aainstitutetext: School of Physics, Korea Institute for Advanced Study, Seoul
02455, Korea

# Explaining ATOMKI, $(g-2)_{\mu}$, and MiniBooNE anomalies with light
mediators in $U(1)_{H}$ extended model

Sumit Ghosh a and Pyungwon Ko<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider $U(1)_{H}$ extensions of Type-I 2HDM plus a singlet scalar
$\phi_{H}$, introducing a new Higgs doublet $H_{2}$ and a singlet $\phi_{H}$
charged under $U(1)_{H}$. The SM Higgs doublet $H_{1}$ as well as all the SM
fermions and three right-handed singlet neutrinos, introduced to generate
nonzero neutrino masses and mixings, are neutral under $U(1)_{H}$. We also
introduce a SM singlet Dirac fermion, charged under $U(1)_{H}$ and utilize it
as sterile neutrinos relevant to the MiniBooNE experiment. The $U(1)_{H}$
symmetry breaks due to the vacuum expectation values of $H_{2}$ and
$\phi_{H}$, leading to the emergence of a light vector boson with a mass of
approximately 17 MeV. This vector boson interacts with fermions through mass
mixing and kinetic mixing process involving other neutral gauge bosons.
Furthermore, alongside the light vector boson, another light scalar particle
with a mass around 10–100 MeV may arise from scalar sector mixing. By
utilizing the gauge couplings of the light vector boson and the Yukawa
couplings of the light scalar, this model can simultaneously provide an
explanation for the Beryllium anomaly observed in the ATOMKI experiment, the
anomalous magnetic moment of charged leptons and the excess of electron-like
events detected at the MiniBooNE experiment.

###### Keywords:

Light mediators, ATOMKI, MiniBooNE, $(g-2)_{\mu}$

††preprint:  KIAS-P23062

##  1 Introduction

The landscape of fundamental particle physics currently stands at an
interesting crossroad. On one hand, it is absolutely certain that the realm of
physics must extend beyond the confines of the Standard Model (SM) despite its
phenomenal success. Yet, on the other hand, the pursuit of these novel
frontiers has proven to be very challenging. This quest for new physics beyond
the SM finds a well-established foothold within the neutrino sector. The
compelling oscillation data of neutrinos Fukuda:1998mi ; Ahmad:2002jz
unequivocally demand the inclusion of non-zero masses for at least two
neutrinos. On the other hand, convincing evidence from cosmology reveals that
a staggering $25\%$ of the Universe’s matter exists in the form of elusive
dark matter (DM) Aghanim:2018eyx , which is not described within the SM. While
the vast majority of terrestrial experiments align with remarkable precision
to the predictions of the SM, a select few results emerge as outliers. These
interesting anomalies demand renewed understanding, challenging the
comprehensive dominion of the SM.

The pursuit of physics beyond the SM (BSM) involves exploring a wide spectrum
of particle masses and interactions. To reconcile BSM theories with the
absence of current experimental findings, two avenues emerge: new particles
might be extremely massive or possess very weak interactions with the SM.
High-energy colliders like the Large Hadron Collider (LHC) play a pivotal role
in directly detecting massive particles at the TeV scale. Complementary low-
energy and/or high-intensity experiments focus on uncovering particles with
lower mass and weak couplings. The prospect of new physics revealing itself
through light and weakly coupled new states is gaining momentum, driven partly
by the LHC’s lack of significant signals at the TeV scale. In this context,
low-energy and high-intensity experiments emerge as vital tools for
investigation. Notably, a handful of intriguing hints pointing to deviations
from the SM have surfaced in some of these low-energy experiments.

Rare nuclear transitions offer a promising avenue to detect new physics at
lower energy scales, as they can be substantially influenced by BSM physics
even if the coupling is extremely weak. While numerous nuclear transitions can
be explored, those of particular interest involve the resonant production of
excited nuclear states, followed by their decay through internal pair creation
(IPC). In IPC, the excited nucleus emits a virtual photon that subsequently
decays into an $e^{+}e^{-}$ pair. This high-statistics phenomenon serves as an
excellent platform for investigating new physics with weak interactions. In
recent times, the ATOMKI pair spectrometer experiment Gulyas:2015mia  has
reported an intriguing observation: an excess of 6.8 $\sigma$ in IPC decays of
excited 8Be(18.15) nuclei Krasznahorkay:2015iga ; Krasznahorkay:2018snd . This
anomaly manifests as noticeable bumps in both the invariant mass and the
angular opening of the $e^{+}e^{-}$ pairs. This stands in stark contrast to
the predictions of electromagnetic (EM) IPC from virtual photons, which
anticipate a smooth and rapidly declining distribution of opening angles. The
presence of this anomalous bump aligns with expectations if a slowly moving
vector boson with a mass of 17 MeV was produced on-shell via the emission of
the excited nucleus, followed by its decay into $e^{+}e^{-}$ pairs. More
recently, analogous anomalous outcomes have surfaced for IPC decays of excited
4He Krasznahorkay:2019lyl ; Krasznahorkay:2021joi  and 12C
Krasznahorkay:2022pxs  nuclei.

The magnetic dipole moment of the muon can be calculated and measured with
exceptional precision, serving as a classic test of the SM Keshavarzi:2021eqa
. Recently, the E989 experiment Muong-2:2015xgu ; Fienberg:2019ddu  at Fermi
National Laboratory (FNAL) achieved the most accurate measurement of the
muon’s anomalous magnetic moment by combining data from its initial three runs
Muong-2:2023cdq . This outcome agrees perfectly with the earlier measurement
from Brookhaven National Laboratory (BNL) Muong-2:2006rrc  and amplifies the
world average discrepancy from the SM prediction to a significance of
$5.1\sigma$. Ongoing efforts aim to clarify the existing theoretical landscape
Colangelo:2022jxc , with an updated prediction incorporating comprehensive
data from lattice calculations potentially leading to a narrower and less
significant deviation. On the experimental front, an upcoming J-PARC
experiment Saito:2012zz  aims to cross-validate these findings. Potential
explanations for this tension may emerge from quantum loop corrections linked
to new electroweak (EW) scale particles around $100$ GeV, characterized by
$\mathcal{O}(1)$ couplings, or from significantly lighter particles in the
range of $10-100$ MeV, exhibiting $\mathcal{O}(10^{-4})$ couplings
Jaeckel:2020dxj .

