$\geq$ 10 GeV, cascade decays open up and start dominating depending on the
exact value of $v_{\Delta}$. Note that the mass splitting $|\Delta m|$ between
any two components of $\Delta$ cannot be larger than $\sim 50$ GeV due to
stringent constraints from electroweak precision data Ghosh:2018drw .

###  8.3 Comment on 4-body Decay of $\Delta$

In addition to the two-body decays given in Eqs. (8.130)-(8.135), there will
also be four-body decay modes of the doubly-charged scalar via the virtual
exchange of $R_{2}$ and $S_{3}$ LQs proportional to the $\mu$ term in Eq.
(2.7): $\Delta^{\pm\pm}\to(\omega^{\pm 2/3})^{\star}(\rho^{\pm 4/3})^{\star}$,
with each LQ decaying to two fermions. These decays will depend on the same
parameters that lead to $\Delta^{\pm\pm}\rightarrow\ell^{\pm}\ell^{\pm}$
decays. The phase space for these decays would appear to be comparable to the
two-body decays, since the latter has a suppression of a loop factor,
$1/(16\pi^{2})^{2}$. We have evaluated these four-body decays of $\Delta^{++}$
semi-analytically following the procedure outline in Ref. Pois:1993ay , as
well as numerically. The two methods gave very similar results. As an example,
for a benchmark values of $m_{\Delta^{++}}=800$ GeV, $m_{R_{2}}=1$ TeV,
$m_{S_{3}}=2$ TeV, $\mu=246$ GeV, $v_{\Delta}=10^{-4}$ GeV, and the values of
the Yukawa couplings given in Fit I (cf. Eq. (7.127)), the four-body decay
width is $2.3\times 10^{-15}$ GeV, which turns out to be much smaller than
that for the dileptonic decay, which is $2\times 10^{-9}$ GeV. As $v_{\Delta}$
is increased, the four-body decay may compete with the dileptonic decay;
however, in this case $\Delta^{++}\rightarrow W^{+}W^{+}$ decay would
dominate. Consequently, the four-body decay of $\Delta^{++}$ can be safely
ignored in our discussions.

###  8.4 Signal Sensitivity

We focus on the small $v_{\Delta}$ region which gives same-sign dilepton final
states from the $\Delta^{\pm\pm}$ decay, because charged leptons with large
transverse momenta can be cleanly identified with good resolution and the
charge of the leptons can be identified with fairly good accuracy at hadron
colliders. For the benchmark fits given in Section 7.2 with normal hierarchy,
the dilepton branching ratios of the $\Delta^{\pm\pm}\to\ell_{i}\ell_{j}$ for
different flavors are as follows:

$\displaystyle{\rm BR}(ee)\ =\ 0\,,\quad{\rm BR}(\mu\mu)\ =\ 0.22\,,\quad{\rm
BR}(\tau\tau)\ =\ 0.23\,,$ $\displaystyle{\rm BR}(e\mu)\ =\ 0.01\,,\quad{\rm
BR}(\mu\tau)\ =\ 0.39\,,\quad{\rm BR}(e\tau)\ =\ 0.16\,.$ (8.139)

For simplicity, we focus on the $\mu\mu$ final states and consider the signal
$pp\to\Delta^{++}\Delta^{--}+X\to\mu^{+}\mu^{+}\mu^{-}\mu^{-}+X$ to derive the
sensitivity at future hadron colliders. The relevant SM background is mainly
from the multi-top and multi-gauge boson production Aaboud:2017qph ;
CMS:2017pet . However, there are several discriminating characteristics of our
signal: (a) the invariant mass distributions for same-sign lepton pair from
the $\Delta^{\pm\pm}$ decay would peak at a mass value much higher than the SM
$Z$ boson mass; and (b) the outgoing leptons will be more energetic compared
to the ones produced in the decay of SM gauge bosons, since these leptons are
produced from heavy particle $\Delta^{\pm\pm}$ decay. To derive the signal
sensitivity, we first implement our model file in FeynRules package
Christensen:2008py , then analyze the cross section for the signal using
MadGraph5aMC@NLO Alwall:2014hca , simulating the hadronization effects with
Pythia8 Sjostrand:2007gs  and detector effects with the Delphes3 package
deFavereau:2013fsa . In order to optimize the signal efficiency over the SM
background, we impose the following basic acceptance criteria:
$p_{T}^{\ell}>15$ GeV for each lepton, pseudorapidity $|\eta^{\ell}|<2.5$ and
a veto on any opposite sign dilepton pair invariant mass being close to the
$Z$ boson mass $|M(\ell^{+}\ell^{-})-m_{Z}|>15$ GeV. In addition, events are
selected such that the invariant mass for same-sign muon pair is higher than
500 GeV. After passing through all these acceptance criteria, we estimate the
required luminosities to observe at least 25 events at different center-of-
mass energies ($\sqrt{s}$=14, 27, 100 $\mathrm{TeV}$). Our results are shown
in Fig. 14. It is clear that for a given luminosity and a given $\sqrt{s}$,
the doubly-charged scalar mass reach in the LQ mode is higher than that in the
DY mode. The mass reach for 3 ab-1 integrated luminosity is summarized in
Table V for different center-of-mass energies.

