# Dark Matter and $(g-2)_{\mu,e}$ in radiative Dirac neutrino mass models

Talal Ahmed Chowdhury<EMAIL_ADDRESS>Department of Physics, University of
Dhaka, P.O. Box 1000, Dhaka, Bangladesh  The Abdus Salam International Centre
for Theoretical Physics, Strada Costiera 11, I-34014, Trieste, Italy  Md.
Ehsanuzzaman<EMAIL_ADDRESS>Department of Physics, University of
Dhaka, P.O. Box 1000, Dhaka, Bangladesh  Shaikh Saad<EMAIL_ADDRESS>Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056
Basel, Switzerland

###### Abstract

The origin of neutrino mass is a mystery, so is its nature, namely, whether
neutrinos are Dirac or Majorana particles. On top of that, hints of large
deviations of the muon and the electron anomalous magnetic moments (AMMs) are
strong evidence for physics beyond the Standard Model. In this work, piecing
these puzzles together, we propose a class of radiative Dirac neutrino mass
models to reconcile $(g-2)_{\mu,e}$ anomalies with neutrino oscillation data.
In this framework, a common set of new physics (NP) states run through the
loops that generate non-zero neutrino mass and, due to chiral enhancement,
provide substantial NP contributions to lepton AMMs. In addition, one of the
three models studied in this work offers a Dark Matter candidate automatically
stabilized by the residual symmetry, whose phenomenology is non-trivially
connected to the other two puzzles mentioned above. Finally, our detailed
numerical analysis reveals a successful resolution to these mysteries while
being consistent with all colliders and cosmological constraints.

##  I Introduction

The Standard Model (SM) of particle physics is the most successful theory in
particle physics that describes the fundamental interactions between
elementary particles. Despite its major triumph, it is not perfect- it cannot
explain the origin of neutrino mass or dark matter (DM). Moreover, the SM is
under scrutiny since its predicted values of the muon and the electron
anomalous magnetic moments111AMM is defined as $a_{\ell}=(g_{\ell}-2)/2$,
where $\ell=e,\mu,\tau$. (AMMs) are in tension with experimental measurements.
There is a longstanding discrepancy in the muon AMM measured at BNL in 2006
Bennett:2006fi . A new measurement performed at the Fermilab Abi:2021gix  in
2021 is in excellent agreement with BNL’s result, and combinedly they
correspond to a large $4.2\sigma$ disagreement with the SM prediction
Aoyama:2020ynm  (for original works, see Refs. Aoyama:2012wk ; Aoyama:2019ryr
; Czarnecki:2002nt ; Gnendiger:2013pva ; Davier:2017zfy ; Keshavarzi:2018mgv ;
Colangelo:2018mtw ; Hoferichter:2019mqg ; Davier:2019can ; Keshavarzi:2019abf
; Kurz:2014wya ; Melnikov:2003xd ; Masjuan:2017tvw ; Colangelo:2017fiz ;
Hoferichter:2018kwz ; Gerardin:2019vio ; Bijnens:2019ghy ; Colangelo:2019uex ;
Blum:2019ugy ; Colangelo:2014qya ):

$\displaystyle\Delta a_{\mu}=(2.51\pm 0.59)\times 10^{-9}.$ (1)

On the other hand, precise measurement of the fine-structure constant using
Cesium atom at the Berkeley National Laboratory Parker:2018vye  in 2018
yields,

$\displaystyle\alpha^{-1}(C_{s})=137.035999046(27).$ (2)

This result corresponds to a negative $2.4\sigma$ deviation of the electron
AMM with respect to the SM value Aoyama:2017uqe :

$\displaystyle\Delta a_{e}=(-8.8\pm 3.6)\times 10^{-13}.$ (3)

These discrepancies are large in magnitude, and the opposite sign between them
is somewhat puzzling and hints towards physics beyond the SM (BSM). For
attempts to solve these discrepancies simultaneously in BSM frameworks, see,
e.g., Refs. Giudice:2012ms ; Davoudiasl:2018fbb ; Crivellin:2018qmi ;
Liu:2018xkx ; Dutta:2018fge ; Han:2018znu ; Crivellin:2019mvj ; Endo:2019bcj ;
Abdullah:2019ofw ; Bauer:2019gfk ; Badziak:2019gaf ; Hiller:2019mou ;
CarcamoHernandez:2019ydc ; Cornella:2019uxs ; Endo:2020mev ;
CarcamoHernandez:2020pxw ; Haba:2020gkr ; Bigaran:2020jil ; Jana:2020pxx ;
Calibbi:2020emz ; Chen:2020jvl ; Yang:2020bmh ; Hati:2020fzp ; Dutta:2020scq ;
Botella:2020xzf ; Chen:2020tfr ; Dorsner:2020aaz ; Arbelaez:2020rbq ;
Jana:2020joi ; Chua:2020dya ; Chun:2020uzw ; Li:2020dbg ; DelleRose:2020oaa ;
Kowalska:2020zve ; Hernandez:2021tii ; Bodas:2021fsy ; Cao:2021lmj ;
Mondal:2021vou ; CarcamoHernandez:2021iat ; Han:2021gfu ; Escribano:2021css ;
CarcamoHernandez:2021qhf ; Chang:2021axw ; Chowdhury:2021tnm ;
Bharadwaj:2021tgp ; Borah:2021khc ; Bigaran:2021kmn ;
PadmanabhanKovilakam:2022FN ; Li:2021wzv ; Biswas:2021dan ; Julio:2022ton ;
Julio:2022bue .

The observation of neutrino oscillations Super-Kamiokande:1998kpq ; Super-
Kamiokande:2001ljr ; SNO:2002tuh ; KamLAND:2002uet ; KamLAND:2004mhv ;
K2K:2002icj ; MINOS:2006foh  was the first conclusive evidence that the SM is
incomplete and must be extended. Although the existence of non-zero neutrino
masses222For an extensive review on this subject, see Ref. Cai:2017jrq . has
been firmly established, the nature of neutrinos, viz. Dirac or Majorana is
still unknown. As widely known, observation of neutrinoless double beta decay
(see, e.g., Ref. Dolinski:2019nrj ) would settle this issue and establish the
Majorana nature of neutrinos; however, all experiments so far have null
results. Similarly, despite the discovery that about eighty percent of the
Universe’s gravitating matter is non-luminous, we are yet to know anything
about the nature of DM (see, e.g., Young:2016ala ).

Figure 1: Schematic diagram demonstrating the link between neutrino mass
generation, lepton $g-2$ anomalies, and Dark Matter in one of the benchmark
models (Model-A) proposed in this work. See text for details.

This work considers neutrinos as Dirac particles, and non-zero neutrino masses
originate from quantum corrections. This framework proposes a simultaneous
solution to the muon and the electron AMMs that is non-trivially linked to the
neutrino mass generation mechanism. New physics (NP) contributions to the
lepton $g-2$ and non-zero neutrino masses arise via one-loop corrections
mediated by a common set of BSM particles. We present a class of radiative
Dirac neutrino mass models that share these same features and study in detail
a particular model (dubbed as Model-A) belonging to this class that also
addresses the DM puzzle and offers rich collider phenomenology. Remarkably,
the stability of the DM is guaranteed by the residual symmetry that emerges
after $U(1)_{B-L}$ gauge symmetry is spontaneously broken. In Model-A, all
these puzzles are deeply intertwined, which is illustrated in Fig. 1.

The paper is organized as follows: in Sec. II, we discuss the general
framework, and in Sec. III, we provide details of the set of models under
investigation. Next, we discuss the experimental constraints in Sec. IV and
carry out a detailed DM phenomenology for Model-A in Sec. V. Finally, we
conclude in Sec. VI.