Neutrino oscillation experiments detect charged leptons resulting from
(anti)neutrino charged current interactions with a target. These experiments
utilize appearance measurements, seeking charged leptons with different
flavors from the initially produced neutrinos. Among these experiments,
MiniBooNE at FNAL MiniBooNE:2008paa  was designed to detect electron
(anti)neutrinos in a muon (anti)neutrino beam. It focused on identifying
electron-like charged current quasi-elastic (CCQE) events originating from
both neutrino and antineutrino modes. Interestingly, a significant surplus of
events was detected in both modes. Recently, after 17 years of data
collection, MiniBooNE updated their results Aguilar-Arevalo:2020nvw ,
revealing that the consistently observed excess in both neutrino and
antineutrino modes now stands at a 4.8 $\sigma$ significance level. This
establishes it as one of the most statistically significant anomaly in the
neutrino sector. The traditional explanations, which leaned on oscillations
driven by mixing with a new eV-scale neutrino, are now greatly challenged and
possibly ruled out. This prompts the consideration of new physics in the
neutrino sector as a potential solution to this excess.

The aforementioned anomalies could find an explanation through a new physics
scenario that links to the SM via lightly connected mediator particles. The
idea of such weakly coupled particles has been a plausible theoretical concept
for a while. A compelling possibility for BSM new physics is the existence of
new gauge groups. One scenario involves the potential survival of a remnant of
grand unification at lower energies, such as the $U(1)_{T3R}$ gauge group
Dutta:2019fxn ; Dutta:2022qvn . Other prospects encompass the gauging of
specific combinations of the accidental $B$ and $L_{i}$ global symmetries of
the SM Foot:1990mn ; He:1990pn ; He:1991qd . If these symmetries break
spontaneously at lower energies, they also exhibit weak coupling. Another
intriguing and well-founded option is $U(1)_{H}$, initially proposed to tackle
the significant flavor-changing neutral current (FCNC) issue in the two Higgs
doublet model (2HDM) Ko:2012hd . In general 2HDMs, sizeable FCNC could occur
through neutral Higgs bosons, often regarded as a phenomenological challenge.
Glashow and Weinberg introduced criteria to address this problem, typically
involving a softly broken $Z_{2}$ symmetry, yielding Type-I, II, X, Y 2HDMs
Glashow:1976nt ; Branco:2011iw . Some years ago, one of the authors of this
paper proposed implementing the $Z_{2}$ symmetry into a continuous $U(1)_{H}$
gauge symmetry, either eliminating or replacing the usual soft $Z_{2}$
breaking through spontaneous $U(1)_{H}$ breaking Ko:2012hd . In the former
scenario, the massless Goldstone boson is absorbed by the $U(1)_{H}$ gauge
boson, resulting in the absence of the typical pseudoscalar in the scalar
spectra. In the latter case, introducing an SM singlet scalar $\phi_{H}$ with
a nonzero $U(1)_{H}$ charge allows $U(1)_{H}$ symmetry breaking through the
nonzero vacuum expectation value (vev) of $\phi_{H}$, inducing the expected
soft $Z_{2}$ breaking effects. These $U(1)_{H}$ extensions of 2HDMs have been
explored in various high-energy contexts, such as Higgs phenomenology
Ko:2013zsa , dark matter within the inert 2HDM Ko:2014uka , and
$E_{6}$-motivated leptophobic $Z_{H}$ Ko:2015fxa ; Ko:2016lai . In this paper,
we explore the low-energy manifestation of $U(1)_{H}$.

The intriguing question arises: can any simple extension of the SM explain all
these observations concurrently? There have been attempts to address one or
two of these anomalies at a time within a model framework in the literature
DelleRose:2017xil ; Bertuzzo:2018itn ; Datta:2020auq ; Dutta:2020scq ;
Ballett:2019pyw , whereas we aim to explain all three simultaneously. To
achieve this, we propose a straightforward expansion of the SM based on the
$U(1)_{H}$ gauge group. This extension introduces an additional SM scalar
doublet and a singlet, both charged within the $U(1)_{H}$ framework. The
expansion of the Higgs sector remains uncomplicated yet well-motivated, tied
to the electroweak sector of the SM. The combined vev of the two doublets
result in the spontaneous breaking of electroweak symmetry, while the vev of
the SM singlet scalar and $U(1)_{H}$ charged doublet lead to the symmetry
breaking of $U(1)_{H}$. Moreover, three right-handed neutrinos are introduced,
with the further assumption that all fermions remain neutral under the
$U(1)_{H}$ gauge group. Interactions between the new gauge boson and fermions
arise from the interplay of mixing among the neutral gauge bosons. The most
notable feature of the model is the emergence of a light scalar particle with
a mass around 10-100 MeV, stemming from scalar sector mixing, and a light
neutral vector boson with a mass around 17 MeV, resulting from neutral gauge
boson sector mixing. Additionally, the presence of sterile neutrinos enables
the realization of small neutrino masses via the type I seesaw mechanism
within this framework. The lightest sterile neutrino could also serve as a
viable dark matter candidate. The parameter space governing the light scalar
and vector boson is outlined by various low-energy experiments. Within this
space, we identify an allowable range to employ the light vector boson in
clarifying the Beryllium anomaly. Furthermore, the excess events observed in
the MiniBooNE experiment can primarily be attributed to the light scalar
particle and a heavy sterile neutrino with a mass around 400 MeV. While both
the light vector and scalar contribute to the muon’s magnetic moments with
opposing signs, fine-tuning between them allows for the attainment of accurate
values.

