# Low Scale Leptogenesis in Singlet-Triplet Scotogenic Model

Labh Singh1, Devabrat Mahanta2, and Surender Verma1
sainilabh5@gmail.comdeva.m2910<EMAIL_ADDRESS>
(1Department of Physics and Astronomical Science, Central University of
Himachal Pradesh, Dharamshala, India
2Department of Physics, Abhayapuri College, Bongaigaon (Assam), India
)

###### Abstract

The scotogenic model presents an elegant and succinct framework for
elucidating the origin of tiny neutrino masses within the framework of the
Standard Model, employing radiative corrections within the domain of the dark
sector. We investigate the possibility of achieving low-scale leptogenesis in
the singlet-triplet scotogenic model (STSM), where dark matter mediates
neutrino mass generation. We initially considered a scenario involving two
moderately hierarchical heavy fermions, N and $\Sigma$, wherein the lepton
asymmetry is generated by the out-of-equilibrium decay of both particles. Our
analysis indicates that the scale of leptogenesis in this scenario is similar
to that of standard thermal leptogenesis and is approximately
$M_{N,\Sigma}\sim 10^{9}$ GeV, which is comparable to the Type-I seesaw case.
Further, we consider the case with three heavy fermions ($N_{1}$, $N_{2}$, and
$\Sigma$) with the hierarchy $M_{N_{1}}<M_{\Sigma}\ll M_{N_{2}}$, which yields
the lower bound on heavy fermions up to 3.1 TeV, therefore significantly
reduce the scale of the leptogenesis up to TeV scale. The only prerequisite is
suppression in the $N_{1}$ and $\Sigma$ Yukawa couplings, which causes
suppressed washout effects and a small active neutrino mass of about $10^{-5}$
eV. This brings about the fascinating insight that experiments aiming to
measure the absolute neutrino mass scale can test low-scale leptogenesis in
the scotogenic model. Further, the hyperchargeless scalar triplet $\Omega$
provides an additional contribution to mass of the $W$-boson explaining CDF-II
result.

Keywords: Leptogenesis; Baryogenesis; Phenomenology; Dark Matter; W-Boson
Mass.

##  1 Introduction

Although the early Universe started with equal amounts of matter and
antimatter, the present Universe contains an excess of matter over antimatter.
To account for this asymmetry, a dynamical mechanism must be identified, one
that can create an imbalance so that after matter and antimatter annihilate, a
surplus of matter remains, giving rise to the visible matter we observe. This
surplus of baryons over antibaryons, is often expressed using the baryon-to-
photon ratio denoted as $\eta_{B}$[1, 2]:

$\eta_{B}=\frac{n_{B}-n_{\bar{B}}}{n_{\gamma}}=6.1\times 10^{-10},$

where the variables $n_{B}$, $\eta_{\bar{B}}$, and $n_{\gamma}$ correspond to
the number densities of baryons, antibaryons, and photons, respectively. There
are substantial evidences suggesting existence of more matter than antimatter
in the Universe. These evidences come, primarily, from two sources: the
abundance of light elements present in the Universe and observations of cosmic
microwave background radiation anisotropies[3, 4, 5]. In order to create this
asymmetry dynamically, certain established prerequisites needed to exist
during the early Universe. These prerequisites are commonly known as
Sakharov’s conditions [6]. These conditions encompass (i) the violation of
baryon number conservation (B), (ii) the occurrence of $\mathrm{C}$ and
$\mathrm{CP}$ violation, and (iii) a departure from thermal equilibrium.
Although the SM falls short in satisfactorily fulfilling all these criteria,
different theories beyond the Standard Model (BSM) have been proposed to offer
a dynamic explanation for the observed baryon abundance in the Universe.
One of the most straightforward mechanisms among these hypotheses entails the
addition of extra massive particles that can annihilate or decay into the SM
particles in a way that meets all the prerequisites for successful
baryogenesis[7, 8]. Another intriguing approach, which also connects to the
physics of the lepton sector, is known as leptogenesis[9]. In its standard
conceptualization, leptogenesis is intricately associated with the Type-I
seesaw mechanism, designed to elucidate the origin of the small neutrino
masses within the SM framework. This mechanism introduces two or more heavy
right-handed fermions with significant Majorana masses. Under the standard
thermal leptogenesis paradigm, these heavy fermions emerge through scattering
interactions within the thermal bath which lead to their CP-violating out-of-
equilibrium decays that catalyze the creation of an initial lepton asymmetry.
Consequently, $B-L$ conserving and $B+L$ violating electroweak sphaleron[10]
process then cause this initial lepton asymmetry translate to the baryon
asymmetry. Leptogenesis relies on CP violation within the lepton sector, which
may be significant, as suggested by some neutrino oscillation experiments[11]
because $CP$ violation in the quark sector is not enough to account for the
requisite baryon imbalance. Also, the fact that lepton asymmetry can be
produced by CP-violating, out-of-equilibrium decays of some heavy BSM fields
involved in well-known seesaw processes[12, 13, 14] is an intriguing feature
of this scenario.

Thermal leptogenesis has a limitation in that it requires an extremely high
scale of right-handed heavy fermions. In other words, for thermal leptogenesis
to be successful, the lightest Majorana fermions must have a minimum mass of
around $10^{9}$ GeV, which is known as the Davisson-Ibarra bound[15]. By
incorporating flavor effects, this lower bound can be brought down to $10^{8}$
GeV. It is however difficult to validate such a large-scale phenomenon in
future collider experiments. This serves as motivation to seek an alternative
to standard thermal leptogenesis that can produce the BAU at a much lower
scale.

The purpose of this paper is to propose a low-scale alternative to standard
thermal leptogenesis without considering degenerate heavy fermions [16].
Keeping this motivation in mind, we revisit the singlet-triplet scotogenic
model[17, 18, 19, 20, 21, 22, 23, 24], which extends the original idea
presented in[25] to make its phenomenology more viable and diverse. A scalar
triplet that is $Z_{2}$-invariant facilitates the mixing between singlet and
triplet fermions. At the tree level, this scalar triplet introduces additional
contributions[26, 27, 28] to the W-mass as observed in CDF-II[29]. The
extended scotogenic model has several important advantages over the minimal
scotogenic model, including the fact that it prevents the unintended
spontaneous breaking of the $Z_{2}$ parity symmetry. This avoidance is
possible due to influence of new couplings emerged in extensions of the
minimal scenario[30]. The lightest $Z_{2}$-odd particle exhibits inherent
stability and, electromagnetically neutral, emerges as a plausible candidate
for dark matter. In our investigation, we have considered the real component
of scalar doublet to be our dark matter candidate. To achieve successful
leptogenesis in the STSM, we take into account the out-of-equilibrium decay of
both heavy singlet (N) and triplet ($\Sigma$) fermion to produce the necessary
lepton asymmetry by considering them to be moderately hierarchical
($M_{N}\lesssim M_{\Sigma}$) and, ultimately, the electroweak sphaleron
transitions that can account for the observed BAU. Leptogenesis has been
examined in previous studies and is a topic that has been covered extensively.
In particular, the decay of singlet fermions has been discussed in Refs.[31,
32, 33, 34, 35, 41, 36, 37, 38, 39, 40] while Refs.[28, 42, 43, 44, 45]
focuses on the investigation of leptogenesis concerning the decay of triplet
fermions. In the scotogenic model, leptogenesis occurs through various
mechanisms that are determined by the mass spectrum in the $Z_{2}$-odd sector.
In our model, an interesting question remains about the nature of the DM
candidate. In the model under consideration, there can be two DM candidates:
fermion DM, a mixed state of singlet and triplet fermion, and the lightest
component of the inert scalar doublet ($\eta$). In the fermionic DM case, the
relic density requires larger Yukawa couplings, which results in huge washout
effects in the process of leptogenesis. However, in the scalar dark matter
scenario, the relic density depends upon the gauge interactions of the $\eta$.
Therefore, realizing thermal leptogenesis with a specific mass spectrum is
impossible in the former scenario. Furthermore, it’s important to note that
when dealing with fermionic dark matter, the significant Yukawa couplings can
readily result in a breach of restrictions concerning lepton flavor
violations. As a result, we decided to make the lightest part of $\eta$ our DM
candidate.

Various studies of thermal leptogenesis in the minimal scotogenic model are
already present in the literature[46, 47, 48, 49]. We investigate the
possibility of achieving low-scale leptogenesis in the phenomenologically rich
STSM, where dark matter mediates neutrino mass generation. We initially
considered a scenario involving two nearly equal heavy fermions, which leads
to a lightest neutrino mass eigenvalue of $m_{1}=0$. In this scenario, lepton
asymmetry is generated by the out-of-equilibrium decay of both particles
denoted as N and $\Sigma$. Our analysis shows that the scale of leptogenesis
in this scenario is similar to that of the standard thermal leptogenesis in
the Type-I seesaw mechanism. In order to reduce the scale of leptogenesis, we
propose addition of two singlet right-handed fermions ($N_{1}$ and $N_{2}$)
along with a triplet fermion $\Sigma$ with a specific mass hierarchy:
$M_{N_{1}}<M_{\Sigma}\ll M_{N_{2}}$. Here, $M_{N_{1}}$, $M_{N_{2}}$, and
$M_{\Sigma}$ represent the respective masses of $N_{1}$, $N_{2}$, and
$\Sigma$. STSM is more phenomenologically rich than the minimal scotogenic
model due to its additional contribution to the $W$-boson mass. In the context
of the minimal scotogenic model, the possibility of TeV scale leptogenesis has
been investigated in Refs. [33, 32]. Within STSM scenario, if low-scale
leptogenesis is feasible, then the gauge interactions of a TeV scale triplet
fermion could be probed in future collider experiments. Lowering the
leptogenesis scale to sub-TeV range enhances the testability of the model in
future collider experiments. Further, we study the phenomenology of scalar DM
where we chose the real component ($\eta^{R}$) of the neutral part of $\eta$
as candidate for scalar DM.

The structure of the paper is as follows: In Section 2, we begin by
introducing the STSM and providing an overview of its essential
characteristics. Section 3 delves into the $W$-boson mass anomaly observed in
the CDF-II results. Section 4 is devoted to the phenomenology of the scalar
DM. In Section 5, we formulate all necessary expressions for studying
leptogenesis in the STSM. Considering the scenario with two heavy fermions,
the analysis shows that this scenario offers no advantage over the standard
thermal leptogenesis in Type-I seesaw framework (Section 5.1). We extend our
model in Section 5.2 by introducing another right-handed singlet fermion to
lower down the scale of leptogenesis. Section 5.3 will briefly present
numerical analysis and discussion of leptogenesis scenario. In Section 6, we
present conclusions of the work reported in this paper.

##  2 Model and Formalism

Particle Content | Generations | Symmetry
---|---|---
|  | $(SU(2)_{L}\times U(1)_{Y}\times Z_{2})$
$L$ | 3 | $(2,-\frac{1}{2},+)$
$\phi$ | 1 | $(2,\frac{1}{2},+)$
$N$ | 1 | $(1,0,-)$
$\Sigma$ | 1 | $(3,0,-)$
$\Omega$ | 1 | $(3,0,+)$
$\eta$ | 1 | $(2,\frac{1}{2},-)$
Table 1: Particle content, generations, and symmetry assignments in the STSM
under $SU(2)_{L}\times U(1)_{Y}\times Z_{2}$ gauge symmetry.

