# Dark matter for Majorana neutrinos in a $\mathbb{Z}_{4}$ symmetry

Leon M.G. de la<EMAIL_ADDRESS>Rolando
<EMAIL_ADDRESS>Patrick J.
<EMAIL_ADDRESS>Eduardo
<EMAIL_ADDRESS>1Instituto de Física,
Universidad Nacional Autónoma de México, A.P. 20-364, Ciudad de México 01000,
México.
2Physics Division, National Center for Theoretical Sciences, National Taiwan
University, Taipei 106319, Taiwan.
3PRISMA+ Cluster of Excellence and Institute for Nuclear Physics, Johannes
Gutenberg-University, 55099 Mainz, Germany.
4Departamento de Física, Centro de Investigación y de Estudios Avanzados del
Instituto Politécnico Nacional.

###### Abstract

In the minimal Scotogenic model with fermionic dark matter, the neutrino
Yukawa matrix is fine-tuned to satisfy current bounds on $\mu\rightarrow
e\gamma$ while producing dark matter through thermal freezeout. One way to
avoid this bound is to open a new annihilation channel for DM with a scalar
field that breaks the lepton number spontaneously. This introduces a physical
Goldstone boson $a$, which can constitute a light relic and opens up the
lepton flavor violating channel $\mu\rightarrow ea$. In this work, we
introduce a model that contains the new annihilation channel for fermionic
dark matter but avoids the introduction of the Goldstone by using a discrete
$\mathbb{Z}_{4}$ instead of a continuous symmetry. In this way, cosmological
constraints on the light relic and experimental limits on lepton flavor
violating decays to the Goldstone are avoided. We analyze the experimental
viability of the model, identifying the region of parameter space where lepton
flavor violating constraints are not saturated, DM has the correct relic
density, and DM direct detection experiments are sensitive to the DM
candidate.

##  I Introduction

The Standard Model (SM) accurately describes the fundamental particles and
their interactions via the strong and electroweak forces. Nevertheless,
massless neutrinos are considered in its formulation, contradicting neutrino
oscillation experiments which establish the existence of at least two massive
neutrinos Davis et al. (1968); Fukuda et al. (1998); Ahmad et al. (2002);
Eguchi et al. (2003); Ahn et al. (2006); Michael et al. (2006); Abe et al.
(2008, 2012); An et al. (2012); Ahn et al. (2012); Abe et al. (2011, 2014).
The mechanism that provides masses to neutrinos remains an open question
within the SM framework.

Meanwhile, observations of large-scale structure formation and the shape of
the Cosmic Microwave Background (CMB) anisotropy power spectrum conclusively
establish the existence of dark matter (DM) in the universe, providing a value
for its relic density given by Aghanim et al. (2020)

$\Omega_{c}h^{2}=0.120\pm 0.0012\text{ at }68\%\ \text{ C.L.}.$ (1)

The SM does not include a viable DM candidate.

It is possible to address both major theoretical shortcomings in SM extensions
where the neutrino masses are generated at the radiative level, and an extra
symmetry both forbids tree-level neutrino mass terms and provides a DM
candidate (see, e.g., Krauss et al. (2003); Cheung and Seto (2004)). In this
class of SM extensions, the smallness of neutrino masses is addressed
naturally due to extra suppression from loop factors. Additionally, the
lightest particle charged under the extra symmetry becomes stable and
constitutes the DM.

The Scotogenic model Ma (2006) is a paradigmatic example of this class of SM
extensions, where a discrete $\mathbb{Z}_{2}$ symmetry is imposed, and
neutrino masses are generated by one-loop processes. In this case, the DM
candidate is the lightest particle charged under the $\mathbb{Z}_{2}$
symmetry. While an additional $\mathbb{Z}_{2}$ symmetry composes the
Scotogenic model, distinct $\mathbb{Z}_{n}$ symmetries can be used to
stabilize the dark sector in a similar framework, leading to different
phenomenology (see, e.g. Srivastava et al. (2019); Yaguna and Zapata (2024)).
For example, a residual $\mathbb{Z}_{2n}$ symmetry can arise from the breaking
of a continuous symmetry already present in the SM, such as
$U(1)_{B-L}$Bonilla et al. (2020a); Centelles Chuliá et al. (2020), giving DM
and neutrino phenomenology similar to the minimal Scotogenic case.

When the DM candidate in the minimal Scotogenic model is the heavy Majorana
fermion, the annihilation channel which controls DM production requires
sizable Yukawa couplings to SM leptons. This produces observably large lepton
flavor violating (LFV) processes such as $\mu\rightarrow e\gamma$, which are
in conflict with the observed DM relic abundance. A more experimentally viable
solution is to introduce a new scalar field that couples to the dark fermions,
dynamically generates their mass, and opens up a new DM annihilation channel
Bonilla et al. (2020b). If the DM mass, in this case, is generated by the
spontaneous breaking of a global symmetry, the Majoron is also generated,
which gives rise to new phenomenology constrained by LFV, Higgs-to-invisible
decays, and possible cosmological constraints on the new light relics De
Romeri et al. (2023).

In this work, we present a new possibility to explain the origin of neutrino
masses and the production of DM: it is possible to dynamically generate light
fermion masses by breaking a discrete $\mathbb{Z}_{4}$ symmetry, which
prevents the generation of neutrino masses at tree-level. Through the
spontaneous breaking of the $\mathbb{Z}_{4}$ symmetry to a $\mathbb{Z}_{2}$
symmetry, the DM is stabilized, and neutrino masses are generated at one-loop
order, as in the Scotogenic model. For a type I seesaw implementation of this
idea see Bhattacharya et al. (2018). The lightest particle, which is odd under
the $\mathbb{Z}_{2}$ subgroup, provides a DM candidate with either bosonic or
fermionic nature. Meanwhile, this scenario avoids the prohibitive LFV
constraints on DM production in the minimal Scotogenic case, as well as the
phenomenological consequences of the Majoron associated with the dynamical
generation of DM masses by the breaking of a continuous global symmetry, as
proposed in previous studies.

This model provides a uniquely economical way to generate the observed
neutrino masses as well as the DM relic abundance, in a way that is
experimentally viable against stringent constraints on an extended lepton
sector. Furthermore, the experimentally viable window can provide sharp
targets which will be explored in near-future direct DM searches at the
electroweak scale.

This paper is organized as follows. In Section II we introduce the model by
enumerating the symmetries and field content. We write down the interaction
terms and calculate the physical spectrum. We identify theoretical and
experimental constraints on the scalar spectrum. With this, in Section III we
enumerate experimental constraints on the model, namely DM observables and
charged LFV limits. We define the region of the parameter space over which we
will perform a numerical scan. In Section IV we present the results of the
scan, identifying the region where the correct DM relic density is obtained
while satisfying other experimental limits. Finally, in Section V we conclude
on the success of the model in presenting a viable DM candidate and generating
neutrino masses and mixings without saturating limits on LFV rates or the
abundance of light relics.

##  II The model

We propose a SM extension that includes three right-handed sterile neutrinos,
$N_{k}$, a scalar, $\eta$, transforming as $(2,1/2)$ under the
$SU(2)_{L}\times U(1)_{Y}$ SM gauge group, and two scalars, $\sigma$ and
$\phi$, which are singlets are the SM gauge group. These new particles
transform non-trivially under an additional $\mathbb{Z}_{4}$ symmetry, with
the $\mathbb{Z}_{4}$ charges listed in Table 1.

|  Fields | $SU(2)_{L}\otimes U(1)_{Y}$ |  $\mathbb{Z}_{4}$
---|---|---|---
Fermions |  |  |
| $L_{i}$ | ($\mathbf{2},{-1/2}$) | $1$
| $e_{R_{i}}$ | ($\mathbf{1},{-1}$) | $1$
| $N_{k}$ | ($\mathbf{1},{0}$) | $i$
Scalars |  |  |
| $H$ | ($\mathbf{2},{1/2}$) | $1$
| $\eta$ | ($\mathbf{2},{1/2}$) | $i$
| $\sigma$ | ($\mathbf{1},{0}$) | $i$
| $\phi$ | ($\mathbf{1},{0}$) | $-1$
Table 1: Particle content of the model with $i\in\\{e,\mu,\tau\\}$ and
$k\in\\{1,2,3\\}$. All the fields listed are $SU(3)_{C}$ singlets. Fields with
imaginary charges under $\mathbb{Z}_{4}$ become odd under $\mathbb{Z}_{2}$
after SSB, and the lightest among them provides a stable DM candidate. The
scalar field $\phi$ is chosen to be real.