The subsequent sections of this paper are structured as follows. In Section 2,
a comprehensive overview of the model is provided. Specifically, Section 2.1
delves into the scalar sector of the model, while Section 2.2 elaborates on
the Yukawa interactions between fermions and scalars. The neutrino masses and
mixing are explored in Section 2.3, followed by an in-depth discussion of
gauge boson mass, mixings, and fermion-gauge interactions in Section 2.4.
Turning our attention to specific phenomena, Section 3 undertakes an
examination of the parameter space associated with the anomalous result
observed in the ATOMKI experiment. An exploration of anomalous magnetic
moments is undertaken in Section 4, while Section 5 delves into the excess
observed in the MiniBooNE experiment. To consolidate our findings, a concise
summary is presented in Section 6.

##  2 Model Description

We extend the SM gauge symmetry with a new abelian gauge group $U(1)_{H}$ and
additional new particles. A summary of the charges of the new particles in the
model is given in Table. 1. Note that the new gauge symmetry does not
contribute to electric charge. This new gauge symmetry is supposed to be
broken at a relatively low energy scale by the condensations of two new scalar
fields, $\phi_{H}$, which is charged under $U(1)_{H}$ but singlet under the SM
gauge symmetry, and new Higgs doublet $H_{2}$ charged under $U(1)_{H}$. In the
following, we will briefly describe the masses and interaction terms for the
scalars, fermions, and gauge bosons.

Symmetry Group | $H_{1}$ | $H_{2}$ | $\phi_{H}$ | $n^{\prime}_{R}$ | $\psi^{\prime}_{L}$ | $\psi^{\prime c}_{R}$
---|---|---|---|---|---|---
$SU(2)_{L}$ | 2 | 2 | 1 | 1 | 1 | 1
$U(1)_{Y}$ | 1/2 | 1/2 | 0 | 0 | 0 | 0
$U(1)_{H}$ | 0 | 1 | $-1$ | 0 | 1 | $-1$
Table 1:  The quantum numbers of the particles under the gauge group
$SU(2)_{L}\times U(1)_{Y}\times U(1)_{H}$ are reported. The other SM fermions
have the usual charges, and none of them have charges under the new gauge
group $U(1)_{H}$.

###  2.1 Scalar sector

The scalar sector comprises two SM doublet fields, denoted as $H_{1}$ and
$H_{2}$, as well as one SM singlet field, $\phi_{H}$. These scalar fields
possess following quantum numbers within the context of the $SU(2)_{L}\times
U(1)_{Y}\times U(1)_{H}$ gauge symmetry:

$H_{1}\sim(2,1/2,0),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
H_{2}\sim(2,1/2,1),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\phi_{H}\sim(1,0,-1)\leavevmode\nobreak\ ,\leavevmode\nobreak\ \,$ (1)

The most comprehensive, renormalizable, and CP-conserving scalar potential can
be expressed in the following manner:

$\displaystyle V(H_{1},H_{2},\phi_{H})$ $\displaystyle=$ $\displaystyle
m^{2}_{1}H_{1}^{\dagger}H_{1}+m^{2}_{2}H_{2}^{\dagger}H_{2}+m_{\phi}^{2}\phi_{H}^{\dagger}\phi_{H}$
(2)
$\displaystyle+\frac{\lambda_{1}}{2}(H_{1}^{\dagger}H_{1})^{2}+\frac{\lambda_{2}}{2}(H_{2}^{\dagger}H_{2})^{2}+\frac{\lambda_{\phi}}{2}(\phi_{H}^{\dagger}\phi_{H})^{2}+\lambda_{3}H_{1}^{\dagger}H_{1}H_{2}^{\dagger}H_{2}+\lambda_{4}H_{1}^{\dagger}H_{2}H_{2}^{\dagger}H_{1}$
$\displaystyle+\lambda_{1\phi}\phi_{H}^{\dagger}\phi_{H}H_{1}^{\dagger}H_{1}+\lambda_{2\phi}\phi_{H}^{\dagger}\phi_{H}H_{2}^{\dagger}H_{2}$
$\displaystyle+m_{12\phi}\left(H_{1}^{\dagger}H_{2}\phi_{H}+\phi_{H}^{\dagger}H_{2}^{\dagger}H_{1}\right)$

In general, all three scalar fields can obtain nonzero vev’s. Therefore, we
take $\braket{H_{1}}=\left(0,v_{1}/\sqrt{2}\right)$,
$\braket{H_{2}}=\left(0,v_{2}/\sqrt{2}\right)$, and
$\braket{\phi_{H}}=v_{\phi}/\sqrt{2}$. The vevs $\braket{H_{1}}$ and
$\braket{H_{2}}$ are responsible for electroweak symmetry breaking, where
$v_{1}^{2}+v_{2}^{2}=v^{2}=246^{2}$ GeV2, and $\tan\beta=v_{2}/v_{1}$. The vev
of $\phi_{H}=v_{\phi}/\sqrt{2}$ breaks the new gauge symmetry of $U(1)_{H}$.
The scale of $v_{\phi}$ can be chosen such that the new vector gauge boson has
a very small mass, preferably in the MeV scale keeping 17 MeV $Z^{\prime}$ for
the ATOMKI anomaly in mind. After the symmetry breaking the scalar fields can
be written as,

$\displaystyle H_{1}$ $\displaystyle\sim\left(\begin{array}[]{c}{G}^{+}\\\
\frac{1}{\sqrt{2}}(v_{1}+\rho_{1}+iG_{0})\end{array}\right),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ H_{2}\sim\left(\begin{array}[]{c}{\phi_{2}}^{+}\\\
\frac{1}{\sqrt{2}}(v_{2}+\rho_{2}+i\eta_{2})\end{array}\right),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (6)
$\displaystyle\phi_{H}$
$\displaystyle\sim\frac{1}{\sqrt{2}}(v_{\phi}+\rho_{\phi}+iG_{0\phi})\leavevmode\nobreak\
.\leavevmode\nobreak\ \,$ (7)

The extremization of the tree level scalar potential leads to the following
conditions:

