{0}}\lesssim 6u$. Having in mind that EWPD sets $u\leq 8$ GeV,
we conclude that $m^{2}_{h_{3}^{0}}\lesssim 25$ GeV and
$m^{2}_{h_{3}^{0}}\lesssim 50$ GeV. This result, obtained resorting to simple
linear algebra theorems, turns out to be in excellent agreement with the
numerical simulations presented in Section 4.

This analytical result also casts a different light on the possibility of
generating naturally small neutrino masses in the 2STM with $\mathcal{O}(1)$
Yukawa couplings and triplet VEVs $u$ of the eV order – we now see that, if CP
is broken spontaneously, there will be two extremely light neutral scalars. In
short, in a scenario where the triplet VEVs are complex and their magnitude is
of the order of the neutrino mass scale, a scalar spectrum with all particles
much heavier than the SM Higgs boson cannot be realised. It should be stressed
that, even if $u$ is close to its 8 GeV upper bound, two of the neutral scalar
masses will be at most a few tenths of GeV. This, however, would require very
tiny triplet Yukawa couplings so that $\mathcal{O}$(eV) neutrino masses are
generated.

From the analysis above, we conclude that the main contribution to the masses
of the three heaviest neutral scalars stem from $\mathcal{H}_{0}$, which has
the block diagonal form shown in Eq. (21) with $\mathcal{A}$ and $\mathcal{B}$
defined in Eq. (36). One of those masses (typically the lightest) correspond
to the 125 GeV SM Higgs mass, while the remaining two are heavier. Barring
accidental and/or special $\tan\beta$ limits, the entries of $\mathcal{A}$ and
$\mathcal{B}$ scale as $\mu_{1}/u$ and $\mu_{1}/v$. Given that $u\ll v$, the
former ratio is much larger than the latter. We can obtain the following
approximate expressions for the neutral masses,

$\displaystyle m_{h^{0}_{4}}^{2}$ $\displaystyle\simeq
2\lambda_{0}v^{2}+2\sqrt{2}\,\mu_{1}uc_{\beta}\dfrac{s_{\theta_{1}-\theta_{2}}}{s_{\theta_{2}}}\,,$
(46) $\displaystyle m_{h^{0}_{5}}^{2}$ $\displaystyle\simeq
m_{h^{0}_{6}}^{2}\simeq-\dfrac{s_{\theta_{2}}}{s_{\theta_{1}-\theta_{2}}}\left[f_{1}(\beta,\theta_{1},\theta_{2})+f_{2}(\beta)\right]M_{\Delta}^{2}\,,$

where, similarly to what we have done for the STM model, we now define
$M_{\Delta}$ as

$M_{\Delta}^{2}\equiv\dfrac{\mu_{1}\,v^{2}}{\sqrt{2}u}\,.$ (47)

The functions $f_{1}$ and $f_{2}$ are given by:

$f_{1}(\beta,\theta_{1},\theta_{2})=\dfrac{s^{2}_{\theta_{1}}}{t_{\beta}s_{\beta}s^{2}_{\theta_{2}}}\quad,\quad
f_{2}(\beta)=\dfrac{1}{c_{\beta}}\,.$ (48)

From Eq. (46) we see that we can obtain a SM-like scalar state – corresponding
to $h^{0}_{4}$ – with a mass close to the electroweak scale provided
$\mu_{1}u\lesssim\mathcal{O}(v^{2})$. We will verify later that this choice is
also compatible with $h^{0}_{4}$ having couplings to gauge bosons and fermions
similar to those of the SM Higgs. Further, we see from the same equation that
if we wish to have the two extra neutral scalars be much heavier than the weak
scale (and thus decouple, easily avoiding current LHC search limits) we need
to ensure $M^{2}_{\Delta}$ is large enough, which pushes us to the regime
where $\mu_{1}/u$ assumes large values. To quantify this statement, having
$M_{\Delta}$ be of the TeV order would require (modulo large
enhancements/suppressions from the angular factors in Eq. (46))
$\mu_{1}\gtrsim 25u$.