##  II Setup

Neutrinos being Dirac in nature requires the presence of right-handed partners
$\nu_{R_{i}}$ ($i=1-3$), which automatically allows for tree-level neutrino
mass via the term $-\mathcal{L}_{y}\supset Y_{\nu}\overline{L}\epsilon
H^{\ast}\nu_{R}$ when the SM Higgs, $H$ acquires a vacuum expectation value
(VEV); here $L$ is the SM lepton doublet, and $\epsilon$ is the Levi-Civita
tensor. This, however, demands $y_{\nu}\sim\mathcal{O}(10^{-11})$ to be
consistent with experimental data, which is seemingly unnatural tHooft:1979rat
since the Yukawa couplings of the charged fermions in the SM are typically in
the range $10^{-6}-1$. On the contrary, it is aesthetically attractive to
generate Dirac neutrino mass radiatively that would naturally require the
corresponding Yukawa couplings typically in the range $10^{-3}-1$. Symmetry
arguments can naturally forbid the aforementioned tree-level term to achieve
this. In this work, we accomplish this by extending the SM gauge symmetry by
$U(1)_{B-L}$ Davidson:1978pm ; Mohapatra:1980qe ; in the literature, various
types of symmetries are imposed to realize radiative Dirac mass, see, e.g.,
Refs. Mohapatra:1987hh ; Mohapatra:1987nx ; Balakrishna:1988bn ; Branco:1978bz
; Babu:1988yq ; Gu:2007ug ; Farzan:2012sa ; Okada:2014vla ; Ma:2016mwh ;
Bonilla:2016diq ; Wang:2016lve ; Ma:2017kgb ; Yao:2017vtm ; Wang:2017mcy ;
Helo:2018bgb ; Reig:2018mdk ; Han:2018zcn ; Kang:2018lyy ; Yao:2018ekp ;
Calle:2018ovc ; CentellesChulia:2018gwr ; Bonilla:2018ynb ; Calle:2018ovc ;
Carvajal:2018ohk ; CentellesChulia:2018bkz ; Ma:2019yfo ; Bolton:2019bou ;
Saad:2019bqf ; Bonilla:2019hfb ; Dasgupta:2019rmf ; CentellesChulia:2019gic ;
CentellesChulia:2019xky ; Jana:2019mez ; Enomoto:2019mzl ; Ma:2019byo ;
Restrepo:2019soi ; Jana:2019mgj ; Nanda:2019nqy ; Wang:2020dbp ;
Borgohain:2020csn ; Mahanta:2021plx ; Bernal:2021ezl ; Biswas:2021kio ;
Calle:2021tez ; De:2021crr ; Bernal:2021ppq ; Mishra:2021ilq .

Gauge anomaly cancellation conditions333As usual, quark fields
$Q_{L}(3,2,1/6),u_{R}(3,1,2/3),d_{R}(3,1,-1/3)$ carry $1/3$ and SM leptons
$L_{L}(1,2,-1/2),\ell_{R}(1,1,-1)$ carry $-1$ charges under $U(1)_{B-L}$
symmetry. The SM Higgs doublet $H(1,2,1/2)$ transforms trivially under
$U(1)_{B-L}$. then require the right-handed neutrinos to carry charges which
are either $\nu_{R_{1,2,3}}=\\{-1,-1,-1\\}$ or $\nu_{R_{1,2,3}}=\\{5,-4,-4\\}$
Montero:2007cd ; Machado:2010ui ; Machado:2013oza . We choose the latter
charge assignment since the former allows the unwanted tree-level term in the
Lagrangian. To spontaneously break $U(1)_{B-L}$, we employ a SM singlet scalar
that carries three units of $B-L$ charge: $\sigma\sim(1,1,0,3)$444Quantum
number presented here is under the gauge group $\textrm{SM}\times
U(1)_{B-L}=SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}\times U(1)_{B-L}$.. Then
non-zero mass for the neutrinos appears through loop diagrams when both the
electroweak (EW) and $U(1)_{B-L}$ symmetries are broken (in our setup, the
only two fields that acquire VEVs are $H$ and $\sigma$). These loop diagrams
originate from ultra-violate (UV) completion of the following unique dimension
five operator:

$\displaystyle-\mathcal{L}_{d=5}=\frac{y_{ij}}{\Lambda}\;\overline{L}_{i}\epsilon
H^{\ast}{\nu_{R}}_{j}\sigma+h.c.,$ (4)

where $i=1-3$ and $j=2,3$. In the following, we very briefly summarize how to
construct UV-complete one-loop models utilizing this $d=5$ operator; for
details, we refer the reader directly to Ref. Jana:2019mgj  (we adopt the
nomenclature used therein).

Figure 2: Two distinct one-loop topologies that lead to Dirac neutrino mass.
See text for details.

One-loop Dirac neutrino mass models can be constructed out of two independent
topologies: T1-i and T1-ii that are shown in Fig. 2. Depending on the Lorentz
structures (i.e, fermion-fermion-scalar or scalar-scalar-scalar interaction)
associated with these vertices, three different diagrams (T1-i-1, T1-i-2, and
T1-i-3) can be drawn for topology T1-i. On the other hand, for T1-ii, there is
a unique diagram labeled as T1-ii-1. Moreover, by interchanging the external
scalar legs of some of these diagrams, in total, eight minimal models can be
fabricated Jana:2019mgj .

In this work, we focus555Conclusions obtained in our work are very general and
applicable to most of the models fabricated from topology T1-i. on the diagram
T1-ii-1 and propose explicit models in light of the muon and the electron
$g-2$. In particular, we formulate three minimal models (labeled as Model-A,
Model-B, Model-C), each of which, in addition to correctly reproducing
neutrino oscillation data, addresses both $(g-2)_{\mu,e}$. Among these three
models, DM can also be incorporated within Model-A. Moreover, this model shows
profound correlations among the neutrino mass, DM, and $(g-2)_{\mu,e}$ as well
as offers rich collider phenomenology. Stunningly, no ad hoc symmetry needs to
be imposed by hand to realize this dark matter; instead, its stability is
assured by a leftover discrete symmetry resulting from the breaking of
$U(1)_{B-L}$ gauge symmetry.

Since topology T1-i in Fig. 2 has a 4-particle vertex, there is a unique
choice of attaching external $H$ and $\sigma$ lines to it to form the $d=5$
operator of Eq. (4). Completion of the neutrino mass diagram then requires (i)
a vector-like666Due to vector-like nature, these fermions do not alter the
anomaly cancellation conditions. Dirac fermion $\psi$ (of three generations)
and (ii) two distinct BSM scalars $\Phi_{1,2}$; see the top diagram in Fig. 3.
With only these new states, corrections to muon and electron AMMs are too
small to be consistent with experimental findings. However, large NP
contributions to lepton $g-2$ can naturally arise within this setup via
chirally enhanced terms that are proportional to vector-like fermion mass by
introducing (iii) the third scalar $\Phi_{3}$; see the bottom diagram in Fig.
3.

Figure 3:  In this class of models, neutrino mass (lepton $g-2$) originates
from the diagram on the top (bottom). The blue line shows the direction of the
iso-doublet flow in an economical fashion. Any other direction of this flow
would correspond to non-minimal model, see text for details.

In Fig. 3, blue lines correspond to the iso-doublet flow drawn to indicate the
most economical way to build this class of models. As shown in Fig. 3, this
choice requires a single BSM scalar field to be iso-doublet (and no BSM iso-
doublet fermion is required). This way, the least number of new degrees of
freedom is introduced in a given model belonging to this class (this is our
minimality criterion). Minimal models of this class then contain four BSM
fields that propagate inside the loops and contribute to neutrino mass as well
as lepton $g-2$, and have the following quantum numbers:

$\displaystyle\psi\sim(1,1,Y,\beta),$ (5)
$\displaystyle\Phi_{1}\sim(1,2,-Y-\frac{1}{2},-\beta-1),$ (6)
$\displaystyle\Phi_{2}\sim(1,1,Y,\beta+4),$ (7)
$\displaystyle\Phi_{3}\sim(1,1,Y+1,\beta+1),$ (8)

where, $Y$ and $\beta$ are the hypercharge and the $B-L$ charge, respectively,
carried by the vector-like fermion. It is important to note that: (a) if
$Y=-1$ then $\beta=2$ is not allowed. In this case, a cubic term of the form
$\mathcal{V}\supset H^{\dagger}\Phi_{1}\sigma$ is allowed, which would lead to
an induced VEV for $\Phi_{1}$ resulting in a tree-level Dirac mass for the
neutrinos via
$\mathcal{L}_{y}\supset\overline{L}\epsilon\Phi_{1}^{\ast}\nu_{R}$.
Furthermore, (b) if $Y=0$ then $\beta=-1$ is not allowed. In this scenario, a
combination of three terms $\mathcal{L}_{y}\supset\overline{L}\epsilon
H^{\ast}\psi_{R}$, $\overline{\psi}_{L}\sigma\nu_{R}$, and
$m_{\psi}\overline{\psi}_{L}\psi_{R}$ in the Lagrangian would generate
neutrino mass via tree-level Dirac seesaw (once VEVs of $H$ and $\sigma$ are
inserted).