The most general scalar potential of this model is given by:

$\displaystyle V=$
$\displaystyle\mu^{2}_{H}\left(H^{\dagger}H\right)+\mu^{2}_{\eta}\left(\eta^{\dagger}\eta\right)+\mu^{2}_{\sigma}\left(\sigma^{*}\sigma\right)+\mu^{2}_{\phi}\left(\phi^{2}\right)$
(2)
$\displaystyle+\lambda_{1}\left(H^{\dagger}H\right)^{2}+\lambda_{2}\left(H^{\dagger}H\right)\left(\eta^{\dagger}\eta\right)+\lambda_{3}\left(H^{\dagger}\eta\right)\left(\eta^{\dagger}H\right)+\lambda_{4}\left(H^{\dagger}H\right)\left(\sigma^{*}\sigma\right)+\lambda_{5}\left(H^{\dagger}H\right)\phi^{2}$
$\displaystyle+\lambda_{6}\left(\eta^{\dagger}\eta\right)^{2}+\lambda_{7}\left(\eta^{\dagger}\eta\right)\left(\sigma^{*}\sigma\right)+\lambda_{8}\left(\eta^{\dagger}\eta\right)\phi^{2}+\lambda_{9}\left(\sigma^{*}\sigma\right)^{2}+\lambda_{10}\left(\sigma^{*}\sigma\right)\phi^{2}$
$\displaystyle+\lambda_{11}\phi^{4}+\bigg{[}\xi\left(H^{\dagger}\eta\right)\sigma\,\phi+\kappa_{1}\left(H^{\dagger}\eta\right)\sigma^{*}+\kappa_{2}\,\sigma^{2}\phi+\rm
h.c.\bigg{]},$

where:

$H=\begin{pmatrix}H^{+}\\\
H^{0}\end{pmatrix}\quad\text{and}\quad\eta=\begin{pmatrix}\eta^{+}\\\
\eta^{0}\end{pmatrix},$ (3)

represent the SM Higgs and the scalar inert doublet, respectively.

The Yukawa interactions, relevant for neutrino masses, compatible with the SM
and $\mathbb{Z}_{4}$ symmetries are:

$\mathbb{L}_{Y}=Y_{p}\overline{L}\,\tilde{\eta}N+Y_{v}\overline{N^{c}}N\phi+\rm
h.c.$ (4)

The coupling coefficients $\xi,\kappa_{1}$, $\kappa_{2}$,
$\lambda_{a}\,(a=1,...,11)$, $Y_{p}$ and $Y_{v}$ in Equations 2 and 4 are
assumed to be real.

In the stable vacuum, both $\phi$ and $H$ acquire non-zero vacuum expectation
values, and the $\mathbb{Z}_{4}$ and electroweak symmetries are spontaneously
broken. Majorana masses are then generated at leading order through one-loop
processes, as shown in Figure 1.

Figure 1: One-loop Feynman diagram for neutrino mass generation. After SSB,
both $\phi$ and $H$ acquire non-zero vacuum expectation values and Majorana
neutrino masses are generated. The particles inside the loop constitute the
dark sector.

###  II.1 Mass Spectrum

We parameterize the neutral scalar fields $H$, $\eta$, $\sigma$, and $\phi$
around the vacuum in terms of their CP-even and CP-odd components as:

$\displaystyle H^{0}$
$\displaystyle\rightarrow\dfrac{1}{\sqrt{2}}\left(v_{H}+h^{0}_{R}+i\,h^{0}_{I}\right),$
(5) $\displaystyle\eta^{0}$
$\displaystyle\rightarrow\dfrac{1}{\sqrt{2}}\left(v_{\eta}+\eta^{0}_{R}+i\,\eta^{0}_{I}\right),$
$\displaystyle\sigma$
$\displaystyle\rightarrow\dfrac{1}{\sqrt{2}}\left(v_{\sigma}+\sigma_{R}+i\,\sigma_{I}\right),$
$\displaystyle\phi$
$\displaystyle\rightarrow\dfrac{1}{\sqrt{2}}\left(v_{\phi}+\phi_{R}\right),$

where $v_{H}$ is the SM Higgs vacuum expectation value. The scalar potential
minimum corresponds to:

$H^{0}\rightarrow\frac{1}{\sqrt{2}}v_{H}\quad\text{and}\quad\phi\rightarrow\frac{1}{\sqrt{2}}v_{\phi},$
(6)

which spontaneously breaks the symmetry $SU(2)_{L}\times
U(1)_{Y}\times\mathbb{Z}_{4}\rightarrow U(1)_{QED}\times\mathbb{Z}_{2}$. The
stable vacuum configuration is given by:

$\displaystyle v_{\eta}$ $\displaystyle=v_{\sigma}=0,$ (7)
$\displaystyle\mu^{2}_{H}$
$\displaystyle=-v_{H}^{2}\,\lambda_{1}-\dfrac{1}{2}v_{\phi}^{2}\,\lambda_{5},$
$\displaystyle\mu^{2}_{\phi}$
$\displaystyle=-v_{\phi}^{2}\,\lambda_{11}-\dfrac{1}{2}v_{H}^{2}\,\lambda_{5}.$

Spontaneous symmetry breaking (SSB) induces mass mixing between the scalars in
the visible sector ($h^{0}_{R}$ and $\phi_{R}$) through the Higgs portal
interaction, $\lambda_{5}$. Mass mixing is also separately induced between the
scalar dark sector ($\eta^{0}_{R}$ and $\sigma_{R}$) and the pseudoscalar dark
sector ($\eta^{0}_{I}$ and $\sigma_{I}$). Meanwhile, the fields with imaginary
charge $i$ under $\mathbb{Z}_{4}$ are odd under the residual $\mathbb{Z}_{2}$
symmetry after SSB, with the lightest particle being a stable DM candidate.

The mass matrix for the visible sector is given by:

$\mathbb{M}_{\text{Visible}}=\begin{pmatrix}2\,v_{H}^{2}\,\lambda_{1}&&v_{H}\,v_{\phi}\,\lambda_{5}\\\
v_{H}\,v_{\phi}\,\lambda_{5}&&2\,v_{\phi}^{2}\,\lambda_{11}\end{pmatrix},$ (8)

which is diagonalized by the $2\times 2$ rotation matrix:

$\mathbb{O}_{\text{Visible}}=\begin{pmatrix}\cos{\theta_{V}}&\sin{\theta_{V}}\\\
-\sin{\theta_{V}}&\cos{\theta_{V}}\end{pmatrix},$ (9)

where:

$\tan{\left(2\,\theta_{V}\right)}=\dfrac{v_{H}\,v_{\phi}\,\lambda_{5}}{v_{\phi}^{2}\,\lambda_{11}-v_{H}^{2}\,\lambda_{1}}.$
(10)

Thus, the rotation between the interaction and mass eigenstates $h$ and
$\varphi$ is given by

$\begin{pmatrix}h\\\
\varphi\end{pmatrix}=\mathbb{O}_{\text{Visible}}\begin{pmatrix}h^{0}_{R}\\\
\phi_{R}\end{pmatrix},$ (11)

with mass eigenvalues:

$\displaystyle m_{h}^{2}$
$\displaystyle=v_{H}^{2}\,\lambda_{1}+v_{\phi}^{2}\,\lambda_{11}-\sqrt{(v_{H}^{2}\,\lambda_{1}-v_{\phi}^{2}\,\lambda_{11})^{2}+(v_{H}\,v_{\phi}\,\lambda_{5})^{2}},$
(12) $\displaystyle m_{\varphi}^{2}$
$\displaystyle=v_{H}^{2}\,\lambda_{1}+v_{\phi}^{2}\,\lambda_{11}+\sqrt{(v_{H}^{2}\,\lambda_{1}-v_{\phi}^{2}\,\lambda_{11})^{2}+(v_{H}\,v_{\phi}\,\lambda_{5})^{2}}.$
(13)

We identify one of these mass eigenstates, $h$ as the physical Higgs field,
with a mass of $m_{h}=125.25\text{ GeV}$.

