#  Long-Lived Doubly Charged Scalars in the Left-Right Symmetric Model:
Catalyzed Nuclear Fusion and Collider Implications

Evgeny Akhmedov<EMAIL_ADDRESS>Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany  P. S. Bhupal Dev<EMAIL_ADDRESS>Department of Physics and McDonnell Center for the Space Sciences, Washington
University, St. Louis, Missouri 63130, USA  Sudip Jana sudip.jana@mpi-
hd.mpg.de Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117
Heidelberg, Germany  Rabindra N. Mohapatra<EMAIL_ADDRESS>Maryland Center
for Fundamental Physics and Department of Physics, University of Maryland,
College Park, Maryland 20742, USA

###### Abstract

We show that the doubly-charged scalar from the $SU(2)_{R}$-triplet Higgs
field in the Left-Right Symmetric Model has its mass governed by a hidden
symmetry so that its value can be much lower than the $SU(2)_{R}$ breaking
scale. This makes it a long-lived particle while being consistent with all
existing theoretical and experimental constraints. Such long-lived doubly-
charged scalars have the potential to trigger catalyzed fusion processes in
light nuclei, which may have important applications for energy production. We
show that it could also bear consequences on the excess of large ionization
energy loss ($dE/dx$) recently observed in collider experiments.

##  I Introduction

Since the discovery of the Standard Model (SM) Higgs boson ATLAS:2012yve ;
CMS:2012qbp , no new particles have emerged from the Large Hadron Collider
(LHC). This might suggest that the current energy and luminosity of the LHC
may not be sufficient for direct exploration of beyond the Standard Model
(BSM) particles. Optimism for potential discoveries relies on the high-
luminosity LHC upgrade or higher-energy colliders. An alternative possibility,
however, is that there are long-lived particles (LLPs) that may yield null
results in mainstream prompt signal searches at collider experiments and yet
are part of new physics. There is certainly mounting interest (both
theoretical and experimental) in unconventional investigative approaches in
the search for the LLPs Fairbairn:2006gg ; Alekhin:2015byh ; Curtin:2018mvb ;
Lee:2018pag ; Alimena:2019zri ; Feng:2022inv ; Knapen:2022afb . In this
context, we note that most of the LLP searches focus on neutral LLPs, being
theoretically motivated by feebly-interacting neutral BSM particles like dark
matter, axions, heavy neutral leptons, and new neutral gauge bosons, whereas
charged LLPs have received limited attention so far. It is worth remembering
that the 1930s’ discovery of muons, unexpected charged long-lived particles
observed by Anderson and Neddermeyer Anderson:1936zz ; Neddermeyer:1937md ,
underscores the importance of vigilance, as breakthroughs may emerge in
unforeseen ways. In this letter, we scrutinize the prospect that the doubly-
charged scalars inherent to the minimal left-right symmetric model (LRSM)
could embody the distinctive characteristics of such LLPs.

Doubly-charged scalars emerge in various BSM scenarios, such as the type-II
seesaw model Magg:1980ut ; Schechter:1980gr ; Lazarides:1980nt ;
Mohapatra:1980yp , left-right symmetric model Pati:1974yy ; Mohapatra:1974gc ;
Senjanovic:1975rk ; Mohapatra:1979ia , radiative neutrino mass model
Zee:1985id ; Babu:1988ki , little Higgs model Arkani-Hamed:2002iiv , $d=7$
neutrino mass models Babu:2009aq ; Bonnet:2009ej , $331$ model Pisano:1992bxx
and the Georgi-Machacek Model Georgi:1985nv . In most instances, they manifest
themselves either as $SU(2)_{L}$ singlets with hypercharge $Y=2$, as seen in
loop models for neutrino masses, or as members of a triplet of $SU(2)_{L}$, as
in the type-II seesaw models. In the LRSM, they can exist as either
$SU(2)_{L}$-singlet or triplet. However, they may also arise from a doublet,
quadruplet, or quintuplet of $SU(2)_{L}$ Babu:2009aq ; Bonnet:2009ej ;
Babu:2019mfe ; Babu:2020hun ; Bhattacharya:2016qsg . If doubly-charged scalars
originate from $SU(2)_{L}$ multiplets other than singlets, the challenge of
conferring longevity becomes particularly intricate, primarily due to
potential interactions with two same-sign $W$ bosons. Additionally, the masses
of charged or neutral partners within the $SU(2)_{L}$ multiplet of doubly-
charged Higgs are tightly constrained by electroweak $\rho$ parameter
contributions, indicating anticipated signals at colliders. Designating
doubly-charged scalars as weak isosinglets classifies them aptly as LLP if
their couplings to lepton pairs are either suppressed or set to zero by some
symmetry. In general, these scalars exhibit couplings to two like-sign charged
leptons with arbitrary flavor content, crucial for their role in generating
neutrino masses Zee:1985id ; Babu:1988ki . Prohibiting their coupling to
leptons diminishes their role in neutrino mass generation. In this context,
the unique standing of the doubly-charged scalar $\Delta^{\pm\pm}_{R}$ within
the LRSM, belonging to the $SU(2)_{L}$-singlet and $SU(2)_{R}$-triplet,
distinguishes it from other conceivable scenarios. In this letter, we show
that we can have a long-lived doubly-charged scalar and generate neutrino
masses simultaneously.

The mass of the $\Delta^{\pm\pm}_{R}$ in the LRSM is commonly expected to be
at the $SU(2)_{R}$ breaking scale. However, we show that due to the existence
of a hidden symmetry, its magnitude is decoupled from the $W_{R}$ scale. In
other words, even for very high $W_{R}$ mass, the $\Delta^{\pm\pm}_{R}$ mass
can be made very low by choice of a symmetry-breaking parameter in the Higgs
potential. We then discuss under what conditions these doubly-charged bosons
can be very long-lived and find constraints on them from current observations.
Furthermore, we demonstrate that although suppressing the leptonic couplings
precludes the use of the type-I seesaw mechanism Minkowski:1977sc ;
Mohapatra:1979ia ; Yanagida:1979as ; Gell-Mann:1979vob ; Glashow:1979nm  to
explain small neutrino masses, the inverse seesaw mechanism Mohapatra:1986aw ;
Mohapatra:1986bd  can be responsible for neutrino mass generation.

We subsequently analyze the consequential phenomenological implications of
long-lived doubly-charged scalars. Recent findings indicate that sufficiently
long-lived doubly-charged scalars can facilitate the fusion of light nuclei,
bearing significant practical implications for energy production
Akhmedov:2021qmr . Investigating their longevity necessary for catalyzed
fusion, we also analyze the collider implications of these almost stable
particles, characterized by distinctive ionization energy loss ($dE/dx$)
behavior Jager:2018ecz ; Altakach:2022hgn . Notably, recent observations by
the ATLAS collaboration indicated an excess in $dE/dx$ for LLP searches
ATLAS:2022pib , although subsequent data have neither confirmed nor ruled it
out thus far ATLAS:2023zxo . Nevertheless, these long-lived doubly-charged
particles offer a compelling explanation for the observed excess
Giudice:2022bpq , making them promising candidates for future LLP discoveries
based on ionization energy loss gradients.