##  III Models

This section discusses three different versions of models belonging to the
class introduced in the previous section. For simplicity, we restrict
ourselves to the case of $|Y|\leq 1$. We label these models as Model-A
($Y=0$), Model-B ($Y=-1$), and Model-C ($Y=+1$) for which the full quantum
numbers of NP states are specified in Table 1. In the following text, we
provide all the necessary details of these models.

Fields | Model-A ($Y=0,\beta\neq-1$) | Model-B ($Y=-1,\beta\neq 2$) | Model-C ($Y=+1$)
---|---|---|---
$\psi$ | $\psi^{0}(1,1,0,\beta)$ | $\psi^{-}(1,1,-1,\beta)$ | $\psi^{+}(1,1,+1,\beta)$
$\Phi_{1}$ | $\phi=\begin{pmatrix}\phi^{0}\\\ \phi^{-}\end{pmatrix}=(1,2,-\frac{1}{2},-\beta-1)$ | $\phi=\begin{pmatrix}\phi^{+}\\\ \phi^{0}\end{pmatrix}=(1,2,+\frac{1}{2},-\beta-1)$ | $\phi=\begin{pmatrix}\phi^{-}\\\ \phi^{--}\end{pmatrix}=(1,2,-\frac{3}{2},\beta-1)$
$\Phi_{2}$ | $S^{0}(1,1,0,\beta+4)$ | $\eta^{-}(1,1,-1,\beta+4)$ | $\eta^{+}(1,1,+1,\beta+4)$
$\Phi_{3}$ | $\eta^{+}(1,1,+1,\beta+1)$ | $S^{0}(1,1,0,\beta+1)$ | $\kappa^{++}(1,1,+2,\beta+1)$
DM? | $\psi^{0}$ | ✗ | ✗

Table 1:  Particle contents of new physics models proposed in this work.

Yukawa interactions:– In three of these models, the new Yukawa part of the
Lagrangian takes the following general form:

$\displaystyle-\mathcal{L}_{Y}$
$\displaystyle=Y^{iJ}_{1}\overline{L}_{L_{i}}\Phi_{1}\psi_{R_{J}}+Y^{Ij}_{2}\overline{\psi}_{L_{I}}\Phi_{2}\nu_{R_{j\neq
3}}$
$\displaystyle+Y^{Ij}_{3}\overline{\psi}_{L_{I}}\Phi_{3}\ell_{R_{j}}+M^{IJ}_{F}\overline{\psi}_{L_{I}}\psi_{R_{J}}.$
(9)

Here $Y_{1,2,3}$ are in general $3\times 3$ arbitrary matrices, and we define
their entries by,

$\displaystyle Y_{1}^{ij}=y_{ij},\;Y_{2}^{ij}=x_{ij},\;Y_{3}^{ij}=z_{ij}.$
(10)

Without loss of generality, we choose to work in a basis where the vector-like
fermion mass matrix is diagonal,

$\displaystyle M_{F}=diag(M_{1},M_{2},M_{3}).$ (11)

From Fig. 3, it can be seen that $Y_{1}$ and $Y_{2}$ are responsible for
neutrino mass generation, whereas, $Y_{1}$ and $Y_{3}$ provide NP
contributions to the lepton $g-2$ that are chirally enhanced. For sizable
Yukawa couplings, lepton flavor violating (LFV) processes provide stringent
constraints on the off-diagonal couplings of these matrices and force them to
take almost diagonal form. To be consistent with the experimental data of
$(g-2)_{e,\mu}$, entries of $Y_{1}$ and $Y_{3}$ coupling matrices are required
to be substantial; hence to suppress LVF, we adopt diagonal textures for these
two matrices. On the other hand, entries of $Y_{2}$ are required to be
somewhat smaller to incorporate correct neutrino mass scale. Hence, for the
rest of the analysis, the Yukawa coupling matrices are chosen to have the
following form:

$\displaystyle
Y_{1}=\scalebox{0.9}{\mbox{$\displaystyle\begin{pmatrix}y_{1}&0&0\\\
0&y_{2}&0\\\
0&0&y_{3}\end{pmatrix}$}},Y_{2}=\scalebox{0.9}{\mbox{$\displaystyle\begin{pmatrix}0&x_{12}&x_{13}\\\
0&x_{22}&x_{23}\\\
0&x_{32}&x_{33}\end{pmatrix}$}},Y_{3}=\scalebox{0.9}{\mbox{$\displaystyle\begin{pmatrix}z_{1}&0&0\\\
0&z_{2}&0\\\ 0&0&z_{3}\end{pmatrix}$}}.$ (12)

Note that, due to different $B-L$ charge assignments of the right-handed
neutrinos, the first column of $Y_{2}$ is zero. For the simplicity of our
work, we treat all couplings to be real.

Scalar interactions:– Owing to the $B-L$ charge assignments, the scalar
potential of this theory takes a simple form. Instead of writing the entire
potential, we only provide the relevant interactions required to generate
neutrino mass as well as lepton AMMs,

$\displaystyle-V_{\nu}$ $\displaystyle\supset\lambda
H\epsilon\Phi_{1}\Phi_{2}\sigma^{\ast}+\mu
H^{\dagger}\Phi_{1}\Phi_{3}+h.c.\;,$ (13)

here, $\epsilon$ is the Levi-Civita tensor. Since the SM Higgs doublet
transforms trivially under $B-L$, it does not mix with the BSM scalars. Its
VEV $\langle H\rangle=v_{H}/\sqrt{2}$ (where $v_{H}=246$ GeV) breaks the
electroweak (EW) symmetry, while VEV of $\sigma$ field
$\langle\sigma\rangle=v_{\sigma}/\sqrt{2}$ breaks the $U(1)_{B-L}$ symmetry.
As a result of these spontaneous symmetry breakings, BSM neutral (charged)
scalars originating from $\Phi_{1,2,3}$ mix as can be seen from Eq. (13). Then
the corresponding mass-squared matrices can be written as,

$\displaystyle V$
$\displaystyle\supset\begin{pmatrix}\phi^{0\ast}&S^{0}\end{pmatrix}\begin{pmatrix}m^{2}_{\phi}&a_{0}\\\
a_{0}&m^{2}_{S}\end{pmatrix}\begin{pmatrix}\phi^{0}\\\ S^{0\ast}\end{pmatrix}$
$\displaystyle+\begin{pmatrix}\phi^{+}&\eta^{+}\end{pmatrix}\begin{pmatrix}m^{2}_{\phi}&a_{+}\\\
a_{+}&m^{2}_{\eta}\end{pmatrix}\begin{pmatrix}\phi^{-}\\\
\eta^{-}\end{pmatrix}$
$\displaystyle+\begin{pmatrix}\phi^{++}&\kappa^{++}\end{pmatrix}\begin{pmatrix}m^{2}_{\phi}&a_{++}\\\
a_{++}&m^{2}_{\kappa}\end{pmatrix}\begin{pmatrix}\phi^{--}\\\
\kappa^{--}\end{pmatrix},$ (14)

where we have defined the following quantities:

Model-A: (no doubly charged scalar) $\displaystyle
a_{0}=-\frac{\lambda}{2}v_{H}v_{\sigma},\;a_{+}=\frac{\mu}{\sqrt{2}}v_{H},\;$
(15) Model-B: (no doubly charged scalar) $\displaystyle
a_{0}=\frac{\mu}{\sqrt{2}}v_{H},\;a_{+}=-\frac{\lambda}{2}v_{H}v_{\sigma},\;$
(16) Model-C: (no neutral scalar) $\displaystyle
a_{+}=-\frac{\lambda}{2}v_{H}v_{\sigma},\;a_{++}=\frac{\mu}{\sqrt{2}}v_{H}.$
(17)