$\displaystyle
m_{1}^{2}+\frac{\lambda_{1}v_{1}^{2}}{2}+\frac{(\lambda_{3}+\lambda_{4})v_{2}^{2}}{2}+\frac{\lambda_{1\phi}v_{\phi}^{2}}{2}$
$\displaystyle=$ $\displaystyle-\frac{m_{12\phi}v_{2}v_{\phi}}{\sqrt{2}v_{1}}$
(8) $\displaystyle
m_{2}^{2}+\frac{\lambda_{2}v_{2}^{2}}{2}+\frac{(\lambda_{3}+\lambda_{4})v_{1}^{2}}{2}+\frac{\lambda_{2\phi}v_{\phi}^{2}}{2}$
$\displaystyle=$ $\displaystyle-\frac{m_{12\phi}v_{1}v_{\phi}}{\sqrt{2}v_{2}}$
(9) $\displaystyle
m_{\phi}^{2}+\frac{\lambda_{\phi}v_{\phi}^{2}}{2}+\frac{\lambda_{1\phi}v_{1}^{2}}{2}++\frac{\lambda_{2\phi}v_{2}^{2}}{2}$
$\displaystyle=$ $\displaystyle-\frac{m_{12\phi}v_{1}v_{2}}{\sqrt{2}v_{\phi}}$
(10)

There are a total of 10 degrees of freedom (d.o.f.), out of which 4
($G^{\pm}$, $G_{0}$, and $G_{0\phi}$) will be absorbed by the massless gauge
bosons to become massive. The remaining d.o.f. correspond to physical scalar
particles in the model. Specifically, 2 d.o.f. are for the singly charged
scalar $h^{\pm}$ (coming from $\phi_{2}^{\pm}$), 1 is for the pseudo-scalar,
which we denote as $s$ (arising from $\eta_{2}$), and the remaining 3 d.o.f.
correspond to the physical scalar particles $h,h_{1},h_{2}$. These arise from
the mixing of the three CP-even states, namely, $\rho_{1},\rho_{2}$, and
$\rho_{\phi}$.

The charged scalar mass square is given by,

$\displaystyle m^{2}_{h^{\pm}}$ $\displaystyle=$
$\displaystyle-\frac{m_{12\phi}v_{1}v_{\phi}}{\sqrt{2}v_{2}}-\frac{\lambda_{4}v_{1}^{2}}{2}$
(11)

where we have utilized the conditions arising from the extremization to
simplify this. The pseudo-scalar mass square term is

$\displaystyle m_{s}^{2}$ $\displaystyle=$
$\displaystyle-\frac{m_{12\phi}v_{1}v_{\phi}}{2\sqrt{2}v_{2}}$ (12)

Our primary focus lies in the mixing of the CP-even scalars, as this process
leads to the emergence of the desired light scalar field. The three neutral
CP-even components undergo mixing amongst themselves, described by the mass
square matrix,

$M^{2}_{\rho}=\left(\begin{array}[]{ccc}\lambda_{1}v_{1}^{2}-\frac{m_{12\phi}v_{2}v_{\phi}}{\sqrt{2}v_{1}}&(\lambda_{3}+\lambda_{4})v_{1}v_{2}+\frac{m_{12\phi}v_{\phi}}{\sqrt{2}}&\lambda_{1\phi}v_{1}v_{\phi}+\frac{m_{12\phi}v_{2}}{\sqrt{2}}\\\
(\lambda_{3}+\lambda_{4})v_{1}v_{2}+\frac{m_{12\phi}v_{\phi}}{\sqrt{2}}&\lambda_{2}v_{2}^{2}-\frac{m_{12\phi}v_{1}v_{\phi}}{\sqrt{2}v_{2}}&\lambda_{1\phi}v_{2}v_{\phi}+\frac{m_{12\phi}v_{1}}{\sqrt{2}}\\\
\lambda_{1\phi}v_{1}v_{\phi}+\frac{m_{12\phi}v_{2}}{\sqrt{2}}&\lambda_{1\phi}v_{2}v_{\phi}+\frac{m_{12\phi}v_{1}}{\sqrt{2}}&\lambda_{\phi}v_{\phi}^{2}-\frac{m_{12\phi}v_{1}v_{2}}{\sqrt{2}v_{\phi}}\end{array}\right)\leavevmode\nobreak\
.\leavevmode\nobreak\ \,$ (13)

This mass square matrix can be diagonalized as $R^{T}M^{2}_{\rho}R$ using a
$3\times 3$ orthogonal matrix $R_{3\times 3}$ that can be parametrized using
three rotation angles.

$\displaystyle R_{3\times 3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}r_{11}&r_{12}&r_{13}\\\
r_{21}&r_{22}&r_{23}\\\
r_{31}&r_{32}&r_{33}\end{array}\right)\leavevmode\nobreak\
,\leavevmode\nobreak\ \,$ (17)

where $r_{ij}$’s are the functions of $sine$ and $cosine$ of the three
rotation angles. The physical states can be defined as
$R^{T}(\rho_{1},\rho_{2},\rho_{\phi})$. The interaction states can be expanded
as superposition of the physical states as,

$\displaystyle\rho_{1}$ $\displaystyle=$ $\displaystyle
r_{11}h+r_{12}h_{2}+r_{13}h_{1}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \,$
$\displaystyle\rho_{2}$ $\displaystyle=$ $\displaystyle
r_{21}h+r_{22}h_{2}+r_{23}h_{1}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \,$
$\displaystyle\rho_{\phi}$ $\displaystyle=$ $\displaystyle
r_{31}h+r_{32}h_{2}+r_{33}h_{1}\leavevmode\nobreak\ .\leavevmode\nobreak\ \,$
(18)

Here $h$ can be identified as the SM Higgs field; $h_{2}$ can be heavy such
that it satisfy the LHC bounds. And we want a scenario where $h_{1}$ is a
light scalar with mass $\mathcal{O}(10)$ MeV, which will play key roles to
explain the muon $(g-2)$ and the MiniBooNE excess (see Secs. 4 and 5,
respectively for details).