At this point, it is convenient to comment on the special case
$\theta_{1}=\theta_{2}$ which, according to the above result, seems to lead to
infinite $h_{5,6}^{0}$ masses. As we mentioned above, this situation
corresponds to $\mu_{1}=\mu_{2}=0$, which means that the U(1) symmetry is now
exact, and a VEV for the doublet yields two zero eigenvalues for the neutral
scalars – one the gauge Goldstone boson, the other a pseudo-Goldstone. This is
an extreme case of non-decoupling, wherein one of the extra scalars is
actually massless, and we will not consider it further.

We now comment on the charged-scalar sector of the 2STM, which comprises two
singly and two doubly-charged physical states (besides the charged massless
Goldstone bosons) denoted by $H_{1,2}^{+}$ and $H_{1,2}^{++}$, respectively.
The corresponding mass matrices are determined, as usual, by the second
derivatives of the scalar potential with respect to the $\delta_{1,2}^{+}$ and
$\delta_{1,2}^{++}$ degrees of freedom – see Eq. (9). Considering the case
$\mu_{1}\gg u$, and Taylor expanding the eigenvalues of the mass matrices in
the small parameter $u/\mu_{1}$, we get the following approximate masses for
the charged physical scalars:

$\displaystyle m^{2}_{H_{2}^{+}}$
$\displaystyle\simeq\frac{m^{2}_{H_{1}^{++}}}{2}\simeq-\dfrac{1}{4}\dfrac{\hat{\lambda}_{1}f_{1}(\beta,\theta_{1},\theta_{2})+\hat{\lambda}_{2}f_{2}(\beta)}{f_{1}(\beta,\theta_{1},\theta_{2})+f_{2}(\beta)}v^{2}\,,$
(49) $\displaystyle m^{2}_{H_{3}^{+}}$ $\displaystyle\simeq
m^{2}_{H_{2}^{++}}\simeq-\dfrac{s_{\theta_{2}}}{s_{\theta_{1}-\theta_{2}}}\left[f_{1}(\beta,\theta_{1},\theta_{2})+f_{2}(\beta)\right]M_{\Delta}^{2}\,,$

with $f_{1,2}$ and $M_{\Delta}^{2}$ defined as in Eqs. (47) and (48),
respectively. These results show that there is a pair of charged scalars
$H_{1}^{+}$ and $H_{1}^{++}$ with similar masses set by the electroweak scale
$v$, while $H_{2}^{+}$ and $H_{2}^{++}$ decouple in the limit of very large
$M_{\Delta}^{2}$.

To conclude, in the case of CP-violating vacua, and assuming $\mu_{1}\gg u$ so
as to have at least some decoupled states, the 2STM scalar mass spectrum
exhibits the following features:

  * •

Besides the 125 GeV SM Higgs $h_{4}^{0}$, there are two light neutral scalars
$h_{2,3}^{0}$ with masses proportional to the triplet VEV parameter $u$
meaning that these particles cannot mix significantly with SM fermions and
gauge bosons. The remaining two scalars $h_{5,6}^{0}$ have similar masses and
scale with $\mu_{1}v^{2}/u$, which is typically very large.

  * •

There are four charged scalars $H_{1,2}^{+}$ and $H_{1,2}^{++}$ with
$m^{2}_{H_{1}^{++}}\simeq 2m^{2}_{H_{1}^{+}}\sim v^{2}$ and
$m^{2}_{H_{2}^{+}}\simeq m^{2}_{H_{2}^{++}}\sim\mu_{1}v^{2}/u$, i.e. one
singly-charged and one doubly-charged scalars have masses of the order of the
electroweak scale, while the remaining two can be much heavier. In addition,
$H_{2}^{+}$ and $H_{2}^{++}$ have masses similar to those of $h_{5,6}^{0}$.