The assignments of fields within the STSM, subject to the $SU(2)_{L}\times
U(1)_{Y}\times Z_{2}$ symmetry, are presented in Table 1. The relevant
invariant Yukawa Lagrangian is given by

$\displaystyle\mathcal{L}=Y^{\alpha\beta}_{L}\overline{L_{\alpha}}\phi\ell_{\beta}+Y^{\alpha}_{N}\overline{L_{\alpha}}i\sigma_{2}\eta
N+Y^{\alpha}_{\Sigma}\overline{L_{\alpha}}C\Sigma^{\dagger}i\sigma_{2}\eta+Y_{\Omega}Tr(\overline{\Sigma}\overline{\Omega})N+\frac{1}{2}M_{N}\overline{N^{c}}N+\frac{1}{2}M_{\Sigma}Tr(\overline{\Sigma^{c}}\Sigma)+h.c.,$
(1)

where $\alpha$, $\beta$=$1,2,3$ denotes the three flavors. The $SU(2)_{L}$
doublet fields $(\phi,\eta)$ and $SU(2)_{L}$ triplet fields $(\Sigma,\Omega)$
can be expressed as

$\phi=\left(\begin{array}[]{c}\phi^{+}\\\
\phi^{0}\end{array}\right),\eta=\left(\begin{array}[]{c}\eta^{+}\\\
\eta^{0}\end{array}\right),\Sigma=\left(\begin{array}[]{cc}\frac{\Sigma^{0}}{\sqrt{2}}&\Sigma^{+}\\\
\Sigma^{-}&-\frac{\Sigma^{0}}{\sqrt{2}}\end{array}\right)\text{ and
}\quad\Omega=\left(\begin{array}[]{cc}\frac{\Omega^{0}}{\sqrt{2}}&\Omega^{+}\\\
\Omega^{-}&-\frac{\Omega^{0}}{\sqrt{2}}\end{array}\right).$

The invariant scalar potential of the model is given by

$\displaystyle V$ $\displaystyle=$ $\displaystyle-
m_{\phi}^{2}\phi^{\dagger}\phi+m_{\eta}^{2}\eta^{\dagger}\eta+\frac{\lambda_{1}}{2}\left(\phi^{\dagger}\phi\right)^{2}+\frac{\lambda_{2}}{2}\left(\eta^{\dagger}\eta\right)^{2}+\frac{m_{\Omega}^{2}}{2}\operatorname{Tr}\left(\Omega^{\dagger}\Omega\right)+\lambda_{3}\left(\phi^{\dagger}\phi\right)\left(\eta^{\dagger}\eta\right)+\lambda_{4}\left(\phi^{\dagger}\eta\right)\left(\eta^{\dagger}\phi\right)$
(2) $\displaystyle+$
$\displaystyle\frac{\lambda_{5}}{2}\left[\left(\phi^{\dagger}\eta\right)^{2}+h.c.\right]+\frac{\lambda^{\eta}}{2}\left(\eta^{\dagger}\eta\right)\operatorname{Tr}\left(\Omega^{\dagger}\Omega\right)+\frac{\lambda_{1}^{\Omega}}{2}\left(\phi^{\dagger}\phi\right)\operatorname{Tr}\left(\Omega^{\dagger}\Omega\right)+\frac{\lambda_{2}^{\Omega}}{4}\operatorname{Tr}\left(\Omega^{\dagger}\Omega\right)^{2}$
$\displaystyle+$
$\displaystyle\mu_{1}\phi^{\dagger}\Omega\phi+\mu_{2}\eta^{\dagger}\Omega\eta.$

In order to maintain the perturbativity of the theory, it is required that all
couplings ($\lambda_{i}$) remain less than or equal to one[30, 50]. Due to the
conservation of $\mathbb{Z}_{2}$ symmetry, the $\eta$ field does not acquire a
vacuum expectation value (VEV). The spontaneous electroweak symmetry breaking
is triggered primarily by the neutral components of the $\phi$ and $\Omega$
fields,

$\left\langle\phi^{0}\right\rangle=\frac{v_{\phi}}{\sqrt{2}},\quad\left\langle\Omega^{0}\right\rangle=v_{\Omega}.$

The scalar spectrum is categorized into two distinct sets: the
$\mathbb{Z}_{2}$-even scalars, namely $\phi^{0}$, $\Omega^{0}$,
$\Omega^{\pm}$, and $\phi^{\pm}$, and the $\mathbb{Z}_{2}$-odd scalars, namely
$\eta^{0}$ and $\eta^{\pm}$. Among these, $\eta^{0}$ has garnered significant
attention as a promising dark matter candidate, with extensive studies
documented in the literature[51, 52, 53, 54, 55, 56, 57]. Within this
framework, the neutral scalars $\phi^{0}$ and $\Omega^{0}$ undergo mixing
through a $2\times 2$ matrix that can be parameterized by the angle $\Theta$,
such that

$\displaystyle\left(\begin{array}[]{l}H_{1}\\\
H_{2}\end{array}\right)=\left(\begin{array}[]{cc}\cos\Theta&\sin\Theta\\\
-\sin\Theta&\cos\Theta\end{array}\right)\left(\begin{array}[]{l}\phi^{0}\\\
\Omega^{0}\end{array}\right),$ (9)

where

$\displaystyle\tan(2\Theta)=\frac{4v_{\Omega}v_{\phi}\left(\sqrt{2}\mu_{1}-2\lambda_{1}^{\Omega}v_{\Omega}\right)}{8\lambda_{2}^{\Omega}v_{\Omega}^{3}-4\lambda_{1}v_{\Omega}v_{\phi}^{2}+\sqrt{2}\mu_{1}v_{\phi}^{2}}.$
(10)

In this theory, the lightest $Z_{2}$-even scalar, $H_{1}$, is identified as
the SM Higgs boson, while the heavier one, $H_{2}$, remains a proposed new
scalar Higgs boson. Similarly, the charged scalars $\phi^{\pm}$ and
$\Omega^{\pm}$ will undergo mixing described by a $2\times 2$ matrix given by

$\displaystyle\left(\begin{array}[]{l}H_{1}^{\pm}\\\
H_{2}^{\pm}\end{array}\right)=\left(\begin{array}[]{cc}\cos\delta&\sin\delta\\\
-\sin\delta&\cos\delta\end{array}\right)\left(\begin{array}[]{c}\phi^{\pm}\\\
\Omega^{\pm}\end{array}\right),$ (17)

with

$\displaystyle\tan(2\delta)=-\frac{4v_{\Omega}v_{\phi}}{v_{\phi}^{2}-4v_{\Omega}^{2}}.$
(18)

It is to be noted that the lightest charged scalar, which is represented by
$H_{1}^{\pm}$, is the Goldstone boson. Also, the theory introduces a novel
charged scalar ($H_{2}^{\pm}$) field. Further, the masses of the $Z_{2}$-odd
scalars $\eta^{0}$ and $\eta^{\pm}$ are given by

$\displaystyle
m_{\eta^{\pm}}^{2}=m_{\eta}^{2}+\frac{1}{2}\lambda_{3}v_{\phi}^{2}+\frac{1}{2}\lambda^{\eta}v_{\Omega}^{2}+\frac{1}{\sqrt{2}}v_{\Omega}\mu_{2},$
(19) $\displaystyle
m_{\eta^{R}}^{2}=m_{\eta}^{2}+\frac{1}{2}\lambda_{3}v_{\phi}^{2}+\frac{1}{2}\left(\lambda_{4}+\lambda_{5}\right)v_{\phi}^{2}-\frac{1}{\sqrt{2}}v_{\Omega}\mu_{2}+\frac{1}{2}\lambda^{\eta}v_{\Omega}^{2},$
(20) $\displaystyle
m_{\eta^{I}}^{2}=m_{\eta}^{2}+\frac{1}{2}\lambda_{3}v_{\phi}^{2}+\frac{1}{2}\left(\lambda_{4}-\lambda_{5}\right)v_{\phi}^{2}-\frac{1}{\sqrt{2}}v_{\Omega}\mu_{2}+\frac{1}{2}\lambda^{\eta}v_{\Omega}^{2}.$
(21)

Within the fermionic sector, the fields $\Sigma^{0}$ and $N$, which possess
$\mathbb{Z}_{2}$-odd properties, undergo mixing governed by Yukawa coupling
$Y_{\Omega}$ and occur in presence of the non-zero vacuum expectation value,
$v_{\Omega}$. In the basis of $\left(\Sigma^{0},N\right)$, the Majorana mass
matrix is given by

$\displaystyle
M_{\chi}=\left(\begin{array}[]{cc}M_{\Sigma}&v_{\Omega}Y_{\Omega}\\\
v_{\Omega}Y_{\Omega}&M_{N}\end{array}\right),$ (24)

which is diagonalized by $2\times 2$ matrix $V(\alpha)$ as

$\displaystyle\left(\begin{array}[]{l}\chi_{1}^{0}\\\
\chi_{2}^{0}\end{array}\right)=V(\alpha)\left(\begin{array}[]{c}\Sigma^{0}\\\
N\end{array}\right),$ (29)

where

$V(\alpha)=\left(\begin{array}[]{cc}\cos\alpha&\sin\alpha\\\
-\sin\alpha&\cos\alpha\end{array}\right).$

Consequently, masses of $\chi^{\pm}$ and $\chi_{i}^{0}$ eigenstates at tree-
level are

$\displaystyle m_{\chi^{\pm}}=M_{\Sigma},$ (30) $\displaystyle
m_{\chi_{1,2}^{0}}=\frac{1}{2}\left(M_{\Sigma}+M_{N}\mp\sqrt{\left(M_{\Sigma}-M_{N}\right)^{2}+4\left(v_{\Omega}Y_{\Omega}\right)^{2}}\right),$
(31)

where the mixing angle $\alpha$ satisfy the relation

$\displaystyle\tan{(2\alpha)}=\frac{2v_{\Omega}Y_{\Omega}}{M_{\Sigma}-M_{N}}.$
(32)

$\Sigma$$N$$<\Omega>$$<\phi>$$<\phi>$$\eta^{0}$$\eta^{0}$$\nu_{L}$$\nu_{L}$
Figure 1: Feynman diagram used to generate neutrino masses at one-loop in
STSM.