The dark sector mass matrix is given by:

$\mathbb{M}_{DM_{X}}=\begin{pmatrix}a&&b_{X}\\\ b_{X}&&c_{X}\end{pmatrix},$
(14)

where:

$\displaystyle a$
$\displaystyle=\dfrac{1}{2}\bigg{[}v_{H}^{2}\,(\lambda_{2}+\lambda_{3})+v_{\phi}^{2}\,\lambda_{8}+2\,\mu_{\eta}^{2}\bigg{]},$
(15) $\displaystyle b_{X}$
$\displaystyle=\dfrac{1}{2}v_{H}\left(\sqrt{2}\,\kappa_{1}+\zeta_{X}\,v_{\phi}\,\xi\right),$
$\displaystyle c_{X}$
$\displaystyle=\dfrac{1}{2}\bigg{[}\zeta_{X}\,2\sqrt{2}\,v_{\phi}\,\kappa_{2}+v_{\phi}^{2}\,\lambda_{10}+v_{H}^{2}\,\lambda_{4}+2\,\mu^{2}_{\sigma}\bigg{]}.$

Here $X=\\{R,I\\}$ denotes scalar $(R)$ and pseudoscalar $(I)$ dark sector
fields, with $\zeta_{R}=1$ and $\zeta_{I}=-1$. Similar to the visible case,
the mass matrices are diagonalized by a $2\times 2$ rotation matrix
$\mathbb{O}_{DM_{X}}$ with:

$\tan{\left(2\,\theta_{X}\right)}=\dfrac{2\,b_{X}}{c_{X}-a}.$ (16)

The rotation between the interaction and mass eigenstates $X_{1}$ and $X_{2}$
is:

$\begin{pmatrix}X_{1}\\\
X_{2}\end{pmatrix}=\mathbb{O}_{DM_{X}}\begin{pmatrix}\eta^{0}_{X}\\\
\sigma_{X}\end{pmatrix},$ (17)

with the corresponding masses:

$\displaystyle m_{X_{1}}^{2}$
$\displaystyle=\dfrac{1}{2}\left(a+c_{X}-\sqrt{(a-c_{X})^{2}+4\,b_{X}^{2}}\right),$
(18) $\displaystyle m_{X_{2}}^{2}$
$\displaystyle=\dfrac{1}{2}\left(a+c_{X}+\sqrt{(a-c_{X})^{2}+4\,b_{X}^{2}}\right).$
(19)

The mass for the charged scalar particle $\eta^{\pm}$ is given by

$m_{\eta^{\pm}}^{2}=\dfrac{1}{2}\left(v_{H}^{2}\,\lambda_{2}+v_{\phi}^{2}\,\lambda_{8}\right)+\mu_{\eta}^{2}.$
(20)

SSB also generates the masses of fermions $N_{k}$ through the Yukawa
interaction:

$m_{N_{k}}=\sqrt{2}\,(Y_{v})_{kk}\,v_{\phi}.$ (21)

We work in the basis where fermions $N_{k}$ do not mix with each other,
implying a diagonal Yukawa matrix $Y_{v}$.

###  II.2 Neutrino Masses

SSB generates light neutrino masses through the one-loop mechanism shown in
Figure 1. The corresponding $3\times 3$ neutrino mass matrix can be written
as:

$\left(\mathbb{M}_{\nu}\right)_{\alpha\beta}=\sum_{i}^{3}\dfrac{(Y_{p})_{\alpha
i}\,(Y_{p})_{i\beta}}{32\pi^{2}}M_{N_{i}}\sum_{A=1}^{4}V_{A}(\theta_{X})V_{A}(\theta_{X})\left[\frac{m_{A}^{2}\log(m_{A}^{2}/M_{N_{i}}^{2})}{m_{A}^{2}-M_{N_{i}}^{2}}\right],$
(22)

where

$\displaystyle m_{1}$ $\displaystyle=m_{R_{1}},\quad\quad m_{2}=m_{R_{2}},$
(23) $\displaystyle m_{3}$ $\displaystyle=m_{I_{1}},\quad\quad
m_{4}=m_{I_{2}};$

and

$\displaystyle V_{1}$ $\displaystyle=\cos\left(\theta_{R}\right),\quad\quad
V_{2}=-\sin\left(\theta_{R}\right),$ (24) $\displaystyle V_{3}$
$\displaystyle=-i\cos\left(\theta_{I}\right),\quad\quad
V_{4}=i\sin\left(\theta_{I}\right).$

###  II.3 Dark Matter Production

After SSB the Majorana fermions $N_{k}$, as well as the neutral scalar and
pseudoscalar dark sector fields, $X_{1}$ and $X_{2}$, are odd under the
remaining $\mathbb{Z}_{2}$ symmetry. Therefore, the lightest of these states
is stable, providing a DM candidate. Here we will consider the phenomenology
of the case in which the DM is the lightest Majorana fermion, $N_{1}$.

We consider thermally produced DM with a mass in the GeV-TeV scale, where a
wealth of current and near-future direct detection experiments are
particularly sensitive. We will characterize the experimental targets of DM
production at the sub-GeV scale in a forthcoming study. At the electroweak
scale, DM annihilates primarily via the channels shown in Figure 2.

DM which is thermally produced by $t$-channel annihilations to SM leptons
through their Yukawa interaction with the scalar dark sector, as shown in
panel (a) of Figure 2, encounters prohibitively strong experimental
constraints from LFV processes Bonilla et al. (2020b); Toma and Vicente
(2014); Vicente and Yaguna (2015). Meanwhile, WIMP DM produced via t-channel
annihilations to SM particles through the Higgs-portal mixing $\lambda_{5}$,
as shown in panel (b) of Figure 2, predicts a large visible mixing angle
$\sin\theta_{V}$, which is excluded by experimental constraints from direct
detection Aalbers et al. (2023) and Higgs-to-invisible decays ATL (2020). The
same is true for WIMP DM thermally produced via $s$-channel annihilations,
shown in panel (d) of Figure 2, except in the experimentally viable resonance
region where either $2m_{N_{1}}\sim m_{h}$ or $2m_{N_{1}}\sim m_{\varphi}$. In
the remaining window, DM is thermally produced through $2\rightarrow 2$
annihilations to scalars, $NN\rightarrow\varphi\varphi$, shown in panel (c) of
Figure 2, with scalars $\varphi$ subsequently decaying to the SM.

DM production through $NN\rightarrow\varphi\varphi$ annihilations exhibits two
distinct parameter regions, the kinematically allowed region where
$m_{N}>m_{\varphi}$, which controls DM production in the majority of the
experimentally viable window, and region where $m_{N}<m_{\varphi}$ and
kinematically suppressed $NN\rightarrow\varphi\varphi$ annihilations,
proceeding at finite temperature, require exponentially enhanced couplings
$Y_{v}\sin\theta_{V}$ to thermally produce the DM relic density, relative to
the kinematically allowed case. Kinematically suppressed
$NN\rightarrow\varphi\varphi$ annihilations produce the DM relic abundance in
a thin mass window $m_{\varphi}/m_{N}\gtrsim 1$ consistent with perturbative
Yukawa couplings $Y_{v}$.

Figure 2: Tree-level DM annihilation channels including, a) $t$-channel
annihilations to SM leptons, b) $t$-channel annihilations to Higgs bosons via
Higgs-portal mixing, c) $t$-channel annihilations to scalars $\varphi$, and d)
$s$-channel annihilations to SM via Higgs-portal mixing.

The results for DM production presented here are obtained by numerical
integration of the Boltzmann equations over the entire experimentally viable
parameter region, using micrOmegas (5.3.41) Bélanger et al. (2018), as
described in Section IV.