  * •

It is interesting to notice that, while for the CP conserving vacuum we
obtained quasi degeneracy between two pairs of singly-charged-doubly-charged
scalar (see Eq. (35)), for the CP-violating vaccum the mass of one pair
remains the same (for $H_{3}^{+}$ and $H_{2}^{++}$), but for the other pair
(for $H_{2}^{+}$ and $H_{1}^{++}$) the doubly-charged scalar mass is
approximately twice of the singly-charged one.

Thus, and unlike the CP-conserving case, there is no region of parameter space
for which all extra scalars – neutral or charged – can be guaranteed to have
masses much above the electroweak scalar, and the 2STM with SCPV has no
decoupling limit. In Table 1 we summarise the properties of the neutral and
scalar mass spectrum for both the CP-conserving and the CP-violating vacuum
configurations.

Before moving to the numerical analysis which will confirm the results of this
section, let us briefly comment on the relation between non-decoupling and
SCPV in the 2STM, when compared to the 2HDM case. In Ref. Nebot:2020niz  it
was shown that, when SCPV occurs in the 2HDM, the extra scalar masses cannot
be made arbitrarily large. Rather, there are upper bounds on those masses set
by the fact that the size of the quartic couplings is limited by
perturbativity and unitarity. That the 2HDM has a decoupling limit in the CP-
conserving case but not in the CP-violating one is due to the different number
of minimisation conditions – two and three, respectively. Since the model has
only three quadratic parameters, a CP-conserving minimum fixes two of them to
be of order $\lambda_{i}v^{2}$, with $\lambda_{i}$ a generic quartic coupling
of limited magnitude. A third quadratic coefficient of the scalar potential
becomes, therefore, a free parameter and can be used to render the extra
scalar masses as heavy as wanted, thus yielding a decoupling regime. However,
in the case of SCPV the extra minimisation equation will force all quadratic
terms in the potential to be of order $\lambda_{i}v^{2}$, which becomes also
the mass scale of all (quadratic) scalar masses and sets an upper bound for
those masses. In the 2STM the extra minimisation equations needed for SCPV –
five equations, as in (13), as opposed to three for a CP-conserving extremum,
as in (15) – do limit the size of an extra two mass parameters, compared to
the CP-conserving case. But there is a difference regarding the 2HDM case: if
one looks at the fully softly-broken potential of Eqs. (10) and (11), one
counts a total of six mass parameters. Even in the SCPV case, the minimisation
conditions will only fix the magnitude of at most five of them, so a priori
one would have a free mass parameter which could drive the extra scalar masses
as high as wanted. But in the 2STM that does not occur because electroweak
precision data requires that the triplet VEVs be small, and that extra
requirement, along with the additional minimisation conditions, forces the
model to be non-decoupling when SCPV occurs.

Mass spectrum | CP-Conserving | CP-Violating
---|---|---
Neutral | $h_{1}^{0}$ | Massless - Goldstone boson
$h_{2}^{0}$ | SM Higgs-like |
$h_{3}^{0}$ |  | Light
$h_{4}^{0}$ | Decoupled | SM Higgs-like
$h_{5}^{0}$ |  |
$h_{6}^{0}$ | Decoupled | Decoupled
Singly-charged | $H_{1}^{+}$ | Massless - Goldstone boson
$H_{2}^{+}$ | Decoupled | Electroweak
$H_{3}^{+}$ | Decoupled | Decoupled
Doubly-charged | $H_{1}^{++}$ | Decoupled | Electroweak
$H_{2}^{++}$ | Decoupled | Decoupled
Table 1: Main features of the neutral and charged-scalar mass spectrum in the
2STM for the CP-conserving and CP-violating vacuum cases. Scalars marked with
the same colours have masses of similar magnitude. Here,“Decoupled” is
applicable to the states for which the mass can be arbitrary large, while
“Electroweak” is used for those whose mass is, typically, around the
electroweak scale.