##  II $\Delta^{\pm\pm}_{R}$ in the minimal Left-Right Symmetric Model

Let us start with a brief review of the minimal LRSM for neutrino masses
Pati:1974yy ; Mohapatra:1974gc ; Senjanovic:1975rk ; Mohapatra:1979ia  and the
uniqueness of the $\Delta^{\pm\pm}_{R}$ boson. The model is based on the gauge
group $SU(3)_{c}\times SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$, with the
quantum numbers of the fermion and Higgs fields given in Table 1. Here and in
what follows, we omit the $SU(2)_{L}$-triplet field $\Delta_{L}$ $(1,3,1,2)$
since we will deal with the effective field theory at lower energies below the
$D$-parity breaking scale and the $\Delta_{L}$ multiplet is expected to remain
at the higher $D$-parity breaking scale in this version Chang:1983fu . This
ensures that the neutrino masses are given by the simple type-I seesaw formula
linking the neutrino mass to the $W_{R}$ boson mass scale Mohapatra:1979ia ,
without any fine-tuning between the type-I and type-II seesaw contributions.
Moreover, this version is more amenable to $SO(10)$ embedding with lighter
$W_{R}$ bosons Chang:1984qr . Most importantly for our discussion here, as
noted in Section I, the doubly-charged component $\Delta_{L}^{\pm\pm}$ cannot
be long-lived because of its gauge coupling to $W$ bosons. Therefore, keeping
it in the theory will only complicate the discussion without affecting our
main results. Nevertheless, we analyze the full scalar potential, including
the $\Delta_{L}$ field in Appendix A.

Table 1: Gauge quantum numbers of all the fields in the minimal LRSM based on $SU(3)_{c}\times SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$. Particles | Quantum numbers
---|---
fermions | $Q_{L}=\begin{pmatrix}u_{L}\\\ d_{L}\end{pmatrix}$ | $(3,2,1,1/3)$
$Q_{R}=\begin{pmatrix}u_{R}\\\ d_{R}\end{pmatrix}$ | $(3,1,2,1/3)$
$\ell_{L}=\begin{pmatrix}\nu\\\ e_{L}\end{pmatrix}$ | $(1,2,1,-1)$
$\ell_{R}=\begin{pmatrix}N\\\ e_{R}\end{pmatrix}$ | $(1,1,2,-1)$
scalars | $\Phi=\begin{pmatrix}\phi_{1}^{0}&\phi_{2}^{+}\\\ \phi_{1}^{-}&\phi_{2}^{0}\end{pmatrix}$ | $(1,2,2,0)$
$\Delta_{R}=\begin{pmatrix}\Delta_{R}^{+}/\sqrt{2}&\Delta_{R}^{++}\\\ \Delta_{R}^{0}&-\Delta_{R}^{+}/\sqrt{2}\end{pmatrix}$ | $(1,1,3,2)$

The Yukawa Lagrangian for the model is given by

$\displaystyle{\cal L}_{Y}$ $\displaystyle=$ $\displaystyle
h_{q}\bar{Q}_{L}\Phi
Q_{R}~{}+~{}h^{\prime}_{q}\bar{Q}_{L}\tilde{\Phi}Q_{R}~{}+~{}h_{\ell}\bar{\ell}_{L}\Phi\ell_{R}$
(1)
$\displaystyle~{}+~{}h^{\prime}_{\ell}\bar{\ell}_{L}\tilde{\Phi}\ell_{R}+f_{R}{\ell}^{T}_{R}\Delta_{R}\ell_{R}~{}+~{}{\rm
H.c.}\,,$

where $\tilde{\Phi}=\sigma_{2}\Phi^{*}\sigma_{2}$ (with $\sigma_{2}$ being the
second Pauli matrix), and $h_{q,\ell}$, $h^{\prime}_{q,\ell}$, $f_{R}$ are the
Yukawa coupling matrices.

In the scalar sector, after the neutral components of the bidoublet $\Phi$ and
$SU(2)_{R}$-triplet $\Delta_{R}$ develop their vacuum expectation values
(VEVs) [cf. Eq. (17)], there remain eight physical massive scalar fields,
namely, four neutral ($h,H_{1}^{0},A_{1}^{0},H_{3}^{0}$) and pairs of singly-
charged ($H^{\pm}$) and doubly-charged ($H^{\pm\pm}\equiv\Delta_{R}^{\pm\pm}$)
scalars; see Appendix A for details. We identify $h$ as the SM-like Higgs
boson with $m_{h}\simeq 125$ GeV [cf. Eq. (19)]. The masses of the remaining
scalar fields are proportional to the $\Delta_{R}$ VEV $v_{R}$ Dev:2016dja :

$\displaystyle m_{H_{1}^{0}}^{2}\simeq m_{A_{1}^{0}}^{2}\simeq
m_{H^{\pm}}^{2}\simeq$ $\displaystyle\ \alpha_{3}v_{R}^{2}\,,$ (2)
$\displaystyle m_{H_{3}^{0}}^{2}\simeq$ $\displaystyle\ 4\rho_{1}v_{R}^{2}\,,$
(3) $\displaystyle m_{H^{\pm\pm}}^{2}\simeq$ $\displaystyle\
4\rho_{2}v_{R}^{2}\,,$ (4)

The neutral Higgs masses $m_{H_{1}^{0},A_{1}^{0}}$ are constrained to be
larger than about 15 TeV to satisfy the flavor-changing neutral current
constraints from $K-\bar{K}$ and $B-\bar{B}$ transitions Zhang:2007da ;
Bertolini:2019out ; Dekens:2021bro . It follows from Eq. (2) that the singly-
charged scalars $H^{\pm}$ must also be equally massive and are, therefore,
inaccessible at the LHC Dev:2016dja . The $H_{3}^{0}$ field can in principle
be much lighter (though at the cost of some fine-tuning), and may have
interesting phenomenological consequences Dev:2016nfr ; Dev:2017dui ;
Dev:2023kzu .

As for the doubly-charged scalars $H^{\pm\pm}$, an interesting point to note
from Eq. (4) is that their mass only depends on the quartic coupling
$\rho_{2}$. This can be seen without doing any explicit calculation as
follows: if $\rho_{2}$ is set to zero in the potential (10) at the tree level,
it is not induced by loops. Therefore, we can tune down the value of
$\rho_{2}$ arbitrarily. The presence of this naturally tunable parameter
allows the mass of $H^{\pm\pm}$ to be not necessarily of the order of the
$v_{R}$ scale so that it can be considerably lighter. The basic reason for
this is that when $\rho_{2}$ and $\alpha_{3}$ are set to zero, the scalar
potential has a higher global $SU(3)$ symmetry, not shared by gauge
interactions, and as a result, the doubly-charged boson is a pseudo-Goldstone
particle with zero tree-level mass. To see this, note that when
$\langle\Delta^{0}_{R}\rangle=v_{R}\neq 0$, the above global $SU(3)$ symmetry
breaks down to $SU(2)$ leaving $8-3=5$ Goldstone bosons. This VEV $v_{R}$ also
breaks the gauge symmetry $SU(2)_{R}\times U(1)_{B-L}$ down to $U(1)_{Y}$. Out
of the five massless scalar bosons, three are absorbed by the three massive
gauge bosons of broken $SU(2)_{R}\times U(1)_{B-L}$ symmetry, leaving two
massless states. These are the doubly-charged states that pick up mass once
the $\rho_{2}$ and $\alpha_{3}$ terms in the potential are switched on since
$\rho_{2}$ and $\alpha_{3}$ explicitly break this $SU(3)$ symmetry. Thus
$\rho_{2}$ and $\alpha_{3}$ play a crucial role in giving $H^{\pm\pm}$ mass in
combination with $v_{R}$ and $\kappa$. Thus, the $H^{\pm\pm}$ could be
considerably lighter than the $v_{R}$ scale since $\rho_{2}$ and $\alpha_{3}$
are symmetry-breaking parameters and could be adjusted to be small, although,
in practice, $\alpha_{3}$ cannot be arbitrarily small in order to satisfy the
FCNC constraints mentioned above.