##  III Experimental Viability

###  III.1 Boundedness conditions

Perturbativity of the theory requires the scalar and Yukawa couplings of
Equations 2 and 4 to satisfy:

$\displaystyle|\lambda_{a}|,|\xi|,|Y_{v}^{kk}|^{2},|Y_{p}^{ij}|^{2}<4\,\pi,$
(25) $\displaystyle\text{with}\quad
a=(1,...,11),\,k=(1,...,3)\,\text{and}\,\,i,j=(1,...,3).$

The condition for the scalar potential to be bounded from below imposes the
following additional constraints in the parameter spectrum Branco et al.
(2012); Lindner et al. (2016):

$\displaystyle\lambda_{1,6,9,11}>0,\quad\quad\lambda_{2}\geq-2\,\sqrt{\lambda_{1}\,\lambda_{6}},\quad\quad\lambda_{4}\geq-2\,\sqrt{\lambda_{1}\,\lambda_{9}},\quad\quad\lambda_{5}\geq-2\,\sqrt{\lambda_{1}\,\lambda_{11}},$
(26)
$\displaystyle\lambda_{7}\geq-2\,\sqrt{\lambda_{6}\,\lambda_{9}},\quad\quad\lambda_{8}\geq-2\,\sqrt{\lambda_{6}\,\lambda_{11}},\quad\quad\lambda_{10}\geq-2\,\sqrt{\lambda_{9}\,\lambda_{11}},\quad\quad\lambda_{2}+\lambda_{3}\geq-2\sqrt{\lambda_{1}\lambda_{6}}.$

###  III.2 Neutrino Oscillation Data

The light neutrino mass matrix can be rewritten as:

$(\mathbb{M}_{\nu})_{\alpha\beta}=\left(Y_{p}^{T}\,\Lambda\,Y_{p}\right)_{\alpha\beta},$
(27)

where the matrix $\Lambda=\text{diag}\left(\Lambda_{i}\right)$, with:

$\displaystyle\Lambda_{i}$
$\displaystyle=\,\dfrac{1}{32\pi^{2}}M_{N_{i}}\sum_{A=1}^{4}V_{A}(\theta_{X})V_{A}(\theta_{X})\left[\dfrac{m_{A}^{2}\log
m_{A}^{2}-M_{N_{i}}^{2}\log M_{N_{i}}^{2}}{m_{A}^{2}-M_{N_{i}}^{2}}\right],$
(28)

and the dark sector mixing parameters $V_{A}\left(\theta_{X}\right)$ are
defined as in Equation 24.

The neutrino mass matrix can be diagonalized as:

$U_{\rm PMNS}^{T}\,\mathbb{M}_{\nu}\,U_{\rm
PMNS}=\hat{m}_{\nu}\equiv\begin{pmatrix}m_{1}&0&0\\\ 0&m_{2}&0\\\
0&0&m_{3}\end{pmatrix},$ (29)

where $U_{\rm PMNS}$ is the PMNS matrix. In the PDG parametrization we can
write the PMNS matrix in terms of three mixing angles and one complex phase:

$U_{\text{PMNS}}=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta_{CP}}\\\
-s_{12}c_{23}-c_{12}s_{13}s_{23}e^{i\delta_{CP}}&c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta_{CP}}&c_{13}s_{23}\\\
s_{12}s_{23}-c_{12}s_{13}c_{23}e^{i\delta_{CP}}&-c_{12}s_{23}-s_{12}s_{13}c_{23}e^{i\delta_{CP}}&c_{13}c_{23}\\\
\end{pmatrix},$ (30)

where we have used the shorthand $c_{ij}=\cos\theta_{ij}$,
$s_{ij}=\sin\theta_{ij}$. Using the Casas-Ibarra parametrization Casas and
Ibarra (2001), the Yukawa matrix $\left(Y_{p}\right)_{i\alpha}$ can be written
as

$Y_{p}=\sqrt{\Lambda}^{-1}R\,\sqrt{\hat{m}_{\nu}}\,U_{\rm PMNS}^{\dagger},$
(31)

where $R$ is a complex orthogonal matrix satisfying $R^{T}R=1$. Using this
parametrization, we are able to use the neutrino oscillation data, the scalar
spectrum of the theory, and the matrix $R$ as input to determine the Yukawa
matrix values. Neutrino oscillation experiments have measured quite precisely
the mixing in the lepton sector, with several global fits agreeing on the
allowed range of mixing angles de Salas et al. (2021); Capozzi et al. (2021);
Esteban et al. (2020). The remaining major unknowns in the three neutrino
oscillation paradigm are the value of the CP violating phase $\delta_{CP}$ and
the sign of the atmospheric neutrino mass squared difference $\Delta
m_{i3}^{2}$. The choice of $R$ does not affect the prediction for mixing
angles and neutrino masses, but it does affect the prediction for the rates of
LFV processes. In this work, we aim to decouple the DM annihilation cross-
section in the early Universe from LFV constraints. In order to achieve this,
our strategy for fitting oscillation data is to set constraints on the scalar
spectrum so that the Yukawa couplings obtained in Equation 31 lead to small
LFV rates without resorting to a fine-tuning of the Yukawa matrix structure
through the choice of $R$. For this purpose we take into account the
cosmological limit on the sum of neutrino masses Alam et al. (2021)

$\sum_{i}m_{\nu_{i}}<0.13\text{ eV}\quad\quad\text{CMB[TT,TE,EE+lowE] + BAO}.$
(32)

###  III.3 Electroweak precision constraints

The enriched scalar sector induces additional contributions to the electroweak
oblique parameters at the one-loop level with respect to the SM. We calculate
these one-loop contributions to S and T following Grimus et al. (2008). These
parameters are tightly constrained by global fits of electroweak precision
data, we consider the ranges: Navas et al. (2024)

$\displaystyle S$ $\displaystyle=$ $\displaystyle-0.05\pm 0.07$ (33)
$\displaystyle T$ $\displaystyle=$ $\displaystyle 0.00\pm 0.06$ (34)

with a positive 93% correlation.

###  III.4 Lepton Flavor Violating processes

The Yukawa interactions, which contribute to the generation of neutrino masses
in this model, also produce LFV radiative decays through charged
$\eta^{\pm}$-$N$ loops, as in the minimal scotogenic model Toma and Vicente
(2014). This leads to experimental constraints on LFV radiative decays,
$l_{i}\rightarrow l_{j}\gamma$, three-body decays, $l_{i}\rightarrow 3l_{j}$,
and $\mu$-$e$ conversion processes, $\mu+N\rightarrow e+N$, measured in
scattering experiments. We consider the strongest bound $l_{i}\rightarrow
l_{j}\gamma$ to set the requirements for the scalar spectrum that leads to
small $Y_{p}$ Yukawa couplings and therefore to a DM phenomenology dominated
by the $Y_{v}$ couplings.
The $l_{i}\rightarrow l_{j}\gamma$ branching ratio is given by

$BR(l_{i}\rightarrow
l_{j}\gamma)=\frac{3(4\pi)^{3}\alpha_{em}}{4G_{F}^{2}}|A_{D}|^{2}BR(l_{i}\rightarrow
l_{j}\nu_{i}\overline{\nu}_{j}),$ (35)

where $l_{i}\rightarrow l_{j}\nu_{i}\overline{\nu}_{j}$ is the SM, $W$
mediated charged lepton decay process, and

$A_{D}=\sum_{a=1}^{3}(Y_{p})_{ai}^{*}(Y_{p})_{aj}\frac{F_{2}(m^{2}_{N_{i}}/m^{2}_{\eta})}{2(4\pi)^{2}m^{2}_{\eta}}\;,\;F_{2}(x)=\frac{1-6x+3x^{2}+2x^{3}-6x^{2}\ln{x}}{6(1-x)^{4}}.$
(36)

The current most sensitive experimental limit is

$BR(\mu\rightarrow e\gamma)<3.1\times 10^{-13},$ (37)

obtained from MEG-II data in combination with MEG data Afanaciev et al.
(2024).

###  III.5 Higgs Decay

The Yukawa interaction between DM and visible scalars in our model predicts an
additional invisible decay channel for the SM Higgs boson, $h\rightarrow NN$,
through the Higgs portal coupling $\lambda_{5}$. Direct searches for invisible
decay channels of the Higgs boson by ATLAS and CMS (e.g. Tumasyan et al.
(2022); ATL (2020); Biekötter and Pierre (2022)) therefore put direct
experimental limits on our model when $m_{h}>2m_{N_{i}}$.