In the model under investigation, the small neutrino masses are generated
radiatively at the one-loop level as shown in Fig. 1. The real triplet scalar
$\Omega$ whose VEV accounts for additional contribution to the $W$-boson mass,
plays a vital role in neutrino mass generation as coupling $Y_{\Omega}$
generates a mixing between $N$ and $\Sigma$ fermions. At one-loop level, the
neutrino mass matrix can be written as

$\displaystyle\left(\mathcal{M}_{\nu}\right)_{\alpha\beta}$
$\displaystyle=\sum_{i=1}^{2}\frac{h_{\alpha i}h_{\beta
i}m_{\chi_{i}^{0}}}{2(4\pi)^{2}}\left[\frac{m_{\eta^{R}}^{2}\ln\left(\frac{m_{i}^{2}}{m_{\eta^{0}}^{2}}\right)}{m_{\chi_{i}^{0}}^{2}-m_{\eta^{R}}^{2}}-\frac{m_{\eta^{I}}^{2}\ln\left(\frac{m_{\chi_{i}^{0}}^{2}}{m_{\eta^{I}}^{2}}\right)}{m_{\chi_{i}^{0}}^{2}-m_{\eta^{I}}^{2}}\right],$
(33) $\displaystyle=\sum_{i=1}^{2}\frac{h_{\alpha i}h_{\beta
i}m_{\chi_{i}^{0}}}{2(4\pi)^{2}}[L_{i}(m_{\eta^{R}}^{2})-L_{i}(m_{\eta^{I}}^{2})],$
(34)

which can be further written as

$\displaystyle=\sum_{i=1}^{2}h_{\alpha
i}\Lambda_{i}\left(h^{T}\right)_{i\beta}=\left(h\Lambda
h^{T}\right)_{\alpha\beta}$ (35)

where $h$ and $\Lambda$ matrices are given by

$h=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}Y_{\Sigma}^{1}&\sqrt{2}Y_{N}^{1}\\\
Y_{\Sigma}^{2}&\sqrt{2}Y_{N}^{2}\\\
Y_{\Sigma}^{3}&\sqrt{2}Y_{N}^{3}\end{array}\right)V^{T}(\alpha),\quad\Lambda=\left(\begin{array}[]{cc}\Lambda_{1}&0\\\
0&\Lambda_{2}\end{array}\right),$

with

$\displaystyle\Lambda_{i}=\frac{m_{\chi_{i}^{0}}}{2(4\pi)^{2}}\left[\frac{m_{\eta^{R}}^{2}\ln\left(\frac{m_{\chi_{i}^{0}}^{2}}{m_{\eta^{R}}^{2}}\right)}{m_{\chi_{i}^{0}}^{2}-m_{\eta^{R}}^{2}}-\frac{m_{\eta^{I}}^{2}\ln\left(\frac{m_{\chi_{i}^{0}}^{2}}{m_{\eta^{I}}^{2}}\right)}{m_{\chi_{i}^{0}}^{2}-m_{\eta^{I}}^{2}}\right].$
(36)

In the scenario where the masses of $\eta^{R}$ and $\eta^{I}$ are equal, i.e.,
$m_{\eta^{R}}=m_{\eta^{I}}$, the resulting neutrino masses are zero, which
leads to the condition $\lambda_{5}=0$, so the lepton number is conserved. One
convenient way to represent the Yukawa couplings, $h_{\alpha i}$, is by
employing the Casas-Ibarra parametrization[58, 59] which turns out to be

$\displaystyle h=U^{*}\sqrt{\widetilde{M}}R\sqrt{\Lambda}^{-1},$ (37)

where $U$ is Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix,
$\widetilde{M}=\operatorname{diag}\left(m_{1},m_{2},m_{3}\right)$ with $m_{i}$
are the neutrino physical masses and $\Lambda$ is given by Eqn. (36).
Additionally, the matrix $R$, which is a $3\times 2$ orthogonal matrix
satisfying $RR^{T}=\mathbb{I}_{3\times 3}$, can be expressed as,

$\begin{aligned} &R=\left(\begin{array}[]{cc}0&0\\\ \cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{array}\right),\end{aligned}\hskip 14.22636pt$

where $\theta$ is a complex angle, and this description is applicable in the
case of normal hierarchy only[17, 18, 19].

##  3 W-Mass Anomaly

It is important to note that the pseudo-scalar $Z_{2}$-even portion solely
comprises the imaginary component of the neutral part of $\phi$ and does not
include any contributions from $\eta$ or $\Omega$. Therefore, $Z$-boson gets
no contribution from $\Omega$. The inclusion of the real scalar triplet,
$\Omega$, with zero hypercharge leads to the contribution in the mass of the
$W$-boson ($M_{W}$). When the field $\Omega$ undergoes spontaneous symmetry
breaking by acquiring a non-zero VEV, it imparts an extra contribution to the
mass of the $W$-boson. This supplementary contribution to the mass of the
$W$-boson serves as a potential explanation for the experimental findings
observed by the CDF-II collaboration [29].

Figure 2: The plot illustrating the correlation between the $M_{W}$ and
$v_{\Omega}$. The horizontal dashed lines represent the SM predictions, the
horizontal dotted lines indicate the 1-$\sigma$ limit based on the CDF-II
measurement, and the vertical solid lines signify the range of $v_{\Omega}$
values that match with the CDF-II outcome.

The pertinent components within the Lagrangian that pertain to the mass of the
$W$-boson, are

$\displaystyle\mathcal{L^{\prime}}=\left(D_{\mu}\phi\right)^{\dagger}\left(D^{\mu}\phi\right)+Tr[\left(D_{\mu}\Omega\right)^{\dagger}\left(D^{\mu}\Omega\right)],$

where the covariant derivative is defined as

$\displaystyle
D_{\mu}=\partial_{\mu}+ig\frac{\sigma_{a}}{2}W^{a}_{\mu}+ig^{\prime}\frac{Y}{2}B_{\mu}.$
(38)

The coupling constants $g$ and $g^{\prime}$ correspond to the gauge symmetries
$SU(2)_{L}$ and $U(1)_{Y}$, respectively.
Considering the non-zero value of $v_{\Omega}$ allows us to deduce the masses
of the gauge bosons resulting from the contribution of $\Omega$. The
expressions for the masses of the $W$ and $Z$ bosons are as

$\displaystyle M_{W}^{2}=(v_{\phi}^{2}+4v_{\Omega}^{2})\frac{g^{2}}{4}\hskip
8.5359pt\text{and}\hskip 8.5359ptM_{Z}^{2}=(g^{2}+g^{\prime
2})\frac{v_{\phi}}{4}.$ (39)

Since $\Omega$ lacks hypercharge, it remains uncoupled to the $Z$-boson, thus
exerting no impact on its mass. The mass of the $W$-boson is now recognized to
be influenced by scalar triplet’s VEV $v_{\Omega}$, prompting the utilization
of the latest CDF-II measurement to establish an upper limit on $v_{\Omega}$.
The measured value of $M_{W}$ obtained by the CDF-II experiment can be
expressed as follows

$\displaystyle(M_{\text{W}}^{2})^{\text{CDF}}-(M_{\text{W}}^{2})^{\text{SM}}=g^{2}v_{\Omega}^{2},$
(40)

where $(M_{\text{W}}^{2})^{\text{SM}}$=$\frac{g^{2}v_{\phi}^{2}}{4}$ is the
mass of $W$-boson predicted by SM and value of $g$= 0.6517[60].
Based on the latest CDF-II outcome and prediction of the SM, it becomes
imperative to ensure, with in the framework, that the impact on the observed
value of $M_{W}$ arises exclusively from the VEV of the real scalar triplet
$\Omega$. Hence, the constraint delineated in Eqn. (40) can be expressed as a
bound on $v_{\Omega}$ which is given by

$\displaystyle 4.8\hskip 2.84544pt\text{GeV}\leq v_{\Omega}\leq 6\hskip
2.84544pt\text{GeV},$ (41)

and represented by the vertical band in Fig. 2. The CDF-II outcome and
prediction by the SM are indicated by dotted and dashed horizontal lines,
respectively. It is to be noted that the range of $v_{\Omega}$ obtained in
Eqn. (41) is in consonance with that of reported in Ref.[19].

##  4 Phenomenology of Scalar Dark Matter

Parameter | Range
---|---
$M_{\Sigma},M_{N}$ | $\left[5\times 10^{3},10^{12}\right]\mathrm{GeV}$
$m_{\eta}^{2}$ | $\left[10^{2},10^{10}\right]\mathrm{GeV}$
$\mu_{i}$ | $[10,10^{5}]\mathrm{GeV}$
$v_{\Omega}$ | $[2,6]\mathrm{GeV}$
$\lambda_{5}$ | $\left[-1,-10^{-5}\right]$
$\lambda_{4}$ | $\left[-1,-10^{-5}\right]$
$\lambda_{2,3}$ | $\left[10^{-5},10^{-1}\right]$
$Y_{\Omega}$ | $\left[10^{-5},10^{-1}\right]$
$\lambda_{i}^{\Omega}$ | $\left[10^{-5},10^{-1}\right]$
$\lambda^{\eta}$ | $\left[10^{-5},10^{-1}\right]$
Table 2: Input parameter’s range used for the numerical analysis.

Figure 3: Left Panel: Correlation between DM Relic density and DM ($\eta^{R}$)
mass, $M_{DM}$ for input values, as shown in Table 2. The grey horizontal band
represents the 3$\sigma$ experimental range of $\Omega h^{2}$ coming from
Planck observation[65, 2]. LEP rule-out region with
$m_{\eta}^{R}<\frac{m_{Z}}{2}$ and $m_{\eta^{+}}\lesssim 70\,\text{GeV}$, is
represented by the shaded region[64]. Right Panel: Correlation between Spin-
independent DM-nucleon cross section vs $M_{DM}$. The green points signify
values within the 3$\sigma$ range of $\Omega h^{2}$, while grey and cyan color
points represent over and under-abundance, respectively.

The model allows for the possibilities of both fermionic and bosonic DM
candidates. In so far, fermionic DM is concerned, the lighter neutral
eigenstates $\chi_{1}^{0}$ and $\chi_{2}^{0}$ are good candidates while for
bosonic DM the lightest particle amongst the components of $\eta$ may serves
as the potential DM candidate. Within the premise that the masses of $N$ and
$\Sigma$ are sufficiently large, the preservation of $Z_{2}$ symmetry
designates lighter of $\eta^{R}$ and $\eta^{I}$, as the promising candidate
for DM. In our investigation, we contemplate $\eta^{R}$ as the preferred DM
candidate, given the condition of $\lambda_{5}<0$. Alternatively, in the
scenario of $\lambda_{5}>$ 0, $\eta^{I}$ would be considered the potential DM
candidate. We implement the model in SARAH[61] and subsequently employed in
SPheno[62] to effectively compute all relevant mass matrices and vertices.
Further, micrOMEGAs-5.0.8[63] served as a critical tool in solving the
Boltzmann equations and evaluating the thermal characteristics of the relic
density of WIMP DM and calculating the cross-section for direct-detection (DD)
of DM particle. In the left panel of Fig. 3, we depict the projected DM relic
density ($\Omega h^{2}$) as a function of the DM mass ($M_{DM}$). The solid
grey line represents the 3$\sigma$ range of relic density (0.1126$\leq\Omega
h^{2}\leq 0.1246$) inferred from data obtained by the Planck satellite[65, 2].
We performed a numerical scan of the input parameters within the defined
intervals outlined in Table 2. In order to ensure a mass of $125$ GeV for the
lighter neutral scalar, $H_{1}$, the parameter $\lambda_{1}$ is adjusted
accordingly.

