aainstitutetext: Department of Mathematical Physics, Institute of Physics,
University of São Paulo,
R. do Matão 1371, São Paulo, SP 05508-090, Brazil bbinstitutetext: Department
of Physics and Enrico Fermi Institute, University of Chicago,
933 East 56th Street, Chicago, Illinois 60637, U.S.A.

# Neutrino Masses in the Mirror Twin Higgs with Spontaneous $\mathbb{Z}_{2}$
Breaking

Pedro Bittar b Davi B. Costa a Gustavo Burdman<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We introduce a mirror twin Higgs model with spontaneous $\mathbb{Z}_{2}$
symmetry breaking that ameliorates the constraints in twin Higgs cosmology
and, at the same time, generates the Standard Model neutrino masses. The model
features an $SU(2)$ triplet with hypercharge $1$ alongside its twin
counterpart. Spontaneous breaking of both $\mathbb{Z}_{2}$ and electroweak
symmetry occurs in the scalar sector. The Standard Model neutrinos acquire
small masses through the type-II seesaw mechanism. In contrast, their twin
counterparts acquire large masses, effectively addressing the dark radiation
problem in mirror twin Higgs scenarios. We study the impact of the model on
the $N_{\rm eff.}$ constraints, as well as on collider phenomenology.

##  1 Introduction

The standard model (SM) of particle physics is a successful quantum field
theory describing the interactions of all known elementary particles. However,
many questions are not addressed by the SM. Two of these are the stability of
the electroweak (EW) scale and the origin of neutrino masses. The Higgs sector
of the SM is quadratically sensitive to ultra-violet (UV) scales. Thus, if the
SM cutoff were to be set high above the electroweak (EW) scale, the
renormalization condition fixing the Higgs boson mass becomes fine tuned. A
solution to the problem is to postulate that the SM cutoff, and therefore new
physics, are not too far above the EW scale, typically not more than a loop
factor above, i.e. at the TeV scale. This so-called the EW hierarchy problem
is a motivation for introducing physics Beyond the Standard Model (BSM) at the
$\operatorname{\text{TeV}}$ scale Craig:2022uua . The traditional solutions to
the HP, Supersymmetry Carena:2002es  and composite Higgs models Panico:2015jxa
, predict the existence of new particles with masses of order
$\operatorname{\text{TeV}}$, some of which have QCD color charge. These are
called SM partners and are responsible for canceling the corresponding SM
contribution to the Higgs potential, making them good targets for collider
resonance searches.

On the other hand, since the discovery of neutrino oscillations
ParticleDataGroup:2022pth ; SNO:2002tuh ; Super-Kamiokande:1998kpq , it is
clear that the SM neutrinos must have nonzero masses, which the SM cannot
accommodate. For instance, the seesaw mechanism is a way to introduce new
physics that would lead to neutrino masses, as well as to an explanation of
their size compared to other fermion masses. There are three incarnations of
the seesaw mechanism, type-I Minkowski:1977sc ; Mohapatra:1979ia ;
Yanagida:1979as ; Gell-Mann:1979vob ; Glashow:1979nm ; Schechter:1980gr ,
type-II Schechter:1980gr ; Mohapatra:1980yp ; Lazarides:1980nt ;
Mandal:2022zmy , and type-III Foot:1988aq . The scale of the new physics can
be as low as the $\operatorname{\text{TeV}}$ and as high as
$10^{15}\operatorname{\text{GeV}}$, depending on the details.

The search for new physics associated with the $\operatorname{\text{TeV}}$
scale at the LHC has not resulted in any discovery so far. In particular,
states with color charges are particularly constrained by the LHC data. Thus,
an interesting possibility are neutral naturalness models as they provide an
alternative solution to the stability of the EW scale that rely on having
colorless partners Batell:2022pzc ; Craig:2022uua ; Burdman:2006tz . This is
the case in twin Higgs models Chacko:2005pe ; Chacko:2005vw ; Barbieri:2005ri
, where the stability of the Higgs boson mass is achieved by the presence of
colorless partners from a twin copy of the SM, supplemented by a discrete
$\operatorname{\mathbb{Z}_{2}}2$ symmetry. These new states are not subject to
the strong constraints from top/gauge partner searches at the LHC
Burdman:2014zta . In the mirror twin Higgs model (MTH), the SM is extended
with a mirror sector that contains a copy of all SM states and gauge
symmetries. The Higgs doublet is a pseudo-Nambu-Goldstone boson (pNGB) of a
spontaneously broken global symmetry. The $\operatorname{\mathbb{Z}_{2}}2$
acts as an exchange symmetry between the SM and twin sectors, ensuring that
the one loop corrections to the Higgs potential respect the original global
symmetry, therefore not reintroducing the UV sensitivity of the Higgs boson
mass present in the SM.

The current experimental bounds from the measurements of the Higgs boson
couplings at the LHC ATLAS:2021vrm ; CMS:2022dwd  and the Higgs invisible
decay width ATLAS:2023tkt  require the $\mathbb{Z}_{2}$ symmetry to be broken
at IR scales. This requirement means the twin sector breaking scale $f$ should
be somewhat higher than the EW vacuum expectation value (VEV), giving
$f/v\gtrsim 3$ Burdman:2014zta . The $\mathbb{Z}_{2}$ symmetry is expected to
be broken either explicitly Chacko:2005pe ; Harigaya:2019shz , spontaneously
Beauchesne:2015lva ; Beauchesne:2021opx , or radiatively Yu:2016bku .
Furthermore, the original MTH is severely constrained by cosmology
Chacko:2018vss ; Chacko:2016hvu ; Craig:2016lyx ; Csaki:2017spo . The presence
of twin-neutrinos and the twin-photon in the original model results in new
contributions to dark radiation at the Big Bang Nucleosynthesis (BBN)
ParticleDataGroup:2022pth ; Fields:2019pfx  and the Cosmic Microwave
Background (CMB) Planck:2018vyg  epochs. These contributions are constrained
by the observed effective number of relativistic degrees of freedom, with the
most recent Planck measurement giving $N_{\rm eff}=2.99_{-0.33}^{+0.34}$,
Planck:2018vyg , and from a recent combination of CMB with baryon acoustic
oscillation (BAO) data from DESI DESI:2024mwx : $N_{\rm eff}=3.10\pm 0.17$.

There are several proposed solutions to the dark radiation problem in the
cosmology of the MTH. One class of models proposes asymmetrically reheating
the SM bath by some preferential decays to SM states Chacko:2016hvu ;
Craig:2016lyx . In this work, we will focus on solutions that give large
masses to twin neutrinos so that they do not contribute to $\Delta N_{eff}$.
There are examples in the literature where large twin neutrino masses are
achieved with soft $\operatorname{\mathbb{Z}_{2}}2$ breaking and by
introducing right-handed neutrinos. Then, using the type-I seesaw mechanism,
one can reproduce light neutrino masses in the SM and heavy twin neutrinos
Abada:2007sm ; Bai:2015ztj ; Batell:2015aha ; Batell:2019ptb ; Bishara:2018sgl
; Holst:2023hff . This method largely reduces the tension with dark radiation
bounds. With the issue of dark radiation addressed, then it is possible to
investigate other MTH features such as the nature of dark matter and
mechanisms of baryogenesis Bittar:2023kdl ; Farina:2015uea ; Farina:2016ndq ;
Beauchesne:2020mih ; Curtin:2021spx ; Curtin:2021alk ; Alonso-Alvarez:2023bat
; Chacko:2021vin ; GarciaGarcia:2015pnn ; Holst:2023hff ; Bodas:2024idn ;
Craig:2015xla ; GarciaGarcia:2015fol ; Cheng:2018vaj ; Hochberg:2018vdo ;
Badziak:2019zys ; Koren:2019iuv ; Terning:2019hgj .

In this paper we present an MTH model with spontaneous breaking of the
$\mathbb{Z}_{2}$ symmetry, which results in large masses for the twin
neutrinos, as well as in the small masses for the SM neutrinos. For this
purpose, we add an $SU(2)$ triplet with hypercharge $1$, as well as its twin
copy. The vacuum expectation value of the scalar sector breaks
$\mathbb{Z}_{2}$ and electroweak symmetry spontaneously. Then, by the type-II
seesaw mechanism, the SM neutrinos get small masses, while their twin
counterparts get large masses controlled by the
$\operatorname{\mathbb{Z}_{2}}2$ breaking parameter, therefore mitigating the
dark radiation problem. In Section 2, we present the model and show the
spontaneous $\mathbb{Z}_{2}$ symmetry breaking pattern. Then, we investigate
the vacuum structure of the theory and its spectrum. We obtain neutrino masses
and discuss their phenomenology in Section 3.1, and discuss LHC bounds in
Section 3.2. Other constraints, such as those from domain wall and the
resulting twin Majoron from the spontaneous breaking of lepton number, are
discussed in Section 3.3. We conclude in Section 4.

##  2 Spontaneous $\mathbb{Z}_{2}$ breaking with the type-II Seesaw MTH

Following Chacko:2005pe , we assume the MTH model where the Higgs is a
(pseudo) Nambu-Goldstone boson (pNGB) of the global symmetry breaking
$SU(4)\times U(1)\rightarrow SU(3)\times U(1)$. The SM Higgs doublet $H_{A}$
is part of a complex scalar bi-doublet

$\displaystyle\begin{gathered}H=\left(H_{A},H_{B}\right)^{T},\vspace{0.2cm}\\\
H_{A}=\left(G_{A}^{+},h_{A}+iG_{A}^{0}\right)^{T},\hskip
28.45274ptH_{B}=\left(G_{B}^{+},h_{B}+iG_{B}^{0}\right)^{T}\end{gathered}$ (3)

which transforms under the fundamental representation of $SU(4)$. The indices
$A,B$ designate the SM particles and twin partners, respectively. The global
symmetry is broken explicitly by gauge and Yukawa interactions; however, due
to the $\operatorname{\mathbb{Z}_{2}}2$ symmetry, the effective potential is
accidentally $SU(4)\times U(1)$ symmetric at the one loop level. We introduce
additional $SU(2)_{A,B}$ triplets to the scalar sector of the theory,
$\Delta_{A,B}$. The SM triplet has hypercharge $Y_{\Delta}=1$, and the triplet
partner has twin hypercharge $\widetilde{Y}_{\Delta}=1$. We can write the
triplet fields as

$\Delta_{A}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}\Delta_{A}^{+}&\sqrt{2}\Delta_{A}^{++}\\\
\sqrt{2}\Delta_{A}^{0}&-\Delta_{A}^{+}\end{array}\right),\qquad\Delta_{B}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}\Delta_{B}^{+}&\sqrt{2}\Delta_{B}^{++}\\\
\sqrt{2}\Delta_{B}^{0}&-\Delta_{B}^{+}\end{array}\right),$ (4)

where $\Delta_{A,B}^{++},\Delta_{A,B}^{+}$ and $\Delta_{A,B}^{0}$ are the
charge/twin charge eigenstates with $Q=(2,1,0)$ respectively. If we ignore
quarks, the most general Lagrangian respecting the $\mathbb{Z}_{2}$ symmetry
is given by