The $h\rightarrow NN$ decay rate is given by:

$\Gamma_{h\rightarrow
NN}=\frac{1}{8\pi}m_{h}Y_{v}^{2}\sin^{2}\theta_{V}Re\left(1-\frac{4m_{N_{1}}^{2}}{m_{h}^{2}}\right)^{3/2}.$
(38)

The current strongest constraints on the branching ratio
$\text{Br}\left(h\rightarrow NN\right)$, obtained from the combined VBF and
ttH production of Higgs bosons at ATLAS ATL (2020), give
$\text{Br}\left(h\rightarrow NN\right)<0.11$ at 95$\%$ confidence level. This
constraint directly translates to an upper bound on the product of Yukawa
coupling and visible mixing angle, $Y_{v}\sin\theta_{V}$ for a given
$m_{N_{1}}$. We project these experimental limits on the physical parameter
region consistent with DM production and neutrino mass generation in Figures 6
through 8 and find that they are subdominant to direct detection constraints
at the GeV-TeV scale.

###  III.6 Direct Detection

The visible mixing of the SM Higgs and the singlet $\phi$ via the Higgs-portal
coupling induces a tree-level contribution to the DM-nucleon spin-independent
elastic scattering cross section, $\sigma_{\textbf{SI}}$. The cross-section is
given by Matsumoto et al. (2019):

$\sigma_{\textbf{SI}}=Y_{v}^{2}\sin{\theta_{V}}^{2}\cos{\theta_{V}}^{2}\frac{m_{N}^{2}m_{\text{nuc}}^{4}f_{N}^{2}}{\pi
v_{H}^{2}(m_{N}+m_{\text{nuc}})^{2}}\left(m_{\varphi}^{-2}-m_{h}^{-2}\right)^{2},$
(39)

where $f_{N}=f_{Tu}+f_{Td}+f_{Ts}+(2/9)f_{TG}$ with $f_{Tu}=0.0153$,
$f_{Td}=0.0191$, $f_{Ts}=0.0447$ and $f_{TG}=0.921$ and $m_{\text{nuc}}$ is
the mass of the nucleon. This sets a limit on the scalar mixing angle as a
function of the DM mass, the scalar mass, and the Yukawa coupling $Y_{v}$.
Roughly, assuming that the mixing angle is small, to avoid the recent LUX-
ZEPLIN (LZ) constraint $\sigma\leq 10^{-47}\text{cm}^{2}$ Aalbers et al.
(2023) we find $Y_{\nu}\sin\theta_{V}<10^{-4}$ for $m_{\phi}=30$ GeV and
$Y_{\nu}\sin\theta_{V}<10^{-3}$ for $m_{\phi}=100$, $500$ GeV. The recent LZ
results give the strongest experimental constraints to the physical parameter
region consistent with DM and neutrino masses, as shown in Figures 6 through
8.

##  IV Numerical Results

We perform a numerical analysis using the computational tools SARAH (versions
4.15.2 and 4.11.0) Staub (2014), SPheno (4.0.2) Porod (2003); Porod et al.
(2014) and micrOmegas (5.3.41) Bélanger et al. (2018). The model is first
implemented in SARAH, which provides analytical expressions for the spectrum
and couplings of the model and generates a source code for SPheno, where the
analytical expressions are evaluated numerically given specific values of the
Lagrangian parameters. We use micrOmegas to solve the Boltzmann equations for
DM production and calculate direct detection cross-sections based on the
numerical outputs obtained from SPheno555For a pedagogical tutorial on these
computational tools, see Vicente (2015)..

With the Majorana fermion, $N_{1}$, constituting the DM, we perform a random
scan over all physical parameters of the model, over the ranges displayed in
Table 2.

Parameter
---
$\lambda_{2,3,4,6,7,8,9,10}\in[10^{-6},1]$
$(Y_{v})_{11}\in[0.05,1]$
$\kappa_{1}\in[10^{-6},1]\,\,\rm GeV$
$\mu^{2}_{\eta},\mu^{2}_{\sigma}\in[10^{7},10^{8}]\,\,\rm GeV^{2}$
$\kappa_{2}\in[10^{-2},1]\,\,\rm GeV$
$\xi=10^{-13}$
$m_{N_{1}}\in[10,2000]\,\,\rm GeV$
Table 2: Parameter ranges of the numerical scan.

We consider the three cases of fixed $m_{\varphi}=30\,\rm GeV$,
$m_{\varphi}=100\,\rm GeV$ and $m_{\varphi}=500\,\rm GeV$, which gives a
representative characterization of DM production in each distinct thermal
freezeout regime over the entire electroweak scale. Over each numerical scan,
the subset of all parameters, which is consistent with boundedness conditions,
perturbativity, neutrino oscillation data, electroweak precision constraints,
and LFV constraints, as described in Section III, is identified. For example,
the Yukawa couplings $(Y_{p})_{ij}$ are fixed to reproduce observed neutrino
oscillation data and to be allowed by LFV constraints. In order to identify
the experimentally viable window for DM production against direct detection
constraints, for each distinct value of $m_{\varphi}$ we consider a different
range for the Higgs-portal mixing $\lambda_{5}$:

$\displaystyle\lambda_{5}$ $\displaystyle\in[10^{-5},10^{-3}]\text{ for
}m_{\varphi}=30\,\rm GeV,$ $\displaystyle\lambda_{5}$
$\displaystyle\in[10^{-4},10^{-2}]\text{ for }m_{\varphi}=100\,\rm GeV\quad
and$ $\displaystyle\lambda_{5}$ $\displaystyle\in[10^{-3},10^{-1}]\text{ for
}m_{\varphi}=500\,\rm GeV.$

We have set $\xi=10^{-13}$ to fix $\theta_{R}\simeq\theta_{I}$ for simplicity
and without loss of generality. Furthermore, the small values of $\kappa_{1}$
lead to $\theta_{R/I}\ll 1$. In this limit, neutrino masses are determined
dominantly by the $R_{1}$ and $I_{1}$ loops. We set a minimum mass difference
between these states of $|m_{R_{1}}-m_{I_{1}}|>0.1\,\rm GeV$, which in Eq.
(31) leads to $Y_{p}<10^{-2}$ for all entries of the matrix without resorting
to fine-tuning through the choice of $R$. This mass difference is controlled
with the parameter $\kappa_{2}$. With this, we obtain $BR(\mu\rightarrow
e\gamma)<10^{-15}$, two orders of magnitude below the current limit set by
MEG-II and one order of magnitude below its planned sensitivity. Crucially,
the Yukawa couplings in this limit can be pushed even lower by tuning the
value of $\kappa_{2}$ to increase the mass splitting $|m_{R_{1}}-m_{I_{1}}|$
without affecting DM phenomenology. The only observable sensitive to this
splitting, besides the neutrino mass matrix, is the oblique parameter $T$.

Results for DM production over data consistent with the experimental limits
described above are shown in Figs. 3-5. The value of the DM relic abundance
predicted by experimentally viable data is shown as a function of DM mass.
Horizontal red lines represent values of the relic density within a $3\sigma$
range of the result observed by Planck Aghanim et al. (2020), identified in
Equation 1. Grey points represent data that are inconsistent with DM
production, predicting DM overabundance, while blue points represent data that
are consistent with the production of at least a fraction of the DM relic
abundance.

Most of the data in Figures 3 through 5 produce the DM abundance through
$t$-channel annihilations to scalars $\varphi$ (shown in panel (c) of Figure
2), which then promptly decay to SM particles. In the region where
$m_{N_{1}}<m_{\varphi}$, DM is produced through kinematically suppressed
$NN\rightarrow\varphi\varphi$ annihilations in only a thin parameter region
where $\frac{m_{N_{1}}}{m_{\varphi}}\sim 1$. At lower values of
$\frac{m_{N_{1}}}{m_{\varphi}}<1$ the kinematic suppression of
$NN\rightarrow\varphi\varphi$ annihilations leads to an overabundance of DM.
There are also thin regions at $2m_{N_{1}}\sim m_{h}$ and $2m_{N_{1}}\sim
m_{\varphi}$ where the DM relic abundance is produced by $s$-channel DM
annhilations (shown in panel (d) of Figure 2) through an on-shell resonance.