##  4 Numerical and phenomenological analysis

Having found analytical arguments for the fact that the 2STM has a non-
decoupling regime when CP is spontaneously broken, we now corroborate this by
performing a numerical study of the model. We also wish to investigate the
possibility that the lighter scalars predicted by the analysis of the previous
section might have escaped detection thus far. For that, we will concentrate
on triplet VEVs of GeV order, which would require suppressed Yukawa couplings
to attain small neutrino masses. Instead, with $\mathcal{O}(1)$ Yukawa
couplings, $\mathcal{O}$(eV) triplet VEVs are needed, implying the existence
of scalars with masses of the same order, a possibility which seems bizarre,
and that deserves further investigation elsewhere. In the following, we define
the mass of a generic scalar state $h$ as $m_{h}$. We start by considering the
$\text{U}(1)$ softly-broken scalar potential of Eqs. (10) and (11), and
generate a large sample of parameters satisfying the following conditions:

  * •

The triplet VEV magnitude $u$ is chosen randomly in the interval
$]0,10]\,\text{GeV}$;

  * •

The VEV of the neutral component of $\Phi$ is $v=\sqrt{246^{2}-2u^{2}}$;

  * •

The angle $\beta=\arctan(u_{2}/u_{1})$ is chosen randomly in
$\beta\in\,]0,\pi/2[$;

  * •

The CP-violating phases $\theta_{1,2}$ are chosen randomly in the interval
$]0,2\pi[$ with $\theta_{1}\neq\theta_{2}$ and $\theta_{1,2}\neq\pi$;

  * •

The quartic couplings, apart from $\lambda_{0}$, are chosen randomly in the
interval $[-10,10]$;

  * •

All scalar squared-masses have to be positive;

  * •

The parameters $\mu_{1}$ and $\lambda_{0}$ are chosen in such a way that the
mass eigenstate $h_{4}^{0}$ corresponds to the 125 GeV SM Higgs boson, for
which Eq. (46) provides an excellent approximation;

  * •

The state $h_{4}^{0}$ has couplings to SM particles - gauge bosons and
fermions - at most $10\%$ deviated from the expected SM values.

From Eq. (46), we can extract the following expression for $\lambda_{0}$:

$\lambda_{0}\approx\dfrac{m_{H}^{2}-2\sqrt{2}\mu_{1}uc_{\beta}\dfrac{s_{\theta_{1}-\theta_{2}}}{s_{\theta_{2}}}}{2v^{2}}\,,$
(50)

in which we have used $m_{h_{4}^{0}}^{2}\equiv
m_{\text{Higgs}}^{2}=(125~{}\text{GeV})^{2}$. From this equation, assuming
$\lambda_{0}$ to be in the real-valued interval $]0,10[$, we get:

$0<m_{H}^{2}-2\sqrt{2}\mu_{1}uc_{\beta}\dfrac{s_{\theta_{1}-\theta_{2}}}{s_{\theta_{2}}}<20\,v^{2}\,.$
(51)

After randomly selecting a value for $\mu_{1}$ within the regime discussed
previously - for which we expect a particle of $\mathcal{O}(v)$ as well as
decoupled mass states -, we assure that Eq. (51) is satisfied and the value
for $\lambda_{0}$ is obtained using Eq. (50). We emphasise that, although we
have used our analytical considerations in order to choose the ranges of
parameters for numerical scans, all masses and couplings were obtained through
the numerical diagonalisation of the exact mass matrices. The selection
criteria mentioned indeed give rise not only to a Higgs-like particle but also
to positive decoupled squared-masses, namely two neutral, one singly- and one
doubly-charged.

|
---|---
Figure 1: Left: Scatter plot of the lightest neutral-scalar masses vs. the triplet VEV $u$, in orange and blue for $m_{h_{2}^{0}}$ and $m_{h_{3}^{0}}$, respectively. The solid (dashed) black line corresponds to the maximum value of the upper bounds given in Eq. (45) at each value of $u$. Right: The same as in the left panel but now as function of $\tan\beta$. |
---|---
Figure 2: Left: scatter plots of $\Delta m_{23}$ vs. $m(h_{2}^{0})$ for values
of the triplet VEV $u\leq 8,6,4,2$ GeV (in orange, purple, cyan and turquoise,
respectively). Right: scatter plot of $m_{H_{1}^{++}}$ vs. $m_{H_{2}^{+}}$ for
$u\leq 8$ GeV.