$\displaystyle\begin{split}-{\cal
L}&=-(DH_{A})^{\dagger}DH_{A}-(DH_{B})^{\dagger}DH_{B}+m^{2}H^{\dagger}H+\lambda(H^{\dagger}H)^{2}+\lambda^{\prime}H^{\dagger}_{A}H_{A}H^{\dagger}_{B}H_{B}\\\
&-(D\Delta_{A})^{\dagger}D\Delta_{A}-(D\Delta_{B})^{\dagger}D\Delta_{B}+M^{2}({\rm
Tr}\Delta_{A}^{\dagger}\Delta_{A}+{\rm
Tr}\Delta_{B}^{\dagger}\Delta_{B})+\lambda_{1}\left({\rm
Tr}\Delta_{A}^{\dagger}\Delta_{A}+\right.\\\ &\left.+{\rm
Tr}\Delta_{B}^{\dagger}\Delta_{B}\right)^{2}+\lambda_{2}\left({\rm
Tr}\Delta_{A}^{\dagger}\Delta_{A}\right)\left({\rm
Tr}\Delta_{B}^{\dagger}\Delta_{B}\right)+\lambda_{3}\left({\rm
Tr}\Delta_{A}^{\dagger}\Delta_{A}\Delta_{A}^{\dagger}\Delta_{A}+{\rm
Tr}\Delta_{B}^{\dagger}\Delta_{B}\Delta_{B}^{\dagger}\Delta_{B}\right)\\\
&+\lambda_{4}H^{\dagger}H\left({\rm Tr}\Delta_{A}^{\dagger}\Delta_{A}+{\rm
Tr}\Delta_{B}^{\dagger}\Delta_{B}\right)+\lambda_{5}\left(H_{A}^{\dagger}\Delta_{A}\Delta_{A}^{\dagger}H_{A}+H_{B}^{\dagger}\Delta_{B}\Delta_{B}^{\dagger}H_{B}\right)\\\
&+\left\\{b^{*}\left(H_{A}^{\dagger}\Delta_{A}H_{A}^{*}+H_{B}^{\dagger}\Delta_{B}H_{B}^{*}\right)+\kappa\left(\ell^{T}_{A}\epsilon\Delta_{A}\epsilon\ell_{A}+\ell^{T}_{B}\epsilon\Delta_{B}\epsilon\ell_{B}\right)+{\rm
h.c.}\right\\}.\end{split}$ (5)

where $\epsilon_{12}=-\epsilon_{21}=1$. In addition to the $SU(4)$ preserving
quadratic term for the doublets with coefficient $m^{2}$, we have an $SU(4)$
preserving triplet mass term with coefficient $M^{2}$, as well as an $SU(4)$
breaking but $\operatorname{\mathbb{Z}_{2}}2$ preserving quartic coupling
$\lambda^{\prime}$. In the third line in (5) above, we have the triplet self-
interactions, whereas, in the fourth line, we present the possible
renormalizable and $\operatorname{\mathbb{Z}_{2}}2$ preserving doublet-triplet
interactions. Finally, in the last line, the first term corresponds to the
cubic doublet-triplet couplings. The last term in (5) denotes the triplet
interactions to the visible and twin left-handed leptons. The last two terms
will be responsible for neutrino masses in both sectors. If we choose the
dimensionful parameter $b$ to be small enough, it is possible to implement the
type-II seesaw model for neutrino masses. The crucial point is that the
spontaneous $\operatorname{\mathbb{Z}_{2}}2$ breaking will result in a
significant VEV for the twin triplet $\Delta_{B}$. In contrast, the visible
sector triplet $\Delta_{A}$ has a practically vanishing VEV. If we can arrange
the potential in (5) to accommodate this situation, we would not only break
the $\operatorname{\mathbb{Z}_{2}}2$ symmetry spontaneously but also give
large masses to the twin neutrinos. At the same time, by a judicious choice of
a value of $b$, we can get small enough visible neutrino masses.

We want to investigate the minimum of the potential associated with this model
and show that this minimum can produce spontaneous $\mathbb{Z}_{2}$ and
electroweak symmetry breaking. However, it is hard to find closed analytical
formulas in full generality. It will be advantageous to start by studying the
case in which $b$ is small compared with the other dimensional parameters of
the model since, in this case, the potential greatly simplifies, and we can
find simple expressions that hold to first order in $b$.

Before analyzing the minima of the scalar potential in this limit, it is
useful to consider a restricted set of operators to see how spontaneous
$\operatorname{\mathbb{Z}_{2}}2$ breaking can be obtained in the theory. For
the twin Higgs quadratic terms, the $\operatorname{\mathbb{Z}_{2}}2$ symmetry
makes the global $SU(4)$ an accidental symmetry at the 1-loop level. However,
$SU(4)$ is expected to be explicitly broken in the quartic terms by gauge
interactions, the Yukawa couplings, and possibly BSM effects Chacko:2005vw .
Then, it is natural to write the $\operatorname{\mathbb{Z}_{2}}2$-preserving
and $SU(4)$-violating operator

$\lambda_{5}\left(H_{A}^{\dagger}\Delta_{A}\Delta_{A}^{\dagger}H_{A}+H_{B}^{\dagger}\Delta_{B}\Delta_{B}^{\dagger}H_{B}\right).$
(6)

If we now assume that the scalar fields acquire VEVs along the neutral field
direction, we can have

$|\langle H_{A}\rangle|^{2}=v^{2},\qquad|\langle
H_{B}\rangle|^{2}=f^{2},\qquad|\langle\Delta_{A}\rangle|^{2}=V^{2}\sim
0,\qquad|\langle\Delta_{B}\rangle|^{2}=F^{2}.$ (7)

We are interested in a potential( with the proper structure to allow for
$V^{2}\sim 0$, as this vacuum structure can lead to small neutrino masses as
we demonstrate in Section 3.1. Taking the vevs of the triplets as in (7), we
obtain the effective $\operatorname{\mathbb{Z}_{2}}2$ breaking term

$m^{2}|H|^{2}+\lambda_{5}\left(H_{A}^{\dagger}\Delta_{A}\Delta_{A}^{\dagger}H_{A}+H_{B}^{\dagger}\Delta_{B}\Delta_{B}^{\dagger}H_{B}\right)\rightarrow
m^{2}h_{A}^{2}+\left(m^{2}+\lambda_{5}F^{2}\right)h_{B}^{2},$ (8)

where $h_{A,B}$ are the radial modes of the doublets. Assuming that both
$m^{2}$ and $\lambda_{5}$ are negative, the overall result is a net increase
of the twin sector breaking scale compared to the SM, as desired. Thus, we see
that including (6) in the lagrangian (5), among other $SU(4)$-preserving and
-violating operators, will allow us to look for phases that have the
appropriate vacuum structure we seek, i.e. that in (7).

Assuming $v\neq 0,f\neq 0$ and $F\neq 0$, we can expand the fields around
their VEVs and minimize the potential to get

$\displaystyle-2bV+m^{2}+2(f^{2}+v^{2})\lambda+(F^{2}+V^{2})\lambda_{4}+V^{2}\lambda_{5}+f^{2}\lambda^{\prime}$
$\displaystyle\,=0,$ (9a) $\displaystyle-
bv^{2}+V\left(M^{2}+F^{2}(2\lambda_{1}+\lambda_{2})+2V^{2}\lambda_{13}+f^{2}\lambda_{4}+v^{2}\lambda_{45}\right)$
$\displaystyle\,=0,$ (9b)
$\displaystyle-2bF+m^{2}+2(f^{2}+v^{2})\lambda+(F^{2}+V^{2})\lambda_{4}+F^{2}\lambda_{5}+v^{2}\lambda^{\prime}$
$\displaystyle\,=0,$ (9c) $\displaystyle-
bf^{2}+F\left(M^{2}+V^{2}(2\lambda_{1}+\lambda_{2})+2F^{2}\lambda_{13}+v^{2}\lambda_{4}+f^{2}\lambda_{45}\right)$
$\displaystyle\,=0,$ (9d)

where we defined $\lambda_{13}=\lambda_{1}+\lambda_{3}$ and
$\lambda_{45}=\lambda_{4}+\lambda_{5}$. From (9b), if $b=0$, we can see that
$V=0$ is an extremum of the potential. Keeping $b=0$, we obtain a relationship
between the different VEVs by subtracting (9c) from (9a),

$F^{2}=\frac{\lambda^{\prime}}{\lambda_{5}}(f^{2}-v^{2}).$ (10)

The critical point with $V=0$ in the $b=0$ limit is

$\displaystyle
v^{2}=\frac{M^{2}(\lambda^{\prime}\lambda_{45}+2\lambda\lambda_{5})-m^{2}(2\lambda^{\prime}\lambda_{13}+\lambda_{5}\lambda_{45})}{2\lambda\left(4\lambda^{\prime}\lambda_{13}+\lambda_{5}^{2}\right)+2\lambda^{\prime}(\lambda^{\prime}\lambda_{13}-\lambda_{4}\lambda_{45})},$
(11a) $\displaystyle
f^{2}=\frac{M^{2}(\lambda^{\prime}\lambda_{4}-2\lambda\lambda_{5})-m^{2}(2\lambda^{\prime}\lambda_{13}-\lambda_{4}\lambda_{5}))}{2\lambda\left(4\lambda^{\prime}\lambda_{13}+\lambda_{5}^{2}\right)+2\lambda^{\prime}(\lambda^{\prime}\lambda_{13}-\lambda_{4}\lambda_{45})},$
(11b) $\displaystyle
F^{2}=\frac{\lambda^{\prime}\left(m^{2}(\lambda_{4}+\lambda_{45})-M^{2}(4\lambda+\lambda^{\prime})\right)}{2\lambda\left(4\lambda^{\prime}\lambda_{13}+\lambda_{5}^{2}\right)+2\lambda^{\prime}(\lambda^{\prime}\lambda_{13}-\lambda_{4}\lambda_{45})}.$
(11c)

Figure 1: Vacuum expectation values of the neutral scalars in the model as a
function of the $\lambda_{5}$ parameter of the potential (5). Through
spontaneous $\operatorname{\mathbb{Z}_{2}}2$ breaking, $\lambda_{5}$ controls
the separation between the SM and twin sector. The VEVs for the doublets
$H_{A}$ and $H_{B}$ are denoted as $v$ and $f$, while the VEVs for the
triplets $\Delta_{A}$ and $\Delta_{B}$ are represented by $V$ and $F$
respectively. The parameters are chosen to have a realistic hierarchy of VEVs,
with $f/v\approx 3$, as shown by the gray dashed line, and reproduce the
correct SM Higgs mass.

As argued before, the spontaneous $\operatorname{\mathbb{Z}_{2}}2$ breaking is
controlled by the $\lambda_{5}$ parameter. Indeed, for $\lambda_{5}\rightarrow
0$, one can see from (11) that $v^{2}\rightarrow f^{2}$. Assuming negative
$\lambda_{5}$, a nonzero value can generate the necessary hierarchy between
the doublets, as shown in Figure 1. The necessary hierarchy between the
doublet VEVs, $f/v\gtrsim 3$, can be achieved for negative $\lambda_{5}$.
Taking a negative $\lambda^{\prime}$, from $\eqref{eq:VEVs_relation}$, the
twin-triplet VEV can be kept larger than $f$ by having
$\lambda^{\prime}<\lambda_{5}$. In the figure, the parameters of the potential
are fixed to have a realistic scalar spectrum, including the
$125\operatorname{\text{GeV}}$ SM Higgs and the scalar masses that are allowed
by experimental constraints. These constraints are imposed in Section 3.2
below.