Figure 3: Relic density as function of $m_{N_{1}}$ for $m_{\phi}=30\rm\,GeV$.
The red horizontal line represents the value of the relic density observed by
Planck within a $3\sigma$ range. Blue points represent solutions with
underabundance of DM relic density, gray points lead to overabundance. The
green vertical line indicates where $m_{N_{1}}=m_{\varphi}$. Figure 4: Relic
density as function of $m_{N_{1}}$ for $m_{\phi}=100\rm\,GeV$. Same color code
as Fig. 3. Figure 5: Relic density as function of $m_{N_{1}}$ for
$m_{\phi}=500\rm\,GeV$. Same color code as Fig. 3.

Our results for the experimental viability of data that produces both the
observed DM relic abundance and generates experimentally consistent neutrino
masses are shown in Figures 6 through 8. Data are shown projected against the
strongest current direct detection constraints from the LZ experiment Aalbers
et al. (2023). The currently strongest experimental constraint on Higgs-to-
invisible decaysATL (2020) is also shown in Figures 6 through 8, although it
is far less sensitive than direct detection limits at this DM mass scale.

The parameter region of our model, which produces the relic abundance of DM to
within $3\sigma$ of the observed value, is indicated by colored regions in
Figures 6 through 8, and each distinct DM freezeout regime is indicated by a
distinct color. The purple region indicates where the relic abundance of DM is
produced by kinematically allowed $NN\rightarrow\varphi\varphi$ annihilations,
the red region indicates where the relic abundance is produced through
kinematically suppressed $NN\rightarrow\varphi\varphi$ annihilations, and
turquoise regions correspond to production of the DM abundance by $s$-channel
DM-to-SM annihilations through an on-shell $h$ or $\varphi$ resonance.

The simultaneous generation of neutrino masses and thermal production of DM
gives a wide and predictive experimentally viable window in the GeV-TeV mass
range, below the sensitivity of current direct detection limits from LZ.
Furthermore, the remaining experimental window above the neutrino floor of
Xenon will be nearly entirely explored by the near-future generation of direct
detection experiments. Concretely, we show the projected sensitivity of
DARWIN/XLZDBaudis (2024).

In particular, for the $m_{\varphi}=30,100$ GeV cases we observe that the
points with viable DM relic density produced through scalar resonances are
excluded by LZ results or within reach of DARWIN. This result is not
surprising, given that the annihilation cross-section in these cases grows
with $\sin\theta_{V}$, as is the case for direct detection. For the
$m_{\varphi}=500$ GeV case, points with viable relic density are found beyond
the reach of DARWIN and beneath the neutrino floor.

Figure 6: Direct detection in the model for $m_{\varphi}=30\,\rm GeV$. All
data from Figure 3 are shown. Colored bands correspond to each regime of DM
production. Purple band indicates where the relic abundance of DM is produced
by kinematically allowed $NN\rightarrow\varphi\varphi$ annihilations, red band
indicates where the relic abundance is produced through kinematically
suppressed $NN\rightarrow\varphi\varphi$, and turquoise bands correspond to
production of the DM abundance by $s$-channel DM-to-SM annihilations through
an on-shell $h$ or $\varphi$ resonance. It is worth noting that the upper line
of the bands corresponds to $\lambda_{5}=10^{-3}$, and the lower line is for
$\lambda_{5}=10^{-5}$, if we continue lowering the values for $\lambda_{5}$,
we will find regions with lower cross-section. The green line represents the
LZ 2022 constraint on SI direct detection Aalbers et al. (2023), the purple
line indicates the limit on Higgs-to-invisible decays ATL (2020), the orange
dashed line represents the DARWIN projection constraint Baudis (2024), the
yellow region is the neutrino floor for Xenon. Figure 7: Direct detection in
the model for $m_{\varphi}=100\,\rm GeV$. All data from Figure 4 are shown.
Same color code as Fig. 6. In this case, the upper line of the bands is for
$\lambda_{5}=10^{-2}$, and the lower line is for $\lambda_{5}=10^{-4}$.
Figure 8: Direct detection in the model for $m_{\varphi}=500\,\rm GeV$. All
data from Figure 5 are shown. Same color code as Fig. 6. In this case, the
upper line of the bands is for $\lambda_{5}=10^{-1}$, and the lower line is
for $\lambda_{5}=10^{-3}$.

##  V Conclusions

We have presented a neutrino mass model with a DM candidate in the spirit of
the Scotogenic model. The purpose of this model is to show the viability of a
simple one-loop neutrino mass model with fermion DM, where LFV constraints are
decoupled from DM phenomenology and no Majoron is introduced. To this end, we
introduce a $\mathbb{Z}_{4}$ symmetry that is broken spontaneously to
$\mathbb{Z}_{2}$. Since no global symmetry, imposed or accidental, is broken,
no Goldstone boson appears in the spectrum. The dark sector fields generate
neutrino masses at the one-loop level after EW and $\mathbb{Z}_{4}$ symmetry
breaking. By setting a minimum value of the dark neutral scalar-pseudoscalar
mass splitting, we can obtain the observed neutrino mass mixing and viable
neutrino spectrum with small neutrino Yukawa couplings $Y_{p}<10^{-2}$, which
guarantees unobservable LFV rates. The field breaking the discrete symmetry
also introduces a channel for DM annihilation in the early Universe, which we
have studied. We found that the new Yukawa coupling can lead to viable DM
relic density in the kinematically allowed/forbidden regimes of
$NN\rightarrow\varphi\varphi$ annihilations or in the scalar resonance regime
$NN\xrightarrow{h,\phi}\text{SM SM}$. We have studied the prospects for DM
detection in direct detection experiments, namely LZ and DARWIN. We observe
that current LZ constraints have excluded part of the parameter space we
consider, while DARWIN will continue probing the model. Some parts of the
parameter space will remain untested after DARWIN and lie beneath the neutrino
floor.
This allows us to conclude that Scotogenic-type models of neutrino masses and
DM need not be in tension with LFV experiments or cosmological constraints on
light relics while having a thermal fermion DM candidate in the GeV-TeV scale.

## Acknowledgements

The work of LMGD was supported by the National Science and Technology Council,
the Ministry of Education (Higher Education Sprout Project NTU-113L104022-1),
and the National Center for Theoretical Sciences of Taiwan. RMR acknowledges
CONAHCYT for funding his MSc studies and the support of the EU Horizon 2020
research and innovation program, STRONG-2020 project, under grant agreement No
824093. PJF thanks IFUNAM for its generous hospitality. EP is grateful for the
support of PASPA-DGAPA, UNAM for a sabbatical leave, and Fundaci’on Marcos
Moshinsky.

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# Outlier-Robust Long-Term Robotic Mapping Leveraging Ground Segmentation

Hyungtae Lim
Massachusetts Institute of Technology

##  I Motivation

Despite the remarkable advancements in deep learning-based perception
technologies [20, 23, 43] and simultaneous localization and mapping (SLAM) [5,
7, 10, 17, 18], one can face the failure of these approaches when robots
encounter scenarios outside their modeled experiences [22] (here, the term
modeling encompasses both conventional pattern finding and data-driven
approaches). In particular, because learning-based methods are prone to
catastrophic failure when operated in untrained scenes [27, 39], there is
still a demand for conventional yet robust approaches that work out of the box
in diverse scenarios [14, 28, 36, 37, 38, 41, 46, 48], such as real-world
robotic services [19] and SLAM competitions [8, 12, 31, 47]. In addition, the
dynamic nature of real-world environments, characterized by changing
surroundings over time and the presence of moving objects, leads to
undesirable data points that hinder a robot from localization and path
planning [15]. Consequently, methodologies that enable long-term map
management [34], such as multi-session SLAM [6, 9, 45, 53] and static map
building [13, 24], become essential.