Figure 4: Variation in the $-\lambda_{5}$ as function of $M_{DM}$. The color
scheme is similar to the right panel of Fig. 3

The distinct dips observed can be understood by examining the annihilation and
coannihilation processes, which are detailed in Appendix A. The initial trough
observed at $M_{\text{DM}}\sim\frac{M_{Z}}{2}$ arises due to annihilation and
coannihilation processes facilitated by the exchange of the $Z$-boson in the
$s$-channel. The subsequent dip, located at $M_{\text{DM}}\sim 60$ GeV, is
linked to annihilation processes mediated through the $s$-channel exchange of
$H_{1}$. In comparison, annihilation mediated by the exchange of $H_{1}$ is
more efficient than that involving the $Z$-boson exchange, primarily due to
the momentum suppression that hampers the latter. As the mass of $\eta^{R}$
increases, the quartic interactions with gauge bosons begin to play a
significant role. For instance, the third dip in $\Omega h^{2}$ is brought on
by the annihilation of DM candidate $\eta^{R}\rightarrow W^{+}W^{-},ZZ$ $via$
quartic couplings for $M_{\text{DM}}\geq$ 80 GeV. Moreover, for
$M_{\text{DM}}\geq$ 120 GeV and $m_{\eta^{R}}\geq$ $m_{t}$ GeV, $\eta^{R}$ has
the capability to undergo annihilation processes yielding two Higgs bosons
($H_{1}H_{1}$) and top-antitop quark pairs ($t\bar{t}$), respectively. Also,
it is noted that, beyond 120 GeV, as $M_{\text{DM}}$ increases, the relic
density rises. This phenomenon is attributed to the suppression of the
annihilation cross-section, which decreases as $\sim\frac{1}{M_{\text{DM}}}$.
Additionally, it is crucial to highlight that in parameter space regions
characterized by a small $\lambda_{5}$, coannihilation processes involving
both $\eta^{I}$ and $\eta^{\pm}$ can take place.
Let us now explore the possibilities for DD of $\eta^{R}$. The interaction
cross-section of DM and nucleons at the tree-level is governed by $\phi$ and
$Z$ portals. Due to non-zero hypercharge of the $\eta$ doublet, the DM-nucleon
interaction through the $Z$-boson can, in principle, surpass the latest
constraints set by DD experiments. However, $\lambda_{5}$ introduces a mass
difference between the CP-odd counterparts, $\eta^{I}$ and $\eta^{R}$.
Consequently, the interaction involving the $Z$-boson is either prevented due
to kinematic constraints or leads to inelastic scattering. Thus, the dominant
contribution to the DM-nucleon interaction occurs $via$ $\phi$ in whole region
of the parameter space. Fig. 3 (right panel) illustrates the relationship
between the elastic scattering cross-section of DM with nucleons in a spin-
independent manner as a function of the mass of DM. The red dashed and dashed-
dotted lines in the plot represent the upper limits set by the XENON1T[66, 67]
and XENONnT[68] collaborations, respectively. Also, the blue line represents
the lower limit, which corresponds to the neutrino floor. In Fig. 4, we have
illustrated the relationship between $\lambda_{5}$ and $M_{DM}$. The DM spans
the entire spectrum of $\lambda_{5}$ values, ranging from -1 to $-10^{-5}$, as
indicated by the green points.

##  5 Leptogenesis in STSM

In this section, we delve into baryogenesis via leptogenesis in STSM,
considering two cases: one with two heavy fermions and the other with three,
aimed at lowering the scale of leptogenesis.

###  5.1 Two heavy fermion case

As previously mentioned, the present model allows for the emergence of a net
lepton asymmetry through the out-of-equilibrium decay of both the $N$ and
$\Sigma$. The decay channels of singlet ($N$) and triplet ($\Sigma$) at tree
and one-loop are shown in Figs. 5 and 6. Here we consider the masses of $N$
and $\Sigma$ to be of similar order such that the decays of both contribute to
leptogenesis. Much like the previously mentioned Davidson-Ibarra bound
utilized in the context of Type-I seesaw leptogenesis, a similar lower limit
can also be deduced in this scenario. The expressions for the $CP$ asymmetries
resulting from the $N$ and $\Sigma$ decays, as shown in Figs. 5 and 6, can be
expressed as

$N$$\eta$$L_{\alpha}$$\Sigma$$\eta$$L_{\alpha}$

Figure 5: Tree-level Feynman diagrams for the decays $\left(N\rightarrow
L_{\alpha}+\eta\right)$ and $\left(\Sigma\rightarrow L_{\alpha}+\eta\right)$.

$\eta$$N$$\eta$$L_{\beta}$$\Sigma$$L_{\alpha}$$\eta$$\Sigma$$\eta$$L_{\beta}$$N$$L_{\alpha}$

Figure 6: Feynman diagrams illustrating the vertex corrections at the one-loop
level in the decays of heavy fermions ($N$ and $\Sigma$).

$\displaystyle\epsilon^{N}=\frac{1}{8\pi\left(Y_{N}^{\dagger}Y_{N}\right)}\sum_{j\neq
i}\operatorname{Im}\left[\left(Y_{N}^{\dagger}Y_{\Sigma}\right)_{ij}^{2}\right]\mathcal{F}\left(\frac{M_{\Sigma}^{2}}{M_{N}^{2}}\right),$
(42)

and

$\displaystyle\epsilon^{\Sigma}=\frac{1}{8\pi\left(Y_{\Sigma}^{\dagger}Y_{\Sigma}\right)}\sum_{j\neq
i}\operatorname{Im}\left[\left(Y_{\Sigma}^{\dagger}Y_{N}\right)_{ij}^{2}\right]\mathcal{F}\left(\frac{M_{N}^{2}}{M_{\Sigma}^{2}}\right),$
(43)

where

$\displaystyle Y_{N}=h_{\alpha 1}=\begin{pmatrix}Y_{N}^{1}\\\ Y_{N}^{2}\\\
Y_{N}^{3}\\\ \end{pmatrix},\hskip 28.45274ptY_{\Sigma}=h_{\alpha
2}=\frac{1}{\sqrt{2}}\begin{pmatrix}Y_{\Sigma}^{1}\\\ Y_{\Sigma}^{2}\\\
Y_{\Sigma}^{3}\\\ \end{pmatrix},$ (44)

and

$\displaystyle\mathcal{F}(x)=\sqrt{x}\left[1+\frac{1}{1-x}+(1+x)\ln\left(\frac{x}{1+x}\right)\right].$
(45)

The relevant Boltzmann equations for leptogenesis can be written as

$\displaystyle\dfrac{dn_{\Sigma}}{dz}$ $\displaystyle=$ $\displaystyle-
D_{\Sigma}(n_{\Sigma}-n_{\Sigma}^{eq})-S_{A}(n_{\Sigma}^{2}-(n_{\Sigma}^{eq})^{2}),$
(46) $\displaystyle\dfrac{dn_{N}}{dz}$ $\displaystyle=$ $\displaystyle-
D_{N}(n_{N}-n_{N}^{eq}),$ (47) $\displaystyle\dfrac{dn_{B-L}}{dz}$
$\displaystyle=$
$\displaystyle-\epsilon^{\Sigma}D_{\Sigma}(n_{\Sigma}-n_{\Sigma}^{eq})-\epsilon^{N}D_{N}(n_{N}-n_{N}^{eq})-W_{\rm
ID}^{\Sigma}n_{B-L}-W_{\rm ID}^{N}n_{B-L}$ (48) $\displaystyle-W_{\Delta
L}n_{B-L}.$

Here, $n_{N}^{eq}$ and $n_{\Sigma}^{eq}$ are the equilibrium number density of
$N$ and $\Sigma$ respectively. Also, $n_{N}$, $n_{\Sigma}$ and $n_{B-L}$ are
the comoving number densities of $N$, $\Sigma$ and $B-L$ asymmetry
respectively. $D_{N}$ and $D_{\Sigma}$ are the decay terms for $N$ and
$\Sigma$ decays which are given by

$\displaystyle D_{N}$ $\displaystyle=$ $\displaystyle
K_{N}z\dfrac{\kappa_{1}(z)}{\kappa_{2}(z)},$ (49) $\displaystyle D_{\Sigma}$
$\displaystyle=$ $\displaystyle
K_{\Sigma}\bigg{(}\dfrac{M_{\Sigma}}{M_{N}}z\bigg{)}\dfrac{\kappa_{1}\bigg{(}\dfrac{M_{\Sigma}}{M_{N}}z\bigg{)}}{\kappa_{2}\bigg{(}\dfrac{M_{\Sigma}}{M_{N}}z\bigg{)}},$
(50)

where $z=\frac{M_{N}}{T}$ with $T$ is temperature of thermal bath and

$\displaystyle K_{N}$ $\displaystyle=$
$\displaystyle\dfrac{\Gamma_{N}}{\textbf{H}(T=M_{N})},$ (51) $\displaystyle\
K_{\Sigma}$ $\displaystyle=$
$\displaystyle\dfrac{\Gamma_{\Sigma}}{\textbf{H}(T=M_{\Sigma})},$ (52)

are the decay parameters for the $N$ and $\Sigma$ decay, respectively.
$\Gamma_{N,\Sigma}$ are the decay widths given by

$\displaystyle\Gamma_{N}$ $\displaystyle=$
$\displaystyle\dfrac{M_{N}}{8\pi}(Y_{N}^{\dagger}Y_{N})\bigg{(}1-\dfrac{m_{\eta}^{2}}{M_{N}^{2}}\bigg{)},$
(53) $\displaystyle\Gamma_{\Sigma}$ $\displaystyle=$
$\displaystyle\dfrac{M_{\Sigma}}{8\pi}(Y_{\Sigma}^{\dagger}Y_{\Sigma})\bigg{(}1-\dfrac{m_{\eta}^{2}}{M_{\Sigma}^{2}}\bigg{)}.$
(54)

In the early Universe, dominated by radiation, the Hubble parameter can be
expressed in terms of temperature as
$H=2\sqrt{\frac{\pi^{3}g_{*}}{45}}\frac{T^{2}}{M_{pl}}$, with $g_{*}=124$ is
the effective number of relativistic degrees of freedom and $M_{Pl}$ is the
Planck mass given by $M_{Pl}=1.22\times 10^{19}$ GeV. The washout term arises
from the combined effect of inverse decays and $L$ violating washout
processes. The total washout term is given by $W_{\textit{\rm
Total}}=W^{N,\Sigma}_{\textit{\rm ID}}+W_{\rm\Delta L}$, where $W_{\rm
ID}^{N}$ and $W_{\rm ID}^{\Sigma}$ are the inverse decay terms for $N$ and
$\Sigma$ decay, respectively and are given by

$\displaystyle W_{\rm ID}^{N}$ $\displaystyle=$
$\displaystyle\dfrac{1}{4}K_{N}z^{3}\kappa_{1}(z),$ (55) $\displaystyle W_{\rm
ID}^{\Sigma}$ $\displaystyle=$
$\displaystyle\dfrac{1}{4}K_{\Sigma}\bigg{(}\dfrac{M_{\Sigma}}{M_{N}}z\bigg{)}^{3}\kappa_{1}\bigg{(}\dfrac{M_{\Sigma}}{M_{N}}z\bigg{)}.$
(56)

Apart from the inverse decays, the other contribution to washout $W_{\rm\Delta
L}$ originates from scatterings that violate the lepton number. The scattering
washouts due to $l\eta\longrightarrow\bar{l}\eta^{*}$,
$ll\longrightarrow\eta^{*}\eta^{*}$ process is given by

$\displaystyle W_{\rm\Delta L}=\frac{0.585\times M_{pl}}{g_{l}\times
g_{*}z^{2}v^{4}}\left(\frac{2\pi^{2}}{\lambda_{5}}\right)^{2}M_{N}\overline{m}_{\varkappa_{N}}^{2}+\frac{0.585\times
M_{pl}}{g_{l}\times
g_{*}z^{2}v^{4}}\left(\frac{2\pi^{2}}{\lambda_{5}}\right)^{2}M_{\Sigma}\overline{m}_{\varkappa_{\Sigma}}^{2}.$
(57)

Here, $g_{l}$ represents the intrinsic degrees of freedom of SM leptons, while
$\overline{m}_{\varkappa}$ is defined as an effective neutrino mass parameter
and in the limit of $m_{1}$=0, given by

$\displaystyle\overline{m}_{\varkappa_{N,\Sigma}}^{2}=\varkappa_{2}^{2}m_{2}^{2}+\varkappa_{3}^{2}m_{3}^{2}$
(58)

with $m_{i}^{s}$ being the light neutrino mass eigenvalues and $\varkappa$ is
defined as:

$\displaystyle\varkappa_{2,3}=\frac{M_{N,\Sigma}^{2}}{8\left(m^{2}_{\eta^{R}}-m^{2}_{\eta^{I}}\right)}[L_{i}(m_{\eta^{R}}^{2})-L_{i}(m_{\eta^{I}}^{2})].$
(59)

The gauge boson mediated scattering term, $S_{A}$, for $\Sigma$ and can be
identified as