When the triplet scalar coupling $b$ is nonzero, the VEV of $\Delta_{A}$ is
non-vanishing and proportional to $b$. Assuming that $b$ is much smaller than
the other scales of the theory, we have:

$\displaystyle
V=b\left(\frac{v^{2}}{F^{2}(\lambda_{2}-2\lambda_{3})+(v^{2}-f^{2})\lambda_{5}}\right)+\mathcal{O}(b^{2}/v)$
(12)

A vacuum expectation value for the triplet contributes to the $T$ parameter
that is constrained by electroweak precision measurements at LEP and the LHC.
These constraints put a bound on the scale of the parameter $b$ as we discuss
in Section 3.2.

One useful relation is obtained by inverting the expressions for $v^{2}$ and
$f^{2}$ in (7) to write $m$ and $M$,

$\displaystyle
m^{2}=\frac{\tilde{\lambda}\lambda_{4}-2\lambda\lambda_{5}}{\lambda_{5}}v^{2}-\frac{\tilde{\lambda}\lambda_{4}+\lambda_{5}(2\lambda+\tilde{\lambda})}{\lambda_{5}}f^{2},$
(13a) $\displaystyle
M^{2}=\frac{2\overline{\lambda}-\lambda_{4}\lambda_{5}}{\lambda_{5}}v^{2}-\frac{2\overline{\lambda}+\lambda_{5}\lambda_{45}}{\lambda_{5}}f^{2}.$
(13b)

Now, we can compute the mass eigenvalues and analyze the spectrum of the
model. The particle content of the model includes two CP-even, two CP-odd, and
six charged $Q={\pm 1,\pm 2}$ fields for each sector. Due to electroweak and
twin-electroweak breaking, the goldstone bosons get eaten by the six physical
gauge and twin-gauge bosons. Thus, the physical fields can be divided as

  * •

CP-even: $h,H,\Delta_{A_{s}}^{0},\delta_{B_{s}}^{0}$.

  * •

CP-odd: $\Delta_{A_{a}}^{0},J_{B}\,\,(\text{Majoron})$.

  * •

Charged:
$\Delta_{A}^{\pm},\Delta_{A}^{\pm\pm},\delta_{B}^{\pm},\Delta_{B}^{\pm\pm}$.

  * •

Goldstone bosons: $G_{A}^{0},G_{A}^{\pm},S_{B}^{0},S_{B}^{\pm}$.

The fields $h,H,\delta_{B_{s}}^{0},J_{B},S_{B}^{0},S_{B}^{\pm}$ and
$\delta_{B}^{\pm}$ are obtained after redefining the states from the
interaction to the physical basis. The fields get rotated as
$\\{h_{A},h_{B},\Delta_{B_{s}}^{0}\\}\rightarrow\\{h,H,\delta_{B_{s}}^{0}\\}$,
$\\{G_{B}^{\pm},\Delta_{B}^{\pm}\\}\rightarrow\\{S_{B}^{\pm},\delta_{B}^{\pm}\\}$
and $\\{G_{B}^{0},\Delta_{B_{a}}^{0}\\}\rightarrow\\{S_{B}^{0},J_{B}\\}$.
Here, we neglect the mixing induced by the $b$ coupling in the CP-even block
and $Q_{B}=\pm 1$ block as the mixing angle is small. However, for the CP-odd
twin block we must take the $b$ term into account since $J_{B}$ is a light
Majoron as we describe below.

In the case of the triplet $\Delta_{A}$, the only mixing term appears due to
the triple scalar coupling in (5) and is proportional to $b$. The neutral
triplet fields are divided between CP even and CP odd components as
$\Delta_{A}^{0}=\Delta_{A_{s}}^{0}+i\Delta_{A_{a}}^{0}$. Since $b$ is much
smaller than other scales of the theory, the mixing angle of $\Delta_{A_{s}}$
with $h_{A}$ is negligible, and the masses are

$\displaystyle
m_{\Delta_{A}^{\pm\pm}}^{2}=\frac{(\alpha+\lambda_{5})f^{2}-\alpha
v^{2}}{2},\quad\alpha\equiv\frac{\lambda^{\prime}(\lambda_{2}-2\lambda_{3})}{\lambda_{5}}$
(14a) $\displaystyle
m_{\Delta_{A}^{\pm}}^{2}=m_{\Delta_{A}^{\pm\pm}}^{2}+\frac{\lambda_{5}}{4}v^{2},$
(14b) $\displaystyle
m_{\Delta_{A_{a}}^{0}}^{2}=m_{\Delta_{A}^{\pm\pm}}^{2}+\frac{\lambda_{5}}{2}v^{2},$
(14c) $\displaystyle m_{\Delta_{A_{s}}^{0}}^{2}=m_{\Delta_{A_{a}}^{0}}^{2}.$
(14d)

In the twin sector, the $\Delta_{B}^{\pm\pm}$ component of the twin triplet
does not get mixed with other scalar components. This is also true when $b=0$
for the $\Delta_{B_{a}}^{0}$ field. Therefore, their masses are obtained
simply by the second derivative of the potential,

$\displaystyle
m_{\Delta_{B}^{\pm\pm}}^{2}=-\frac{2\lambda^{\prime}\lambda_{3}+\lambda_{5}^{2}}{2\lambda_{5}}f^{2}+\frac{\lambda^{\prime}\lambda_{3}}{\lambda_{5}}v^{2},$
(15) $\displaystyle m_{\Delta_{B_{a}}^{0}}^{2}=0\quad(\text{if $b=0$}).$ (16)

While the doubled charged scalar has a mass of order
$\operatorname{\text{TeV}}$, the massless pseudoscalar twin state
$\Delta_{B_{a}}^{0}$ is a Majoron - the nambu goldstone (NGB) boson of
spontaneously broken lepton number. In the absence of the $b$ coupling, lepton
number is conserved in the theory as $\Delta_{A},\Delta_{B}$ have lepton/twin-
lepton number $L=2$ and $\widetilde{L}=2$ respectively. When $\Delta_{B}$
acquires a VEV, twin lepton number is spontaneously broken and a twin Majoron
is generated only in the twin sector. When the triple scalar coupling is
nonzero, the twin pseudoscalars $\\{G_{B}^{0},\Delta_{B_{a}}^{0}\\}$ mix,
leading to the mass matrix,

$\mathcal{M}^{2}(\\{G_{B}^{0},\Delta_{B_{a}}^{0}\\})=\begin{pmatrix}2\sqrt{2}bF&-\sqrt{2}bf\\\
-\sqrt{2}bf&\frac{bf^{2}}{\sqrt{2}F}\end{pmatrix}.$ (17)

We can rewrite the fields in the physical basis

$\displaystyle
S_{B}^{0}=\cos\alpha\,G_{B}^{0}+\sin\alpha\,\Delta_{B_{a}}^{0},$ (18)
$\displaystyle J_{B}=\cos\alpha\,\Delta_{B_{a}}^{0}-\sin\alpha\,G_{B}^{0},$
(19)

to find the masses of $S_{B}^{0}$ and $J_{B}$ and the mixing angle $\alpha$,

$\displaystyle m_{S_{B}^{0}}^{2}=0$ (20) $\displaystyle
m_{J_{B}}^{2}=b\frac{(f^{2}+4F^{2})}{\sqrt{2}F}$ (21) $\displaystyle\sin
2\alpha=-\frac{fF}{f^{2}+4F^{2}}~{},$ (22)

where the $S_{B}^{0}$ is the NGB eaten by the twin gauge sector, and $J_{B}$
is the pseudo-NGB associated with the spontaneous breaking of lepton number,
the physical Majoron. We discuss the phenomenological consequences of the twin
Majoron in Section 3.

The mass matrix of the CP even fields $\\{h_{A},h_{B},\Delta_{B_{s}}^{0}\\}$
is given by

$\displaystyle\mathcal{M}^{2}({\\{h_{A},h_{B},\Delta_{B_{s}}^{0}\\}})=\begin{pmatrix}2\lambda
v^{2}&(2\lambda+\lambda^{\prime})vf&(\lambda_{4}\eta)vf\\\
(2\lambda+\lambda^{\prime})vf&2f^{2}\lambda&(\lambda_{45}\eta)f^{2}\\\
(\lambda_{4}\eta)vf&(\lambda_{45}\eta)f^{2}&2\lambda_{13}\eta^{2}f^{2}\end{pmatrix},$
(23)

where we have defined
$\eta^{2}\equiv\frac{\lambda^{\prime}(1-\xi)}{\lambda_{5}}=\frac{F^{2}}{f^{2}}+\mathcal{O}(b^{2}/f^{2})$
and $\xi\equiv\frac{v^{2}}{f^{2}}$. In general, the expressions for the mass
eigenvalues are complicated and not very illuminating. However, the mixing
matrix is constrained by the Higgs invisible decay width measurements, as we
discuss in Section 3. To limit the mixing of the Higgs, we need to tune the
quartic couplings that control the off-diagonal terms of (23),

$\displaystyle 2\lambda+\lambda^{\prime}\lesssim\frac{v^{2}}{f^{2}},\hskip
28.45274pt\lambda_{4}\eta\lesssim\frac{v^{2}}{f^{2}}.$ (24)

After tuning the couplings as in (24), the SM Higgs is aligned with the
$h_{A}$ state up to corrections of order $v/f$. Then, we can redefine $h_{B}$
and $\Delta_{B_{s}}^{0}$ as the physical states $H$ and $\Delta_{B_{s}}^{0}$
by

$\displaystyle H=\cos\theta\,h_{B}+\sin\theta\,\Delta_{B_{s}}^{0},$ (25)
$\displaystyle\delta_{B_{s}}^{0}=\cos\theta\Delta_{B_{s}}^{0}-\sin\theta
h_{B},$ (26)

Therefore, up to order $v/f$, we have the following masses and mixing angle
$\theta$,

$\displaystyle m_{h}^{2}=2\lambda v^{2}+\mathcal{O}(v^{2}\sqrt{\xi})$ (27)
$\displaystyle
m_{H}^{2}=f^{2}\left[(\lambda+\eta^{2}\lambda_{13})-\sqrt{(\lambda-\eta^{2}\lambda_{13})^{2}+\eta^{2}\lambda_{5}^{2}}\right]+\mathcal{O}(v^{2}\sqrt{\xi})$
(28) $\displaystyle
m_{\delta_{B_{s}}^{0}}^{2}=f^{2}\left[(\lambda+\eta^{2}\lambda_{13})+\sqrt{(\lambda-\eta^{2}\lambda_{13})^{2}+\eta^{2}\lambda_{5}^{2}}\right]+\mathcal{O}(v^{2}\sqrt{\xi})$
(29) $\displaystyle\sin
2\theta=\frac{\eta\lambda_{5}}{\sqrt{(\lambda-\eta^{2}\lambda_{13})^{2}+\eta^{2}\lambda_{5}^{2}}}+\mathcal{O}(v^{2}\sqrt{\xi}).$
(30)