Figure 14: Required integrated luminosities for different center-of-mass energies of the $pp$ collider to observe at least 25 events for the signal $pp\to\Delta^{++}\Delta^{--}+X\to\mu^{+}\mu^{+}\mu^{-}\mu^{-}+X$ in the LQ and DY production modes.  Production |  $\Delta^{\pm\pm}$ mass reach for ${\cal L}=3$ ab-1
---|---
Channel |  $\sqrt{s}=14$ TeV |  $\sqrt{s}=27$ TeV |  $\sqrt{s}=100$ TeV
LQ-mode | 1.1 TeV | 2.0 TeV | 6.2 TeV
DY-mode | 0.9 TeV | 1.3 TeV | 2.9 TeV
Table V: Comparison of the doubly-charged scalar mass reach in the LQ and DY
modes (with same-sign di-muon pair final states only) for 3 ab-1 integrated
luminosity.

Once we identify the doubly-charged scalar from the multi-lepton signal, the
next step is to distinguish the underlying model. In order to identify whether
the $\Delta^{\pm\pm}$’s come from the $S_{3}$ LQ decay, accompanied by the
$\omega^{2/3}$ LQs, we can consider the decay chain given in Fig. 11, i.e.

$\displaystyle pp\ \to\ \rho^{4/3}\rho^{-4/3}\ \to\
\omega^{-2/3}\Delta^{++}\omega^{2/3}\Delta^{--}\ \to\
\ell^{+}\ell^{+}\ell^{-}\ell^{-}+\tau^{+}\tau^{-}+b\bar{b}\,.$ (8.140)

In this case, the right combination of the $b\tau$ invariant mass peaks at the
$\omega^{2/3}$ LQ mass, if it is produced on-shell from the $\Delta$ decay.
Considering the fact that the benchmark fits in our model give 54% branching
ratio of $\omega^{2/3}$ to $b\tau$ (cf. Table III), and taking into account
the $b$-tagging and $\tau$-identification efficiencies of $\sim 70\%$ each, we
find that at least 25 signal events in the channel given by Eq. (8.140) can be
obtained with 3 ab-1 luminosity for the $S_{3}$ LQ masses up to 1.5, 2.4 and
5.5 TeV respectively at $\sqrt{s}=14$, 27 and 100 TeV. Hence, it is possible
to independently test the unified description of $B$-anomalies, muon $g-2$ and
neutrino masses in our model at future colliders.

##  9 Conclusion

We have presented a radiative neutrino mass model involving TeV-scale scalar
leptoquarks $R_{2}$ and $S_{3}$, which can simultaneously explain the
$R_{D^{(\star)}}$, $R_{K^{(\star)}}$, as well as muon $g-2$ anomalies, all
within $1\,\sigma$ CL, while being consistent with neutrino oscillation data,
as well as all flavor and LHC constraints. The $R_{2}$ LQ is responsible for
the $R_{D^{(\star)}}$ and $(g-2)_{\mu}$, while the $S_{3}$ LQ explains the
$R_{K^{(\star)}}$ anomaly. The model also features a scalar quadruplet
$\Delta$, which is required for the radiative neutrino mass generation. The
same trilinear $\Delta^{\star}R_{2}S_{3}$ coupling that is responsible for
neutrino mass also leads to interesting collider signatures in the $S_{3}$ and
$\Delta$ decays that can be probed in the forthcoming run of the LHC.
Similarly, the same Yukawa couplings responsible for the chirally-enhanced
contribution to $\Delta a_{\mu}$ give rise to new contributions to the SM
Higgs decays to muon and tau pairs, with the modifications to the
corresponding branching ratios being at 2-6% level, which could be tested at
future hadron colliders, such as HL-LHC and FCC-hh.

###### Acknowledgements.

We thank Wolfgang Altmannshofer for useful discussions. The work of KB and AT
are supported in part by U.S. Department of Energy Grant Number DE-SC 0016013.
The work of BD is supported in part by U.S. Department of Energy under Grant
No. DE-SC0017987 and by a Fermilab Intensity Frontier Fellowship. This work is
also supported by the Neutrino Theory Network Program Grant No. DE-
AC02-07CHI11359.

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