$\displaystyle
S_{A}=\left(\frac{0.032\sqrt{g_{*}}M_{pl}}{g^{2}_{\Sigma}M_{\Sigma}}\right)\left(\frac{I(z)}{z\times\kappa_{2}\bigg{(}\dfrac{M_{\Sigma}}{M_{N}}z\bigg{)}^{2}}\right),$
(60)

where

$\displaystyle
I(z)=\int_{4}^{\infty}\sqrt{x}\kappa_{1}\bigg{(}z\sqrt{x}\bigg{)}\hat{\sigma}_{A}(x)\,dx\
.$ (61)

Here, the $\hat{\sigma}(x)$ is cross-section for the gauge boson mediated
processes given by[45]

$\displaystyle\hat{\sigma}_{A}=\frac{6g^{4}}{72\pi}\left[\frac{45}{2}r(x)-\frac{27}{2}r(x)^{3}-\left\\{9\left(r(x)^{2}-2\right)+18\left(r(x)^{2}-1\right)^{2}\right\\}\ln\left(\frac{1+r(x)}{1-r(x)}\right)\right],$
(62)

where $r(x)=\sqrt{1-4/x}$. Eqn. (62) includes $\Sigma$, $\Sigma$ → all
possible fermion doublets, Higgs, gauge bosons. After solving the Boltzmann
equations mentioned in Eqns. (46), (47), and (48), the relation

$\displaystyle\eta_{B}=\frac{3g_{*}^{0}}{4g_{*}}a_{sphl}n_{B-L}\simeq
9.2\times 10^{-3}n_{B-L},$ (63)

converts $B-L$ asymmetry, $n_{B-L}$, into the observed baryon-to-photon ratio,
prior to electroweak sphaleron freeze-out. Here $a_{sphl}=\frac{8}{23}$ is the
sphaleron conversion factor and $g_{*}^{0}$ represents the effective count of
relativistic degrees of freedom during the epoch of recombination.

###  5.2 Three heavy fermions case

Here we consider two singlet heavy fermions ($N_{1}$ and $N_{2}$) with one
triplet fermion $\Sigma$. $M_{N_{1}}$, $M_{N_{2}}$ and $M_{\Sigma}$ are the
masses of $N_{1}$, $N_{2}$ and $\Sigma$, respectively. Due to the presence of
an additional singlet fermion, the Yukawa matrix $h$ takes the form

$\displaystyle h$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{2}}\begin{pmatrix}\sqrt{2}Y_{N_{1}}^{1}&Y_{\Sigma}^{1}&\sqrt{2}Y_{N_{2}}^{1}\\\
\sqrt{2}Y_{N_{1}}^{2}&Y_{\Sigma}^{2}&\sqrt{2}Y_{N_{2}}^{2}\\\
\sqrt{2}Y_{N_{1}}^{3}&Y_{\Sigma}^{3}&\sqrt{2}Y_{N_{2}}^{3}\end{pmatrix}V^{T}(\alpha).$
(64)

Here, we perform the analysis assuming the hierarchy $M_{N_{1}}\lesssim
M_{\Sigma}\ll M_{N_{2}}$. Since $N_{2}$ is assumed to be much heavier compared
to the other two fermions, the lepton asymmetry generated from the decay of
$N_{2}$ is washed-out by the inverse decays of $N_{1}$ and $\Sigma$.
Effectively, only $N_{1}$ and $\Sigma$ will generate the lepton asymmetry. The
relevant CP asymmetry parameter due to the decay of $N_{1}$ and $\Sigma$ are
found to be

$\displaystyle\epsilon_{N_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi\left(Y_{N_{1}}^{\dagger}Y_{N_{1}}\right)}\left[\operatorname{Im}\left[\left(Y_{N_{1}}^{\dagger}Y_{\Sigma}\right)^{2}\right]\mathcal{F}\left(\frac{M_{\Sigma}^{2}}{M_{N_{1}}^{2}}\right)+\left(Y_{N_{1}}^{\dagger}Y_{\Sigma}\right)^{2}\mathcal{F}\left(\frac{M_{\Sigma}^{2}}{M_{N_{1}}^{2}}\right)\right],$
(65) $\displaystyle\epsilon_{\Sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi\left(Y_{\Sigma}^{\dagger}Y_{\Sigma}\right)}\left[\operatorname{Im}\left[\left(Y_{\Sigma}^{\dagger}Y_{N_{1}}\right)^{2}\right]\mathcal{F}\left(\frac{M_{N_{1}}^{2}}{M_{\Sigma}^{2}}\right)+\left(Y_{\Sigma}^{\dagger}Y_{N_{2}}\right)^{2}\mathcal{F}\left(\frac{M_{N_{2}}^{2}}{M_{\Sigma}^{2}}\right)\right].$
(66)

The relevant Boltzmann equations in this scenario are

$\displaystyle\dfrac{dn_{N_{1}}}{dz}$ $\displaystyle=$ $\displaystyle-
D_{N}\left(n_{N_{1}}-n^{eq}_{N_{1}}\right),$ (67)
$\displaystyle\dfrac{dn_{\Sigma}}{dz}$ $\displaystyle=$ $\displaystyle-
D_{\Sigma}(n_{\Sigma}-n_{\Sigma}^{eq})-S_{A}(n_{\Sigma}^{2}-(n_{\Sigma}^{eq})^{2}),$
(68) $\displaystyle\dfrac{dn_{B-L}}{dz}$ $\displaystyle=$
$\displaystyle-\epsilon_{N_{1}}D_{N_{1}}(n_{N_{1}}-n_{N_{1}}^{eq})-\epsilon_{\Sigma}D_{\Sigma}(n_{\Sigma}-n_{\Sigma}^{eq})-W_{\rm
ID}^{N_{1}}n_{B-L}-W_{\rm ID}^{\Sigma}n_{B-L}$ (69) $\displaystyle-W_{\Delta
L}n_{B-L}.$

Here, the decay terms for $N_{1}$ and $\Sigma$ are given by

$\displaystyle D_{N_{1}}$ $\displaystyle=$ $\displaystyle
K_{N_{1}}\dfrac{\kappa_{1}(z)}{\kappa_{2}(z)},$ (70) $\displaystyle
D_{\Sigma}$ $\displaystyle=$ $\displaystyle
K_{\Sigma}\left(\dfrac{M_{\Sigma}}{M_{N_{1}}}z\right)\dfrac{\kappa_{1}\left(\dfrac{M_{\Sigma}}{M_{N_{1}}}z\right)}{k_{2}\left(\dfrac{M_{\Sigma}}{M_{N_{1}}}z\right)}.$
(71)

and inverse decay terms for $N_{1}$ and $\Sigma$ are defined as

$\displaystyle W_{\rm ID}^{N_{1}}$ $\displaystyle=$
$\displaystyle\dfrac{1}{4}K_{N_{1}}z^{3}\kappa_{1}(z),$ (72) $\displaystyle
W_{ID}^{\Sigma}$ $\displaystyle=$
$\displaystyle\dfrac{1}{4}K_{\Sigma}\left(\dfrac{M_{\Sigma}}{M_{N_{1}}}z\right)^{3}\kappa_{1}\left(\dfrac{M_{\Sigma}}{M_{N_{1}}}z\right).$
(73)

Figure 7: Comoving number density of $N$ (upper left panel), $\Sigma$ (upper
right panel) and $B-L$ (lower panel) with different values of $M_{N}$. In the
lower panel, the grey horizontal line represents the experimentally required
value for the $B-L$ asymmetry.

Figure 8: Comoving number density of $N$ (upper left panel), $\Sigma$ (upper
right panel) and $B-L$ (lower panel) with different values of $\lambda_{5}$.
In the lower panel, the grey horizontal line represents the experimentally
required value for the $B-L$ asymmetry.

Figure 9: Comoving Number density of $N_{1}$ (top left panel plot), $\Sigma$
(top right panel plot) and $B-L$ (lower panel plot) with $z=M_{N_{1}}/T$ for
different values of $\lambda_{5}$. In the lower panel, the grey horizontal
line represents the experimentally required value for the $B-L$ asymmetry.

Figure 10: Comoving Number density of $N_{1}$ (upper left panel), $\Sigma$
(upper right panel) and $B-L$ (lower panel) with $z=M_{N}/T$ for different
values of $M_{N_{1}}$. In the lower panel, the grey horizontal line represents
the experimentally required value for the $B-L$ asymmetry. Figure 11:
Evolution of $B-L$ asymmetry for $N_{1}+\Sigma$ (red dashed line) and $N_{1}$
(blue line) as a function of $z=M_{N_{1}}/T$. The grey horizontal line
represents the experimentally required value for the $B-L$ asymmetry.

###  5.3 Numerical Analysis and Discussion

We study the leptogenesis in the STSM where we first consider the minimal
scenario with two moderately hierarchical ($M_{N}\lesssim M_{\Sigma}$) heavy
fermions, $N$ and $\Sigma$. This results in the lightest neutrino remaining
massless. In order to satisfy Sakharov’s third condition, the lepton asymmetry
must be generated by the out-of-equilibrium decay of heavy fermions. This
asymmetry is then converted into the baryon asymmetry $via$ ($B+L$) violating
sphaleron processes. As the Universe evolves, characterized by a decrease in
temperature or increase in $z$, $N$ and $\Sigma$ begin decaying into SM
particles reducing their respective comoving number densities ($n_{N}$ and
$n_{\Sigma}$). Consequently, this process occurs concomitantly with an
escalation in lepton asymmetry $n_{B-L}$, as depicted in Figs. 7 and 8. At
fixed $\lambda_{5}=-1$ and $M_{\Sigma}=10^{0.5}M_{N}$, $n_{N}$ and
$n_{\Sigma}$ are very close to $n^{eq}_{N}$ and $n^{eq}_{\Sigma}$ as shown in
the upper panel of Fig. 7. Due to the presence of only two heavy fermions
generating masses of the neutrinos, the Yukawa couplings for the heavy
fermions are large enough such that the decay and inverse decays keep their
abundances very close to their equilibrium abundance. This is explained in
detail in Appendix B. Due to the same reason, we are necessarily in a strong
washout region ($K_{N,\Sigma}>>1$) with two heavy fermions. This leads to a
high-scale leptogenesis scenario where $M_{N,\Sigma}\geq 10^{9}$ GeV. In Fig.
7, one can see that $n_{\Sigma}$ deviates more from its equilibrium abundance
with the increase in $M_{\Sigma}$ near $z\simeq 1$. With the increase in
$M_{\Sigma}$, the rate of gauge boson-mediated annihilations for $\Sigma$
decreases. This makes $\Sigma$ to deviate more from its equilibrium abundance.
It can, also, be seen that with the increase in $M_{N,\Sigma}$ the $B-L$
asymmetry increases. This is due to the suppression of inverse decay washouts
with the increase in $M_{N,\Sigma}$. In Fig. 8 we show the evolution of
$n_{N}$, $n_{\Sigma}$ and $n_{B-L}$ with $z=M_{N}/T$ for different values of
$\lambda_{5}$. With the decrease in $\lambda_{5}$ the Yukawa couplings for
$N,\Sigma$ increases through the CI parameterisation of Eqn. (37). Due to the
increase in Yukawa coupling the decay and inverse decay rates increase
resulting in $n_{N,\Sigma}$ to remain closer to their equilibrium abundance.
Similarly from the lower panel plot of Fig. 8 it can be seen that with the
decrease in $\lambda_{5}$ the $B-L$ asymmetry first increases, however, after
a certain small value of $\lambda_{5}$ it starts decreasing with further
decrease in $\lambda_{5}$. For the small value of $\lambda_{5}$ the Yukawa
coupling becomes large enough to increase the CP asymmetry resulting in an
increase in $B-L$ asymmetry. However, beyond a certain large value of the
Yukawa coupling the washouts increase which dominates over the increase in CP
asymmetry. As a result, the $B-L$ asymmetry decreases with the decrease in
$\lambda_{5}$.