The mass matrix with mixing fields $G_{B}^{\pm}$ and $\Delta_{B}^{\pm}$ is

$\mathcal{M}^{2}({\\{G_{B}^{\pm},\Delta_{B}^{\pm}\\}})=\begin{pmatrix}-\frac{1}{2}\lambda_{5}f^{2}&\lambda_{5}\eta
vf\\\ \lambda_{5}\eta vf&\hskip
5.69046pt\lambda^{\prime}f^{2}\left(1-\xi\right)\end{pmatrix}$ (31)

Now, we can rotate the fields to the mass basis,

$\displaystyle
S_{B}^{\pm}=\cos\varphi\,G_{B}^{\pm}+\sin\varphi\,\Delta_{B}^{\pm},$ (32)
$\displaystyle\delta_{B}^{\pm}=\cos\varphi\,\Delta_{B}^{\pm}-\sin\varphi\,G_{B}^{\pm},$
(33)

to find the mass eigenvalues and mixing angle

$\displaystyle m_{S_{B}^{\pm}}^{2}=0,$ (34) $\displaystyle
m_{\delta^{\pm}}^{2}=-\frac{\lambda_{5}+2\lambda^{\prime}}{4}f^{2}+\frac{\lambda^{\prime}}{2}v^{2},$
(35) $\displaystyle\sin 2\varphi=\frac{2\sqrt{2}\lambda_{5}\eta
f^{2}}{(\lambda_{5}+2\lambda^{\prime})f^{2}-2\lambda^{\prime}v^{2}}.$ (36)

Finally, the masses of the eaten SM and neutral twin Goldstone bosons are
zero, as expected.

$m_{G_{A}^{0}}^{2}=m_{G_{A}^{\pm}}^{2}=m_{G_{B}^{0}}^{2}=0.$ (37)

We conclude this section by commenting on how realistic the model is regarding
the range of possible parameters and how commonly occurring the symmetry-
breaking behavior described above is. As discussed above, some parameters must
be tuned so that the CP even states do not mix too much. These conditions are
given by (24). Because of this, the quartic coupling $\lambda$ should be close
to the SM value up to order $v/f$. Additionally, both $\lambda^{\prime}$ and
$\lambda_{5}$ must be negative to obtain a lighter SM VEV. The model also
requires that the dimensionfull parameter b should be small so that the SM
triplet VEV is much below the EW scale, $V\ll v$. However, since setting b to
zero restores lepton number symmetry in the theory, we see that a small value
for b is still technically natural tHooft:1979rat , and radiatively stable.
Given these conditions, it is straightforward to obtain a realistic spectrum
of scalar and neutrino masses compatible with all constraints on the model.
Table 1 shows one benchmark point we use throughout the paper.

Parameter | Value |  Particle |  Mass
---|---|---|---
$m^{2}$ | $-(310\,\mathrm{GeV})^{2}$ | $m_{\nu_{A}}$ | $\sim 0.12\,\mathrm{eV}$
$M^{2}$ | $-(640\,\mathrm{GeV})^{2}$ | $m_{J_{B}}$ | $180\,\mathrm{MeV}$
$\lambda$ | $0.13$ | $m_{\nu_{B}}$ | $280\,\operatorname{\text{MeV}}$
$\lambda^{\prime}$ | $-0.15$ | $m_{h}$ | $125\,\mathrm{GeV}$
$\lambda_{1}$ | $0.98$ | $m_{\delta_{B}^{\pm}}$ | $370\,\mathrm{GeV}$
$\lambda_{2}$ | $0.8$ | $m_{H}$ | $370\,\mathrm{GeV}$
$\lambda_{3}$ | $-0.58$ | $m_{\Delta^{\pm\pm}_{B_{s}}}$ | $775\,\mathrm{GeV}$
$\lambda_{4}$ | $0.11$ | $m_{\delta^{0}_{B_{s}}}$ | $894\,\mathrm{GeV}$
$\lambda_{5}$ | $-0.071$ | $m_{\Delta^{0}_{A_{s,a}}}$ | $1\,\operatorname{\text{TeV}}$
$b$ | $10\,\mathrm{keV}$ | $m_{\Delta^{\pm}_{A_{s}}}$ | $\sim 1\,\operatorname{\text{TeV}}$
$\kappa$ | $\sim 4\crossproduct 10^{-4}$ | $m_{\Delta^{\pm\pm}_{A_{s}}}$ | $\sim 1\,\operatorname{\text{TeV}}$
Table 1: Benchmark point chosen that satisfy all the constraints. On the left,
we show the values of the parameters of the potential, and on the right, the
resulting mass spectrum of the model.

Just as all other twin Higgs models, this one solves the little hierarchy
problem, but it also maintains (almost ) exactly111We will need to introduce a
very small explicit breaking of the $\operatorname{\mathbb{Z}_{2}}2$ symmetry
in order to avoid a domain wall problem. This is addressed in Section 3.3. the
$\operatorname{\mathbb{Z}_{2}}2$ symmetry in the potential. Additionally,
including triplets with couplings to neutrinos equips us with the means to
address the neutrino mass problem through the type-II seesaw model. In the
next section, we show how the light SM neutrino masses can be obtained and how
the twin neutrinos are kept heavy as a consequence of having a large twin
triplet VEV. Then, we proceed to explore other phenomenological aspects of the
model.

##  3 Phenomenology

In this section, we examine the phenomenological implications of the model. We
consider a combination of neutrino and collider phenomenology from triplet
scalars in type-II seesaw models alongside novel features within the twin
sector, including the presence of the light twin Majoron impacting cosmology.
Finally, we discuss the formation of a domain wall network and show that it is
very simple to implement a way of avoiding this problem.

###  3.1 Neutrino and Twin Neutrino Masses

As we have seen, the twin triplet $\Delta_{B}$ acquires a large vacuum
expectation value of the order of the $\operatorname{\text{TeV}}$ scale, while
$\Delta_{A}$ has a much smaller VEV proportional to the $b$ coupling. As shown
in Figure 2, this hierarchy allows us to have large twin-neutrino masses and
small SM neutrino masses. After EW and $\mathbb{Z}_{2}$ spontaneous symmetry
breaking we have

$\displaystyle\mathcal{L}$
$\displaystyle\supset\frac{V}{\sqrt{2}}\kappa_{ij}\overline{\nu_{A}^{c}}_{,i}\nu_{A,j}+\frac{F}{\sqrt{2}}\kappa_{ij}\overline{\nu_{B}^{c}}_{,i}\nu_{B,j}~{},$
(38)

where $i,j$ are flavor indices.

Figure 2: Neutrino mass generation from the interaction with the triplets. On
the SM side (diagram on the left), the type-II seesaw mechanism is
implemented. The SM neutrinos, $\nu_{A}$, are much lighter than the twin
neutrinos since, in addition to the triplet mass, there is an extra
suppression factor coming from the $b$ coupling. On the twin neutrino side,
the VEV of the triplet is much larger and results in a large twin neutrino
mass directly (diagram on the right).

Therefore, we can write the following neutrino and twin-neutrino mass
matrices,

$\displaystyle M_{\nu_{A}}=2\kappa V=2\kappa b\frac{v^{2}}{m_{\Delta}^{2}},$
(39) $\displaystyle M_{\nu_{B}}=2\kappa F=2\kappa
f\sqrt{\frac{\lambda^{\prime}}{\lambda_{5}}(1-\xi)}.$ (40)

where $m_{\Delta}^{2}\equiv
F^{2}(\lambda_{2}-2\lambda_{3})+(v^{2}-f^{2})\lambda_{5}$.

For the SM neutrinos, we can express the free parameters in $\kappa$ as
functions of the neutrino oscillation parameters and the triplet VEV,

$\displaystyle\kappa=\frac{1}{2V}(U_{PMNS})^{*}m_{\nu}^{\rm diag}(U_{PMNS})$
(41)

The PMNS matrix in terms of the oscillation parameters and Majorana phases is
given by

$U_{\text{PMNS}}=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}&c_{23}c_{13}\end{pmatrix}\begin{pmatrix}e^{i\alpha_{1}/2}&0&0\\\
0&e^{i\alpha_{2}/2}&0\\\ 0&0&1\end{pmatrix}$ (42)

with $c_{ij}=\cos{\theta_{ij}}$, $s_{ij}=\sin{\theta_{ij}}$, and $\delta$,
$\alpha_{1}$, and $\alpha_{2}$ are the CP-violating phase and Majorana phases,
respectively. For the Normal Hierachy (NH) of neutrino masses, $m_{1}\ll
m_{2}<m_{3}$, the full set of oscillation parameters is

$\displaystyle\text{NH:}\quad\\{m_{\nu_{1}},\Delta m^{2}_{21},\Delta
m^{2}_{31},\theta_{12},\theta_{13},\theta_{23},\delta,\alpha_{1},\alpha_{2}\\},$
(43) $\displaystyle m_{\nu_{2}}=\sqrt{m_{\nu_{1}}+\Delta m^{2}_{21}},\qquad
m_{\nu_{3}}=\sqrt{m_{\nu_{1}}+\Delta m^{2}_{31}}.$ (44)

Conversely, the parameters of the Inverted Hierachy (IH), $m_{3}\ll
m_{2}<m_{1}$, are

$\displaystyle\text{IH:}\quad\\{m_{\nu_{3}},\Delta m^{2}_{21},\Delta
m^{2}_{32},\theta_{12},\theta_{13},\theta_{23},\delta,\alpha_{1},\alpha_{2}\\},$
(45) $\displaystyle m_{\nu_{1}}=\sqrt{m_{\nu_{3}}+\Delta m^{2}_{32}+\Delta
m^{2}_{21}},\qquad m_{\nu_{3}}=\sqrt{m_{\nu_{3}}+\Delta m^{2}_{32}}.$ (46)

These parameters can be determined from the combined analysis of the latest
neutrino oscillation data as shown in Table 2. Even with complete knowledge of
the oscillation parameters, oscillation data does not fix the overall neutrino
scale. To estimate the effective scale of neutrino masses, we can use the
upper bound value from Planck data from fitting the $\Lambda$CDM model with a
free $\sum m_{\nu}$ parameter,

$\displaystyle m_{\nu}^{2}={\sum}_{i}|U^{ei}_{\rm
PMNS}|^{2}m_{i}^{2}<0.12\operatorname{\text{eV}}.$ (47)

| Normal Ordering (best fit) |  Inverted Ordering ($\Delta\chi^{2}=2.7$)
---|---|---
|  bfp $\pm 1\sigma$ |  $3\sigma$ range |  bfp $\pm 1\sigma$ |  $3\sigma$ range
$\sin^{2}\theta_{12}$ | $0.304_{-0.012}^{+0.013}$ | $0.269\to 0.343$ | $0.304_{-0.012}^{+0.013}$ | $0.269\to 0.343$
$\sin^{2}\theta_{23}$ | $0.570_{-0.024}^{+0.018}$ | $0.407\to 0.618$ | $0.575_{-0.021}^{+0.017}$ | $0.411\to 0.621$
$\sin^{2}\theta_{13}$ | $0.02221_{-0.00062}^{+0.00068}$ | $0.02034\to 0.02430$ | $0.02240_{-0.00062}^{+0.00062}$ | $0.02053\to 0.02436$
$\delta_{CP}/^{\circ}$ | $195_{-25}^{+51}$ | $107\to 403$ | $286_{-32}^{+27}$ | $192\to 360$
$\dfrac{\Delta m^{2}_{21}}{10^{-5}\operatorname{\text{eV}}^{2}}$ | $7.42_{-0.20}^{+0.21}$ | $6.82\to 8.04$ | $7.42_{-0.20}^{+0.21}$ | $6.82\to 8.04$
$\dfrac{\Delta m^{2}_{3l}}{10^{-3}\operatorname{\text{eV}}^{2}}$ | $+2.514_{-0.027}^{+0.028}$ | $+2.431\to+2.598$ | $-2.497_{-0.028}^{+0.028}$ | $-2.583\to-2.412$
Table 2: Neutrino oscillation parameters taken from the most recent global
analysis from the NuFit collaboration Esteban:2020cvm . The results are shown
without adding the Super Kamiokande atmospheric data.