##  III Long-lived doubly-charged scalars

Our interest lies in the existence of a long-lived doubly-charged scalar
within a gauge theory. As mentioned in the introduction, a doubly-charged
scalar arising from $SU(2)_{L}$ singlet is crucial for achieving longevity. We
show here that the $H^{\pm\pm}$ coming from the $SU(2)_{R}$-triplet fittingly
qualifies to be long-lived.

(a) (b) (c)

Figure 1: Representative Feynman diagrams for dominant decay modes of
$H^{\pm\pm}_{R}$.  Figure 2: Lifetime of doubly-charged scalars in the Yukawa
coupling versus mass plane, with iso-contours representing different
$H^{\pm\pm}$ lifetimes. The almost vertical dashed lines show the current
lower limits on the $H^{\pm\pm}$ mass from prompt and LLP searches at the LHC.
Red dot-dashed and dashed lines respectively correspond to two important
lifetime benchmarks relevant to the catalyzed fusion mechanism:
$\tau=9.5\times 10^{11}$ sec and $\tau=\tau_{U}=4.34\times 10^{17}$ sec (age
of the Universe). The cyan dashed line (near the top) indicates the
perturbative limit on the Yukawa coupling.  Figure 3: Lifetime of doubly-
charged scalars in the $W_{R}$ mass versus the $H^{\pm\pm}$ mass plane. Iso-
contours denote different $H^{\pm\pm}$ lifetimes. Red dot-dashed and dashed
lines correspond to two important lifetime benchmarks relevant to the
catalyzed fusion mechanism: $\tau=9.5\times 10^{11}$ sec and $\tau=\tau_{U}$.
Black dashed lines indicate the lower limits on $H^{\pm\pm}$ and $W_{R}$
masses from LLP searches and $W_{R}$ searches at the LHC, respectively.

The possible decay modes of $H^{\pm\pm}$ in the minimal LRSM are shown in Fig.
1. The diagram (a) arises from the Yukawa coupling $f_{R}$ in Eq. (1), which
leads to the decay of $H^{\pm\pm}$ into a pair of same-sign charged leptons,
with the partial decay width

$\displaystyle\Gamma(H^{\pm\pm}\to
l_{R}^{\pm}l_{R}^{\pm})\simeq\frac{f_{R}^{2}}{8\pi}m_{H^{\pm\pm}}\,.$ (5)

In fact, this decay mode has been used by the LHC experiments to set a lower
limit on the doubly-charged scalar mass: $m_{H^{\pm\pm}}>900$ GeV
ATLAS:2022pbd . However, this limit is valid only for prompt decay of
$H^{\pm\pm}$ into a pair of isolated leptons. For $f_{R}\lesssim 10^{-8}$, Eq.
(5) indicates that a TeV-scale $H^{\pm\pm}$ is sufficiently long-lived and
decays outside the detector (see Ref. Dev:2018tox  for a detailed discussion
on this). In this case, the mass limit is weaker, at about 700 GeV, as we
derive below. We first assess the required smallness of the Yukawa coupling
$f_{R}$ for achieving long-lived behavior of $H^{\pm\pm}$. As depicted in Fig.
2, for a TeV mass of $H^{\pm\pm}$ and a very heavy $W_{R}$ mass (so that
diagrams (b) and (c) in Fig. 1 are not relevant), a Yukawa coupling of
$f_{R}\sim 10^{-20}$ is needed to attain a lifetime of $H^{\pm\pm}$ around
$10^{17}$ sec [cf. Eq. 5]. If we impose a discrete $Z_{2}$ symmetry under
which $\Delta_{R}\to-\Delta_{R}$ and all other fields are even, then such
Yukawa couplings are forbidden in Eq. (1). In this case, the only coupling
that leads to the $H^{\pm\pm}$ decay is its $SU(2)_{R}$ gauge coupling via
which it decays to two (on/off-shell) $W_{R}$ bosons, which subsequently decay
to hadrons and leptons [see Fig. 1(b)]. The corresponding decay rate is given
in Appendix B. For $m_{H^{\pm\pm}}\ll m_{W_{R}}$, this scales as $v_{R}^{-6}$.
Therefore, significantly increasing the $W_{R}$ mass relative to the
$H^{\pm\pm}$ mass suppresses this decay rate, thus resulting in an extended
lifetime for $H^{\pm\pm}$. Suppression of $W_{L}-W_{R}$ mixing is necessary to
weaken the $H^{\pm\pm}\to W^{\pm}_{L}W^{\pm}_{L}$ mode, and a high $W_{R}$
mass achieves this objective. In Fig. 3, we present a plot of $W_{R}$ mass
against $H^{\pm\pm}$ mass for various lifetimes of $H^{\pm\pm}$. For instance,
with $m_{H^{\pm\pm}}\simeq$ TeV and $m_{W_{R}}\sim 10^{10}$ GeV, the resulting
lifetime of $H^{\pm\pm}$ exceeds the current age of the Universe,
$\tau_{U}=4.34\times 10^{17}$ sec.

##  IV Neutrino mass generation

Setting the Yukawa coupling of $\Delta_{R}$-triplet with leptons to zero
renders the type-I seesaw Minkowski:1977sc ; Mohapatra:1979ia ;
Yanagida:1979as ; Gell-Mann:1979vob ; Glashow:1979nm  for neutrino masses
inoperative. To address this, we extend the model by introducing a right-
handed scalar doublet $\chi_{R}$ and three singlet fermions $n=(n_{1,2,3})$.
The Yukawa Lagrangian is then expanded to include the following additional
terms:

$\displaystyle{\cal
L}^{\prime}_{Y}~{}=~{}f^{\prime}_{a}\bar{\ell}_{a,R}\chi_{R}n_{a}+\mu_{ab}n_{a}n_{b}+{\rm
H.c.}\,,$ (6)

where $\mu_{ab}$ is a small lepton-number-breaking parameter adjusted to fit
neutrino masses. Now, there are additional new terms in the Higgs potential as
follows:

$\displaystyle
V^{\prime}(\chi_{R},\Delta_{R},\Phi)~{}=-\mu^{2}_{\chi}\chi^{\dagger}_{R}\chi_{R}+\lambda_{R}(\chi^{\dagger}_{R}\chi_{R})^{2}$
$\displaystyle\qquad+\lambda^{\prime}_{R}(\chi^{\dagger}_{R}\chi_{R}){\rm
Tr}(\Delta^{\dagger}_{R}\Delta_{R})+\rho{{}^{\prime\prime}}\chi^{\dagger}_{R}\Delta^{\dagger}_{R}\Delta_{R}\chi_{R}$
$\displaystyle\qquad+\rho{{}^{\prime}}\chi^{\dagger}_{R}\Delta_{R}\Delta^{\dagger}_{R}\chi_{R}+\alpha^{\prime}\chi^{\dagger}_{R}\Phi^{\dagger}\Phi\chi_{R}\,.$
(7)

We do not include the $\chi_{L}$ in the theory for the same reason that we do
not include $\Delta_{L}$, i.e. because we break $D$-parity at a much higher
scale. Another possible term $\mu_{\chi}\chi_{R}\Delta_{R}\chi_{R}$ is not
included since it is forbidden by the discrete symmetry
$\Delta_{R}\to-\Delta_{R}$ which we impose to forbid leptonic coupling of
$\Delta_{R}$.