We generate a phenomenologically interesting scalar mass spectrum by selecting
appropriate values for the parameters in Eq. (2) (see for example Ref.
Dutta:2020scq ; Dutta:2022fdt  for similar treatment). Specifically, we choose
mass parameters of the order of $\sim\mathcal{O}(100)$ GeV and dimensionless
coupling parameters around $\sim\mathcal{O}(0.1)$. As an example, we consider
the following vacuum expectation values (vevs): $v_{1}=238$ GeV; $v_{2}=62$
GeV; and $v_{\phi}=10$ GeV. This parameter selection leads to the following
scalar spectrum: $m_{h}=125$ GeV; $m_{h_{2}}=500$ GeV; and
$m_{h_{1}}=\mathcal{O}(10-100)$ MeV.

The SM Higgs boson $h$ can decay into a pair of $h_{1}$ particles,
contributing to the invisible decay channel of the SM Higgs. Experimental
searches for such invisible decay channels at the LHC have yielded null
results so far, constraining the decay branching fraction to $Br(h\rightarrow
invisible)<0.15$ at a $95\%$ confidence level CMS:2023sdw ; ATLAS:2023tkt .
For the chosen parameter set, the coupling $hh_{1}h_{1}$ is of the order of
$\mathcal{O}(0.1)$, and the corresponding branching fraction is approximately
$\mathcal{O}(0.01)$.

###  2.2 Fermion sector

In the fermion sector we introduce three SM singlet right handed fields
$n_{R}^{\prime}$ which are singlet under $U(1)_{H}$ as well. In addition to
this we introduce a pair of Weyl fermion $\psi_{L,R}$ which are charged under
$U(1)_{H}$ with charge 1 but singlet under SM. The new fermionic fields posses
following quantum numbers within the context of the $SU(2)_{L}\times
U(1)_{Y}\times U(1)_{H}$ symmetry:

$n^{\prime}_{R}\sim(1,0,0),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\psi^{\prime}_{L}\sim(1,0,1),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \psi^{\prime
c}_{R}\sim(1,0,-1)\leavevmode\nobreak\ ,\leavevmode\nobreak\ \,$ (19)

The charges of $\psi^{\prime}_{L}$ and $\psi^{\prime}_{R}$ are chosen in such
a way to cancel the anomalies in the model. The Lagrangian responsible for
fermion masses and mixing can be given by,

$\displaystyle-\mathcal{L}_{Y}$ $\displaystyle=$
$\displaystyle\bar{q}^{\prime}_{L_{i}}(y^{\prime}_{d})_{ij}d^{\prime}_{R_{j}}H_{1}+\bar{q}^{\prime}_{L_{i}}(y^{\prime}_{u})_{ij}u^{\prime}_{R_{j}}\tilde{H_{1}}$
(20)
$\displaystyle+\bar{l}^{\prime}_{L_{i}}(y^{\prime}_{e})_{ij}e^{\prime}_{R_{j}}H_{1}+\bar{l}^{\prime}_{L_{i}}(y^{\prime}_{n})_{ij}n^{\prime}_{R_{j}}\tilde{H_{1}}+\frac{1}{2}\bar{n}^{\prime
c}_{R_{i}}{M}^{\prime}_{ij}n^{\prime}_{R_{j}}+m_{\psi}\bar{\psi}^{\prime}_{R}\psi^{\prime}_{L}$
$\displaystyle+\bar{n}^{\prime
c}_{R_{i}}(y^{\prime}_{1\psi})_{i}\psi_{R}^{\prime}\phi^{\dagger}_{H}+\bar{\psi}^{\prime}_{L}(y_{2\psi}^{\prime})_{j}n^{\prime}_{R_{j}}\phi_{H}+H.c.$

where primed fermions are interaction basis fermions; $i$ and $j$ are the
family indices, $i,j=1,2,3$. The first two terms give down-type and up-type
quark masses and their interactions with physical scalars. The next four terms
give charged lepton masses, Dirac mass term for neutrinos, Majorana mass term
for right handed neutrinos, and mass term for the fermion $\psi$ respectively.
Note that there is no Yukawa type term for the second doublet $H_{2}$ as it is
charged under the new gauge group. Therefore our model is an $U(1)_{H}$
extension of the Type I 2HDM. Though dimension five non-renormalizable term
could be possible, but we do not consider them. In general, all the Yukawa
matrices are $3\times 3$ complex matrices, but for simplicity we assume that
they are real.

The Yukawa matrices can be diagonalized using biunitary transformation and the
Majorana mass matrix can be diagonalized using unitary transformation as
follows,

$\displaystyle y_{d}$ $\displaystyle=$ $\displaystyle
U^{\dagger}_{d_{L}}y^{\prime}_{d}U_{d_{R}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{with}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (y_{d})_{ij}=(y_{1d})_{ii}\delta_{ij}\leavevmode\nobreak\
,\leavevmode\nobreak\ \,$ (21) $\displaystyle y_{u}$ $\displaystyle=$
$\displaystyle U^{\dagger}_{u_{L}}y^{\prime}_{u}U_{u_{R}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{with}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (y_{u})_{ij}=(y_{u})_{ii}\delta_{ij}\leavevmode\nobreak\
,\leavevmode\nobreak\ \,$ (22) $\displaystyle y_{e}$ $\displaystyle=$
$\displaystyle U^{\dagger}_{e_{L}}y^{\prime}_{e}U_{e_{R}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{with}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (y_{e})_{ij}=(y_{1e})_{ii}\delta_{ij}\leavevmode\nobreak\
,\leavevmode\nobreak\ \,$ (23) $\displaystyle y_{n}$ $\displaystyle=$
$\displaystyle
U^{\dagger}_{\nu_{L}}y^{\prime}_{n}U_{n_{R}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{with}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (y_{n})_{ij}=(y_{1n})_{ii}\delta_{ij}\leavevmode\nobreak\
,\leavevmode\nobreak\ \,$ (24) $\displaystyle M$ $\displaystyle=$
$\displaystyle U^{\dagger}_{n_{R}}M^{\prime}U_{n_{R}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{with}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ M_{ij}=M_{ii}\delta_{ij}\leavevmode\nobreak\
,\leavevmode\nobreak\ \,$ (25)