Under all the above considerations, we have computed the neutral-scalar
spectrum for a large sample of the parameter space. Concerning the two
lightest scalars $h_{2}^{0}$ and $h_{3}^{0}$, in the left (right) panel of
Fig. 1 we show a scatter plot of their masses against $u$ ($\tan\beta$) in
orange and blue for $m_{h_{2}^{0}}$ and $m_{h_{3}^{0}}$, respectively. The
solid (dashed) lines correspond to the theoretical upper bounds found in Eq.
(45) for $m_{h_{3}^{0}}$ ($m_{h_{2}^{0}}$). We observe that the numerical
scans lead to upper bounds which are in excellent agreement with the
approximate analytical prediction. From the right panel, we notice that
$\tan{\beta}\simeq 1$ is required to reach $m_{h_{2}^{0}}$ near its maximum
value. In Fig. 2 we show a scatter plot for the mass difference $\Delta
m_{23}\equiv m_{h_{3}^{0}}-m_{h_{2}^{0}}$ vs. the lightest scalar mass
$m_{h_{2}^{0}}$ for $u\leq 8,6,4,2$ GeV, in blue, purple, cyan and turquoise,
respectively. We see that, as $m_{h_{2}^{0}}$ increases, the mass difference
$\Delta m_{23}$ decreases, being very small for $m_{h_{2}^{0}}$ close to its
maximum. Concerning the validity of the approximations deduced in the previous
section for the charged-scalar masses, the numerical results plotted in the
right panel of Fig. 2 confirm the relation
$m_{H_{1}^{++}}\approx\sqrt{2}\,m_{H_{2}^{+}}$ stemming from Eq. (49). This
plot also illustrates that, since the quartic parameters
$\widehat{\lambda}_{1}$ and $\widehat{\lambda}_{2}$ are limited, those masses
are upper bounded. In fact, making once more use of Weyl’s inequalities, such
bounds can be found. Namely, considering
$\text{max}\left(-\widehat{\lambda}_{1},-\widehat{\lambda}_{2}\right)=10$, we
get:

$\displaystyle m_{H_{2}^{+}}$
$\displaystyle\lesssim\sqrt{\text{max}\left(-\widehat{\lambda}_{1}/4,-\widehat{\lambda}_{2}/4\right)}v\simeq
390\,\text{GeV}\,,$ (52) $\displaystyle m_{H_{1}^{++}}$
$\displaystyle\lesssim\sqrt{\text{max}\left(-\widehat{\lambda}_{1}/2,-\widehat{\lambda}_{2}/2\right)}v\simeq
550\,\text{GeV}\,,$ (53)

which is in very good agreement with the results in Fig. 2 (right panel).

We now address some phenomenological aspects related to those scalars which do
not decouple, namely $h_{2}^{0}$, $h_{3}^{0}$, $H_{2}^{+}$ and $H_{1}^{++}$
(see Table 1). In particular, we wish to investigate whether such low-mass
scalars are at all compatible with existing searches in accelerators. Remember
that, although the 125 GeV Higgs boson was discovered at LHC, bounds on its
mass and couplings had already been obtained, for instance, at LEP. In spite
of the fact that no charged scalars $H^{+}$ have been detected, measurements
of observables such as $b\rightarrow s\gamma$ have been used to impose tight
constraints on the mass and couplings of $H^{+}$. It is therefore of interest
to analyse the possibility that, though the SCPV minimum implies light scalars
in the 2STM, these new particles might have escaped detection due to their
suppressed interactions with SM particles. Consider that, though the extra
neutral scalars are the result of mixing with the corresponding neutral fields
of the doublet, that mixing is highly suppressed since $u_{1,2}\ll v$.
Therefore, since the extra neutral mass eigenstates do have Yukawa
interactions with the fermions, we expect them to be highly suppressed.