We choose the vacuum as follows: $\langle\chi^{0}_{R}\rangle=u_{R}$ with
$u_{R}\sim$ TeV and $\langle\Delta^{0}_{R}\rangle=v_{R}\gg$ TeV. There are
several implications of this extension of the model: (i) First, once
$\langle\chi^{0}_{R}\rangle\neq 0$, the three neutral fermions
$(\nu,\nu_{R},n)$ mix with each other and via inverse seesaw Mohapatra:1986aw
; Mohapatra:1986bd  give small mass to the neutrinos; (ii) There are new terms
that break the global $SU(3)$ symmetry in addition to those mentioned earlier
explaining smaller $H^{\pm\pm}$ mass, e.g. the $\rho^{\prime}$ and
$\rho^{\prime\prime}$ couplings in the above potential. Since $u_{R}\sim$ TeV,
they only give TeV scale masses to the $H^{\pm\pm}$. Thus, even in the
presence of the right-handed doublet Higgs field, one can have a TeV scale
$H^{\pm\pm}$ with much higher values of $W_{R}$ mass simply by tuning down the
symmetry breaking parameters $\rho_{2}$. The decay of $\chi^{0}_{R}\to
H^{++}H^{--}$ is given by the coupling $\rho^{\prime\prime}$, which we can
choose of order one. This will then help in producing a pair of long-lived
doubly-charged scalars at colliders. We will refer to the neutral scalar field
$\chi^{0}_{R}$ simply as $\chi^{0}$. The coupling $\rho^{\prime}$, on the
other hand, contributes to the mass of $\chi^{0}$, i.e.
$m_{\chi^{0}}^{2}\simeq\rho^{\prime}v^{2}_{R}$. In the following collider
analysis, we fix the ratio $m_{\chi^{0}}/v_{R}=10^{-6}$ in order to keep
$\chi^{0}$ field at the TeV scale while having $v_{R}$ much larger, as needed
to explain the $dE/dx$ excess.

##  V Collider implications

(a) (b) (c)

Figure 4: Representative Feynman diagrams for dominant production modes of
$H^{\pm\pm}$ at the LHC.

Here, we investigate the phenomenological implications of long-lived doubly-
charged scalars in collider experiments. If these LLPs exist within the
accessible energy range of the colliders, they may yield null results in
prompt signal searches. However, their presence could lead to different
signals like disappearing tracks and displaced vertices, depending on their
lifetime, decay modes, and decay length relative to the detector.111Collider
implications of neutral/charged LLPs in the context of other neutrino mass
models have been widely discussed; see e.g., Refs. deCampos:2005ri ;
Dev:2016nfr ; Dev:2017dui ; Ghosh:2018drw ; Jana:2018rdf ; Abada:2018sfh ;
Dev:2018tox ; Arbelaez:2019cmj ; Das:2019fee ; Jana:2020qzn ; Frank:2019nid ;
R:2020odv ; MATHUSLA:2020uve ; Hirsch:2021wge ; Cottin:2022nwp ; Ghosh:2023tur
. Collider experiments from LEP OPAL:2003zpa  to Tevatron CDF:2005hfo , LHC
LHCb:2015ujr ; CMS:2016kce ; ATLAS:2022pib ; ATLAS:2023zxo ; ATLAS:2023esy ;
CMS:2023mny , as well as dedicated detectors like MoEDAL Acharya:2020uwc ;
MoEDAL:2023ost , have searched for the signatures of stable and long-lived
multiply charged particles across diverse energy scales, exploring a wide
range of possibilities for their existence. If the charged LLP lives long
enough to traverse the entire detector without decaying, it leaves a distinct
ionizing track in the inner detector, from which a large ionization energy
gradient ($dE/dx$) signal is expected. It is noteworthy to highlight recent
findings by the ATLAS collaboration, which observed an excess of 7 events
characterized by a substantial energy loss gradient ($dE/dx>2.4$ MeV g-1 cm2),
with a known background of $0.7\pm 0.4$ events, resulting in a local (global)
significance of $3.6\sigma$ (3.3$\sigma$) ATLAS:2022pib . As we show below,
this excess can be explained in our model setup.

In the minimal LRSM, there exist two dominant production modes for doubly-
charged scalars at the LHC: (a) quark-fusion process through $s$–channel
$Z/\gamma$ exchange (Drell-Yan) Gunion:1989in ; FileviezPerez:2008jbu , and
(b) photon fusion processes Babu:2016rcr ; see Fig 4. In the extended version
for neutrino mass generation discussed above, there is an additional gluon-
fusion production channel via $\chi^{0}$ mixing with the SM Higgs boson [cf.
Fig 4 (c)]. This latter diagram turns out to be crucial for explaining the
$dE/dx$ excess. Before going into that discussion, we first derive a lower
limit on the $H^{\pm\pm}$ mass following the recent ATLAS analysis
ATLAS:2023zxo  of heavy, long-lived, multi-charged particle search at the
$\sqrt{s}=13\mathrm{TeV}$ LHC with an integrated luminosity of 139 fb-1. After
incorporating our model file into the FeynRules package Alloul:2013bka , we
calculate the cross-section for the Drell–Yan plus photon fusion pair
production mode of $H^{\pm\pm}$ at the 13 TeV LHC using the Monte Carlo event
generator MadGraph5aMC@NLO Alwall:2014hca . Comparing it with the observed
upper limit on the same Drell–Yan plus photon fusion cross section from ATLAS,
we obtain a lower limit of 700 GeV on the mass of $H^{\pm\pm}$, which is shown
by the vertical dashed line in the $f_{R}\lesssim 10^{-8}$ region of Fig. 2.

However, our analysis indicates that the pair production of long-lived doubly-
charged scalars through the Drell-Yan and photon fusion processes yields a
relatively small boost factor $\beta\lesssim 0.7$, where $\beta(=v/c)$ denotes
the velocity of the ionizing particle within the detector. On the other hand,
all $dE/dx$ excess events display a relatively high $\beta\approx 1$
ATLAS:2022pib . To account for this observed excess and maintain consistency
with time-of-flight measurements, a highly boosted, long-lived doubly-charged
particle is necessary Giudice:2022bpq . Therefore, the pair production of
long-lived doubly-charged scalars within the minimal LRSM, facilitated by
quark and photon fusion processes, is unlikely to explain the observed
substantial ionization energy loss gradient ATLAS:2022pib . However, we have
identified an additional production mechanism for such long-lived doubly-
charged scalars within the LRSM involving the gluon fusion process [cf. Fig. 4
(c)]. Due to mixing with SM Higgs, this process resonantly produces a neutral
$\chi^{0}$, which subsequently decays into a pair of long-lived doubly-charged
scalars. In this scenario, the long-lived doubly-charged scalars can attain
significant boosts, providing a potential explanation for the observed $dE/dx$
excess while maintaining consistency with time-of-flight measurements.

One might wonder whether any of the neutral scalar fields in the minimal LRSM
can also qualify to act as the parent resonance instead of $\chi^{0}$. As
previously discussed, the masses of the bidoublet scalars
($m_{H_{1}^{0},A_{1}^{0}}$) are constrained to be above 15 TeV or so to adhere
to the FCNC constraints Zhang:2007da , so they cannot be resonantly produced
at the LHC. On the other hand, it is possible to have the mass scale of
$m_{H_{3}^{0}}$ significantly lower than the $v_{R}$ scale, depending on the
tunable quartic coupling $\rho_{1}$ [cf. Eq. (3)] Dev:2016nfr ; Dev:2017dui .
In this context, it is crucial to assess the radiative stability of such a
configuration. In the SM, a lower bound of 5 GeV exists for the SM Higgs boson
mass Linde:1975sw ; Weinberg:1976pe , considering only contributions from
gauge bosons and neglecting fermionic contributions in the Coleman-Weinberg
effective potential Coleman:1973jx . Given that we are setting the Yukawa
coupling $f_{R}$ to zero in our scenario, a lower bound on $m_{H_{3}^{0}}$
emerges when considering contributions solely from right-handed gauge bosons
in the Coleman-Weinberg effective potential. Analogous scenarios, as
demonstrated in Ref. Holthausen:2009uc , establish a lower bound of 900 GeV
when considering a neutral scalar originating from an $SU(2)_{R}$-doublet.
Consequently, as long as $m_{H_{3}^{0}}\gtrsim$ TeV, it satisfies these
theoretical constraints. However, the cubic coupling $H_{3}^{0}H^{++}H^{--}$
is proportional to $2\sqrt{2}(\rho_{1}+2\rho_{2})v_{R}$, while the cubic
coupling $H_{3}^{0}hh$ is proportional to $\alpha_{1}v_{R}/\sqrt{2}$
Dev:2016dja . Setting the $v_{R}$ scale to a high-value results in small
values for $\rho_{1,2}$ to maintain the masses of $m_{H_{3}^{0},H^{\pm\pm}}$
at TeV scale. The important point here is that despite the cubic coupling’s
dependence on $v_{R}$, this configuration favors the dominance of the di-Higgs
branching ratio mode in the decay of $H_{3}^{0}$, making the $H^{++}H^{--}$
branching mode significantly small. This is attributed to the mass-coupling
relations, specifically expressed as