In the presence of three fermions ($N_{1}$,$N_{2}$,$\Sigma$), the Yukawa
structure changes and the lightest active neutrino mass ($m_{1}$) becomes an
important parameter as discussed in the Appendix C. Here we choose the
hierarchy $M_{N_{1}}\lesssim M_{\Sigma}<M_{N_{2}}$ among the heavy fermions.
With vanishingly small lightest active neutrino mass, the Yukawa coupling for
the lightest fermion $N_{1}$ becomes small enough to make a weak washout
leptogenesis scenario ($K_{N_{1}}<1$). This results in a decrease of the
leptogenesis scale from $10^{10}$ GeV to sub-$\rm TeV$ scale. In Fig. 9 we
show the evolution of $n_{N_{1}},n_{\Sigma}$ and $n_{B-L}$ with
$z=M_{N_{1}}/T$ for different values of $\lambda_{5}$. Similar behaviour of
the $n_{N_{1}}$, $n_{\Sigma}$ and $n_{B-L}$ can be seen with the change in
$\lambda_{5}$ as in the case of the two heavy fermions. In Fig. 10 we show the
evolution of $n_{N_{1}}$, $n_{\Sigma}$ and $n_{B-L}$ with $z=M_{N_{1}}/T$ for
different values of $M_{N_{1}}$ with $M_{\Sigma}=10^{0.5}M_{N_{1}}$ for fixed
benchmark values of other relevant parameters. Similar to two fermion cases,
we can see the increase in asymmetry with the increase in $M_{N_{1}}$ and
$M_{\Sigma}$. Furthermore, previous investigations indicate that the lightest
heavy fermion is the sole contributor to the $B-L$ asymmetry, while heavier
fermions are disregarded due to their substantial washout effects. However,
our study suggests that when we choose moderately hierarchical heavy fermion
masses, the heavier fermions can significantly contribute to the $B-L$
asymmetry. The evolution of the $B-L$ asymmetry for the $N_{1}+\Sigma$ (red
dashed line) and $N_{1}$ only (blue line) is depicted in Fig. (11). For later
case, the $B-L$ asymmetry remains very low compared to the required value of
$n_{B-L}$. However, considering the hierarchy of the heavy fermions
($M_{N_{1}}\lesssim M_{\Sigma}<M_{N_{2}}$), the triplet fermion can contribute
significantly contribute to the final $B-L$ asymmetry which is, also, evident
from the Fig. (11).
Despite the lack of signatures of $\Sigma$ beyond 1 TeV at LHC, it is still
feasible to explore the triplet fermions in final states that involve multiple
leptons and fat-jets. These final states are more refined compared to the
typical LHC searches and can be extended up to 10 TeV[69, 70]. At 95$\%$
confidence level, the ATLAS experiment at LHC has excluded the triplet fermion
with mass below 790 GeV[71]. We have successful leptogenesis around the sub-
TeV scale. Collider experiments already constrain light triplet fermions, but
future experiments may test viable parameter space.

Benchmark points yielding correct $n_{B-L}$: For two heavy fermion case:
$M_{N}=7.97\times 10^{9}$ GeV, $\lambda_{5}$=-0.1 and $M_{N}=6.30\times
10^{9}$ GeV, $\lambda_{5}$=-0.3.
For three heavy fermion case:
$M_{N_{1}}=1513.56$ GeV, $\lambda_{5}$=-0.1 and $M_{N_{1}}=1995.26$ GeV,
$\lambda_{5}$=-0.2.

##  6 Summary

We examined the singlet-triplet scotogenic model (STSM) in which dark sector
particles run in the loop and generate tiny neutrino masses at one loop level.
The particle content of the model includes the heavy $SU(2)_{L}$ singlet
$(N)$, triplet $(\Sigma)$ fermions. Scalar sector is extended by a $SU(2)_{L}$
doublet scalar $\eta$ and a hyperchargeless $SU(2)_{L}$ triplet $\Omega$. We
examined the prospects of the model meeting the recent $W$-boson mass
measurements of the CDF-II collaboration at tree-level through hyperchargeless
scalar triplet $\Omega$. Also, we studied the phenomenology of DM where the
real component of $\eta$ acts as a viable WIMP-type DM candidate.
In our study, we explored the possibility of producing baryogenesis at the TeV
scale through leptogenesis in the STSM. The correct lepton asymmetry is
generated through the out-of-equilibrium decay of both singlet (N) and triplet
fermions. This lepton asymmetry is then converted into the baryon asymmetry
via sphaleron processes that accidentally conserve $B-L$ but violate $B+L$
symmetry. In our analysis, we are examining two scenarios: one involving two
heavy fermions and the other involving three, and we explicitly distinguish
between the two cases. First of all, we deal with two heavy fermions, namely N
and $\Sigma$, which result in the vanishing lightest neutrino mass eigenvalue
($m_{1}=0$). Both heavy fermions have a moderately hierarchical mass, and
their out-of-equilibrium decay contributes to the final $B-L$ asymmetry. As
light neutrino masses require a large value of the Yukawa coupling through
Casas-Ibarra parameterization, the scale of leptogenesis is estimated to be
around $10^{10}$ GeV, which is similar to the thermal leptogenesis in the
Type-I seesaw scenario. In order to lower the scale of leptogenesis to around
sub-TeV scale, we introduce an additional heavy fermion and consider three
fermion cases ($N_{1}$, $N_{2}$, $\Sigma$). As a result, the lightest neutrino
mass now has a non-zero value ($m_{1}\neq 0$). In the scotogenic model, with
three heavy Majorana fermions, the scale of leptogenesis depends heavily on
the value of the lightest neutrino mass ($m_{1}$). For vanishingly small
values of $m_{1}$ ($\lesssim 10^{-5}$ eV), the impact of washouts is
negligible. Therefore reducing the $m_{1}$ beyond this range results in
lowering the leptogenesis scale. This mechanism requires leptogenesis to end
before electroweak sphaleron drop out of equilibrium. The analysis yields a
minimum mass estimate of around $3$ TeV for the lightest right-handed fermion
which is quite small as compared to the mass bound of $\sim 10^{9}$ GeV in
standard thermal leptogenesis. Moreover, the quadratic coupling $\lambda_{5}$
plays a significant role in bridging DM and leptogenesis. Future collider
experiments may offer insights into the gauge interactions of a TeV-scale
triplet fermion.

Acknowledgments
L. S. acknowledges the financial support provided by Central University of
Himachal Pradesh. The authors, also, thank Sushant Yadav for the helpful
discussions.

##  Appendix A Relevant annihilation and coannihilation diagrams contributing
to the relic abundance of $\eta^{R}$

$\eta^{R}$$\eta^{I}$$\l^{-},\overline{q}$$\l^{+},q$$Z$$\eta^{R}$$\eta^{I}$$\l^{-},\overline{q}$$\l^{+},q$$W^{\pm}$$\eta^{R}$$\eta^{R}$$W^{-}$$W^{+}$$\eta^{R}$$\eta^{I}$$W^{-}$$W^{+}$$Z$$\eta^{R}$$\eta^{I}$$\l^{-},\overline{q}$$\l^{+},q$$H_{1},H_{2}$$\eta^{R}$$\eta^{I}$$\nu$$l^{\pm}$$H_{1,2}^{\pm}$$W^{\pm}$$\eta^{I}$$W^{+}$$\eta^{R}$$\eta^{\pm}$$\eta^{R}$$\eta^{R}$$Z$$Z$$l^{\pm}$$\eta^{R/I}$$l^{\pm}$$\eta^{R}$$\chi^{\pm}$$\eta^{R}$$\eta^{R}$$W^{-}$$W^{+}$$H_{1},H_{2}$$\nu$$\eta^{R/I}$$\nu$$\eta^{R}$$\chi_{\sigma}$$H_{1},H_{2}$$\eta^{R}$$H_{1},H_{2}$$\eta^{R}$$H_{1}$$\nu$$\eta^{\pm}$$l^{\pm}$$\eta^{R}$$\chi^{\pm}$$W^{-}$$\eta^{R}$$W^{+}$$\eta^{R}$$\eta^{\pm}$$l^{\pm}$$\eta^{\pm}$$\nu$$\eta^{R}$$\chi_{\sigma}$$\eta^{R}$$\eta^{R}$$H_{1},H_{2}$$H_{1},H_{2}$$H_{1}$

##  Appendix B Two heavy fermion case

When we consider two heavy fermions $N$ and $\Sigma$ the lightest active
neutrino becomes massless and the CI parametrisation gives the Yukawa
couplings for the heavy fermions

$\displaystyle h=U^{*}\sqrt{\widetilde{M}}R\sqrt{\Lambda}^{-1}.$ (74)

Where $\widetilde{M}=diag(m_{1},m_{2},m_{3})$ is the active neutrino mass
matrix. To generate a non-zero CP asymmetry $\epsilon^{N}$ and
$\epsilon^{\Sigma}$ the following $R$ matrix is chosen

$R=\begin{pmatrix}0&0\\\ \cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{pmatrix}.$ (75)

The explicit form of the Yukawa couplings is then found to be

$\displaystyle h=\begin{pmatrix}\Lambda_{1}^{-1/2}(\sqrt{m_{2}}\cos\theta
u_{21}^{*}+\sqrt{m_{3}}\sin\theta
u_{31}^{*})&\Lambda_{2}^{-1/2}(-\sqrt{m_{2}}\sin\theta
u_{21}^{*}+\sqrt{m_{3}}\cos\theta u_{31}^{*})\\\
\Lambda_{1}^{-1/2}(\sqrt{m_{2}}\cos\theta u_{22}^{*}+\sqrt{m_{3}}\sin\theta
u_{32}^{*})&\Lambda_{2}^{-1/2}(-\sqrt{m_{2}}\sin\theta
u_{22}^{*}+\sqrt{m_{3}}\cos\theta u_{32}^{*})\\\
\Lambda_{1}^{-1/2}(\sqrt{m_{2}}\cos\theta u_{23}^{*}+\sqrt{m_{3}}\sin\theta
u_{33}^{*})&\Lambda_{2}^{-1/2}(-\sqrt{m_{2}}\sin\theta
u_{23}^{*}+\sqrt{m_{3}}\cos\theta u_{33}^{*})\end{pmatrix}.$ (76)

From Eqn. (76) it can be seen that the Yukawa couplings are dependent on
$m_{2}$ and $m_{3}$. With the normal ordering of light neutrino mass, we don’t
have freedom over $m_{2}$ and $m_{3}$ to make the Yukawa coupling smaller
irrespective of the value of $\theta$. Here $u_{ij}$s are the elements of the
PMNS matrix. The only way to make the Yukawa couplings smaller is to increase
the scalar coupling $\lambda_{5}$. Since $\lambda_{5}$ has a perturbative
limit of $4\pi$, the Yukawa couplings can not be made arbitrarily small by
increasing the value of $\lambda_{5}$. Therefore we are always in a strong
washout region with two heavy fermions.