Furthermore, we diagonalize these matrices as,

$\displaystyle
M^{2}_{x}=O_{x}\;\textrm{diag}\\{(M^{x}_{1})^{2},(M^{x}_{2})^{2}\\}\;O^{T}_{x},$
(18) $\displaystyle O_{x}=\begin{pmatrix}\cos\theta_{x}&-\sin\theta_{x}\\\
\sin\theta_{x}&\cos\theta_{x}\end{pmatrix},$ (19) $\displaystyle\sin
2\theta_{x}=\frac{2a_{x}}{(M^{x}_{1})^{2}-(M^{x}_{2})^{2}},$ (20)

where we use the notation: $x=\\{0,+,++\\}$ and $M_{1}>M_{2}$. We denote these
two mass eigenstates $S^{x}_{1,2}$ by (i) $S^{0}_{i}\equiv S_{i}$ for neutral,
(ii) $S^{+}_{i}\equiv\eta_{i}$ for singly-charged, and (iii)
$S^{++}_{i}\equiv\kappa_{i}$ for doubly-charged scalars. Explicitly, the
flavor and the mass eigenstates are related via the following identities:

neutral: $\displaystyle\phi^{0\ast}=c_{\theta_{0}}S_{1}-s_{\theta_{0}}S_{2},$
(21) $\displaystyle S^{0}=s_{\theta_{0}}S_{1}+c_{\theta_{0}}S_{2},$ (22)
singly-charged:
$\displaystyle\phi^{+}=c_{\theta_{+}}\eta_{1}-s_{\theta_{+}}\eta_{2},$ (23)
$\displaystyle\eta^{+}=s_{\theta_{+}}\eta_{1}+c_{\theta_{+}}\eta_{2},$ (24)
doubly-charged:
$\displaystyle\phi^{++}=c_{\theta_{++}}\kappa_{1}-s_{\theta_{++}}\kappa_{2},$
(25)
$\displaystyle\kappa^{++}=s_{\theta_{++}}\kappa_{1}+c_{\theta_{++}}\kappa_{2}.$
(26)

It is important to note that due to the simplified form of the scalar
potential, there is no mass splitting between $Re[\phi^{0}]$ and
$Im[\phi^{0}]$ components; this is why, neutral scalar cannot serve as a
viable DM candidate in Model-A and -B (Model-C does not contain any neutral
scalar within $\Phi_{1,2,3}$). Consequently, the only model that provides a DM
candidate is Model-A (see Sec. V for details), which is a Dirac fermion DM
(Model-B and Model-C do not contain electrically neutral BSM fermion).

Neutrino mass:– The leading contributions to neutrino masses in this theory
appear at the one-loop order, as shown in Fig. 3 (Feynman diagram on the top).
In this Feynman diagram, BSM neutral (singly-charged) scalars and fermions run
through the loop in Model-A (Model-B and Model-C). It is straightforward to
compute the neutrino mass formula, which is given by,

$\displaystyle M^{ij}_{\nu}$ $\displaystyle=\frac{\sin
2\theta_{x}}{8\pi^{2}}\left(Y^{ik}_{1}\right)^{\ast}M_{F_{k}}\left(Y^{kj}_{2}\right)^{\ast}$
$\displaystyle\times\bigg{\\{}\frac{M^{2}_{S^{x}_{2}}\log\left[\frac{M_{F_{k}}^{2}}{M^{2}_{S^{x}_{2}}}\right]}{M^{2}_{F_{k}}-M^{2}_{S^{x}_{2}}}-\left(S^{x}_{2}\to
S^{x}_{1}\right)\bigg{\\}}$ $\displaystyle\equiv
Y_{1}^{\ast}\hat{M}Y_{2}^{\ast},$ (27)

where,

$\displaystyle\hat{M}=\hat{M}_{i}\delta_{ij},$ (28)
$\displaystyle\hat{M}_{k}=\frac{\sin
2\theta_{x}}{8\pi^{2}}M_{F_{k}}\bigg{\\{}\frac{M^{2}_{S^{x}_{2}}\log\left[\frac{M_{F_{k}}^{2}}{M^{2}_{S^{x}_{2}}}\right]}{M^{2}_{F_{k}}-M^{2}_{S^{x}_{2}}}-\left(S^{x}_{2}\to
S^{x}_{1}\right)\bigg{\\}},$ (29)

and $x=0,+,+$ for Model-A, -B, -C, respectively. Since neutrinos are Dirac
particles, it is simple to solve for the Yukawa couplings $x_{ij}\in Y_{2}$ in
terms of neutrino observables that are known quantities and in terms of
couplings $y_{i}$ (to be determined from lepton $g-2$) as follows:

$\displaystyle x_{ij}=\frac{U_{ij}m_{j}}{y_{i}\hat{M}_{i}},$ (30)

here

$\displaystyle m_{1}=0,\;\;m_{2}=\sqrt{\Delta
m^{2}_{\textrm{sol}}},\;\;m_{3}=\sqrt{\Delta m^{2}_{\textrm{atm}}},$ (31)

and $U$ is the left-rotation matrix that diagonalizes the neutrino mass matrix
(recall that Dirac neutrino mass matrix is not symmetric), i.e.,

$\displaystyle
U^{\dagger}m_{\nu}m_{\nu}^{\dagger}U=\left(m^{\textrm{diag}}_{\nu}\right)^{2}.$
(32)

The solution given in Eq. (30) corresponds to normal mass ordering for
neutrinos. Analogously, the solution for inverted ordering can be trivially
constructed by relabelling the $B-L$ charges of the right-handed neutrinos,
which would correspond to the third column being zero in Eq. (12) instead of
the first column.

Note that, due to the non-universal charge assignments of the right-handed
neutrinos, one of them carrying five units of $B-L$ charge remains massless
(as well as the lightest SM neutrino). However, within this framework, non-
zero $m_{1}$ is generated via dimension-7 operator777An UV-completion of this
dimension-7 operator requires two more copies of $\Phi_{2}$-like fields:
$\Phi_{2}^{\prime}(1,1,Y,\beta-5)$ and
$\Phi_{2}^{\prime\prime}(1,1,Y,\beta-2)$. of the form
$\mathcal{L}_{7}\supset\overline{L}_{L}H^{\ast}\nu_{R}\sigma^{\ast}\sigma^{\ast}/\Lambda^{3}$.

Figure 4: Demonstration of the required Yukawa coupling to reproduce the
experimentally observed lepton AMMs within $1\sigma$ values are presented as a
function of vector-like fermion mass $M_{F}$. For the masses of two of the
scalars running in the loop, we take $M_{F}+100$ GeV and $M_{F}+150$ GeV,
respectively. Here, for illustration, we have only plotted the dominant
chiraly enhanced contribution; however, for numerical analysis, we have
consider the full expressions.

Lepton magnetic dipole moment:– The NP contributions to lepton AMMs in this
theory appear at the one-loop order, as shown in Fig. 3 (Feynman diagram on
the bottom). These contributions are typically large due to vector-like
fermion mass insertion in the loop. The outgoing photon in this Feynman
diagram is emitted either by an internal scalar or fermion, or by both scalar
and fermion, depending on the model. In Model-A, -B, and -C, scalars
(fermions) running in the loop are singly-charged (neutral), neutral (singly-
charged), and doubly-charged (singly-charged) states, respectively. It is
straightforward to evaluate the contribution arising from BSM states, which
yields,

$\displaystyle\Delta a_{\ell}$
$\displaystyle=-\frac{m_{\ell}}{4\pi^{2}}\bigg{\\{}Re\left(Y^{L\ast}_{kl}Y^{R}_{kl}\right)\frac{M_{k}}{M^{2}_{S^{x}_{b}}}G[r_{kb}]$
$\displaystyle+\left(|Y^{L}_{kl}|^{2}+|Y^{R}_{kl}|^{2}\right)\frac{m_{\ell}}{M^{2}_{S^{x}_{b}}}\widetilde{G}[r_{kb}]\bigg{\\}},$
(33)

where summation over the BSM scalars and fermions must be understood.
Furthermore, we have defined,