$\displaystyle\frac{{\rm Br}(H_{3}^{0}\to hh)}{{\rm Br}(H_{3}^{0}\to
H^{++}H^{--})}=\frac{\sin^{2}\vartheta(\frac{v_{R}}{\kappa})^{2}}{4(1+\frac{2\rho_{2}}{\rho_{1}})^{2}}\,,$
(8)

where $\vartheta$ represents the mixing between the SM Higgs $h$ and
$H_{3}^{0}$. Because the same mixing parameter is responsible for the
$H_{3}^{0}$ production via gluon fusion, it cannot be arbitrarily small. In
fact, to reproduce the observed value of the cross-section for the $dE/dx$
excess, we need $\sin\theta\gtrsim 0.1$ (see below), which implies from Eq.
(8) that the $H_{3}^{0}\to hh$ mode will always be dominant. Due to these
interconnected mass-coupling relationships, $H_{3}^{0}$ does not qualify as a
suitable parent particle responsible for the observed excess in $dE/dx$.

As we discussed in Section IV, within the inverse seesaw extension of the
LRSM, we have the neutral component of the doublet scalar $\chi^{0}$, which
can also mix with the SM Higgs, characterized by the mixing angle $\theta$,
via the $\alpha^{\prime}$ term in the potential (7). The intriguing outcome of
this mixing is the resonant production of $\chi^{0}$ at the LHC through the
gluon fusion process [cf. Fig. 4 (c)]. The magnitude of the mixing angle
$\theta$ is restricted from Higgs signal strength data obtained at the LHC,
which currently permits $\sin~{}\theta<0.33$ from a recent CMS analysis
CMS:2022dwd .222A slightly stronger bound of $\sin~{}\theta<0.26$ is obtained
from the ATLAS analysis ATLAS:2022vkf . In our analysis, we assume that the
VEV $\langle\chi^{0}_{R}\rangle$ is significantly smaller than the right-
handed symmetry-breaking scale $v_{R}$ and place it in the TeV range. As
explained above, this ensures that the cubic coupling $\chi^{0}H^{++}H^{--}$
remains unsuppressed, allowing $\chi_{R}$ to predominantly decay into
$H^{++}H^{--}$, and not to $hh$. Consequently, following resonant production
at the LHC, $\chi^{0}$ primarily undergoes decay into $H^{++}H^{--}$. In this
scenario, the doubly-charged scalars exhibit a substantial boost, as required
to explain the observed $dE/dx$ excess while maintaining agreement with time-
of-flight measurements.

Figure 5: The cross-section times branching ratio for resonant production of
$\chi^{0}$ and decay into a pair of long-lived doubly-charged scalars in the
LRSM at the $\sqrt{s}=13$ TeV LHC via gluon fusion process as a function of
the mass of $\chi^{0}$ for different values of mixing with the SM Higgs. The
green band represents the preferred range, which explains the observed $dE/dx$
excess at 95% CL ATLAS:2022pib ; Giudice:2022bpq . The blue dashed curve
represents the experimental upper limit on the doubly-charged scalar pair-
production cross-section, assuming $m_{\chi^{0}}=2m_{H^{\pm\pm}}$.

Fig. 5 illustrates the cross-section,
$\sigma(pp\to\chi^{0})\times\mathrm{Br}(\chi^{0}\to H^{++}H^{--})$, at the
$\sqrt{s}=13$ TeV LHC as a function of the mass of $\chi^{0}$ for various
values of the mixing angle $\theta$. The green band represents the preferred
range of cross-section (0.10–0.19 fb), which explains the observed $dE/dx$
excess at 95$\%$ CL ATLAS:2022pib ; Giudice:2022bpq . Notably, for $\chi^{0}$
masses exceeding approximately 3 TeV, the production cross-section experiences
a sharp decline, rendering it insufficient to account for the observed excess,
even for the largest possible value of $\sin\theta_{\rm max}\simeq 0.3$
(corresponding to the red curve in Fig. 5). Also shown in Fig. 5 (blue dashed
curve) is the 95% CL upper bound on the cross section $\sigma(pp\to
H^{++}H^{--})$ from the recent ATLAS analysis ATLAS:2023zxo  for a fixed ratio
of $m_{\chi^{0}}/m_{H^{\pm\pm}}=2$. It is clear that the $dE/dx$ preferred
region from the previous ATLAS analysis ATLAS:2022pib  is still consistent
with the latest ATLAS analysis ATLAS:2023zxo  in our model for
$m_{\chi^{0}}>2$ TeV.

Figure 6: Model parameter space in $m_{H^{++}}$ vs $m_{\chi^{0}}$ plane to
explain the $dE/dx$ excess at 95% CL (blue-shaded region). Current LHC data
from LLP searches ATLAS:2023zxo  excludes the black-hatched region. The red-
hatched region is excluded by the LHC data on the Higgs signal strength
observables CMS:2022dwd . The red dashed line corresponds to the lifetime
$\tau=\tau_{U}=4.34\times 10^{17}$ sec. The two black dashed horizontal lines
correspond to mixing values $\sin~{}\theta=0.20$ and 0.26.

Fig. 6 depicts a comprehensive view of the allowed parameter space in the
$m_{H^{++}}$ versus $m_{\chi^{0}}$ plane to explain the $dE/dx$ excess. The
blue-shaded region represents the 95% CL preferred region, taking into account
the number of events necessary for reproducing the best-fit $dE/dx$ excess
signal Giudice:2022bpq :

$\displaystyle N_{\mathrm{ev}}\left(pp\rightarrow\chi^{0}\rightarrow
H^{++}H^{--}\right)=\mathcal{L}\,\varepsilon\,\sigma(pp\to\chi^{0})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\mathrm{Br}(\chi^{0}\to
H^{++}H^{--})\,,$ (9)