We further define,

$\displaystyle y_{1\psi}$ $\displaystyle=$ $\displaystyle
U^{\dagger}_{n_{R}}y^{\prime}_{1\psi}\leavevmode\nobreak\
,\leavevmode\nobreak\ \,$ (26) $\displaystyle y_{2\psi}$ $\displaystyle=$
$\displaystyle y^{\prime}_{2\psi}U_{n_{R}}\leavevmode\nobreak\
,\leavevmode\nobreak\ \,$ (27)

Here $U$’s are $3\times 3$ Unitary matrices which can be used to define the
fermions in the mass basis as follows,

$\displaystyle\left(d_{{L/R}}\right)_{i}$ $\displaystyle=$
$\displaystyle\left(U^{\dagger}_{d_{L/R}}\right)_{ij}\left(d^{\prime}_{{L/R}}\right)_{j}\leavevmode\nobreak\
,\leavevmode\nobreak\ \,$ (28) $\displaystyle\left(u_{{L/R}}\right)_{i}$
$\displaystyle=$
$\displaystyle\left(U^{\dagger}_{u_{L/R}}\right)_{ij}\left(u^{\prime}_{{L/R}}\right)_{j}\leavevmode\nobreak\
,\leavevmode\nobreak\ \,$ (29) $\displaystyle\left(e_{{L/R}}\right)_{i}$
$\displaystyle=$
$\displaystyle\left(U^{\dagger}_{e_{L/R}}\right)_{ij}\left(e^{\prime}_{{L/R}}\right)_{j}\leavevmode\nobreak\
,\leavevmode\nobreak\ \,$ (30) $\displaystyle\left(\nu_{{L}}\right)_{i}$
$\displaystyle=$
$\displaystyle\left(U^{\dagger}_{\nu_{L}}\right)_{ij}\left(\nu^{\prime}_{{L}}\right)_{j}\leavevmode\nobreak\
,\leavevmode\nobreak\ \,$ (31) $\displaystyle\left(n_{{R}}\right)_{i}$
$\displaystyle=$
$\displaystyle\left(U^{\dagger}_{n_{R}}\right)_{ij}\left(n^{\prime}_{{R}}\right)_{j}$
(32)

Now Eq. (20) can be rewritten as,

$\displaystyle-\mathcal{L}_{Y}$ $\displaystyle=$
$\displaystyle{m_{d}}_{i}\bar{d}_{i}d_{i}+\frac{{m_{d}}_{i}}{v_{1}}\bar{d}_{i}d_{i}\rho_{1}+{m_{u}}_{i}\bar{u}_{i}u_{i}+\frac{{m_{u}}_{i}}{v_{1}}\bar{u}_{i}u_{i}\rho_{1}+{m_{e}}_{i}\bar{e}_{i}e_{i}+\frac{{m_{e}}_{i}}{v_{1}}\bar{e}_{i}e_{i}\rho_{1}$
(34)
$\displaystyle+{m_{\nu_{D}}}_{i}(\bar{\nu}_{L_{i}}n_{R_{i}}+\bar{n}_{R_{i}}\nu_{L_{i}})+\frac{{m_{\nu_{D}}}_{i}}{v_{1}}(\bar{\nu}_{L_{i}}n_{R_{i}}+\bar{n}_{R_{i}}\nu_{L_{i}})\rho_{1}$
$\displaystyle+\frac{1}{2}M_{i}\left(\bar{n}^{c}_{R_{i}}n_{R_{i}}+\bar{n}_{R_{i}}n^{c}_{R_{i}}\right)+m_{\psi}\bar{\psi}\psi$
$\displaystyle+\frac{(y_{1\psi})_{i}v_{X}}{\sqrt{2}}\left(\bar{n}^{c}_{R_{i}}\psi_{R}+\bar{\psi}_{R}n^{c}_{R_{i}}\right)+\frac{(y_{2\psi})_{i}v_{X}}{\sqrt{2}}\left(\bar{\psi}_{L}n_{R_{i}}+\bar{n}_{R_{i}}\psi_{L}\right)$
$\displaystyle+\frac{(y_{1\psi})_{i}}{\sqrt{2}}\left(\bar{n}^{c}_{R_{i}}\psi_{R}+\bar{\psi}_{R}n^{c}_{R_{i}}\right)\rho_{\phi}+\frac{(y_{2\psi})_{i}}{\sqrt{2}}\left(\bar{\psi}_{L}n_{R_{i}}+\bar{n}_{R_{i}}\psi_{L}\right)\rho_{\phi}$

where we have used the definitions:
$m_{f_{i}}=(y_{f})_{ii}\delta_{ij}v_{1}/\sqrt{2}$ with $f=d,u,e$; and
$m_{{\nu_{D}}_{i}}=(y_{n})_{ii}\delta_{ij}v_{1}/\sqrt{2}$. Now let us
decompose the lagrangian for neutrino sector and rest of the fermions as
follows,

$-\mathcal{L}_{Y}=-\mathcal{L}_{Y}(f=d,u,e)-\mathcal{L}_{Y}(\nu,n,\psi)$ (35)

here the first term can be expressed compactly as,

$-\mathcal{L}_{Y}(f=d,u,e)=m_{f_{i}}\bar{f}_{i}f_{i}+\bar{f}_{i}(y_{f\phi})_{i}f_{i}\phi$
(36)

where $\phi=h,h_{2},h_{1}$; and the couplings $(y_{f\phi})_{i}$ are defined
as,

$\displaystyle(y_{fh})_{i}$ $\displaystyle=$
$\displaystyle\frac{m_{f_{i}}}{v_{1}}r_{11}$ (37)
$\displaystyle(y_{fh_{2}})_{i}$ $\displaystyle=$
$\displaystyle\frac{m_{f_{i}}}{v_{1}}r_{12}$ (38)
$\displaystyle(y_{fh_{1}})_{i}$ $\displaystyle=$
$\displaystyle\frac{m_{f_{i}}}{v_{1}}r_{13}$ (39)

Note that there is no tree level flavor changing scalar current and also there
is no charged scalar current or pseudo scalar current. The scalars has
couplings to SM fermions proportional to their mass.