$\displaystyle r_{kb}=\frac{M^{2}_{k}}{M^{2}_{S^{x}_{b}}},$ (34)
$\displaystyle G[r]=f[r]+Q_{\psi}g[r],$ (35)
$\displaystyle\widetilde{G}[r]=\widetilde{f}[r]+Q_{\psi}\widetilde{g}[r],$
(36)

with $Q_{\psi}=0,-1,+1$ for Model-A, -B, -C, respectively, and,

$\displaystyle f[r]=2\widetilde{g}[r]=\frac{r^{2}-1-2r\ln r}{4(r-1)^{3}},$
(37) $\displaystyle g[r]=\frac{r-1-\ln r}{2(r-1)^{2}},$ (38)
$\displaystyle\widetilde{f}[r]=\frac{2r^{3}+3r^{2}-6r+1-6r^{2}\ln
r}{24(r-1)^{4}}.$ (39)

Finally, the re-defined Yukawa couplings $(Y^{\ast}_{L},Y_{R})$ appearing in
Eq. (33) are given by,

$\displaystyle\textrm{Model-A}:$
$\displaystyle\eta_{1}:(Y_{1}c_{\theta_{+}},Y_{3}s_{\theta_{+}}),\;\;\eta_{2}:(-Y_{1}s_{\theta_{+}},Y_{3}c_{\theta_{+}}),$
(40) $\displaystyle\textrm{Model-B}:$ $\displaystyle
S_{1}:(Y_{1}c_{\theta_{0}},Y_{3}s_{\theta_{0}}),\;\;S_{2}:(-Y_{1}s_{\theta_{0}},Y_{3}c_{\theta_{0}}),$
(41) $\displaystyle\textrm{Model-C}:$
$\displaystyle\kappa_{1}:(Y_{1}c_{\theta_{++}},Y_{3}s_{\theta_{++}}),\;\;\kappa_{2}:(-Y_{1}s_{\theta_{++}},Y_{3}c_{\theta_{++}}).$
(42)

In Fig. 4, we present beyond SM contributions to the muon and the electron
anomalous magnetic moments arising in three versions of models under
consideration. It is clear from this plot that the required large
contributions as observed in the experiments can be naturally provided within
this framework without requiring large Yukawa couplings. Furthermore, opposite
signs of the muon and the electron AMMs can be incorporated via an appropriate
choice of the signs of the associated Yukawa couplings.

##  IV Experimental constraints

This section briefly describes the phenomenological implications of the
proposed models and the current experimental bounds on the BSM states, along
with future collider prospects.

LHC bounds on scalars and fermions:– Model-A contains a DM candidate (see Sec.
V for details) via which it can be tested in colliders. Specifically, the
singly charged scalars can be efficiently pair-produced at the LHC through the
$s$-channel $\gamma/Z$ exchange. Once pair-produced, each of them decays into
a DM and a SM lepton, i.e., $pp\to\ell^{+}\ell^{-}+\not{E}_{T}$. This process
mimics the standard slepton searches carried out by ATLAS as well as CMS
collaborations Aad:2014yka ; Sirunyan:2018nwe ; Sirunyan:2018vig  and non-
observation of any such processes lead to a bound of $m_{S^{\pm}_{i}}\geq 450$
GeV Sirunyan:2018nwe .

On the contrary, Model-B/C does not contain a DM candidate. Consequently,
collider probes of these models are distinct from Model-A. Following Model-A,
we assume that BSM scalars are heavier than BSM fermionic states in Model-B/C.
Then pair-produced singly (singly and doubly) charged scalars in Model-B
(Model-C) decay into a pair of SM lepton ($\nu\nu$ or $\ell^{+}\ell^{-}$
depending on singly or doubly charged scalar) and a pair of BSM singly charged
fermions ($\psi^{+}\psi^{-}$). In fact, the singly charged fermions also get
pair-produced through the $s$-channel $\gamma/Z$ exchange, which provides the
relevant bounds for these models. Note, however, that for a general charge
assignment with an arbitrary value of $\beta$, a renormalizable coupling
responsible for the decay of these fermions may not be present; hence
$\psi^{\pm}$ are expected to be long-lived.

To make them decay, we fix the $B-L$ charge such that $\beta=-1$ for
Model-B/C, therefore a mixing term of the form $\mathcal{L}\supset
m^{\prime}\overline{\psi}_{L}\ell_{R}$ is allowed. Its contribution to SM
lepton masses can be fully neglected if $\epsilon\ll 1$, where we define
$m^{\prime}\equiv\epsilon v_{H}/\sqrt{2}$. Through this mixing, the vector-
like leptons will promptly decay if $\epsilon\gtrsim 2\times 10^{-7}$
Bhattiprolu:2019vdu . For a quasi-stable vector-like lepton, assuming chargino
like interactions, ATLAS search ATLAS:2019gqq  provides a bound of
$m_{\psi^{\pm}}\geq 750$ GeV Bhattiprolu:2019vdu . On the other hand, for the
prompt decay scenario, each of the pair-produced vector-like lepton decays
into $\psi\to h\ell$, $\psi\to Z\ell$, and $\psi\to W\nu_{\ell}$, for which
currently there is no collider bound. However, HL-LHC (14 TeV with integrated
luminosity of $3ab^{-1}$) will probe these processes and should be able to
exclude up to about $m_{\psi^{\pm}}\geq 460$ GeV for the first two generations
OsmanAcar:2021plv  and $m_{\psi^{\pm}}\geq 600$ GeV for tau-like $\psi$
Bhattiprolu:2019vdu .

Electroweak precision constraints:– Neutrino mass and BSM contributions to the
lepton anomalous magnetic moments heavily depend on the splitting between the
neutral (and charged) BSM scalars; hence electroweak precision measurements
provide stringent constraints on the model parameters. The so-called
$T$-parameter is the most crucial among these oblique parameters, which we
compute following Refs. Peskin:1990zt ; Barbieri:2006dq ; Grimus:2008nb ;
Funk:2011ad  and allow it to vary within the range given by
ParticleDataGroup:2020ssz ,

$\displaystyle\Delta T=0.03\pm 0.12.$ (43)

LEP bounds on vector boson:– Since we study gauged $B-L$ extension of the SM,
this theory contains a heavy gauge boson $Z^{\prime}$, which is significantly
constrained from the non-observation of any direct signature at LEP and LHC.
This gauge boson couples to quarks as well as leptons, thus $Z^{\prime}$ can
be produced and searched for at the LEP via $s$-channel $e\overline{e}\to
Z^{\prime}\to f\overline{f}$ processes. To assure LEP-II bound LEP:2003aa , we
impose

$\displaystyle\frac{M_{Z^{\prime}}}{g^{\prime}}>6.94\textrm{TeV at $95\%$
C.L.},$ (44)

which is derived from effective four-lepton operator Carena:2004xs  and valid
for $M_{Z^{\prime}}\gg 200$ GeV.

LHC bounds on vector boson:– Furthermore, at hadron colliders, $Z^{\prime}$
can be efficiently produced in the $s$-channel due to its couplings to quarks,
which would show up as resonances in the invariant mass distribution of the
decay products. Searching for a massive resonance at the LHC decaying into di-
lepton/di-jet imposes stringent limits on the respective production cross-
section. The current data from 13 TeV LHC search for a heavy resonance
decaying into two leptons (assuming a $100\%$ branching ratio) via $pp\to
Z^{\prime}\to\ell^{+}\ell^{-}$ provides the tightest constraint of
$M_{Z^{\prime}}>4.9$ TeV. This bound is somewhat relaxed for other branching
ratios, which is depicted in Fig. 8 using the current data (ATLAS
ATLAS:2019erb  and CMS CMS:2021ctt ) and future projections (HL-LHC
Ruhr:2016xsg  and FCC-hh Helsens:2019bfw ) in the coupling versus mass plane
Padhan:2022fak .