where ${\cal L}$ is the integrated luminosity (taken here as 139 fb-1) and
$\varepsilon$ is the signal efficiency factor (taken here to be 20%).
Following Ref. Giudice:2022bpq , we take $N_{\mathrm{ev}}=5$ to derive our
best-fit region in Fig. 6. We ensure that these events originate from
anomalously large ionizing tracks attributed to rapidly moving ($\beta\approx
1$), long-lived doubly-charged scalars $H^{\pm\pm}$ and are consistent with
the observed $m_{dE/dx}$ distribution Giudice:2022bpq . Furthermore, we fix
${v_{R}}/{m_{\chi^{0}}}=10^{6}$ (thus inherently fixing the $W_{R}$ mass) to
optimize the scenario, enabling us to achieve a doubly-charged scalar lifetime
$\tau>\tau_{U}$ within the parameter space of our interest, as shown by the
red dashed line. Notably, within the mass range of $\chi^{0}$ from 2.2 TeV to
3.1 TeV and $m_{H^{++}}$ from 700 GeV up to $m_{\chi^{0}}/2$, a considerable
parameter space exists, yielding a substantial ionization energy loss gradient
capable of explaining the $dE/dx$ excess. Outside the blue region, either the
production cross-section proves inadequate to explain the observed events, or
the $H^{\pm\pm}$ particles fail to attain the requisite boost. It was shown in
Ref. Giudice:2022bpq  that if we also take the $p_{T}$ distribution into
account, combined with the $m_{dE/dx}$, the optimal range for the mass of the
parent particle $\chi^{0}$ is found to be above $3$ TeV. However, in order to
achieve a sufficient cross-section above $3$ TeV in our Higgs mixing scenario,
a larger mixing angle $\theta$ is required, which is excluded by the current
Higgs signal strength data from the LHC, as shown by the red hatched region in
Fig. 6. Here we have used the CMS analysis CMS:2022dwd  which gives
$\sin\theta<0.33$; using the ATLAS analysis ATLAS:2022vkf  would give
$\sin\theta<0.26$ (also indicated in Fig. 6). The presented region in our
analysis specifically yields a favorable fit to the $m_{dE/dX}$ distribution.
Notably, it requires the Higgs mixing angle $\sin\theta$ to be above 0.15,
which can be conclusively tested in future precision Higgs data at the HL-LHC
or at future colliders deBlas:2019rxi , should the $dE/dx$ excess persist at
LHC Run-3.

##  VI $H^{--}$–catalyzed fusion mechanism

Long-lived doubly-charged particles have the potential to catalyze the fusion
of light nuclei, presenting a significant possibility for applications in
energy production Akhmedov:2021qmr . Specifically, the negatively-charged
$H^{--}$ can establish atomic bound systems with light element nuclei,
including deuterium, tritium, or helium. Such atomic systems will be
characterized by very small Bohr radii, which will essentially eliminate the
necessity for the light nuclei within them to overcome the Coulomb barrier in
order to undergo fusion. There are several articles in the literature
discussing the potential of generating energy through nuclear fusion with
hypothetical heavy, long-lived or stable singly-charged Zeldovich:1958 ;
Ioffe:1979tv ; Rafelski:1989pz ; Hamaguchi:2006vp ; Hamaguchi:2012px  or
fractionally charged Zweig:1978sb  particles. However, these approaches
encountered catalytic poisoning challenges, making them unsuitable for
practical energy production. Some studies Ioffe:1979tv  proposed reactivating
catalyst particles, but this would require neutron beams significantly
exceeding the current capabilities of nuclear reactors.

This study focuses on catalyzing light nuclei fusion with doubly-charged
$H^{--}$ particles within the LRSM, highlighting its potential as a feasible
energy source. Catalytic poisoning, in this case, has a very small
probability, and each $H^{--}$ can catalyze up to $3.5\times 10^{9}$ deuterium
fusion cycles, producing $\sim 7\times 10^{4}$ TeV of energy before being
knocked out of the catalytic process due to sticking to the produced 6Li
nuclei. Unstable $H^{--}$ have to be produced at accelerators; the catalytic
process is then energetically favorable for $H^{--}$ lifetimes above
$9.5\times 10^{11}$ sec, but in general, requires reactivation and reuse of
the catalyst particles. For stable or practically stable $H^{--}$
($\tau>\tau_{U}=4.34\times 10^{17}$ sec), there should exist a terrestrial
population of relic $H^{--}$ bound to nuclei and thus forming exotic atoms;
they can be extracted from continental crust or marine sediments at low energy
cost instead of being produced at accelerators. No reactivation of the
catalyst particles is necessary in this case.

In Fig. 3, we show by red dot-dashed and dashed lines the two important
benchmark lifetimes ($\tau=9.5\times 10^{11}$ sec and $\tau=4.34\times
10^{17}$ sec, respectively) relevant for the catalyzed fusion mechanism in the
minimal LRSM. These correspond to the minimum $SU(2)_{R}$-breaking scales of
$v_{R}\gtrsim 4\times 10^{5}$ TeV and $3\times 10^{6}$ TeV, respectively.
Thus, we find that the doubly-charged scalars in the minimal LRSM are a
promising candidate for the catalyzed fusion mechanism.

We note that the available experimental searches of exotic atoms on the earth
allow for the existence of stable doubly-charged particles in abundances
suitable for their practical use for catalyzed fusion, provided that their
mass exceeds 1.2 TeV Smith:1982qu ; see Ref. Akhmedov:2021qmr  for more
details. Constraints on massive charged relics have also been derived from
astrophysical observations, such as abundances of old white dwarfs
Fedderke:2019jur . These constraints, however, are rather weak for moderately
light charged relics with masses of interest to us ($10^{3}$ – $10^{5}$ GeV)
and are model-dependent Fedderke:2019jur ; Chuzhoy:2008zy ; they do not
challenge the prospects of using catalyzed fusion for energy production.
Fusion of light nuclei catalyzed by doubly negatively charged scalars may also
play an important role in big-bang nucleosynthesis AP .

##  VII Conclusions

Doubly-charged particles frequently emerge in BSM scenarios. However, they do
not typically qualify as long-lived particles. Here we have shown that the
doubly-charged scalars originating from the $SU(2)_{R}$-triplet Higgs field
within the minimal Left-Right Symmetric Model can be long-lived. By virtue of
a concealed symmetry in the scalar potential, the mass of these doubly-charged
particles (denoted here by $H^{\pm\pm}$) can be significantly lower than the
$SU(2)_{R}$-breaking scale, rendering them sufficiently long-lived (even
cosmologically stable), while adhering to all prevailing theoretical and
experimental constraints. This unique feature positions these $H^{\pm\pm}$
particles as ideal candidates for catalyzed fusion processes in light nuclei,
offering promising prospects for energy production. Incorporating neutrino
mass through the inverse seesaw mechanism and simultaneously allowing the
existence of long-lived doubly-charged scalars enables us to provide an
explanation for the recently observed ionization energy loss ($dE/dx$) excess
in the LHC data. Consequently, this model may carry substantial implications
for heavy stable charged particle searches in future collider experiments, and
can be independently tested in future precision Higgs data.

## Acknowledgments

EA is grateful to Goran Senjanović for an illuminating email correspondence.
BD thanks Yongchao Zhang for useful discussion, and acknowledges the local
hospitality of MPIK where this work was originally conceived. BD and SJ
acknowledge the Fermilab Theoretical Physics Department, where part of this
work was done, for their warm hospitality. BD and SJ also wish to thank the
Center for Theoretical Underground Physics and Related Areas (CETUP*) and the
Institute for Underground Science at SURF for hospitality and for providing a
stimulating environment. The work of BD was partly supported by the U.S.
Department of Energy under grant No. DE-SC 0017987.