Figure 3: Upper bound on the ratio $(g_{hZZ}/g_{hZZ}^{\text{SM}})^{2}$ at
$95\%$ confidence level, for $m_{h_{2,3}^{0}}\geq 10~{}\text{GeV}$, from
combined LEP data. The solid black (blue dashed) line stands for the observed
(expected) bound. The green (yellow) bands correspond to the $68\%$ C.L.
($95\%$ C.L.) regions for the expected bound. In orange and blue we show the
$m_{h_{2}^{0}}$ and $m_{h_{3}^{0}}$ vs. vs $(g_{hZZ}/g_{hZZ}^{\text{SM}})^{2}$
points, for the considered mass range. These results were obtained considering
$|u|<8~{}\text{GeV}$, and the LEP bounds were taken from Ref. Barate:2003sz .

We start by computing the $hZZ$ couplings for $h\equiv h_{2,3}$. At LEP, the
main production process of the Higgs was Higgsstrahlung,
$e^{+}e^{-}\rightarrow Z\rightarrow Zh$, whereby a Higgs is emitted by a
virtual $Z$ boson. Data from LEP was used to set upper bounds on the
$hZZ$-coupling in non-standard models. We numerically compute the ratio
$g_{hZZ}/g_{hZZ}^{\text{SM}}$, where $g_{hZZ}$ is the non-standard $hZZ$
coupling and $g_{hZZ}^{\text{SM}}$ the SM one. Within our numerical scan, such
ratio regarding the SM-like Higgs particle must be at most $10~{}\%$ deviated
from the expected SM value. Our results are depicted in Fig. 3 for masses
above $10~{}\text{GeV}$. The fact that $m_{h_{2}^{0}}>10\,\text{GeV}$ also
restricts $\Delta m_{32}$, as seen in the left panel of Fig. 2. This plot
shows that $m_{h_{2}^{0}}>10$ GeV automatically implies $m_{h_{3}^{0}}\lesssim
40$ GeV. This justifies the absence of results near the theoretical upper
bound for the lighter neutral masses for $|u|<8\,\text{GeV}$. Notice that a
large fraction of the points lie below the upper bounds from LEP searches on
low-mass ranges and, thus, such light scalars could have escaped detection at
LEP. However, we also see that there is a sizeable portion of the points which
are excluded by the same dataset. From now on, we will only consider the
parameter space region below the LEP bound.

Turning now to the $H_{2}^{+}$ and $H_{1}^{++}$ charged scalars, we start by
looking at the Yukawa couplings of the lightest charged scalar $H_{2}^{\pm}$
(recall $H_{1}^{\pm}$ is the massless charged Goldstone boson) with the up-
($u$) and down-type ($d$) quarks. In general, these can be written as

$-\mathcal{L}_{H_{2}^{\pm}}=\dfrac{\sqrt{2}\,V_{ud}^{\text{CKM}}}{v}X_{2}H_{2}^{+}\overline{u}\left(m_{d}P_{R}-m_{u}P_{L}\right)d\,+\,\text{H.c.}\quad,$
(54)

in which $V_{ud}^{\text{CKM}}$ is the corresponding element of the Cabibbo-
Kobayashi-Maskawa (CKM) matrix. Since originally only the doublet $\Phi$
couples to quarks, the charged Yukawa interactions are obtained from the terms
featuring $\phi^{+}$, the upper $\Phi$ component. The real parameter $X_{2}$
is therefore the fraction of the $H_{2}^{+}$ mass eigenstate in $\phi^{+}$,
i.e.

$\phi^{+}=X_{1}H_{1}^{+}+X_{2}H_{2}^{+}+X_{3}H_{3}^{+}\,.$ (55)

|
---|---
Figure 4: Left: Allowed points in the $(X_{2}^{-1},m_{H_{2}^{+}})$ plane – see
Eqs. (54) and (55). The hatched region corresponds to the fraction of the
parameter space excluded by $b\rightarrow s\gamma$ constraints, as reported in
Ref. Arbey:2017gmh . The points for which the lightest doubly-charged scalar
mass $m_{H_{1}^{++}}$ is above (below) the ATLAS lower bound of 220 GeV
extracted from $W^{\pm}W^{\pm}$ searches are shown in blue (orange) – see text
for more details. Right: $X_{2}^{-1}$ vs. the triplet VEV parameter $u\lesssim
8$ GeV.