Cosmological constraints on vector boson:– Since neutrinos are Dirac particles
in our scenario, the existence of right-handed neutrinos $\nu_{R}$ is implied.
Since these right-handed neutrinos are mass degenerate with left-handed
neutrinos $\nu_{L}$, they could contribute to the effective number of
relativistic degrees of freedom $N_{eff}$ in the early Universe. In the case
of purely SM interactions, the contributions are completely negligible.
However, in our model, $\nu_{R}$ have gauge interactions with $Z^{\prime}$,
through which they can be in thermal equilibrium with the SM plasma in the
early Universe via $s$-channel
$f\overline{f}\leftrightarrow\nu_{R}\overline{\nu}_{R}$ processes that
increases $N_{eff}$. Cosmological data, however, requires that $\nu_{R}$s
decouple from the SM plasma much earlier than the $\nu_{L}$. To be specific,
Planck 2018 data Planck:2018nkj ; Planck:2018vyg  requires that $\nu_{R}$s
must decouple at temperatures greater than $T>0.6$ GeV Abazajian:2019oqj .

The best current measurement implies $N_{eff}=2.99\pm 0.17$ Planck:2018nkj ;
Planck:2018vyg , which is in complete agreement with the SM prediction
$N^{SM}_{eff}=3.045$ Mangano:2005cc ; Grohs:2015tfy ; deSalas:2016ztq . Then a
$2\sigma$ C.L. corresponds to $\Delta N_{eff}<0.285$, which we adopt in our
analysis. Future sensitives of CMB-S4 experiments will reach a precision of
$\Delta N_{eff}\sim 0.03$ CMB-S4:2016ple ; Abazajian:2019eic  that can probe
NP scale up to about 50 TeV Abazajian:2019oqj ; Luo:2020sho . For heavy
$Z^{\prime}$, i.e., $M_{Z^{\prime}}>20$ GeV, assuming that $\nu_{R}$ decouples
before $T\sim 0.6$ GeV, one obtains $M_{Z^{\prime}}/g^{\prime}>11.4$ TeV, for
the case of the standard $B-L$ theory Luo:2020sho  (this limit becomes much
stronger in the lower mass range, see, e.g., Ref. Abazajian:2019oqj ). We re-
scale this bound using the unconventional charge assignment of the right-
handed neutrinos in our theory, and the corresponding Planck 2018 bound is
presented in Fig. 9.

Figure 5: Dark matter annihilation into SM leptons.

##  V Dark Matter

In this theory, the Diracness of neutrinos is protected by the $U(1)_{B-L}$
symmetry. Remarkably, this same symmetry is also responsible for stabilizing
the DM candidate. The spontaneous symmetry breaking of $U(1)_{B-L}$ by the VEV
of $\sigma$ leaves a residual discrete symmetry $\mathcal{Z}_{D}$ (such that
$\omega^{D}=1$) and stabilizes the DM. In particular, stability is guaranteed
if $D>2$ as well as if $D$ is even Bonilla:2018ynb . In such a case, all SM
fields transform as even powers of $\omega$, and the lightest particle
transforming as an odd power of $\omega$ will be automatically stable. For
concreteness, for Model-A, we fix $\beta=-1/2$ in our analysis, and the
corresponding charges of all fields under the residual symmetry are presented
in Table 2. As can be seen from this Table, all particles in the dark sector
($\psi,\phi,S^{0},\eta^{+}$) carry charges that are odd powers of $\omega$.

Fields | $U(1)_{B-L}$ | $U(1)_{B-L}\to\mathcal{Z}_{18}$
---|---|---
$Q_{L},u_{R},d_{R}$ | $\frac{1}{3}$ | $\omega^{2}$
$L_{L},\ell_{R}$ | $-1$ | $\omega^{12}$
$\nu_{R}$ | $-4,5$ | $\omega^{12}$
$\psi$ | $-\frac{1}{2}$ | $\omega^{15}$
$H$ | $0$ | $\omega^{0}$
$\sigma$ | $3$ | $\omega^{0}$
$\phi^{\ast}$ | $\frac{1}{2}$ | $\omega^{3}$
$S^{0}$ | $\frac{7}{2}$ | $\omega^{3}$
$\eta^{+}$ | $\frac{1}{2}$ | $\omega^{3}$

Table 2: Charges of particles under the residual symmetry in Model-A with
$\beta=-1/2$. The dark sector transforms as an odd power of $\omega$, whereas
the rest of the particles transform as even powers of $\omega$. See text for
details.

The physical states in the dark sector of the Model-A consists of three Dirac
fermions: $\psi_{i},\,(i=1-3)$, two singly-charged scalars, $\eta_{i}$ and two
complex neutral scalars, $S_{i}$ with $i=1,2$. The DM candidate is the
lightest Dirac fermion, which we choose to be $\psi_{2}\equiv\chi$. A typical
DM annihilation channel in our model is demonstrated in Fig. 5. Before delving
into the thermal freeze-out scenario of the DM, we delineate the relevant
parameter space and constraints set by the DM direct detection experiments.

### DM Direct Detection and Parameter Space

DM Direct Detection:– The fermionic DM, being charged under $U(1)_{B-L}$, will
scatter with the nucleon via the exchange of $Z^{\prime}$ at the tree-level,
and in the limit of zero momentum transfer one can deduce the following
effective interaction term in the Lagrangian between the DM and the nucleon,

$-{\cal L}\supset\frac{g^{\prime
2}}{2M_{Z^{\prime}}^{2}}\overline{\chi}\gamma_{\mu}\chi\overline{N}\gamma^{\mu}N\;,$
(45)

where, $N$ is the nucleon. The spin independent scattering cross-section
associated with this effective interaction is given as,

$\sigma_{\mathrm{SI}}=\frac{1}{4\pi}\frac{g^{\prime
4}\mu_{r}^{2}}{4M_{Z^{\prime}}^{4}}\;,$ (46)

where, the reduced mass is
$\mu_{r}=\frac{m_{N}M_{\mathrm{DM}}}{m_{N}+M_{\mathrm{DM}}}$, and the cross-
section is insensitive to the DM mass. Consequently, for a fixed DM mass, the
limits on the $\sigma_{\mathrm{SI}}$ set by the currently operating DM direct
detection experiments will constrain the region of two dimensional
$M_{Z^{\prime}}-g^{\prime}$ parameter plane as shown in Fig. 8.

DM Parameter Space:– We want to correlate the constraints on the electron and
muon magnetic dipole moments with the parameter space associated with the DM
relic density. Hence, based on the choice of the Yukawa matrices given in Eq.
(12), and the scalar masses and mixing angles given in Eq. (20), we choose the
parameter space for the DM relic density analysis which is spanned by
$M_{\mathrm{DM}}$, $M_{F_{1}}$, $M_{\eta_{1}}$, $M_{\eta_{2}}$, $M_{S_{1}}$,
$M_{S_{2}}$, $\theta_{+}$, $\theta_{0}$, $Y_{1_{ee}}$, $Y_{1_{\mu\mu}}$,
$Y_{3_{ee}}$, $Y_{3_{\mu\mu}}$, $g^{\prime}$ and $M_{Z^{\prime}}$. Besides,
the Yukawa couplings $Y_{2}$ being connected with the neutrino mass generation
are much smaller than $Y_{1}$ and $Y_{3}$, and therefore don’t play any
significant role in the DM relic density analysis.

### DM Relic Density

We consider the standard thermal freeze-out to achieve the correct DM relic
density, $\Omega h^{2}=0.12\pm 0.001$ ($68\%$ C.L.) observed by Planck
Planck:2018vyg . The dominant (co)annihilation processes which control the DM
relic density for our considered DM parameter space, where $Y_{1}$ and $Y_{3}$
Yukawa matrices being diagonal, are enumerated in the following.

  * •

$\overline{\chi}\,\chi\rightarrow\mu^{+}\mu^{-}\,\,\mathrm{and}\,\,\overline{\nu}_{\mu}\nu_{\mu}$
via the exchange of $\eta^{+}_{1,2}$ and $S_{1,2}$ at the t-channel,
respectively.

  * •

$\overline{\chi}\,\chi\rightarrow\overline{q}q,\,l^{+}l^{-},\,\overline{\nu}_{l}\nu_{l}$
i.e. to the SM quark ($q$), charged lepton ($l$) and neutrino ($\nu_{l}$)
pairs via the exchange of $Z^{\prime}$ at the s-channel.