To get a small scale of neutrino masses in (39), we need the combination of
the $\kappa$ parameter and $V$ to be of order of $\operatorname{\text{eV}}$.
Using (12) in (39), we can parametrically write,

$b\simeq 10\text{
keV}\left(\frac{m_{\Delta}}{1\operatorname{\text{TeV}}}\right)\left(\frac{\kappa_{max}}{4\times
10^{-4}}\right)$ (48)

As a benchmark point, we choose the largest eigenvalue of the coupling matrix,
$\kappa_{max}$, to be of order $10^{-4}$, which implies $b\sim 10\text{ keV}$.
As we discuss in Section 3.3, this choice minimizes the effective number of
relativistic degrees of freedom as both the Majoron and the twin neutrinos are
non-relativistic at decoupling.

We can also constrain the off-diagonal entries of $\kappa$ and the minimum
value of $V$ by using data on flavor violation tests Ilakovac:1994kj ;
Barrie:2022ake ; Cheng:2022jyi ; Alves:2022yav ; Alves:2023znq . Following
Cheng:2022jyi , we can calculate the effective interaction due to the exchange
of a heavy $\Delta_{A}^{\pm\pm}$ particle,

$\mathcal{L}_{\rm
eff}(\Delta_{A}^{\pm\pm})=\frac{(M_{\nu_{A}})_{ij}(M_{\nu_{A}}^{*})_{kl}}{4m_{\Delta}^{2}V^{2}}\left[\overline{e_{L}}_{,l}\gamma^{\mu}e_{L,j}\right]\left[\overline{e_{L}}_{,k}\gamma^{\mu}e_{L,i}\right].$
(49)

Then, we can calculate the branching ratio of the processes
$l_{i}^{i}\rightarrow l_{j}^{+}l_{k}^{-}l_{l}^{-}$.

$\text{BR}(l_{i}^{i}\rightarrow
l_{j}^{+}l_{k}^{-}l_{l}^{-})=\frac{1}{8(1+\delta_{kl})}\frac{m_{l_{i}}^{5}\tau_{i}}{192\pi^{3}}\left|\frac{(M_{\nu_{A}})_{ij}(M_{\nu_{A}}^{*})_{kl}}{m_{\Delta}^{2}V^{2}}\right|^{2},$
(50)

where $\tau_{i}$ is the lifetime of $l_{i}$. Besides the flavor violating
decays to $3$ leptons, muon and tau decays to electrons and photons are also
important and can constrain the parameters of the model. The loop induced
branching ratio of $l_{i}^{-}\rightarrow l_{j}^{-}\gamma$ due to exchange of
intermediate $\Delta^{+}$ and $\Delta^{++}$ is given by

$\text{BR}(l_{i}^{-}\rightarrow
l_{j}^{-}\gamma)=\frac{m_{l_{i}}^{5}\alpha_{\rm
em}\tau_{i}}{(192\pi^{2})^{2}}\left(\frac{9|(M_{\nu_{A}}^{\dagger}M_{\nu_{A}})_{ji}|}{2m_{\Delta}^{2}V^{2}}\right)^{2}$
(51)

In Table 3, we list the lepton flavor violation bounds for different
combinations of the couplings $\kappa$. The bounds on the squared matrix
$\kappa^{\dagger}\kappa$ can be converted into a lower bound on the scale of
$V$ by using the following relation

$\left|(\kappa^{\dagger}\kappa)_{ij}\right|=\frac{1}{2V}\left|U^{i2}_{\rm
PMNS}U^{j2*}_{\rm PMNS}\Delta m_{21}^{2}+U^{i3}_{\rm PMNS}U^{j3*}_{\rm
PMNS}\Delta m_{3l}^{2}\right|.$ (52)

Process | Branching ratio bound | Constraint
---|---|---
$\mu^{-}\to e^{+}e^{-}e^{-}$ | $1.0\times 10^{-12}$ | $\left|\kappa^{*}_{ee}\kappa_{\mu e}\right|<2.36\times 10^{-5}\left(\frac{m_{\Delta}}{1\operatorname{\text{TeV}}}\right)^{2}$
$\tau^{-}\to e^{+}e^{-}e^{-}$ | $2.7\times 10^{-8}$ | $\left|\kappa^{*}_{ee}\kappa_{\tau e}\right|<9.08\times 10^{-3}\left(\frac{m_{\Delta}}{1\operatorname{\text{TeV}}}\right)^{2}$
$\tau^{-}\to e^{+}e^{-}\mu^{-}$ | $1.8\times 10^{-8}$ | $\left|\kappa^{*}_{e\mu}\kappa_{\tau e}\right|<5.24\times 10^{-3}\left(\frac{m_{\Delta}}{1\operatorname{\text{TeV}}}\right)^{2}$
$\tau^{-}\to e^{+}\mu^{-}\mu^{-}$ | $1.7\times 10^{-8}$ | $\left|\kappa^{*}_{\mu\mu}\kappa_{\tau e}\right|<7.21\times 10^{-3}\left(\frac{m_{\Delta}}{1\operatorname{\text{TeV}}}\right)^{2}$
$\tau^{-}\to\mu^{+}e^{-}e^{-}$ | $1.5\times 10^{-8}$ | $\left|\kappa^{*}_{ee}\kappa_{\tau\mu}\right|<6.77\times 10^{-3}\left(\frac{m_{\Delta}}{1\operatorname{\text{TeV}}}\right)^{2}$
$\tau^{-}\to\mu^{+}\mu^{-}e^{-}$ | $2.7\times 10^{-8}$ | $\left|\kappa^{*}_{\mu e}\kappa_{\tau\mu}\right|<6.42\times 10^{-3}\left(\frac{m_{\Delta}}{1\operatorname{\text{TeV}}}\right)^{2}$
$\tau^{-}\to\mu^{+}\mu^{-}\mu^{-}$ | $2.1\times 10^{-8}$ | $\left|\kappa^{*}_{\mu\mu}\kappa_{\tau\mu}\right|<8.01\times 10^{-3}\left(\frac{m_{\Delta}}{1\operatorname{\text{TeV}}}\right)^{2}$
$\mu\to e\gamma$ | $4.2\times 10^{-13}$ | $\left|\left(\kappa^{\dagger}\kappa\right)_{e\mu}\right|<2.36\times 10^{-4}\left(\frac{m_{\Delta}}{1\operatorname{\text{TeV}}}\right)^{2}$
$\tau\to e\gamma$ | $3.3\times 10^{-8}$ | $\left|\left(\kappa^{\dagger}\kappa\right)_{e\tau}\right|<1.55\times 10^{-1}\left(\frac{m_{\Delta}}{1\operatorname{\text{TeV}}}\right)^{2}$
$\tau\to\mu\gamma$ | $4.4\times 10^{-8}$ | $\left|\left(\kappa^{\dagger}\kappa\right)_{\mu\tau}\right|<1.79\times 10^{-1}\left(\frac{m_{\Delta}}{1\operatorname{\text{TeV}}}\right)^{2}$
Table 3: Lepton flavor violation decay constraints for
$|\kappa_{ij}\kappa^{*}_{kl}|$, $\left|\kappa^{\dagger}\kappa\right|_{ji}$ and
$V$. Table adapted from Cheng:2022jyi . Obtaining a lower bound for $V$
involves numerically searching for the minimum value of (52) within the
$3\sigma$ range for the mixing parameters presented in Table 2.

Then, using the oscillation parameters from Table 2 and the bounds from Table
3, one can numerically search for the minimum value of (52) in the $3\sigma$
confidence limit and solve the inequality for $V$. Using the data from
$\mu\rightarrow e\gamma$ we obtain the most stringent bound for $V$ as

$V>0.625\mathrm{eV}\left(\frac{1\operatorname{\text{TeV}}}{m_{\Delta}}\right).$
(53)

Due to the number of free parameters in $\kappa$, we can reproduce all of the
necessary constraints from neutrino phenomenology. Because of this, we do not
consider an in-depth analysis of the possible neutrino textures for $\kappa$
and leave the discussion for other papers that focus on type-II seesaw models
more specifically Abada:2007ux ; ParticleDataGroup:2022pth .

Regarding the twin neutrinos, their masses are much larger than SM neutrinos.
Large twin neutrinos alleviate the excess of radiation during BBN and CMB in
cosmological history. The Standard Model and twin sectors are kept in
equilibrium by the Higgs portal interactions up to temperatures around $T_{\rm
D}\sim 3\operatorname{\text{GeV}}$. As discussed in Chacko:2016hvu , if the
Higgs portal set the decoupling temperature between the two sectors, the
effective number of relativistic degrees of freedom in the early universe
would be much above the present bounds from Planck Planck:2018vyg . As in
Farina:2015uea , we assume the best-case scenario where the decoupling
temperature is between the two QCD phase transitions. Twin QCD would already
have condensed at temperatures around $\sim 200\operatorname{\text{MeV}}$
while the SM quarks and gluons are still relativistic degrees of freedom.222A
lower decoupling temperature can be achieved by including extra portals to the
twin sector, i.e., kinetic mixing or effective operators at the twin Higgs
cutoff. Then, assuming the benchmark in Table 1, the twin neutrinos are heavy
and already non-relativistic at BBN and CMB times as well as when the two
sectors decouple. We define the number of effective relativistic degrees of
freedom from the energy density in radiation during BBN,

$\displaystyle\rho_{\rm
rad}=\rho_{\gamma_{A}}+\rho_{\nu_{A}}+\rho_{\gamma_{B}}\equiv\left[1+\frac{7}{8}N_{\rm
eff}\left(\frac{T_{\nu_{A}}}{T_{\gamma_{A}}}\right)^{4}\right]\rho_{\gamma_{A}},$
(54)

where $T_{\nu_{A}}$ and $T_{\gamma_{A}}$ are the temperatures of SM neutrinos
and photons respectively. Using that
$\frac{T_{\nu_{A}}}{T_{\gamma_{A}}}=\left(\frac{4}{11}\right)^{1/3}$, we can
obtain the number of extra degrees of freedom by subtracting the SM
contribution, $N_{\rm eff}^{\rm SM}\simeq 3.046$.