##  Appendix A Scalar Sector of the LRSM

In the minimal LRSM, the most general renormalizable scalar potential,
invariant under parity, is given by Deshpande:1990ip ; Maiezza:2016ybz ;
Dev:2018xya

$\displaystyle V(\Phi,\Delta_{L},\Delta_{R})=$ $\displaystyle-\mu^{2}_{1}{\rm
Tr}(\Phi^{\dagger}\Phi)-\mu^{2}_{2}\left[{\rm
Tr}(\Phi^{\dagger}\tilde{\Phi})+{\rm
Tr}(\Phi\tilde{\Phi}^{\dagger})\right]-\mu^{2}_{3}\left[{\rm
Tr}(\Delta^{\dagger}_{L}\Delta_{L})+{\rm
Tr}(\Delta^{\dagger}_{R}\Delta_{R})\right]$
$\displaystyle+\lambda_{1}\left[{\rm
Tr}(\Phi^{\dagger}\Phi)\right]^{2}+\lambda_{2}\left\\{\left[{\rm
Tr}(\Phi^{\dagger}\tilde{\Phi})\right]^{2}+\left[{\rm
Tr}(\Phi\tilde{\Phi}^{\dagger})\right]^{2}\right\\}$
$\displaystyle+\lambda_{3}{\rm Tr}(\Phi^{\dagger}\tilde{\Phi}){\rm
Tr}(\Phi\tilde{\Phi}^{\dagger})+\lambda_{4}{\rm
Tr}(\Phi^{\dagger}\Phi)\left[{\rm Tr}(\Phi\tilde{\Phi}^{\dagger})+{\rm
Tr}(\Phi^{\dagger}\tilde{\Phi})\right]$
$\displaystyle+\rho_{1}\left\\{\left[{\rm
Tr}(\Delta^{\dagger}_{L}\Delta_{L})\right]^{2}+\left[{\rm
Tr}(\Delta^{\dagger}_{R}\Delta_{R})\right]^{2}\right\\}+\rho_{2}\left[{\rm
Tr}(\Delta_{L}\Delta_{L}){\rm
Tr}(\Delta^{\dagger}_{L}\Delta^{\dagger}_{L})+{\rm
Tr}(\Delta_{R}\Delta_{R}){\rm
Tr}(\Delta^{\dagger}_{R}\Delta^{\dagger}_{R})\right]$
$\displaystyle+\rho_{3}{\rm Tr}(\Delta_{L}^{\dagger}\Delta_{L}){\rm
Tr}(\Delta_{R}^{\dagger}\Delta_{R})+\rho_{4}\left[{\rm
Tr}(\Delta_{L}\Delta_{L}){\rm
Tr}(\Delta^{\dagger}_{R}\Delta^{\dagger}_{R})+{\rm
Tr}(\Delta_{R}\Delta_{R}){\rm
Tr}(\Delta^{\dagger}_{L}\Delta^{\dagger}_{L})\right]$
$\displaystyle+\alpha_{1}{\rm Tr}(\Phi^{\dagger}\Phi)\left[{\rm
Tr}(\Delta^{\dagger}_{L}\Delta_{L})+{\rm
Tr}(\Delta^{\dagger}_{R}\Delta_{R})\right]+\left\\{\alpha_{2}e^{i\delta_{2}}\left[{\rm
Tr}(\Phi^{\dagger}\tilde{\Phi}){\rm Tr}(\Delta^{\dagger}_{L}\Delta_{L})+{\rm
Tr}(\Phi\tilde{\Phi}^{\dagger}){\rm
Tr}(\Delta^{\dagger}_{R}\Delta_{R})\right]+{\rm H.c.}\right\\}$
$\displaystyle+\alpha_{3}\left[{\rm
Tr}(\Phi\Phi^{\dagger}\Delta_{L}\Delta_{L}^{\dagger})+{\rm
Tr}(\Phi^{\dagger}\Phi\Delta_{R}\Delta_{R}^{\dagger})\right]+\beta_{1}\left[{\rm
Tr}(\Phi\Delta_{R}\Phi^{\dagger}\Delta_{L}^{\dagger})+{\rm
Tr}(\Phi^{\dagger}\Delta_{L}\Phi\Delta_{R}^{\dagger})\right]$
$\displaystyle+\beta_{2}\left[{\rm
Tr}(\tilde{\Phi}\Delta_{R}\Phi^{\dagger}\Delta_{L}^{\dagger})+{\rm
Tr}(\tilde{\Phi}^{\dagger}\Delta_{L}\Phi\Delta_{R}^{\dagger})\right]+\beta_{3}\left[{\rm
Tr}(\Phi\Delta_{R}\tilde{\Phi}^{\dagger}\Delta_{L}^{\dagger})+{\rm
Tr}(\Phi^{\dagger}\Delta_{L}\tilde{\Phi}\Delta_{R}^{\dagger})\right].$ (10)

Due to the left-right symmetry, all 18 parameters $\mu^{2}_{1,2,3}$,
$\lambda_{1,2,3,4}$, $\rho_{1,2,3,4}$, $\alpha_{1,2,3}$ and $\beta_{1,2,3}$
are real, and the only phase associated with $\alpha_{2}$ is explicitly shown.
Minimizing the potential with respect to the VEVs given in Eq. (17), we can in
fact remove many of these parameters.

Following the convention in Ref. Zhang:2007da , the VEVs of the scalar fields
are given by

$\displaystyle\langle\Phi\rangle=\left(\begin{array}[]{cc}\kappa&0\\\
0&\kappa^{\prime}e^{i\delta}\end{array}\right)\,,\quad\langle\Delta_{L}\rangle=\left(\begin{array}[]{cc}0&0\\\
v_{L}e^{i\theta}&0\end{array}\right)\,,\quad\langle\Delta_{R}\rangle=\left(\begin{array}[]{cc}0&0\\\
v_{R}&0\end{array}\right)\,.$ (17)

In terms of the scalar fields, expanding them around the corresponding VEVs,
we get 20 degrees of freedom in total, which can be separated into the
neutral, singly-charged and doubly-charged components, as follows:

$\displaystyle{\textrm{Neutral}}:$
$\displaystyle\left\\{\Re({\phi_{1}^{0}}),\Re(\phi_{2}^{0}),\Re(\Delta_{L}^{0}),\Re(\Delta_{R}^{0}),\Im({\phi_{1}^{0}}),\Im(\phi_{2}^{0}),\Im(\Delta_{L}^{0}),\Im(\Delta_{R}^{0})\right\\}\,.$
$\displaystyle{\textrm{Singly-charged}}:$
$\displaystyle\left\\{\phi_{1}^{\pm},\phi_{2}^{\pm},\Delta_{L}^{\pm},\Delta_{R}^{\pm}\right\\}\,.$
$\displaystyle{\textrm{doubly-charged}}:$
$\displaystyle\left\\{\Delta_{L}^{\pm\pm},\Delta_{R}^{\pm\pm}\right\\}\,.$
(18)

In the physical mass basis, two neutral components and two pairs of singly-
charged components are identified as the Goldstone modes which are responsible
for the masses of the two neutral ($Z,Z_{R}$) and two pairs of charged
($W^{\pm},W_{R}^{\pm}$) gauge bosons, respectively. Thus, we are left with 14
physical scalar fields.

Due to the large number of parameters involved, the general expressions for
the mass eigenvalues are complicated. However, a few simplifying assumptions
can be made here. First, the $\Delta_{L}$ VEV is restricted to be small from
electroweak precision data, and in particular, from $\rho$-parameter
constraint: $v_{L}\lesssim 2$ GeV Hollik:1986gg ; Akeroyd:2005gt . In
practice, since $v_{L}$ contributes to the neutrino masses, it cannot be much
larger than the eV scale (without introducing too much fine-tuning).
Therefore, it is a natural choice to ignore $v_{L}$ (compared to the VEVs
$\kappa$ and $v_{R}$) and the quartic couplings $\beta_{i}$ in Eq. (10) while
deriving the mass formulas. Similarly, in light of the third-generation
fermion masses, we expect $\xi=\kappa^{\prime}/\kappa\lesssim
m_{b}/m_{t}\simeq 0.03$ (again unless we allow for fine-tuning). Finally,
given the experimental lower limit of a few TeV on the $W_{R}$ mass
CMS:2021dzb ; ATLAS:2023cjo , we have $\varepsilon=\kappa/v_{R}\ll 1$.
Furuthermore, the CP observables require that the phase $\delta\ll 1$
Zhang:2007da .