When the mass splitting between the DM candidate and another dark sector
particle carrying same dark charge is small, the coannihilation channels also
become important. In our case, the dominant coannihilation channels involving
the DM candidate $\chi$ and other dark sector particles: $\psi_{1}$,
$\eta^{+}_{1,2}$ and $S_{1,2}$ are,

  * •

$\chi\,\overline{\psi}_{1}\rightarrow\mu^{-}e^{+},\,\nu_{\mu}\overline{\nu}_{e}$,
$\chi\,\eta^{-}_{1,2},\,\chi\,S^{*}_{1,2}\rightarrow\overline{X^{\prime}}X$
where $X,\,X^{\prime}$ are the SM particles, and the conjugated channels.

  * •

$\psi_{1}\,\eta^{-}_{1,2},\,\psi_{1}S^{*}_{1,2},\,\eta^{+}_{i}\,\eta^{-}_{j},\,S_{i}\,S^{*}_{j},\,\eta^{+}_{i}\,S^{*}_{j}\rightarrow\overline{X^{\prime}}X$
where $i,\,j=1,\,2$, and the relevant conjugated channels.

As the DM relic density involves multiple (co)annihilation channels, which
also depend on the multi-dimensional parameter space, we consider two
scenarios to capture the dynamics in a simplified manner. First, we consider
the mass of the $Z^{\prime}$ quite heavy compared to DM and other BSM
particles so that the contribution of the $Z^{\prime}$ mediated channels in
the thermal freeze-out become negligible. We call it the $Z^{\prime}$
decoupled scenario, and in this case, the thermal freeze-out of the DM is
controlled by the BSM Yukawa sector of the Model-A. In the second scenario,
dubbed here as BSM Yukawa + $Z^{\prime}$ gauge boson scenario, we take into
account both the simultaneous contributions of the BSM Yukawa sector and the
$Z^{\prime}$ gauge bosons to determine the DM relic density.

Figure 6: The correlation between the mass of the DM candidate $F_{2}$,
$M_{DM}$ and the DM relic density $\Omega h^{2}$ for the parameter space which
satisfy the constraints on the electron and muon magnetic moments, $\Delta
a_{e}$ and $\Delta a_{\mu}$ and T-parameter $\Delta T$ for the $Z^{\prime}$
decoupled scenario. The black horizontal line represents the correct DM relic
density determined by Planck.

$Z^{\prime}$ decoupled scenario:– Although we consider the $Z^{\prime}$
contribution to be negligible in determining the DM relic density for this
scenario as mentioned above, the relevant parameter space is still large
enough to disentangle the effects of the BSM Yukawa couplings, $Y_{1}$ and
$Y_{3}$, the mass-splittings between the DM and the charged scalars
$\eta_{1,2}^{+}$ and neutral scalars $S_{1,2}$, and the scalar mixing angles
$\theta_{+}$ and $\theta_{0}$ on the DM relic density concretely just by
scanning over the parameters randomly. Therefore, we consider a few benchmark
points of the parameter space and discuss the impact of the variations of the
parameters on the DM relic density.

First we consider a benchmark point where we fixed the following parameters as
follows,

$\displaystyle
M_{F_{1}}=M_{DM}+1\,\mathrm{GeV},\,M_{\eta_{1}}=M_{S_{1}}=M_{DM}+100\,\mathrm{GeV},$
$\displaystyle
M_{\eta_{2}}=M_{S_{2}}=M_{DM}+150\,\mathrm{GeV},\,\theta_{+}=\theta_{0}=0.7,$
$\displaystyle Y_{3_{ee}}=0.1,\,Y_{3_{\mu\mu}}=1,$ (47)

and vary only the Yukawa couplings, $Y_{1_{ee}}\in(0.01,1)$ and
$Y_{1_{\mu\mu}}\in-(0.1,1)$ randomly to select the points which simultaneously
satisfy the constraints on the electron and muon magnetic moments $\Delta
a_{e}$ and $\Delta a_{\mu}$ and T-parameter $\Delta T$. Afterwards, we
calculate the relic density of the DM $F_{2}$ for each of these selected
points using micrOMEGAsv$5.2$ Belanger:2020gnr  with the model files generated
by FeynRules Alloul:2013bka .

Figure 7: The dependence of the DM relic density on the variation of
$Y_{3_{\mu\mu}}$ for a fixed mass splitting (left fig.) and on the mass
splittings between the DM and charged and neutral scalars for a fixed value of
$Y_{3_{\mu\mu}}$ (right fig.). Here we vary $Y_{1_{\mu\mu}}\in-(0.1,1)$. For
simplicity, in this presentation, we only consider the parameter points
satisfying the constraints on the muon magnetic moment $\Delta a_{\mu}$ and
the T parameter $\Delta T$ simultaneously. Again the black horizontal line
represents the correct DM relic density. The respective collider bounds are
not shown in this plots, see text for details.

Subsequently, for the parameter set given in Eq. (47), the correct DM relic
density is obtained for $M_{DM}=310$ GeV which is excluded by the LHC limits,
and for $1.4-1.7$ TeV mass range as seen in Fig. 6. To illustrate it, first,
we write down the representative annihilation cross-section at low-velocity
approximation that is associated with the DM, $\chi$ annihilating into the SM
leptons (charged and neutrinos) via the exchange of charged ($\eta_{1,2}$) and
neutral ($S_{1,2}$) scalars at t-channel,

$\sigma
v_{0}=\frac{M_{DM}^{2}\left(\tilde{Y}^{2}_{1_{a}}+\tilde{Y}^{2}_{3_{a}}\right)^{2}}{32\pi\left(M_{DM}^{2}+M_{\tilde{S}_{i}}^{2}\right)^{2}}\;,$
(48)

where the terms having small masses of the final-state leptons are neglected,
and $\tilde{Y}_{1_{a}}$ and $\tilde{Y}_{3_{a}}$ are the Yukawa couplings of
electron or muon sector (here denoted by $a$) redefined by absorbing the
appropriate scalar mixing angles associated with charged or neutral scalar
(here expressed with $\tilde{S}_{i}$). As we choose $Y_{3_{\mu\mu}}=1$ for our
parameter set in Eq. (47), when the DM mass is in $100-680$ GeV range, the
term like $\tilde{Y}_{3_{\mu\mu}}^{4}M_{DM}^{2}$ in the numerator of Eq. (48)
dominates the cross-section, and therefore we see the narrow band for that
mass range. In contrast, when $M_{DM}>680$ GeV, the DM mass is large enough to
make the terms like $\tilde{Y}_{1_{\mu\mu}}^{4}M_{DM}^{2}$ in the numerator of
Eq. 48 also comparable to the terms with $Y_{3_{\mu\mu}}$, and thus we see the
band at larger masses in the DM relic density plot as $|Y_{1_{\mu\mu}}|$ takes
value in the range $(0.1,1)$. Besides, as the mass-splittings are small ($150$
GeV) when the DM mass is close to $O(\mathrm{TeV})$, the coannihilation
channels also contribute significantly to the DM relic density, so the
simplified Eq. 48 is not applicable for higher DM mass ranges with small mass-
splittings.

Figure 8: Correlation between the mass of the $Z^{\prime}$ gauge boson
$M_{Z^{\prime}}$ and the DM relic density $\Omega h^{2}$ for four benchmark
points given in Table 3.

In Fig. 7 (left), we can see that if $Y_{3_{\mu\mu}}$ is set to 0.5 instead of
1, the annihilation cross-section decreases for a wide range of the DM mass,
and for this reason, the DM relic density remains overabundant for DM mass
close to TeV scale as opposed to the case with $Y_{3_{\mu\mu}}=1$ for which it
remains underabundant for similar DM mass range. In addition, when we increase
the mass-splittings between the DM and the charged and neutral scalars, for
lower mass range, the denominator in Eq. 48 becomes more significant, and the
annihilation cross-section decreases, which results in the overabundant DM.
Besides, larger mass-splittings suppress the coannihilation processes during
the thermal freeze-out resulting in the overabundance of the DM for masses in
the TeV range.