$\Delta N_{\rm eff}=N_{\rm eff}-N_{\rm eff}^{\rm
SM}=\left[\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}\right]^{-1}\left(\frac{T_{\gamma_{B}}}{T_{\nu_{A}}}\right)^{4}.$
(55)

Finally, using conservation of entropy, we can calculate the ratio of the
temperatures of twin photons by SM neutrinos as
$\frac{T_{\gamma_{B}}}{T_{\nu_{A}}}=\left(\frac{g_{*S}^{B}(T_{\rm
dec})}{g_{*S}^{A}(T_{\rm dec})}\right)^{1/3}$. At decoupling between the SM
and twin sectors, the latter contains only the twin photon and twin electrons.
Conversely, the SM has many more degrees of freedom, with $g_{*S}^{A}(T_{\rm
dec})\approx 61.75$ after $c\overline{c}$ annihilation. Therefore, we have

$\Delta N_{\rm
eff}=\left[\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}\right]^{-1}\left(\frac{2+\frac{7}{8}\times
2\times 2}{61.75}\right)^{4/3}\approx 0.18.$ (56)

The $\Delta N_{\rm eff}$ value above has a small tension with the most recent
value determined by Planck Planck:2018vyg

$\Delta N_{\rm eff.}=2.99_{-0.33}^{+0.34}~{}.$ (57)

A recent determination from the DESI collaboration including both CMB and BAO
results in DESI:2024mwx :

$\Delta N_{\rm eff.}=3.10\pm 0.17~{}.$ (58)

So we see that the model is compatible with the $N_{\rm eff}$ data within
$1~{}\sigma$. Future CMB will have a much greater sensitivity to this quantity
and therefore will be a significant test to this model. So, we conclude that
having large twin neutrino masses and a decoupling temperature between the two
QCDs is still compatible with the available data. One last comment concerns
the role of the twin Majoron in $\Delta N_{\rm eff}$. For the benchmark point
in Table 1, the twin Majoron is already non-relativistic at the twin-SM
decoupling. However, it is possible to have lighter twin Majorons by choosing
a smaller scale for $b$. In Section 3.3 we comment on this possibility.

###  3.2 Collider Bounds

We divide the collider phenomenology of the model into the usual searches in
type-II seesaw models for EW triplets Ashanujjaman:2021txz ; ATLAS:2017xqs ;
Aoki:2011pz ; FileviezPerez:2008jbu ; Fuks:2019clu  on the one hand, and the
collider phenomenology of the twin sector Burdman:2014zta ; Craig:2015pha  on
the other. For the EW triplets, we look for direct production and decay of the
extra scalars at the LHC and their indirect effects in the SM precision
measurements. For the invisible twin states, collider experiments can probe
modifications in the Higgs couplings induced by the heavy sector and put
bounds on the invisible decay width of the Higgs to twin states. We summarize
the constraints at the end of the section and describe the bounds in the
following paragraphs.

Direct searches for type-II seesaw triplets with EW quantum numbers are
usually performed assuming the Drell-Yan production modes:

$\displaystyle q\overline{q}^{\prime}$ $\displaystyle\rightarrow\hskip
5.69054pt{W^{\pm}}^{*}$ $\displaystyle\rightarrow\Delta^{\pm\pm}\Delta^{\mp},$
$\displaystyle\hskip 5.69054pt\Delta^{\pm}\Delta^{0},$ $\displaystyle\hskip
5.69054pt\Delta_{s}^{\pm}\Delta_{a}^{0},$ (59a) $\displaystyle
q\overline{q}^{\prime}$ $\displaystyle\rightarrow\gamma^{*}/Z^{*}$
$\displaystyle\rightarrow\Delta^{\pm\pm}\Delta^{\mp\mp},$ $\displaystyle\hskip
5.69054pt\Delta^{\pm}\Delta^{\mp},$ $\displaystyle\hskip
5.69054pt\Delta_{s}^{0}\Delta_{a}^{0},$ (59b)

where we dropped the $A$ labels of the fields as these are all SM states.
Additionally, for non-zero $b$ coupling, the final states of (59) have a small
doublet component due to mixing, which is relevant to describe their decays.
From (14), the coupling $\lambda_{5}$ controls the mass splitting of the
visible triplets. However, $\lambda_{5}$ is generically small in our model
since $\lambda_{5}<\lambda^{\prime}$ needs to be imposed to keep $F>f$. As
discussed, the mixing of visible and twin states must be tuned as in (24) to
avoid a large invisible Higgs decay width. We comment on the mixing bounds
later in this section, but the constraint (24) leads to

$\lambda_{5}<\lambda^{\prime}\ll\frac{\lambda v^{2}}{f^{2}}$ (60)

which implies

$\Delta m^{2}\equiv
m_{\Delta_{A}^{\pm}}^{2}-m_{\Delta_{A}^{\pm\pm}}^{2}=m_{\Delta_{A}^{0}}^{2}-m_{\Delta_{A}^{\pm}}^{2}=\frac{\lambda_{5}}{4}v^{2}\ll
m_{\Delta_{A}^{\pm}}^{2}.$ (61)

Therefore, the direct triplet searches relevant to us are the ones focused on
the degenerate case, where $\Delta m^{2}=0$ is imposed. Generally, in this
case, the decays of the scalars can be divided for a small triplet VEV scale,
with $V\lesssim 10^{-4}\operatorname{\text{GeV}}$ and a large VEV, $V\gtrsim
10^{-4}\operatorname{\text{GeV}}$:

$\displaystyle V\lesssim
10^{-4}\operatorname{\text{GeV}}:\Delta^{\pm\pm}\rightarrow l^{\pm}l^{\pm},$
$\displaystyle\quad\Delta^{\pm}\rightarrow l^{\pm}\nu,\quad$
$\displaystyle\Delta_{s,a}\rightarrow\nu\nu,$ (62) $\displaystyle V\gtrsim
10^{-4}\operatorname{\text{GeV}}:\Delta^{\pm\pm}\rightarrow W^{\pm}W^{\pm},$
$\displaystyle\quad\Delta^{\pm}\rightarrow W^{\pm}(Z/h),\quad$
$\displaystyle\Delta_{s}\rightarrow\mathrm{VV}/hh,$
$\displaystyle\quad\Delta_{a}\rightarrow hZ$ (63)

where $\mathrm{V}=W^{\pm},Z$. The most constraining decay channels involve the
doubly charged scalar $\Delta^{\pm\pm}$ as these produce multiple charged
final states and are a smoking gun signal for the model. The analysis is
divided between the multilepton searches for smaller triplet VEV and
multiboson searches for larger $V$. We point to Ashanujjaman:2021txz  for a
detailed description of the production and decay modes in the context of the
type-II seesaw model, the multilepton CMS analysis CMS:2019lwf  and multiboson
ATLAS analysis ATLAS:2021jol .

Using the $139\text{ fb}^{-1}$ LHC dataset, in the case of $V\lesssim
10^{-4}\operatorname{\text{GeV}}$ and $\Delta m^{2}=0$, the CMS multiplepton
analysis constrains the mass of the doubly charged scalar to be
$m_{\Delta_{A}^{\pm\pm}}>950\operatorname{\text{GeV}}$ at 95% confidence
level. For $V\gtrsim 10^{-4}\operatorname{\text{GeV}}$, the ATLAS multiboson
analysis is able to constrain
$m_{\Delta_{A}^{\pm\pm}}>400\operatorname{\text{GeV}}$ at 95% confidence
level.333The CMS and ATLAS bounds from Refs. CMS:2019lwf  and ATLAS:2021jol
used here correspond to the recasting performed for the type-II seesaw model
in Ref. Ashanujjaman:2021txz  using these data. In both cases, our model can
reproduce the allowed masses as indicated by the benchmark point of Table 1.

The triplet scalars introduce a new source of custodial symmetry breaking at
tree- and loop-level. This breaking manifests itself as a contribution to the
W-boson mass and a modification of the SM $\rho$ parameter that enters several
EW precision observables. For a triplet $(T,Y)=(1,2)$ and a doublet
$(T,Y)=(1/2,1)$, the tree-level $\rho$ parameter is Gunion:1989we

$\rho_{\rm
tree}=\left.\frac{m_{W}^{2}}{m_{Z}^{2}\cos\theta_{W}^{2}}\right|_{\rm
tree}=\frac{{\sum}_{T,Y}\left[4T(T+1)-Y^{2}\right]v_{i}^{2}}{{\sum}_{T,Y}2Y_{i}^{2}v_{i}^{2}}=1-\frac{2V^{2}}{v^{2}+4V^{2}}.$
(64)

The $\rho$ parameter is tightly constrained by LEP data to be
$\rho=1.00038(20)$ ParticleDataGroup:2022pth ; ALEPH:2013dgf . Using (64) and
the experimental limit, we can bound the triplet VEV to be below $V\lesssim
1.1\operatorname{\text{GeV}}$ at $95\%$ confidence level, which is orders of
magnitude above what we need it to be to obtain the correct SM neutrino
masses.

Additionally, $m_{W}^{2}$ get a radiative contribution from the
$\Delta^{\pm\pm},\Delta^{\pm}$ intermediate state loop. Generically, the
oblique contributions to the W-boson mass can be written as in Maksymyk:1993zm
,

$m_{W}^{2}=m_{W{\rm,SM}}^{2}\left[1-\frac{\alpha_{\rm
EM}S}{2(c_{W}^{2}-s_{W}^{2})}+\frac{c_{W}^{2}\alpha_{\rm
EM}T}{(c_{W}^{2}-s_{W}^{2})}+\frac{\alpha_{\rm EM}U}{4s_{W}^{2}}\right],$ (65)

where $S,T$ and $U$ are the Peskin-Takeuchi oblique parameters Peskin:1991sw
and $s_{W}^{2}=\sin^{2}\theta_{W}\simeq 0.23$. Including the triplet
contributions, the oblique parameters are Mandal:2022zmy ; Heeck:2022fvl

$\displaystyle S_{\rm loop}$
$\displaystyle\simeq-\frac{(2-4s_{W}^{2}+5s_{W}^{4})m_{Z}^{2}}{30\pi
m_{\Delta^{\pm}}^{2}}+\lambda_{5}\frac{v^{2}}{6\pi m_{\Delta^{\pm}}^{2}}\simeq
3\times
10^{-3}\frac{\lambda_{5}-0.04}{(m_{\Delta^{\pm}}/\operatorname{\text{TeV}})^{2}}\,,$
(66) $\displaystyle T_{\rm loop}$
$\displaystyle\simeq\frac{v^{2}\lambda_{5}^{2}}{192\pi^{2}\alpha_{\text{EM}}m_{\Delta^{\pm}}^{2}}\simeq\frac{4\times
10^{-3}}{(m_{\Delta^{\pm}}/\operatorname{\text{TeV}})^{2}}\,\lambda_{5}^{2}\,,$
(67) $\displaystyle U_{\rm loop}$
$\displaystyle\simeq\frac{(2-4s_{W}^{2}+5s_{W}^{4})m_{Z}^{2}-2m_{W}^{2}}{30\pi
m_{\Delta^{\pm}}^{2}}\simeq\frac{-2\times
10^{-5}}{(m_{\Delta^{\pm}}/\operatorname{\text{TeV}})^{2}}\,.$ (68)

As it is usual, the $U$ parameter contribution is more suppressed than the
other two since it is associated to a dimension eight operator. The global fit
reported by the PDG ParticleDataGroup:2022pth  constrains the values of the
oblique parameters to be $S=-0.02\pm 0.10$, $T=0.03\pm 0.12$ and $U=0.01\pm
0.11$444Recent measurements by the CDF collaboration suggest that the best-fit
value is $S=(0.17,0.27)$ assuming $U=0$, which is in tension with the previous
analysis. This result was interpreted in the context of type-II seesaw models
Heeck:2022fvl ; Ghosh:2022zqs ; Butterworth:2022dkt . However, for our model,
the corrections have the opposite sign to explain the CDF best-fit value..We
can see from (66), (67) and (68) that the loop contributions in this model to
$S$,$T$ and $U$ are negligibly small provided that the triplet scalars live
close to the $\operatorname{\text{TeV}}$ scale.