##  Appendix C Three heavy fermion case

With the presence of another heavy fermion ($N_{1}$, $\Sigma$, $N_{2}$) the
minimal $R$ matrix to generate a non-zero CP asymmetry become

$R=\begin{pmatrix}1&0&0\\\ 0&\cos\theta&\sin\theta\\\
0&-\sin\theta&\cos\theta\end{pmatrix},$ (77)

and the Yukawa matrix can be found to be

$\displaystyle
h=\begin{pmatrix}\sqrt{m_{1}}\Lambda_{1}^{-1/2}u^{*}_{11}&\Lambda_{1}^{-1/2}(\sqrt{m_{2}}\cos\theta
u_{21}^{*}+\sqrt{m_{3}}\sin\theta
u_{31}^{*})&\Lambda_{2}^{-1/2}(-\sqrt{m_{2}}\sin\theta
u_{21}^{*}+\sqrt{m_{3}}\cos\theta u_{31}^{*})\\\
\sqrt{m_{1}}\Lambda_{1}^{-1/2}u^{*}_{12}&\Lambda_{1}^{-1/2}(\sqrt{m_{2}}\cos\theta
u_{22}^{*}+\sqrt{m_{3}}\sin\theta
u_{32}^{*})&\Lambda_{2}^{-1/2}(-\sqrt{m_{2}}\sin\theta
u_{22}^{*}+\sqrt{m_{3}}\cos\theta u_{32}^{*})\\\
\sqrt{m_{1}}\Lambda_{1}^{-1/2}u^{*}_{13}&\Lambda_{1}^{-1/2}(\sqrt{m_{2}}\cos\theta
u_{23}^{*}+\sqrt{m_{3}}\sin\theta
u_{33}^{*})&\Lambda_{2}^{-1/2}(-\sqrt{m_{2}}\sin\theta
u_{23}^{*}+\sqrt{m_{3}}\cos\theta u_{33}^{*})\end{pmatrix}.$ (78)

From Eqn. (78) it can be seen that the Yukawa couplings for the lightest of
the three heavy fermions ($N_{1}$) can be made small by choosing the
vanishingly small lightest active neutrino mass $m_{1}$. This makes the decay
and the inverse decay widths smaller, resulting in a weak washout leptogenesis
scenario. Therefore, the leptogenesis scale can be significantly lower
compared to the two generations of heavy fermions.

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# Exploring Moral Principles Exhibited in OSS: A Case Study on GitHub Heated
Issues

Ramtin Ehsani  Drexel UniversityPhiladelphia, PAUSA<EMAIL_ADDRESS>,
Rezvaneh Rezapour  Drexel UniversityPhiladelphia, PAUSA
<EMAIL_ADDRESS>and  Preetha Chatterjee  Drexel
UniversityPhiladelphia, PAUSA<EMAIL_ADDRESS>
(2023)

###### Abstract.

To foster collaboration and inclusivity in Open Source Software (OSS)
projects, it is crucial to understand and detect patterns of toxic language
that may drive contributors away, especially those from underrepresented
communities. Although machine learning-based toxicity detection tools trained
on domain-specific data have shown promise, their design lacks an
understanding of the unique nature and triggers of toxicity in OSS
discussions, highlighting the need for further investigation. In this study,
we employ Moral Foundations Theory to examine the relationship between moral
principles and toxicity in OSS. Specifically, we analyze toxic communications
in GitHub issue threads to identify and understand five types of moral
principles exhibited in text, and explore their potential association with
toxic behavior. Our preliminary findings suggest a possible link between moral
principles and toxic comments in OSS communications, with each moral principle
associated with at least one type of toxicity. The potential of MFT in
toxicity detection warrants further investigation.

moral principles, toxicity, open source, textual analysis

††conference: The 31st ACM Joint European Software Engineering Conference and
Symposium on the Foundations of Software Engineering; 11 - 17 November, 2023;
San Francisco, USA††copyright: acmcopyright††journalyear: 2023††doi:
XXXXXXX.XXXXXXX††price: 15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs: Software
and its engineering Software organization and properties

##  1\. Introduction

Open source software (OSS) projects have become crucial for the digital
advancement of society, with over 90% of companies leveraging open source
(Git, 1 10). However, many new and longstanding OSS projects fail, in part due
to toxic interactions and uncivil language, which serve as a barrier to
contribution and lead to negative emotions and isolation (Steinmacher et al.,
2015; Qiu et al., 2019; Gerosa et al., 2021). Thus, understanding toxicity in
OSS has gained increasing attention in recent years (Cheriyan et al., 6 21;
Sarker et al., 2020, 2022; Ferreira et al., 7 07, 5 23; Egelman et al., 0 01).
Uncivil features such as name-calling, frustration, and impatience were found
in more than half of non-technical emails in the Linux Kernel mailing list
(Ferreira et al., 0 13), swearing and personal offense were identified as the
main reasons for offense in several communities (Cheriyan et al., 6 21), and
GitHub issue threads contain various types of toxic behaviors (Miller et al.,
2022). Although researchers outside the Software Engineering (SE) domain have
designed several techniques to automatically detect toxicity in online forums
(Sheth et al., 4 21; Maity et al., 1 01; Pascual-Ferrá et al., 4 26), directly
applying those tools to SE is ineffective due to several factors, including
the unique nature of SE text (Raman et al., 2020; Sarker et al., 2020; Imran
et al., 2022). More recently, machine learning-based tools have shown promise
in detecting toxicity when trained on domain-specific data (Sarker et al.,
2022), however it remains crucial to address the fundamental issue of their
design lacking an understanding of the unique nature and triggers of toxicity
in OSS (Miller et al., 2022; Qiu et al., 2022).

Language is an indispensable tool to convey information between people, and
analyzing it could play a significant role in understanding the challenges
faced in OSS communities (Sanei et al., 3 18; Ferreira et al., 8 20; Sajadi et
al., 2023). Linguistic styles and word choices can help in analyzing and
understanding people’s thoughts, opinions, and feelings (Triandis, 1989; Boyd
et al., 2022; Rezapour et al., 2019; Chatterjee et al., 2021). People’s values
and personal norms affect their (spontaneous) attitude, decision-making
process, and what they perceive as good or bad, and moral or immoral (Haidt,
2001; Rezapour et al., 2021). Previous research in psychology has addressed
the connection between moral principles and hate and showed that morality is a
key feature of hatred; hate is connected to core moral beliefs and higher
levels of moral emotions (e.g., contempt, anger, disgust) (Pretus et al.,
2018; Atari et al., 2021). Considering the importance of effective
communication, we believe that analyzing moral values as exhibited in OSS
texts can provide valuable insights into the underlying beliefs and values
that influence communication styles in this community, especially in spreading
toxicity.

In this paper, we aim to investigate the moral principles exhibited in GitHub
and explore their relationship with toxicity. To that end, we leverage Moral
Foundation Theory (MFT), a social psychological theory that assumes individual
judgments are influenced by emotional and cognitive appraisals, referred to as
intuitions or foundations (Graham et al., 2013; Haidt and Joseph, 2004). We
explore a dataset of 100 toxic issue threads labeled with their representative
natures (e.g., ‘Insulting’, ‘Arrogant’) (Miller et al., 2022), to analyze
moral principles and how they are exhibited in the context of OSS project
communications. We map the observed moralities in each thread to their
corresponding natures of toxicity to understand the relationship and co-
occurrence. Our findings reveal that toxic issue threads exhibit moral
principles, with each moral principle being associated with at least one type
of toxicity. This suggests that toxic comments are associated with moral
principles, and that the Moral Foundations Theory (MFT) may be useful in
detecting toxicity. To apply MFT to the domain of SE, it would be necessary to
adapt it, and we believe that doing so has the potential to enhance our
understanding and detection of toxicity in SE communications.

##  2\. Moral Foundations Theory

With the goal of investigating instances on GitHub that represent moral
principles exhibited in the text, we leverage Moral Foundations Theory (MFT)
(Graham et al., 2013). MFT classifies human behavior into five fundamental
principles that represent contrasting values:

Care/harm: This principle is based on our general dislike of suffering,
whether for ourselves or others, related to our evolution as mammals. It
underlies the virtues of kindness, compassion, and gentleness, and it condemns
cruelty and aggression.

Fairness/cheating: This principle relates to justice and rights, and is linked
to the evolutionary process of reciprocal altruism.

Loyalty/betrayal: This principle promotes patriotism, heroism, trust, and
self-sacrifice for the group, rooted in our history as tribal beings. It
values the virtue of ”One for all, and all for one,” while considering acts of
betrayal towards social structures as immoral.

Authority/subversion: This principle is shaped by our primate history of
hierarchical social structure, resulting in virtues such as leadership,
followership, and deference to authority/traditions. It may view dissent
against authority as immoral.

Sanctity/degradation: This principle arises from the psychology of disgust and
contamination, where the concept of humans striving to live in a noble and
elevated way is the key. It reflects the idea of the body as a temple that can
be corrupted by immorality.

##  3\. Methodology

To understand how moral principles manifest in an OSS project’s textual
communication and its relation to toxicity, we analyze the GitHub developer
interactions. Our focus is on GitHub heated issues, which often result from
conflicts, as these issues can provide valuable insights into the ethical and
moral considerations of participants. It should be noted that we are only
analyzing textual representations of moral values, and our analysis does not
represent the moral values of people engaging in these discussions.

Moral Principle — Def. |  As Exhibited in Issue Threads
---|---
Care/harm — Protecting versus hurting others |  Care: Developer communications exhibiting kindness, e.g., experienced contributors shielding newcomers from criticism. Harm: Developers exchanging insults in issue threads, either in response to feature requests or bug fixes, or due to trolling or experience from past interactions (Miller et al., 2022)).
Fairness/cheating — Cooperation/ trust/ just versus cheating in interaction with objects and people |  Fairness: Developers and contributors respecting one another’s rights (e.g., having the right to open threads and ask for features/fixes, get the help they need). Cheating: Disregarding developers’ rights by imposing unrealistic expectations (e.g., unfeasible project timeline), or by discriminating against them (e.g., based on gender) and not addressing their concerns. (van Breukelen et al., 2023).
Loyalty/betrayal — Ingroup commitment (to coalitions, teams, brands) versus leaving group |  Loyalty: Acts of self-sacrifice and altruism for the community, ranging from dedicating time to participating in issues threads to defending developers and the project itself against criticisms. Betrayal: OSS communities, like all social communities, view acts of betrayal as a breach of community morale. Betrayal is observed when developers urge others to abandon a project, or exclude them from the community.
Authority/subversion — Adhering to the rules of hierarchy versus challenging hierarchies |  Authority: OSS communities often have guidelines that must be followed, e.g., Code of Conduct: Authorities enforcing CoC or any admin-privileged acts (minimizing comments, closing issues, etc.) to ban or censor the users. Subversion: Developers trying to rebel against authority, and questioning contributors’ ability to lead the community.
Sanctity/degradation — Behavioral immune system versus spontaneous reaction |  Sanctity: Communicating with temperance and respect for one another, and encouraging others to do so as well. Degradation: Developers expressing their hatred (disgust) toward a certain package, system, or code in the project. This behavior is targeted at code (objects) rather than people.

Table 1. Moral Principles in GitHub Heated Issues

Dataset. We analyze a benchmark dataset of 100 toxic GitHub issue comment
threads curated by Miller et al. (Miller et al., 2022), in which toxicity is
defined as ”rude, disrespectful, or unreasonable language that is likely to
make someone leave a discussion”. The dataset includes information on the
issue link, author, trigger, target, and the nature of toxicity for each
thread of comments. Out of the 100 issue threads, 20 contain comments that
were either removed or the project along with its issues were taken down from
GitHub. The 5 categories of toxic instances in this dataset are: Insulting,
Entitled, Arrogant, Trolling, and Unprofessional.