The Lagrangian for neutrino sector can be written as

$\displaystyle-\mathcal{L}_{Y}(\nu,n,\psi)$ $\displaystyle=$
$\displaystyle{m_{\nu_{D}}}_{i}(\bar{\nu}_{L_{i}}n_{R_{i}}+\bar{n}_{R_{i}}\nu_{L_{i}})+\frac{1}{2}M_{i}\left(\bar{n}^{c}_{R_{i}}n_{R_{i}}+\bar{n}_{R_{i}}n^{c}_{R_{i}}\right)+m_{\psi}\bar{\psi}\psi$
(40)
$\displaystyle+m_{{1n\psi}_{i}}\left(\bar{n}^{c}_{R_{i}}\psi_{R}+\bar{\psi}_{R}n^{c}_{R_{i}}\right)+m_{{2n\psi}_{i}}\left(\bar{\psi}_{L}n_{R_{i}}+\bar{n}_{R_{i}}\psi_{L}\right)$
$\displaystyle+\frac{{m_{\nu_{D}}}_{i}}{v_{1}}(\bar{\nu}_{L_{i}}n_{R_{i}}+\bar{n}_{R_{i}}\nu_{L_{i}})\rho_{1}$
$\displaystyle+\frac{m_{{1n\psi}_{i}}}{v_{X}}\left(\bar{n}^{c}_{R_{i}}\psi_{R}+\bar{\psi}_{R}n^{c}_{R_{i}}\right)\rho_{\phi}+\frac{m_{{2n\psi}_{i}}}{v_{X}}\left(\bar{\psi}_{L}n_{R_{i}}+\bar{n}_{R_{i}}\psi_{L}\right)\rho_{\phi}$

where we have defined, $m_{{1n\psi}_{i}}=(y_{1\psi})_{i}v_{\phi}/\sqrt{2}$ and
$m_{{2n\psi}_{i}}=(y_{2\psi})_{i}v_{\phi}/\sqrt{2}$. The terms in the first
two lines are responsible for neutrino masses and mixing while the term in the
third and fourth lines give interaction of neutrinos with physical scalar
particles.

###  2.3 Neutrino masses and mixings

Part of Eq. (40) that gives rise to neutrino masses and mixings is given by,

$\displaystyle-\mathcal{L}^{mass}_{Y}(\nu,n)$ $\displaystyle=$
$\displaystyle{m_{\nu_{D}}}_{i}(\bar{\nu}_{L_{i}}n_{R_{i}}+\bar{n}_{R_{i}}\nu_{L_{i}})+\frac{1}{2}M_{i}\left(\bar{n}^{c}_{R_{i}}n_{R_{i}}+\bar{n}_{R_{i}}n^{c}_{R_{i}}\right)+m_{\psi}\bar{\psi}\psi$
(45)
$\displaystyle+m_{{1n\psi}_{i}}\left(\bar{n}^{c}_{R_{i}}\psi_{R}+\bar{\psi}_{R}n^{c}_{R_{i}}\right)+m_{{2n\psi}_{i}}\left(\bar{\psi}_{L}n_{R_{i}}+\bar{n}_{R_{i}}\psi_{L}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\bar{\nu}^{c}_{L_{i}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \bar{n}_{R_{i}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\bar{\psi}^{c}_{L}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \bar{\psi}_{R}\right)\leavevmode\nobreak\
M^{\prime}_{\nu}\leavevmode\nobreak\ \left(\begin{array}[]{c}\nu_{L_{i}}\\\
n^{c}_{R_{i}}\\\ \psi_{L}\\\ \psi^{c}_{R}\end{array}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\bar{\nu}_{a_{i}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\bar{\nu}_{s_{j}}\right)M_{\nu}\left(\begin{array}[]{c}\nu_{a_{i}}\\\
\nu_{s_{j}}\end{array}\right)\leavevmode\nobreak\ ,\leavevmode\nobreak\ \,$
(48)

where the $8\times 8$ neutrino mass matrix $M^{\prime}_{\nu}$ is given by,

$M^{\prime}_{\nu}=\left(\begin{array}[]{cccc}0&{{m^{T}_{\nu_{D}}}_{i}}&0&0\\\
{{m_{\nu_{D}}}_{i}}&M_{i}&m_{{2n\psi}_{i}}&m_{{1n\psi}_{i}}\\\
0&m_{{2n\psi}_{i}}&0&m_{\psi}\\\
0&m_{{1n\psi}_{i}}&m_{\psi}&0\end{array}\right)\leavevmode\nobreak\
,\leavevmode\nobreak\ \,$ (49)

We diagonalize the neutrino mass matrix using an Unitary matrix $U_{\nu}$ as

$U^{T}_{\nu}M^{\prime}_{\nu}U_{\nu}=M_{\nu}=diag(m_{1},m_{2},....m_{8})$ (50)

We also define mass states as

$\left(\begin{array}[]{c}\nu_{L_{i}}\\\ n^{c}_{R_{i}}\\\ \psi_{L}\\\
\psi^{c}_{R}\end{array}\right)=U_{\nu}\left(\begin{array}[]{c}\nu_{a_{i}}\\\
\nu_{s_{j}}\end{array}\right)$ (51)

where $i=1,2,3$ and $j=1,2,3,4,5$. Therefore we have three active neutrino
states which we identify as the SM light neutrinos while we have 5 sterile
neutrinos which we identify as heavy neutrinos.