Going to the twin sector, collider bounds are similar to the ones imposed in
the usual implementations of the twin Higgs mechanism Batell:2022pzc . The
vacuum misalignment between the two doublet VEVs induces a modification of the
Higgs couplings to fermions and gauge bosons. The present experimental limits,
derived from the Higgs coupling measurements at the LHC ATLAS:2021vrm ;
CMS:2022dwd , constrain $f/v$ to be greater than approximately 3
Burdman:2014zta . Additionally, measurements on the invisible Higgs decay
width, ATLAS:2022yvh ; CMS:2022qva , constrain the mixing with the twin states
via the Higgs portal interactions. Roughly, the invisible Higgs width requires
$f\gtrsim 2v$, which can be translated to a tuning in the parameters of the
off-diagonal terms of the CP-even Higgs mass matrix (23). Because of this, we
need to impose the conditions

$\displaystyle 2\lambda+\lambda^{\prime}\lesssim\frac{v^{2}}{f^{2}}\sim
0.1,\hskip 28.45274pt\lambda_{4}\eta\lesssim\frac{v^{2}}{f^{2}}\sim 0.1.$ (69)

Finally, we summarize the bounds from accelerator experiments,

$\displaystyle\mathbf{m_{\Delta_{A}^{\pm\pm}}\geq 950\textbf{ GeV}}$
$\displaystyle\text{ at $95\%$ C.L. for }V\lesssim
10^{-4}\operatorname{\text{GeV}}\text{ (CMS \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{CMS:2019lwf}{\@@citephrase{(}}{\@@citephrase{)}}})},$ (70)
$\displaystyle\mathbf{m_{\Delta_{A}^{\pm\pm}}\geq 400\textbf{ GeV}}$
$\displaystyle\text{ at $95\%$ C.L. for }V\gtrsim
10^{-4}\operatorname{\text{GeV}}\text{ (ATLAS \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{ATLAS:2021jol}{\@@citephrase{(}}{\@@citephrase{)}}})},$
(71) $\text{(LFVT \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Cheng:2022jyi}{\@@citephrase{(}}{\@@citephrase{)}}})
}\mathbf{0.7\mathbf{eV}\leq V\leq 1.1\mathbf{GeV}}\text{ (EWPT
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Ashanujjaman:2021txz}{\@@citephrase{(}}{\@@citephrase{)}}})}\text{
at $95\%$ C.L.}.$ (72)

LFVT are the lepton flavor violation tests discussed in the previous section,
and EWPT are the electroweak precision tests discussed above.

###  3.3 Twin Majoron Bounds and Domain Wall Decays

We now turn to two other aspects of the cosmological history of the model.
First, we derive bounds on a light relativistic twin Majoron, and then we
address the potential problem with the formation of a domain wall network due
to the spontaneous breaking of a discrete symmetry, here the
$\operatorname{\mathbb{Z}_{2}}2$ exchange symmetry.

The phenomenology of a SM triplet Majoron has been extensively studied in the
literatureMa:1998dx ; Ma:2017xxj ; Gelmini:1980re ; Schechter:1981cv ;
Ma:2000xh ; Diaz:1998zg ; deSPires:2003wwk , and results in constrains mostly
from measurements of the $Z$ boson width. However, in our case, the Majoron is
only part of the twin sector where there are no bounds from EW precision
measurements. Therefore, most of the constraints on the twin Majoron arise
from its thermal history. In Section 3.1, we computed $\Delta N_{\rm eff}$
without including the twin Majoron. This was justified for the benchmark value
in Table 1, since $J_{B}$ is non-relativistic by the time of SM-twin sector
decoupling. However, for lower values of $b$ and higher $\kappa$, the twin
Majoron becomes lighter and can be relativistic at decoupling or BBN and CMB
times. Thus, we need to reconsider the case of a lighter twin Majoron, and
compute its lifetime and decay channels to estimate its effect on cosmology.

We can integrate out the heavy degrees of freedom above the
$\operatorname{\text{GeV}}$ scale to obtain the effective Lagrangian for the
Majoron

$\mathcal{L}_{J_{B}}=c_{\gamma}J_{B}F_{\mu\nu}\widetilde{F}^{\mu\nu}+c_{\psi,i}J_{B}\overline{\psi}_{i}\gamma_{5}\psi_{i}+c_{\nu,ij}J_{B}\overline{\nu_{i}^{c}}\nu_{j},$
(73)

where $\psi=(\psi_{L},\psi_{R})^{T}$ are the electron, muon, and first
generation of quarks Dirac spinors. The couplings of the Majoron with the
photon and gauge bosons are generated from the twin Higgs Yukawa coupling due
to the mixing with the pseudoscalar doublet $G_{B}^{0}$. In terms of the
parameters of the potential, we can write

$\displaystyle(c_{\nu})_{ij}=\kappa_{ij}\cos\alpha=\frac{2F\kappa_{ij}}{\sqrt{f^{2}+4F^{2}}}$
(74)
$\displaystyle(c_{\psi})_{i}=y_{\psi_{i}}\sin\alpha=\frac{fy_{\psi_{i}}}{\sqrt{f^{2}+4F^{2}}},$
(75)

where $y_{\psi_{i}}$ is the $i^{\rm th}$ light fermion yukawa coupling and
$\alpha$ the mixing angle defined in (22). The photon coupling comes from the
loops of the charged fermionic fields,

$c_{\gamma}=\frac{e^{2}}{32\pi^{2}}{\sum}_{i}\frac{y_{\psi_{i}}\sin\alpha}{m_{\psi_{i}}}=\frac{N_{\psi}e^{2}}{32\pi^{2}v^{2}}\frac{f}{\sqrt{f^{2}+4F^{2}}}.$
(76)

Here, $N_{\psi}$ is the number of charged fermions that are heavier than the
Majoron. Then, the decay rate of each channel is given by

$\displaystyle\Gamma(J_{B}\rightarrow\nu_{B}\nu_{B})=\frac{\sum_{ij}|\kappa_{ij}|^{2}m_{J_{B}}}{8\pi}\frac{F^{2}}{f^{2}+4F^{2}}\sqrt{1-\frac{4m_{\nu_{B}}^{2}}{m_{J_{B}}^{2}}},$
(77) $\displaystyle\Gamma(J_{B}\rightarrow
l^{+}_{B,i}l^{-}_{B,i})=\frac{y_{l}^{2}m_{J_{B}}}{4\pi}\frac{f^{2}}{f^{2}+4F^{2}}\sqrt{1-\frac{4m_{l_{B,i}}^{2}}{m_{J_{B}}^{2}}},$
(78)
$\displaystyle\Gamma(J_{B}\rightarrow\gamma_{B}\gamma_{B})=\frac{N_{\psi}^{2}\alpha_{e}^{2}}{64\pi^{3}}\frac{m_{J_{B}}^{3}}{v^{2}}\frac{f^{2}}{f^{2}+4F^{2}}.$
(79)

Out of the possible Majoron decay channels, the smallest partial width is twin
electrons and muons, as these are proportional to the lepton Yukawa couplings.
The neutrino decay can be sizable depending on the value of $\kappa$. However,
as discussed below, the twin neutrinos are heavier than the twin Majoron for
most of the allowed parameter space, making the decay channel forbidden.
Therefore, we conclude that the twin Majoron decays to photons with a
branching ratio of approximately one. As a result, the Majoron lifetime can be
approximated by

$\tau_{J_{B}}\simeq
1\text{s}\left(\frac{1.1\operatorname{\text{MeV}}}{m_{J_{B}}}\right)^{3}\left(\frac{\xi}{0.1}\right)\frac{f^{2}+4F^{2}}{(2.1\operatorname{\text{TeV}})^{2}}.$
(80)

Thus, a Majoron that is lighter than $1\operatorname{\text{MeV}}$ would decay
during BBN, injecting entropy into the twin sector. This would lead to an
increase in $\Delta N_{\rm eff}$ from the entropy injections, which would
effectively increase the twin sector temperature. In our case, we consider
$m_{J_{B}}\gtrsim 1\operatorname{\text{MeV}}$, so the contributions due to
entropy injection are unimportant. However, we must consider the contribution
to $\Delta N_{\rm eff}$ from having relativistic twin Majorons and neutrinos
during and after the SM-twin decoupling.

Previously, we assumed that twin Majoron and twin neutrinos were relativistic
by the time of decoupling between the SM and twin sectors. This rendered
$\Delta N_{\rm eff}\simeq 0.18$ as shown in (56). If the Majoron is much
lighter than the twin decoupling temperature but heavier than $\sim
1\operatorname{\text{MeV}}$, it contributes to $g_{*,S}^{B}(T_{dec})$,
increasing the temperature ratio that enters (55). Therefore, we have

$\Delta N_{\rm eff}\simeq 4.4\left(\frac{2+\frac{7}{8}\times 2\times
2+1}{61.75}\right)^{4/3}\approx 0.21,\quad(1\operatorname{\text{MeV}}\lesssim
m_{J_{B}}\lesssim 100\operatorname{\text{MeV}}).$ (81)

Figure 3: Twin Majoron against twin neutrino masses for fixed values of SM
neutrino masses. The chosen benchmark point (BP) from Table 1 gives the most
favorable scenario for $\Delta N_{\rm eff}\simeq 0.18$. Each quadrant reflects
calculated $\Delta N_{\rm eff}$ values assuming relativistic/non-relativistic
twin Majoron and twin neutrinos. The exclusion bound from BBN comes from late
twin neutrino decays that increase $\Delta N_{\rm eff}$ to values greater than
$0.7$, which is incompatible with data ParticleDataGroup:2022pth ;
DESI:2024mwx .