Taking all this into account, we can do a perturbative expansion in the small
parameters $\varepsilon,\xi$ and $\alpha$ and only keep the terms up to
quadratic order. For the neutral scalar sector, the eigenvalues for the six
physical mass eigenstates are thus given by Maiezza:2016ybz

$\displaystyle m_{h}^{2}=$
$\displaystyle\left(4\lambda_{1}-\frac{\alpha_{1}^{2}}{\rho_{1}}\right)\kappa^{2}\,,$
(19) $\displaystyle m_{H_{1}^{0}}^{2}=$
$\displaystyle\alpha_{3}(1+2\xi^{2})v_{R}^{2}+4\left(2\lambda_{2}+\lambda_{3}+\frac{4\alpha_{2}^{2}}{\alpha_{3}-4\rho_{1}}\right)\kappa^{2}\,,$
(20) $\displaystyle m_{H_{2}^{0}}^{2}=$
$\displaystyle(\rho_{3}-2\rho_{1})v_{R}^{2}\,,$ (21) $\displaystyle
m_{H_{3}^{0}}^{2}=$ $\displaystyle
4\rho_{1}v_{R}^{2}+\left(\frac{\alpha_{1}^{2}}{\rho_{1}}-\frac{16\alpha_{2}^{2}}{\alpha_{3}-4\rho_{1}}\right)\kappa^{2}\,,$
(22) $\displaystyle m_{A_{1}^{0}}^{2}=$
$\displaystyle\alpha_{3}(1+2\xi^{2})v_{R}^{2}+4(\lambda_{3}-2\lambda_{2})\kappa^{2}\,,$
(23) $\displaystyle m_{A_{2}^{0}}^{2}=$
$\displaystyle(\rho_{3}-2\rho_{1})v_{R}^{2}\,.$ (24)

Here $h$ can be readily identified as the SM-like Higgs boson, with its mass
being proportional to the VEV $\kappa$ and independent of $v_{R}$, i.e.
$m_{h}\simeq 125$ GeV. $H_{2}^{0}$ and $A_{2}^{0}$ are just the mass
eigenstates of $\Re(\Delta_{L}^{0})$ and $\Im(\Delta_{L}^{0})$ respectively,
whereas $H_{3}^{0}$ is mostly composed of $\Re(\Delta_{R}^{0})$ with a small
mixing with $h$.

In the singly-charged sector, the mass eigenvalues are

$\displaystyle m_{H_{1}^{\pm}}^{2}=$
$\displaystyle\alpha_{3}(1+2\xi^{2})v_{R}^{2}+\frac{1}{2}\alpha_{3}\kappa^{2}\,,$
(25) $\displaystyle m_{H_{2}^{\pm}}^{2}=$
$\displaystyle(\rho_{3}-2\rho_{1})v_{R}^{2}+\frac{1}{2}\alpha_{3}\kappa^{2}\,.$
(26)

The doubly-charged mass eigenvalues are given by

$\displaystyle m_{H_{1}^{\pm\pm}}^{2}=$
$\displaystyle(\rho_{3}-2\rho_{1})v_{R}^{2}+\alpha_{3}\kappa^{2}\,,$ (27)
$\displaystyle m_{H_{2}^{\pm\pm}}^{2}=$ $\displaystyle
4\rho_{2}v_{R}^{2}+\alpha_{3}\kappa^{2}\,,$ (28)

where $H_{1,2}^{\pm\pm}$ are essentially the mass eigenstates of
$\Delta_{L,R}^{\pm\pm}$ with no mixing between them up to ${\cal
O}(\varepsilon^{2})$.

It is easy to see from the above equations for Higgs boson masses that in the
$D$-parity conserving version of the model, the $H^{\pm}_{1}$ is lighter than
$H^{\pm\pm}_{1}$, because $\alpha_{3}$ has to be positive (since the heavy
bidoublet mass is proportional to $\alpha_{3}$). Therefore $H^{\pm\pm}_{1}$
decays quite fast as $H^{\pm\pm}_{1}\to H^{\pm}_{1}+W^{\pm*}$ (via the gauge
coupling and independent of $v_{L}$) and does not qualify as a long-lived
boson. This leaves $H^{\pm\pm}_{2}$ (denoted in the text by $H^{\pm\pm}$) as
the only long-lived doubly-charged particle in the theory.

In the $D$-parity broken version, $\Delta_{L}^{++}$ and $\Delta^{0,+}_{L}$
have additional contribution to their masses proportional to the $D$-parity
breaking scale $m_{P}$ and become very heavy since $m_{P}\gg v_{R}$. In the
low-energy version of this theory, therefore the $\Delta_{L}$ field can be
integrated out and we can drop mass eigenstates
$H_{2}^{0},A_{2}^{0},H_{1}^{\pm\pm},H^{\pm}_{1}$ from our consideration. The
eigenvalues of the remaining eight mass eigenstates are given in Eqs. (2)-(4),
where we have kept only the leading-order terms, and have renamed the charged
scalars ($H^{\pm}\equiv H_{1}^{\pm}$, $H^{\pm\pm}\equiv H_{2}^{\pm\pm}$) by
dropping the subscripts for brevity.

##  Appendix B $H^{++}$ decays via right-handed gauge bosons

The partial width for the decay of $H^{++}$ into the right-handed gauge bosons
[cf. Fig. 1(b)] is given by Dev:2018tox

$\displaystyle\Gamma(H^{++}\to W_{R}^{+(*)}W_{R}^{+(*)}\to
f\bar{f}^{\prime}f^{\prime\prime}\bar{f}^{\prime\prime\prime})\simeq\frac{m_{H^{++}}^{3}}{16\pi^{3}v_{R}^{2}}\int_{0}^{m_{H^{++}}^{2}}dp\int_{0}^{(m_{H^{++}}-\sqrt{p})^{2}}dq$
$\displaystyle\times\lambda^{1/2}(p,q,m_{H^{++}}^{2})\left[\lambda(p,q,m_{H^{++}}^{2})+\frac{12pq}{m_{H^{++}}^{4}}\right]\left[\frac{m_{W_{R}}\Gamma_{W_{R}}}{(p-m_{W_{R}}^{2})^{2}+m_{W_{R}}^{2}\Gamma_{W_{R}}^{2}}\right]\left[\frac{m_{W_{R}}\Gamma_{W_{R}}}{(q-m_{W_{R}}^{2})^{2}+m_{W_{R}}^{2}\Gamma_{W_{R}}^{2}}\right]\,,$
(29)

where $\Gamma_{W_{R}}\simeq\frac{g_{R}^{2}}{4\pi}m_{W_{R}}$ is the total decay
width of the $W_{R}$ boson and
$\lambda(x,y,z)\equiv\left(1-\frac{x}{z}-\frac{y}{z}\right)^{2}-\frac{4xy}{z^{2}}$.
Since $m_{W_{R}}^{2}=\frac{1}{2}g_{R}^{2}v_{R}^{2}$, in the limit of
$m_{H^{++}}\ll m_{W_{R}}$, the decay rate scales as
$m_{H^{++}}^{7}/v_{R}^{6}$, as expected for a four-body decay.

Finally, the one-loop diagram in Fig. 1(c) is a higher-order correction to the
tree-level diagram in Fig. 1(a), and therefore, is expected to be sub-
dominant. The way to see this is as follows: It crucially depends on the
Majorana mass of the right-handed neutrino in the loop. Therefore, in the
limit of $f_{R}=0$ which implies $m_{N}=0$, this diagram must vanish. A
detailed computation of the loop diagram (following Ref. Zeleny-Mora:2021tym )
indeed confirms this to be true. Since we are working in this limit, we do not
consider this diagram in our lifetime analysis.

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