Procedure. To understand the characteristics and patterns of morality in the
context of SE, we qualitatively analyze 100 issue threads and their context,
using thematic analysis (Braun and Clarke, 2006), and following the
trustworthiness criteria (credibility, transferability, dependability, and
confirmability) as recommended by previous research (Lincoln and Guba, 5 04;
Nowell et al., 0 03). The data analysis process was conducted iteratively and
reflectively, following recommendations from qualitative analysis (Lewis,
2015). To enhance our understanding of the developers’ interactions and the
contextual factors influencing their communication, we first carefully read
through all the issue comment threads. Next, the goal was to associate moral
values (as defined in Section 2) to each comment in the threads, if
applicable. Given the substantial differences between OSS communications and
other social media platforms such as Twitter and Reddit, we adapted the
original definitions of morality to align them with the context of software
engineering. Specifically, we identify key concepts of each morality type,
observe patterns in the dataset, and use them to create the mapping of
morality categories. Throughout the analysis, careful notes were taken to
capture patterns of morality observed in the comments. This process was
repeated multiple times, with constant additions to the notes and
observations, and ongoing reevaluation of the mappings in each iteration.

Note that in the process of labeling morality to the issue comments, we
consciously disregard the toxicity labels to mitigate bias in our decisions,
so that we do not associate a specific type of toxicity with a certain type of
morality. The result of our mapping between the morality categories and their
manifestation in GitHub heated issues is presented in Table 1.

##  4\. Preliminary Observations

In this section, we discuss our observations for the moral principles as
exhibited in the GitHub heated issues. We analyze the issue comment threads in
our data sample to identify instances where moral principles were expressed in
the text. Out of 695 issue comments across 100 threads, 135 exhibit at least
one form of moral principle. In this analysis, people who open the issue
threads to find fixes for problems are referred to as users, and people who
try to resolve the issues and close the threads as contributors. We observed
unique patterns specific to each of the identified roles.

Care/Harm. A total of 48 comments exhibit this principle. Pertaining to this
category, following are the behaviors we observed: (CH1) users who use
derogatory language or insults towards contributors while seeking assistance
or requesting new features; (CH2) users insulting contributors mainly by
trolling, possibly due to differences in ideology, opinions, etc.; (CH3)
contributors responding using insults toward users.

The first type (CH1) was the most common in our dataset (32 out of 48
instances). For example, in one thread, a user opens an issue saying that the
name of the project is duplicated and it should change while using insults:
”This name is in use and should be changed immediately. Yes, I have seen the
other issues. Yes, I am opening a new one because f**k you Microsoft. You are
merely trying to cast a shadow on other, truly open source projects…”.

The second type (CH2) was also seen in a number of cases (5 out of 48
instances). For example, in one instance, a user opens an issue saying:
”Revenue. F**k you guys”, and does not add anything else to the thread.
Threads similar to this are mostly the result of trolling, but they can also
be the result of differences in ideology or based on previous interactions
(Miller et al., 2022).

For the third type (CH3), we saw an interesting pattern of contributors
responding in a detrimental manner towards the users. This type is rare
compared to other types in this category. In one instance, a user opens an
issue asking: ”Where to f**k python2? Why, when I give the brew install python
command, python3 is installed, not python2. Developers are you stoned there?”,
and surprisingly, one of the contributors responds with: ”I recommend not
having intercourse with EOL software. Python 2 is no longer included in
Homebrew”. The user then responds: ”You have an unfinished raw product with
some problems. What can be your attitude? Treating users like sh*t”. This
issue thread exhibits the first and last patterns (CH1 and CH3).

Fairness/Cheating. We observed behaviors associated with this category in a
total of 19 comments, which include: (FC1) users having unrealistic
expectations from the contributors; (FC2) contributors failing to address
user’s issues due to unjustified reasons.

Among the instances of this category, 7 out of 19 instances exhibit the first
type of communication pattern. In one of the instances representing the first
type (FC1), the user has opened an issue asking about a problem in the
debugger. Contributors of the projects try to help the user by providing tips
on how the problem can be solved, but as the conversation goes on, the user
gets frustrated and says: ”@¡USER1¿ @¡USER2¿ Too busy to respond?”, which in
the end, leads to the issue thread being locked. This is an example of
unrealistic expectations about response time causing conflict.

For the second type (FC2), for instance, this is how one of the contributors
responds to a user’s issue: ”How would be this a priority, the app must be
ready to be shipped in a few days and basically, no one uses the smartphone in
landscape mode except for watching media”.

Another type of behavior related to Fairness would be discrimination (based on
race, culture, gender, religion, etc). In our dataset, we did not observe any
instances of discrimination. Analysis of additional data may reveal
discriminatory behavior in OSS, as such occurrences have been observed in
previous studies (van Breukelen et al., 2023; Cheriyan et al., 6 21).

Loyalty/Betrayal. The observed patterns for this category of morality were
found in 23 instances of the dataset: (LB1) users actively promoting and
inciting rebellion against a project, encouraging others to switch to an
alternative project (most common with 15 instances); (LB2:) contributors
excluding users from the project.

To better understand these behaviors, we provide examples for each type. For
instance, in one of the issue threads, a user claims that SSL Insecure is not
being respected in the project: ”Whoever will find this issue and gets pissed
off, because the author doesn’t bother to fix it for years, ditch mitmproxy
and use SSL SPLIT”. One of the developers responds with: ”I’m glad you have
found something that works for you!”. In this instance, both LB1 and LB2 were
observed, because the user is encouraging people to leave this project behind,
and the contributor also encourages the user to not use the project. In
another instance, a user requests the removal of the slur filter in the app,
however, one of the contributors says: ”If you don’t like it, fork it. Stop
bothering us about it, we will never fully remove the slur filter”, which
counts as excluding the user from the community.

Figure 1. Morality and Toxicity: Examples of their Co-occurrence as Exhibited
in GitHub

Authority/Subversion. OSS communities, like any other community, often have
specific rules and guidelines that everyone has to follow. GitHub projects use
the Code of Conduct (CoC) in an attempt to promote their expectations and
standards of ethical behavior within the community (Tourani et al., 7 02). The
establishment of such rules and hierarchies often leads to the dynamic
interplay of moralities such as authority and subversion in OSS
communications. A total of 30 instances with this principle were found in our
dataset. The patterns observed for this category are: (AS1) users trying to
rebel against authority, and questioning developers’ ability to lead the
community. (AS2) contributors (authorities) enforcing Code of Conduct when
thought necessary or any other admin-privileged acts to ban, censor, or
silence users (most common with 16 instances).

In one of the instances, a user has opened an issue, claiming that the project
has stolen his code: ”Don’t be a d*ck as the source code was stolen at that
point and implemented on this”. The contributor enforces authority by saying:
”please be civilized and refrain from profanities as required by our Code of
Conduct, or I’ll ban you from all Falconry projects. This is the last
warning”. And the user challenges the authority by saying: ”I’m not a bot but
if you wish to ignore me go ahead and I’ll get a lawyer”. Additionally, there
were instances where the users questioned the capabilities of contributors.
For example, a user opened an issue asking about how to do a certain thing
within the system. As the conversation goes on, the user responds to one of
the contributors: ”…I would hate to have you in charge of any security
issues”.

Sanctity/Degradation. The most commonly observed pattern in this category, in
a total of 42 instances, was observed when: (SD1) users express their hate
(disgust) toward a certain package, system, or code that the project is using,
most likely due to their personal preferences. This principle is targeted at
the code/system rather than people (targeting people represents Care/Harm).

For example, a user opened an issue in the project expressing how much the
document is poorly written, and in a sense, he is disgusted with it: ”…in
fact, you write a sh*t doc. I’m a real man, it’s my feeling of your holy sh*t
doc”. Or in another instance, a user opens an issue explaining his problem
with the software: ”…I just tried reinstalling you buggy, sh*tty software for
the third time…”.

After gathering our observations of the exhibited patterns of morality in
GitHub communications, we associated the toxicities presented in our dataset
with the moral principles in each issue thread. The results generated for this
mapping are shown in Figure 1. Based on this mapping, we observe that every
type of toxicity is associated with at least one form of morality. Entitlement
is mostly associated with the principle of Fairness/Cheating, Trolling with
Sanctity/Degradation, Arrogant to Loyalty/Betrayal, Unprofessional with
Sanctity/Degradation, and Insulting with Care/Harm.

The relationship between the concepts of toxicity and moral principles can be
observed in their respective definitions. For example, entitlement comes from
disregarding people’s rights for personal gain. This aligns with the
Fairness/Cheating definitions given in previous sections. The relationship
between Authority/Subversion and insulting threads implies that using insults
causes the authorities to intervene more in the issues compared to the other
types of toxicity. The moralities most frequently observed in toxic threads
are Sanctity/Degradation and Care/Harm, which were primarily expressed through
insults directed towards code or individuals.

##  5\. Implications and Challenges

We found that toxic behaviors can be categorized and linked to various types
of moral principles, indicating the potential for utilizing moral principles
to detect toxicity in OSS communications. To this end, numerous Moral
Foundations Dictionaries (MFD) can be integrated into models to detect moral
values in SE texts, as demonstrated in other fields (Rezapour et al., 2019;
Kennedy et al., 2022). Recent studies have explored the association between
moral values and the sense of belonging in OSS communities (Trinkenreich et
al., 2 22). Adapting existing morality dictionaries to SE and incorporating
them into detection tools could facilitate a better understanding of several
human values in OSS, including toxicity. It is worth noting that our study
only examined the association between moral principles and toxicity in OSS
communities, and did not explore any potential causal relationships between
the two. While our findings suggest that moral principles are associated with
toxic interactions, it is possible that other factors may also contribute to
toxic behavior in OSS communities.

We acknowledge that inferring human values based on textual communication data
has its perils. For instance, previous studies show high error rate in
detecting developer personalities based on textual data (van Mil et al., 7 15;
Calefato and Lanubile, 7 31). We emphasize that we are only analyzing the
textual representation of moral principles, and we are not associating
morality values with the people involved in the discussions. Our findings of
the exhibited moralities and their association with toxicity are limited to a
relatively small dataset of GitHub issue threads. Further analysis on a large
and diverse dataset is necessary to generalize the findings. Detecting
morality even in social contexts (e.g., Twitter) is a hard task, mostly in
distinguishing co-occurring moral principles (Hoover et al., 1 01). We
attempted to address that challenge by mapping and creating clear definitions
of moral principles in the domain of SE (Table 1). Another important point to
consider is the potential impact of cultural and social differences on the
interpretation and expression of moral principles in OSS communities. Our
study focused on English texts, and it is possible that different cultures and
languages may have different conceptions and expressions of moral principles.
Therefore, future research should aim to investigate the role of cultural and
linguistic factors in shaping the expression and interpretation of moral
principles in OSS communities.

##  6\. Conclusion

This paper investigates moral principles and their relation with toxicity in
OSS. Due to the use of domain-specific words and jargon, the majority of
toxicity detection tools do not correctly identify toxic language in OSS.
Therefore, there is a need to augment the current methods with informative
features, and to gain a comprehensive understanding of the triggers of
toxicity and its underlying causes. In this work, we leverage a toxicity-
labeled dataset of GitHub to analyze the key concepts of five moral principles
and identify exhibited patterns in the dataset. We also mapped the nature of
toxicity to the observed types of moral principles in each thread. Our
findings revealed that Sanctity/Degradation and Care/Harm were the most
frequently observed moral principles, and were often associated with insulting
threads. Overall, our study represents an initial step towards a deeper
understanding of the role of moral principles in shaping community dynamics in
OSS, and we hope that it will inspire further research in this important area.
Consistent with Miller et al.’s (Miller et al., 2022) data distribution
policy, we will share our annotated dataset upon request for research
purposes.

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