If instead of the Majoron, the neutrinos are relativistic at decoupling,
$\Delta N_{\rm eff}\simeq 0.41$. Alternatively, if both particles are
relativistic below the decoupling temperature, then $\Delta N_{\rm eff}\simeq
0.48$. Finally, suppose the twin neutrinos are lighter than roughly the BBN
temperature. In that case, their decays will reheat the twin sector rendering
a large effective number of extra neutrino species, $\Delta N_{\rm eff}\simeq
0.7$. While having a light twin Majoron with $\Delta N_{\rm eff}\simeq 0.21$
is still not excluded by data Planck:2018vyg ; DESI:2024mwx , having twin
neutrinos below the decoupling temperature is in severe tension with
cosmological data. In Figure 3, we plot the twin Majoron and twin neutrino
masses for fixed values of the SM neutrino masses. The benchmark point chosen
in Table 1 gives the best case scenario for $\Delta N_{\rm eff}\simeq 0.18$,
as both light twin particles are non-relativistic. Each quadrant corresponds
to the $\Delta N_{\rm eff}$ values calculated above.

Finally, we address the issue of the formation of domain walls in the model.
Because of the model’s exact $\operatorname{\mathbb{Z}_{2}}2$ symmetry, the
other vacua obtained by changing the values of $v\rightarrow f$ and
$V\rightarrow F$ have the same depth as the one we considered. Therefore, as
the universe cools down, the scalar fields can relax to any of these
equivalent vacua, potentially leading to a domain wall problem
Zeldovich:1974uw . The formation of the domain wall network is model-dependent
as we need to specify the inflationary and reheating dynamics during the
radiation domination era. As assumed in Beauchesne:2021opx , if the Hubble
parameter during inflation and the reheating temperature are below the model’s
phase transition scale, then the domain walls are not formed, and the model is
cosmologically consistent. On the other hand, if we suppose inflation and
reheating happened before the twin-phase transition, we need a mechanism to
collapse the different domains to the vacuum considered throughout the paper,
i.e. (7).

As shown in Ref. Batell:2019ptb , one way to achieve the collapse of the
domain wall network is to introduce a very small amount of explicit
$\operatorname{\mathbb{Z}_{2}}2$ breaking. For example, we can assume the
following soft-$\operatorname{\mathbb{Z}_{2}}2$ breaking term

$V_{\cancel{\operatorname{\mathbb{Z}_{2}}2}}=\mu_{1}^{2}\big{(}|H_{A}|^{2}-|H_{B}|^{2}\big{)}+\mu_{2}^{2}\big{(}\tr\Delta_{A}^{\dagger}\Delta_{A}-\tr\Delta_{B}^{\dagger}\Delta_{B}\big{)},$
(82)

where $\mu_{1,2}^{2}\ll|m^{2}|,|M^{2}|$ is a positive
$\operatorname{\mathbb{Z}_{2}}2$ constant. This term effectively selects the
vacuum we are interested in by lowering the value of the potential at the
correct minimum. The potential contribution in the true minimum, which is
defined in (7), is

$\left.\langle V_{\cancel{\operatorname{\mathbb{Z}_{2}}2}}\rangle\right|_{\rm
true}=\mu_{1}^{2}\big{(}v^{2}-f^{2}\big{)}+\mu_{2}^{2}\big{(}V^{2}-F^{2}\big{)}<0,$
(83)

We can obtain the $\operatorname{\mathbb{Z}_{2}}2$ symmetric minimum by
changing $v\leftrightarrow f$ and $V\leftrightarrow F$. The contribution to
the potential is

$\left.\langle V_{\cancel{\operatorname{\mathbb{Z}_{2}}2}}\rangle\right|_{\rm
false}=\mu_{1}^{2}\big{(}f^{2}-v^{2}\big{)}+\mu_{2}^{2}\big{(}F^{2}-V^{2}\big{)}>0.$
(84)

Therefore, the true minimum has the correct VEV hierarchy, $V\ll v<f\lesssim
F$ needed for our model. Following Vilenkin:1981zs ; Batell:2019ptb , we can
estimate how much the potential has to be shifted to make the false domains
decay before they can dominate the energy density of the early universe. This
requires

$\Delta V\gtrsim\frac{\sigma^{2}}{M_{\rm pl}^{2}}$ (85)

where $\sigma$ is the surface energy density of the wall, and $M_{\rm
pl}\simeq 1.2\times 10^{19}\operatorname{\text{GeV}}$. The surface density can
be estimated as $\sigma\sim V_{0}^{1/2}f^{2}$, where $V_{0}\sim\lambda f^{2}$
or $V_{0}\sim\lambda_{1}F^{2}$ is the height of the potential barrier between
the two minima Kitajima:2015nla , depending on which quartic term is larger.
Then, we can extract a bound on the $\operatorname{\mathbb{Z}_{2}}2$ breaking
parameters $\mu_{1,2}$ by imposing (85), with the shift in the potential given
by $\Delta V\simeq 2\left.\langle
V_{\cancel{\operatorname{\mathbb{Z}_{2}}2}}\rangle\right|_{\rm true}$.

$\mu_{1,2}^{2}\gtrsim\max\left[\frac{\lambda f^{4}}{M_{\rm
pl}^{2}},\frac{\lambda_{1}F^{4}}{M_{\rm pl}^{2}}\right].$ (86)

Imposing the condition (85), we finally obtain that $\mu_{1,2}\simeq
0.1\text{meV}$. Thus, we see that the explicit
$\operatorname{\mathbb{Z}_{2}}2$ breaking needed to be introduced is much
smaller than any scale in the problem and it would have no observable
consequences in either the spectrum or the couplings of the model presented
here. Therefore, we see that it is justified to completely ignore the explicit
$\operatorname{\mathbb{Z}_{2}}2$ breaking effects, and the way the model was
presented is enough to make sensible predictions without the need to consider
effects of order $\mu_{1,2}/\Lambda$, where here $\Lambda$ stands for any
other scale in the theory, i.e. $v$,$f$, $F$, etc.

##  4 Conclusions

We presented an extension of the mirror twin Higgs model where the
$\operatorname{\mathbb{Z}_{2}}2$ symmetry that relates the SM particles to the
mirror sector is spontaneously broken. To achieve
$\operatorname{\mathbb{Z}_{2}}2$ spontaneous symmetry breaking, we introduce
two $SU(2)$ triplets, one for the SM and one for the twin sector. The
parameters of the scalar potential are fixed to give a particular hierarchy of
VEVs, with $V\ll v<f\lesssim F$, where $V=\langle\Delta_{A}\rangle,v=\langle
H_{A}\rangle,f=\langle h_{B}\rangle$ and $F=\langle\Delta_{B}\rangle$. The
$SU(2)_{A,B}$ triplets are introduced to implement the type-II seesaw
mechanism to give SM and twin neutrino masses. The masses of the neutrinos are
proportional to the triplet VEV of the corresponding sector. Because of the
VEV hierarchy obtained, the SM neutrinos acquire small masses proportional to
$V$, while the twin neutrinos are much heavier, at the scale $F$. The lifting
of the twin neutrino masses considerably ameliorates potential conflict with
the cosmological bounds on extra radiation during BBN and CMB times. If twin
neutrinos are heavier than $\sim 100\operatorname{\text{MeV}}$ they do not
contribute to the number of relativistic degrees of freedom at decoupling of
the SM and twin sectors. The resulting contribution to the effective number of
extra neutrino species is $\Delta N_{\rm eff}\simeq 0.18$, given almost
exclusively by the twin photon contribution. This is within $1\sigma$
confidence level of the most recent measurements Planck:2018vyg ;
ParticleDataGroup:2022pth ; DESI:2024mwx .

One implication of the proposed mechanism is the appearance of a twin Majoron,
the pNGB of spontaneously broken twin lepton number. The Majoron does not
appear in the SM sector since the explicit breaking of SM lepton number is of
the same order as the triplet VEV, $V$. For the twin sector, the explicit
breaking induced by the triplet scalar coupling $b$ is much smaller than $F$.
The most relevant constraints on the twin Majoron come from bounds on $\Delta
N_{\rm eff}$, as discussed in Section 3.3. For a twin Majoron lighter than
$\sim 100\operatorname{\text{MeV}}$, there is an additional contribution to
the effective number of relativistic degrees of freedom at decoupling, which
renders $\Delta N_{\rm eff}\simeq 0.21$. While slightly larger than the best-
case scenario with non-relativistic twin Majorons, $\Delta N_{\rm eff}$ is
still within $1\sigma$ confidence level from the measurements cited above.

As emphasized in Section 3, the model has several distinct experimental
signatures, ranging from neutrino and collider phenomenology to cosmology.
Neutrino oscillation data and neutrino mass measurements constrain the SM
sector’s triplet couplings to neutrinos and triplet VEV scale. Additionally,
SM neutrino masses determine the relationship between the twin Majoron and
twin neutrinos, as indicated in Figure 3. Then, we can calculate the
additional contributions to $\Delta N_{\rm eff}$ as discussed in the previous
section. Regarding collider phenomenology, the main constraints come from
direct searches for visible $SU(2)$ triplets and bounds on the Higgs couplings
and mixing. Direct searches for the doubly charged Higgs constrain
$\Delta_{A}^{\pm\pm}$ mass to be larger than $950\operatorname{\text{GeV}}$
for $V\lesssim 10^{-4}\operatorname{\text{GeV}}$, and larger than
$400\operatorname{\text{GeV}}$ for $V\gtrsim
10^{-4}\operatorname{\text{GeV}}$. Bounds from the $\rho$ parameter restrict
the triplet VEV to be $V\lesssim 1\operatorname{\text{GeV}}$, which is orders
of magnitude above what is necessary for realistic SM neutrino masses. On the
other hand, lepton flavor violation tests require a lower bound of $V\gtrsim
0.7\operatorname{\text{eV}}$ to best fit the oscillation parameter data.
Additionally, as usual in twin Higgs models, the combination of the doublet
VEVs must satisfy $f/v\gtrsim 3$ to be compatible with measurements of the
Higgs couplings. Finally, in cosmology, if the scale of inflation is much
above the scale of EW and $\operatorname{\mathbb{Z}_{2}}2$ spontaneous
symmetry breaking, then a small source of $\operatorname{\mathbb{Z}_{2}}2$
explicit breaking must be introduced to avoid a domain wall problem. We have
shown that the necessary explicit $\operatorname{\mathbb{Z}_{2}}2$ breaking
parameter is much smaller than any other scale in the problem and, therefore,
does not change any of the predictions of the model in any observable way.

In sum, the model has several correlated signals from various sources,
including neutrino oscillation data, collider searches, and cosmological
bounds. The future of these experiments CMB-S4:2016ple ; DUNE:2022aul ; Hyper-
Kamiokande:2018ofw ; Mlynarikova:2023bvx  could significantly test this
particular implementation of the twin Higgs model. From the theoretical side,
we have shown that the mirror twin Higgs is very versatile, adeptly
accommodating both the spontaneous breaking of the
$\operatorname{\mathbb{Z}_{2}}2$ symmetry and the generation of neutrino
masses. It remains one of the most compelling models of physics beyond the
standard model that addresses the hierarchy problem.

###### Acknowledgements.

The authors thank Kfir Blum for early collaboration in this project. We also
acknowledge the support of FAPESP grant 2019/04837-9 and CAPES grant
88887.816450/2023-00.

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