Furthermore, to capture the impact of $Z^{\prime}$ on the DM relic density,
first, we consider four benchmark points presented in Table. 3 which
simultaneously satisfy the constraints on the electron and muon magnetic
moments $\Delta a_{e}$ and $\Delta a_{\mu}$ and T-parameter $\Delta T$ in the
$Z^{\prime}$ decoupled scenario.

Benchmark | $M_{F_{2}}$ | $M_{F_{1}}$ | $M_{\eta_{1}}$ | $M_{\eta_{2}}$ | $M_{S_{1}}$ | $M_{S_{2}}$ | $\theta_{+}$ | $\theta_{0}$ | $Y_{1_{ee}}$ | $Y_{1_{\mu\mu}}$ | $Y_{3_{ee}}$ | $Y_{3_{\mu\mu}}$ | $\Omega h^{2}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
I | 500 GeV | 501 GeV | 961.46 GeV | 1357.04 GeV | 858.75 GeV | 1085.4 GeV | 0.05 | 0.56 | 0.09 | -1.33 | 0.67 | 0.36 | 0.12
II | 500 GeV | 501 GeV | 1008.3 GeV | 1983.3 GeV | 979.83 GeV | 1408 GeV | 0.04 | 0.02 | 0.09 | -1.07 | 0.3 | 0.54 | 0.326
III | 3000 GeV | 3001 GeV | 3160.5 GeV | 3582 GeV | 3077 GeV | 3672.2 GeV | 0.42 | 0.41 | 0.09 | -0.96 | 0.78 | 0.73 | 0.121
IV | 3000 GeV | 3001 GeV | 3124.6 GeV | 3562 GeV | 3205.6 GeV | 3611.4 GeV | 0.6 | 0.46 | 0.06 | -1.11 | 0.84 | 0.45 | 0.312
Table 3: Benchmark points in the $Z^{\prime}$ decoupled scenario for
$M_{DM}=500$ GeV and $M_{DM}=3000$ GeV.

For benchmark point I and II of Table 3 with $M_{DM}=500$ GeV, the thermal
freeze-out of the DM is dominated by the processes,
$\overline{\chi}\chi\rightarrow\overline{\nu}_{\mu}\nu_{\mu},\,\mu^{+}\mu^{-}$.
On the other hand, for benchmark point III and IV with $M_{DM}=3000$ GeV, the
thermal freeze-out processes are dominated by the coannihilation, for example,
by $S_{1}S^{*}_{1}\rightarrow W^{+}W^{-},\,Z\,Z$ and
$\eta^{+}_{1}\eta^{-}_{1}\rightarrow W^{+}W^{-},\,Z\,Z$, respectively.

BSM Yukawa sector + $Z^{\prime}$ gauge boson:– Now we calculate the DM relic
density for the each benchmark points in Table 3 by varying the mass of the
$Z^{\prime}$ gauge boson $M_{Z^{\prime}}\in(0.5,20)$ TeV and the gauge
coupling $g^{\prime}\in(0.02,1)$.

Figure 9: Correlation between $M_{Z^{\prime}}$ and $g^{\prime}$ for four
benchmark points. For $m_{DM}=500$ GeV (left fig.), the cyan and blue points
correspond to $(M_{Z^{\prime}},\,g^{\prime})$ pairs that satisfy the
constraints on DM relic density for benchmark I and benchmark II points in the
$Z^{\prime}$ decoupled scenario, respectively, whereas for $m_{DM}=3$ TeV
(right fig.), the yellow-green and brown points represent those with same
attribute for benchmark III and benchmark IV points, respectively. The shaded
regions are ruled out by current collider bounds, cosmological constraints,
and from dark matter direct detection. Future colliders will probe regions
inside green and purple lines (not shaded).

From Fig. 8, we can see that for a fixed $M_{Z^{\prime}}$, the increase of the
gauge coupling $g^{\prime}$ decreases the DM relic abundance determined in the
$Z^{\prime}$ decoupled scenario. Therefore, even if the BSM Yukawa sector sets
an overabundant DM relic density, one can achieve its observed value by
adjusting the $Z^{\prime}$ mass and coupling $g^{\prime}$ as seen in Fig. 8
(upper and lower right figures). Moreover, we can see that the lower the value
of the gauge coupling $g^{\prime}$, the lower the minimum value of the
$M_{Z^{\prime}}$ where the $Z^{\prime}$’s contribution to the DM relic
abundance becomes negligible. Besides, we see the dip in the DM relic
abundance due to the resonant enhancement of DM annihilation processes
involving the exchange of $Z^{\prime}$ at the s-channel when
$M_{Z^{\prime}}\sim 2M_{DM}$.

In Fig. 9, we present the correlation between $M_{Z^{\prime}}$ and
$g^{\prime}$ for our four benchmark points with all relevant constraints set
by the collider searches and DM direct detection experiments, the observed DM
relic density, and cosmological limit on the extra radiation. By correlating
the Figs. 8 (upper left and right) and Fig. 9 (left), for $M_{DM}=500$ GeV, we
see that the current limits provided by ATLAS and CMS have already ruled out
the $(M_{Z^{\prime}},\,g^{\prime})$ values for which the $Z^{\prime}$ gauge
boson give significant contributions to the DM relic abundance. Again the
close inspection of Figs. 8 (lower left and right) and Fig. 9 (right) for
$m_{DM}=3$ TeV indicates that apart from the small parameter space close to
the resonance point around $M_{Z^{\prime}}\sim 6$ TeV, almost all the
$(M_{Z^{\prime}},\,g^{\prime})$ values are ruled out by LEP, LHC, DM direct
detection (e.g. PandaX-4T PandaX-4T:2021bab  which has set the most stringent
limit on the spin independent DM-nucleon cross-section for 10 GeV - 10 TeV DM
mass range), and Planck constraint on $\mathrm{N}_{\mathrm{eff}}$.
Furthermore, rest of the parameter space where $Z^{\prime}$ contributions
could play an important role in determining DM relic abundance will be fully
probed by the HL-LHC and future collider like FCC-hh.

Since the DM particle is inducing the anomalous magnetic dipole moment via
mass-insertion, the chirality flip enhancement is strongly correlated to the
mass generation mechanism of the associated lepton and causes related loop
contributions to its mass. Hence, if $M_{DM}\gtrsim\mathcal{O}(3)$ TeV,
typically, fine-turning of a large degree is required to adjust the lepton
mass correctly. For details, see, e.g., Athron:2021iuf  and references
therein. This is why in this work, we explored DM phenomenology in the range
$100\,\mathrm{GeV}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}M_{DM}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}3$
TeV.

##  VI Conclusion

The Fermilab’s Muon $g-2$ experiment has recently confirmed the longstanding
tension of the muon AMM. Furthermore, the recent precise measurement of the
electron AMM at the Berkeley Lab shows deviations from the theoretical
prediction. These two anomalies together strongly hint towards physics beyond
the Standard Model. Besides, the origin of neutrino mass remains a mystery
even after the groundbreaking discovery of neutrino oscillation about twenty-
five years ago. Moreover, even though we know DM exists, we do not yet know
what it is at a fundamental level.

This work proposes a class of radiative Dirac neutrino mass models where
neutrino mass arises at a one-loop level. Furthermore, NP states that
participate in neutrino mass generation also run through the loops and
significantly contribute to $(g-2)_{\mu,e}$. These large contributions arise
due to chirality enhancements required to simultaneously explain the
$(g-2)_{\mu}$ and $(g-2)_{e}$ data. For completeness, we have studied three
benchmark models, one of which (Model-A) offers a Dark Matter candidate whose
stability is naturally protected without imposing additional symmetries by
hand. For Model-A, we have performed a detailed numerical analysis to
investigate the correlations among the common model parameters accommodating
neutrino oscillation data, the muon, the electron g-2, and dark matter relic
abundance. Parameters that generate non-zero neutrino mass also play a non-
trivial role in explaining the muon and the electron $g-2$ simultaneously;
furthermore, these same parameters take part in dark matter annihilations to
the SM particles in reproducing the correct relic abundance. This model is
subject to numerous constraints arising from colliders and cosmology, through
which it can be probed in the current and upcoming experiments. A detailed
study of lepton flavor violation and electric dipole moments of the electron
and the muon and their possible links to the puzzles resolved in this work is
left for future work.

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