Therefore, to achieve a robust long-term robotic mapping system that can work
out of the box, first, I propose (i) fast and robust ground segmentation to
reject the ground points, which are featureless and thus not helpful for
localization and mapping [21, 25]. Then, by employing the concept of graduated
non-convexity (GNC) [50, 51], I propose (ii) outlier-robust registration with
ground segmentation that overcomes the presence of gross outliers within the
feature matching results [26, 27], and (iii) hierarchical multi-session SLAM
that not only uses our proposed GNC-based registration but also employs a GNC
solver to be robust against outlier loop candidates. Finally, I propose (iv)
instance-aware static map building that can handle the presence of moving
objects in the environment [29] based on the observation that most moving
objects in urban environments are inevitably in contact with the ground.

##  II Towards Robust Perception and Mapping

###  II-A Noise-Robust Ground Segmentation as Preprocessing

(a) Patchwork++ [21]

(b) Proposed

Figure 1: (a)-(b) Performance comparison between Patchwork++ [21] and our
proposed approach, which rejects the reflected noises and thus successfully
filters ground points. Light green and red bins represent that normal vectors
(arrows) of the estimated planes that are sufficiently upright and too tilted,
respectively. Blue points denote wrongly rejected ground points, i.e., false
negatives (best viewed in color).

(a) Source and target

(b) W/o GS [26]

(c) W/ GS [27]

Figure 2: (a) Example of source (red) and target clouds (green) in the KITTI
dataset, where the left-bottom text represents the pose discrepancy in each
scene. (b)-(c) Registration results before (W/o GS) and after the application
of ground segmentation (W/ GS), where the blue points denote the warped source
cloud by the estimated pose. Our proposed approach [27] successfully estimates
the relative pose even though the pose discrepancy between viewpoints of
source and target clouds is far distant. Black dashed boxes highlight the non-
ground objects that have to be tightly aligned (note that, in (c), these
objects are well-aligned). The solid red and green boxes indicate that
algorithms failed and succeeded, respectively (best viewed in color).

First, we propose a fast and robust region-wise ground segmentation [21, 25]
to be employed as a preprocessing step for instance segmentation [32] or
robust registration [27]. In recent years, while ground segmentation methods
with high precision and recall in complex urban environments have been
proposed [4, 30, 35, 40, 49], it has been observed that these approaches
sometimes fail due to the noisy points close to the bottom of the ground (Fig.
1(a)). Thus, to address this issue, we filter the noisy points by reusing
previously estimated ground planes of neighboring bins when estimating the
ground plane of the target bin in a cascaded manner. Our ground segmentation
operates over 100 Hz for a point cloud acquired by a 64-channel LiDAR sensor
while being robust against these noisy points (Fig. 1(b)); thus, it can be
easily employed as a preprocessing step with little computational burden on
the subsequent algorithms.

Figure 3: Our proposed long-term mapping pipeline. (a) Registration result of
our outlier-robust registration framework when estimating the relative pose
between two distant viewpoints of point clouds. Our approach robustly
estimates the relative pose despite gross outliers (red lines). (b)-(c) Before
and after the application of our proposed registration in the loop closing
module of SC-LeGO-LOAM [14]. Our proposed registration significantly reduces
the number of false positive loops, enhancing the mapping result (see the
upper zoomed boxes). (d) Our multi-session SLAM results by taking two
trajectories from our single-session SLAM as inputs. Note that we leverage our
outlier-robust registration to initially align the data via map-to-map
registration. (e) Example of results by the proposed static map building
approach. Because the points from moving objects are rejected at an instance-
level, traces of pedestrians and vehicles are successfully filtered out. Note
that all the approaches are designed to be learning-free and robust; thus, the
proposed approaches can be easily employed in various robot platforms and
sensor configurations.

###  II-B Outlier-Robust Registration for Loop Closing in Single-Session SLAM
and Initial Alignment of Multi-Session SLAM

LiDAR point clouds have sparse characteristics, which means that the regions
closer to the sensor origin are denser, and those further away become
dramatically sparser [25]. The issue leads to spurious correspondences when
the viewpoint difference between two point clouds becomes distant, which may
result in the failure of the registration process (Fig. 2(b)). That is, as the
viewpoint difference between two point clouds becomes distant, the
geometrically distinguishable non-ground points are partially overlapped, and
moreover, those overlapped areas are too sparse to establish correct
correspondences. Meanwhile, the ground points near the sensor origin are
dominant and geometrically indistinguishable; thus, the feature descriptors
from them become ambiguous, resulting in gross outliers [27].

To tackle this problem in a simple yet effective manner, ground segmentation
is exploited as an outlier pruning step and then a GNC [51]-based robust
global registration is applied, which estimates pose while rejecting the
outlier measurements simultaneously [3, 52], overcoming up to 70-80% of
outliers and providing faster speed compared with previous works [11, 33].
Consequently, we have demonstrated that our proposed method shows a high
success rate when distant source and target clouds are given, particularly,
our proposed approach enhances the PGO results compared with the baseline
approach [14] (see Figs. 3(b) and (c)).

Beyond scan-to-scan registration, the proposed method is also applicable to
map-to-map registration for initial alignment, owing to its learning-free
nature and scalability. Therefore, our approach enhances the performance of
multi-session SLAM by providing a precise initial alignment (Fig. 3(d)).

During PGO in both single-session and multi-session SLAM procedures, a GNC
solver is also employed [1]. By doing so, we can finally achieve consistent
and accurate mapping results by suppressing the impact of false positives
inter- and intra-loops.

###  II-C Instance-Aware Static Map Building

Finally, static map building is performed using the scan data and the
optimized poses. As an extension of our previous work [24], which only detects
the traces of moving objects and removes them without the instance-level
consideration, we propose instance-level dynamic points removal [29], designed
to precisely reject points from moving objects while preserving as many static
points as possible. This approach, based on the observation that dynamic
objects move over time and temporarily occupy space, allows us to detect
traces of dynamic objects by comparing geometrical discrepancies, i.e., the
min-max height difference within a grid, between the $t$-th scan and the map
cloud. As a result, we can construct a static map that is directly usable for
localization and path planning, as shown in Fig. 3(e).

##  III Future Research Direction:
Semantics-Aware Long-Term Mapping

(a)

(b)

Figure 4: Ongoing research topics: (a) semantics-aware registration output in
the MulRan dataset [16] using the semantic segmentation network trained in the
SemanticKITTI dataset [2] and (b) scene graph-based mapping result in our
campus with zero-shot predictions [42].

So far my focus has been on the conventional geometric estimation approaches,
yet I believe integrating semantic information into the geometrical perception
is crucial for developing more intelligent and adaptive robotic systems to
significantly enhance the robustness and scene understanding capabilities. I
especially advocate for approaches that can robustly perform estimation
despite the performance degradation of semantic segmentation in zero-shot
situations, i.e., when using the network in unmodeled scenes.

To this end, I embark on two directions of research to integrate semantic
information: (i) semantics-aware registration (Fig. 4(a)) and (ii) scene
graph-based mapping framework (Fig. 4(b)) using zero-shot predictions from the
semantic segmentation network [42, 44]. Consequently, these semantics-aware
mapping frameworks would enable robots to navigate and interact with their
surroundings in a more intelligent and context-aware manner, significantly
boosting their autonomy and effectiveness in performing a wide array of tasks.

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# Shayona@SMM4H’23: COVID-19 Self diagnosis classification using BERT and
LightGBM models

Rushi Chavda1, Darshan Makwana1, Vraj Patel2, Anupam Shukla2
1Indian Institute of Technology, Bombay
2National Institute of Technology, Surat

## Abstract

This paper describes approaches and results for shared Task 1 and 4 of
SMMH4-23 by Team Shayona. Shared Task-1 was binary classification of english
tweets self-reporting a COVID-19 diagnosis, and Shared Task-4 was Binary
classification of English Reddit posts self-reporting a social anxiety
disorder diagnosis. Our team has achieved the highest f1-score 0.94 in Task-1
among all participants. We have leveraged the Transformer model (BERT) in
combination with the LightGBM model for both tasks.

## Introduction

In recent years, the surge in worldwide social media utilization has provided
researchers with fresh prospects to extract health-related data[7], which can
ultimately be harnessed to enhance public health. The Social Media Mining for
Health Applications (SMM4H) Shared Tasks address natural language processing
(NLP) challenges of using social media data for health informatics, including
informal, colloquial expressions, misspellings, noise, data sparsity,
ambiguity, and multilingual posts[9]. In its 2023 call, 5 shared tasks were
given. We had participated in Task-1 and Task-4.