The structure of the Yukawa interactions in Eq. (54) is identical to those
obtained in the type-I 2HDM for the single charged scalar which appears in
such model, if one makes the correspondence $X_{2}\equiv 1/(\tan\beta)^{\rm
2HDM}$, where $(\tan\beta)^{\rm 2HDM}$ differs from the $\tan\beta$ we have
been considering – $(\tan\beta)^{\rm 2HDM}$ is the ratio of the 2HDM doublet
VEVs, whereas in our work $\tan\beta$ is the ratio of triplet VEVs. With this
correspondence established, we can use the bounds for the type-I 2HDM charged-
scalar interactions obtained in Ref. Arbey:2017gmh , which are presented as
exclusion regions in the $(m_{H^{+}},\tan\beta)$ plane. We plot our results in
the $(m_{H_{2}^{+}},X_{2}^{-1})$ plane as shown in the left panel of Fig. 4,
where the hatched region originate from the most restrictive bounds coming
from $B$ physics, namely from $b\rightarrow s\gamma$ measurements. We thus see
that the lightest charged scalar $H_{2}^{+}$ is not at all constrained by that
data. In 2HDM language, we would say that the whole scanned parameter space
corresponds to values of $\tan\beta$ above roughly 20. We also show in blue
(orange) the points for which the doubly-charged scalar mass $m_{H_{1}^{++}}$
is above (below) the experimental lower bound of $220\,\text{GeV}$ extracted
from $W^{\pm}W^{\pm}$ ATLAS searches Aaboud:2019 . Using Eq. (49), such bound
corresponds to $m_{H_{2}^{+}}>(220\,\text{GeV})/\sqrt{2}\simeq
156\,\text{GeV}$. From the right panel of Fig. 4, where $X_{2}^{-1}$ is
plotted against $u$, we see that $X_{2}^{-1}\gtrsim 20$ stems from the upper
limit on $u\lesssim 8$ GeV. Restricting ourselves to neutral scalar masses
above 10 GeV, which implies $u\gtrsim 2$ GeV (see Fig. 1), we can see from the
same plot that there is also an upper bound $X_{2}^{-1}\lesssim 100$.

##  5 Conclusions

Extensions of the SM with $Y=2$ triplet scalar fields are well motivated since
small neutrino masses can be generated via the type II seesaw mechanism. In
this work, we investigated the possibility that one might also obtain CP
violation in the leptonic sector via spontaneous symmetry breaking in such
models. We have found that a minimum of two triplet fields are required for
SCPV to occur, in a model including a continuous global symmetry, softly
broken by three terms with real coefficients. For comparison, SCPV for a
theory invariant under a global continuous symmetry is not possible in the
2HDM Ferreira:2010hy  or in the STM, even with generic soft-breaking terms
added to the potential. It is, however, already possible in a 3HDM – for
instance, with a U(1) symmetry under which two of the doublets remain
unchanged. Considering a global U(1) symmetry in the scalar sector allowed us
to simplify the potential and, when extended to the fermion sector, it could
perhaps generate interesting textures in the neutrino mass matrix. An in-depth
analysis of the scalar sector revealed an interesting aspect of the model:
when SCPV occurs, the 2STM has no decoupling limit, and two neutral scalars
are predicted to have masses lower than the electroweak scale and of the order
of the triplet VEVs $u$. We were able to obtain this result using matrix
theory theorems to estimate relations between the eigenvalues of the mass
matrices, and verified it through a numerical scan of the model’s parameter
space. This is a result which follows a similar conclusion for SCPV in the
2HDM, as shown in ref. Nebot:2020niz  and is due to the fact that the extra
minimisation conditions that a CP-violating minimum must obey limit the size
of more of the scalar potential’s mass parameters. In the 2STM, however, the
existence of non-decoupling when SCPV occurs is also caused by the smallness
of the triplets VEVs, necessary for compliance with electroweak precision
data. But SCPV is the key ingredient to bring about non-decoupling, since we
also verified that CP-conserving minima of the model do not share this non-
decoupling regime – in such a vacuum of the 2STM, even small triplet VEVs do
not imply the existence of scalars with masses below the electroweak scale.

We therefore conclude that attempting to simultaneously explain neutrino
masses and leptonic CP violation via spontaneous breaking of the CP symmetry
in the 2STM would imply the triplets having VEVs of $\mathcal{O}({\rm eV})$,
and consequently the existence of two neutral scalars with masses of the same
order. Those scalars would have interactions with the known fermions and gauge
bosons, and in particular with the neutrinos. The quark interactions should be
highly suppressed due to the fact that they all arise from the Yukawa term in
the lagrangean involving the doublet $\Phi$ – and since the SM Higgs-like
state $h^{0}_{4}$ is essentially aligned with the neutral, real component of
$\Phi$, all quark Yukawa interactions of the lighter scalars, $h^{0}_{2,3}$,
will be suppressed by off-diagonal elements of the rotation matrix between
weak and mass scalar eigenstates, which are at least of order
$\mathcal{O}(u/v)\sim 10^{-11}$. The gauge interactions of the lighter scalars
are not necessarily suppressed, since they will include contributions from the
triplet covariant derivative terms. However, we have shown, through our
numerical scans, that the lighter scalars can indeed have interactions with
gauge bosons highly suppressed – our scan had triplet VEVs of order GeV, but
the conclusion obviously holds for VEVs of much smaller orders of magnitude.
Of course, if these neutral scalars have masses of the eV order, their decays
to electroweak gauge bosons or fermions are anyway kinematically forbidden. It
would seem, then, that these neutral scalars could be thought of as possible
Dark Matter candidates. But in a reversal of the usual reasoning, one would
expect that the decays to neutrinos of these lighter scalars would be
relevant. In fact, for the neutrino mass explanation arising in this model to
be “natural”, one would expect the scalar-neutrino Yukawa couplings to be of
order $\mathcal{O}(1)$, which would therefore make the decay channel
$h^{0}_{2,3}\rightarrow\nu\bar{\nu}$ be significant, maybe even dominant,
precluding their stability, and thus invalidating their validity as Dark
Matter candidates.

The SCPV 2STM could however be a valid description of particle physics if the
triplet VEVs, while being small enough so that the model complies with
electroweak precision data, are of GeV order. A numerical scan of the model
then shows that there are regions of parameter space for which the lighter
scalars would have evaded detection at LEP; and for which the charged and
doubly charged scalars would have masses and couplings such that they would
not yet have been discovered. This raises the possibility that lighter scalars
than the 125 GeV Higgs boson may be discovered at LHC or elsewhere – one such
possibility (though not possible in this model, due to the masses and
couplings involved for the lighter scalars) would be the possible excess at 96
GeV observed by CMS Sirunyan:2018aui , which can be interpreted in several
ways in BSM theories Fox:2017uwr ; Biekotter:2017xmf ; Haisch:2017gql ;
Liu:2018ryo ; Biekotter:2019kde ; AguilarSaavedra:2020wmg . A full study of
the phenomenology of this model is therefore of interest to fully ascertain
the likelyhood of its being fully probed at LHC or future colliders.

###### Acknowledgements.

This work is supported by Fundação para a Ciência e a Tecnologia (FCT,
Portugal) through the projects UIDB/00777/2020, UIDP/00777/2020,
UIDB/00618/2020, CERN/FIS-PAR/ 0004/2019, CERN/FISPAR/0014/2019, PTDC/FIS-
PAR/29436/2017 and by HARMONIA project’s contract UMO-2015/18/M/ST2/00518. The
work of B.L.G. is supported by the FCT grant SFRH/BD/139165/2018.

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