In Task-1, In the context of monitoring COVID-19 experiences via Twitter, this
research addresses the challenge of automated binary classification. The task
involves discerning self-reported COVID-19 diagnoses (”1”) from non-diagnostic
mentions (”0”), enabling real-time, large-scale analysis. This problem focuses
on distinguishing explicit diagnoses rather than mere user experiences.

Task-4 was also a binary classification problem, Exploring patients’
perspectives and medical needs through social media analysis offers insights
into disease journeys. Social media listening (SML) has the potential to
advance disease understanding and therapy development[2]. For this task, we
used a dataset extracted from the subreddit r/socialanxiety. The challenge is
to build a classifier that correctly identifies patients that report having a
positive or probable diagnosis of social anxiety disorder (positive cases
labeled as ‘1’) from patients that report not having a diagnosis or the
presence of a diagnosis is unlikely or unclear (negative cases labeled as
‘0’).

We know Transformer models are outperforming in all Natural Language
downstream tasks[4], we have leveraged BERT model
(digitalepidemiologylab/covid-twitter-bert-v2[6]) from hugging face and
further fine tuned it using given training data on binary classification task,
and then used its embeddings to train LightGBM model. The selection of
LightGBM is mostly a result of its enormous success since its launch in 2017.
The benefits it has over other algorithms support this accomplishment[5]. In
fact, LightGBM is well-known for its rapid training speed, high accuracy, and
minimal memory usage, as well as for supporting parallel, distributed, and GPU
Learning[5]. Going ahead in this paper we have shown comparison between our
different approaches, and future work. Both tasks provide a binary
classification problem, so we have used the same model architecture in both
the tasks. In The Discussion section we have provided potential Improvements
for Task-1 and Task-4.

Figure 1: Figure 1: An overview of the proposed method

## Data

In Task-1, our dataset comprised a total of 7,600 annotated tweets for
training, 400 for validation, and 10,000 for testing our model. Each entry
contained three columns: tweet ID, tweet text, and a binary label (0 or 1).
Notably, within the training set of 7,600 tweets, 1,334 were labeled as 1,
indicating the presence of self-reports of COVID diagnosis.

For Task-4, we worked with a dataset consisting of 6,090 annotated tweets for
training, 680 for validation, and 1,347 for testing our model. Similar to
Task-1, each entry featured three columns: ID, text, and a binary class label
(0 or 1). In this dataset as well, 1,334 tweets were labeled as 1, denoting
the occurrence of self-reports of COVID diagnosis. Data preprocessing steps
were employed to clean and prepare the text data. These steps involved
filtering out non-alphabet characters, tags, emojis, and URLs, as they
typically carry minimal relevance for text comprehension. we also removed
stopwords and punctuations.

## System

We have used the BERT (bidirectional encoder representations from
transformers) model from hugging face. We have used
“digitalepidemiologylab/covid-twitter-bert-v2” BERT model from hugging face,
which is specially trained on COVID-19 Twitter data and more suitable for
Task-1[3]. Our method leverages the power of BERT to capture complex
linguistic patterns and the efficiency of LightGBM to optimize feature space
and classification[4].

For Task-1, we conducted experiments with three methods

  1. 1.

RoBERTa model for sequence classification[9]

  2. 2.

BERT model fine-tuned for classification

  3. 3.

BERT model with the integration of LightGBM[4]

In the first method, we used pre-trained RobertaForSequenceClassification
model from hugging face[8], we used weights of roberta-base model, we fine-
tuned the model with preprocessed training data, and using this, we achieved
an f1-score of 0.83 on validation data.

In our second approach, we leveraged a pre-trained BERT model trained on
COVID-related tweets[3]. We extended this model with two fully connected
layers: [1024, 256] and [256, 1]. These layers progressively distilled the
features for our binary classification task. During training, we computed
predictions through a forward pass and gauged their accuracy with BCE loss. In
back propagation, gradients were calculated concerning the loss and propagated
backward. Using the Adam optimizer, which merges adaptive learning rates and
momentum, model parameters were adjusted according to their impact on loss
reduction. This dynamic fine-tuning process enabled the model to refine its
predictions iteratively. We achieved a f1-score of 0.89 on the validation set
using this approach.

LightGBM is a robust gradient-boosting framework that excels at handling
complex tasks involving tree-based learning techniques. Combined with BERT, a
model renowned for capturing intricate semantic information from text, we
unlock the synergistic potential of both methods. BERT’s strength lies in its
ability to understand the meaning of words and phrases in context, while
LightGBM’s forte is navigating through high-dimensional data and optimizing
complex problems.

In our third experimental approach, we seamlessly integrated LightGBM with the
BERT model, as illustrated in Figure 1. Here’s how we did it step by step:

  1. 1.

BERT Fine-Tuning: First, we fine-tuned the BERT model following the same
process as in our second experiment (Experiment 2). However, crucially, we did
not freeze the model parameters during this phase. This allowed the BERT
model’s parameters to be updated and adapted to our specific task.

  2. 2.

Hidden [CLS] Token Extraction: Next, we extracted the Hidden [CLS] tokens from
the last three layers of the BERT model. This choice was inspired by the
approach outlined in the work of Eessa et al. [4]. These extracted tokens
served as the input for our LightGBM model.

  3. 3.

Hyperparameter Optimization: We employed hyperparameter optimization
techniques using the Optuna framework[1] to ensure optimal performance. This
step fine-tuned the parameters of our LightGBM model, enabling it to make the
most accurate predictions.

The results were impressive. Our LightGBM model and hyperparameter tuning
outperformed all other methods we explored. Specifically, it achieved an F1
score of 0.93 on the validation dataset and an even higher F1 score of 0.94 on
the test dataset for Task 1, making it the top-performing model for this task.
Encouraged by this success, we extended our approach to Task 4, another binary
classification problem. Here, our model delivered an F1-score of 0.8 on the
test data, reaffirming its versatility and effectiveness.

## Results

| RoBERTa | BERT | BERT+LightGBM
---|---|---|---
Validation | 0.83 | 0.89 | 0.93
Test | — | — | 0.94
Table 1: Task-1 results for different experiments.

Table 1 shows the performance of different models used in Task-1. With the
hybrid model of BERT and LightGBM, we were able to reach a Precision, Recall,
and f1-score of 0.949, 0.938, and 0.943 respectively on the test set. Since
the hybrid model had the highest f1-score among all other methods in Task-1,
we used the same for Task-4 as well, and since Task-1 and Task-4 both are
binary classification problems, we kept the model architecture the same for
Task-4 as well. This resulted in a Precision, Recall, and f1-score of 0.756,
0.871, and 0.809 respectively on the test set.

## Discussion

This paper presents the participation of the Shayona team in SMM4H’23. For
both tasks the evaluative parameter of the competition was f1-score and We got
highest f1-score in the Task-1. We observed that the gradient boosted
algorithm (LightGBM) shows the best results. The LightGBM model can
effectively aggregate the information from the concatenated [CLS] tokens.
LightGBM is a gradient boosting framework that can learn complex interactions
between features. It might be identifying patterns and relationships within
the concatenated tokens that are not readily apparent from a single layer’s
representation.

Due to time and computation constraints, we were unable to explore more, but
we could try changing the number of layers from which we are extracting hidden
[CLS] tokens. We could also try with different layer dimensions. Due to the
fact that we took a hugging face BERT model which was pre-trained on a model
that was trained using COVID-19-related tweeter data rather than Task-4
specific data, we were unable to attain a good f1-score in Task-4. If we
employ a BERT model, pre-trained on Task-4 related data then the f1-score
could be improved.

## Conclusions

Our research showcases a hybrid BERT and LightGBM model that excels in the
binary classification problem of COVID-19 self-diagnosis (Task-1). By
integrating BERT’s language understanding with LightGBM’s feature
optimization, their approach achieved an impressive f1-score of 0.94 in
Task-1, highlighting the synergy of transformer embeddings and gradient
boosting for effective health-related classification.

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