The absolute masses of neutrinos remain unknown; instead, we have information
about the differences in their squared masses, denoted as $\Delta
m^{2}_{ij}\equiv m_{\nu_{a_{i}}}^{2}-m_{\nu_{a_{j}}}^{2}$. In the case of the
normal hierarchy, where $m_{\nu_{a_{1}}}\ll m_{\nu_{a_{2}}}<m_{\nu_{a3}}$,
global fits of neutrino oscillation data deSalas:2020pgw  yield two distinct
mass squared differences: $\Delta m_{21}^{2}=7.506\times 10^{-5}$ eV2 and
$\Delta m_{31}^{2}=2.55\times 10^{-3}$ eV2. Assuming that the lightest active
neutrino is massless, we can deduce the following neutrino masses for active
states: $m_{\nu_{a_{i}}}=(0.0,0.0087,0.0505)$ eV. Simultaneously, we aim to
generate one sterile neutrino with masses of the order of
$m_{\nu_{s_{1}}}\sim\mathcal{O}(10)$ keV and one with mass
$m_{\nu_{s_{2}}}\sim\mathcal{O}(100)$ MeV to solve MiniBooNE excess events
problem. One set of benchmark values capable of producing this intriguing
sterile neutrino mass spectrum can be derived with mixing parameters on the
order of $\mathcal{O}(10^{-4}-10^{-5})$ Dutta:2020scq .

The last three terms in Eq. (40) introduces interactions between the neutrinos
and the scalars. Specifically, it generates interaction terms like
$y_{\nu_{i}}\bar{\nu}_{a_{i}}\nu_{s_{i}}h_{1}$ and
$y_{s_{i}}\bar{\nu}_{s_{i}}\nu_{s_{i}}h_{1}$. Because of neutrino mixing, the
couplings $y_{\nu}$ and $y_{s}$ are proportional to $m_{{{1,2}n\psi}_{i}}$,
which are unbounded and can therefore be treated as free parameters.
Therefore, scalar couplings to neutrinos can be large compared to the other
fermion couplings. The light scalar field, $h_{1}$, would predominantly decay
into $e^{+}e^{-}$ and $\bar{\nu}_{s_{1}}\nu_{s_{1}}$ pairs. We can adjust the
couplings $y_{e{h_{1}}}$ and $y_{s_{1}}$ in such a way that we achieve
approximately $Br(h_{1}\rightarrow\bar{\nu}_{s_{1}}\nu_{s_{1}})\simeq 0.95$
and $Br(h_{1}\rightarrow e^{+}e^{-})\simeq 0.05$, with the decay occurring
promptly inside the detectors. For example, if we choose coupling
$y_{s_{1}}\sim\mathcal{O}(10^{-}5)$ and $m_{\nu_{s_{1}}}\sim\mathcal{O}(10)$
keV, then the lifetime of $h_{1}$ becomes $10^{-14}$ sec.

###  2.4 Gauge sector

We denote the field strenth tensor of $U(1)_{Y}$ and $U(1)_{H}$ as
$B_{\alpha\beta}$ and $X_{\alpha\beta}$ respectively. The kinetic energy can
be expressed in the form,

$\mathcal{L}_{kinetic}=-\frac{1}{4}B_{\alpha\beta}B^{\alpha\beta}-\frac{1}{4}X_{\alpha\beta}X^{\alpha\beta}-\frac{2\alpha}{4}B_{\alpha\beta}X^{\alpha\beta}$
(52)

where $\alpha$ is a physical parameter that gives the strength of kinetic
mixing between the two Abelian fields and the requirement of positive kinetic
energy requires that $|\alpha|<1$. After diagonalizing the kinetic term by a
non-unitary transformation of the fields, the covariant derivative can be
expressed in terms of the transformed fields as Foot:1991kb ,

$\displaystyle
D_{\rho}\mathbb{I}_{2}=\partial_{\rho}\mathbb{I}_{2}+i\frac{g}{2}\tau^{a}W^{a}_{\rho}+ig^{\prime}YB_{\rho}\mathbb{I}_{2}+i\left(\frac{g_{X}}{2}Q_{H}+g^{\prime}_{X}\right)X_{\rho}\mathbb{I}_{2}$
(53)

where $g,g^{\prime}$ and $g_{X}$ are the coupling constants of
$SU(2)_{L},U(1)_{Y}$ and $U(1)_{H}$ gauge groups respectively with values
$g=0.64,g^{\prime}=0.34$;
$g^{\prime}_{X}=-g^{\prime}\alpha/\sqrt{1-\alpha^{2}}$ denotes the gauge
coupling mixing between the two $U(1)$ groups; $W^{a}_{\rho},B_{\rho}$ and
$X_{\rho}$ are the corresponding gauge bosons in interaction basis; and $Y$ is
the hypercharge and $Q_{H}$ is the charge under $U(1)_{H}$ gauge group. Now
the effect of the kinetic mixing is embedded in the definition of the gauge
covariant derivative.

Gauge boson masses and mixing can be calculated from the kinetic terms of the
scalar fields. While $H_{1}$ is uncharged under the new gauge group,
$|D_{\mu}H_{1}|^{2}$ contributes to the SM gauge boson masses and the new
gauge boson mass term through the kinetic mixing effect. On the other hand,
both $H_{2}$ and $\phi_{H}$ are charged under $U(1)_{H}$ and therefore
contribute not only to the SM gauge boson masses but also to the new gauge
boson mass directly.

Therefore the terms of the Lagrangian giving gauge boson masses and mixing is
,

$\mathcal{L}\subset\left(D^{\rho}H_{1}\right)^{\dagger}\left(D_{\rho}H_{1}\right)+\left(D^{\rho}H_{2}\right)^{\dagger}\left(D_{\rho}H_{2}\right)+\left(D^{\rho}\phi_{H}\right)^{\dagger}\left(D_{\rho}\phi_{H}\right)$
(54)

The charged gauge boson mass gets contribution from both $|D_{\mu}H_{1}|^{2}$
and $|D_{\mu}H_{2}|^{2}$ and is given by, $m_{W^{\pm}}^{2}=g^{2}v^{2}/4$. Now
collecting all the mass terms for the neutral gauge bosons, the mass squared
term can be written as,

$\mathcal{L}^{0,gauge}_{mass}=\frac{1}{2}\left(B^{\rho}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ W_{3}^{\rho}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ X^{\rho}\right)\leavevmode\nobreak\
\left(M^{0}_{gauge}\right)^{2}_{3\times 3}\leavevmode\nobreak\
\left(\begin{array}[]{c}B_{\rho}\\\ W_{3\rho}\\\
X_{\rho}\end{array}\right)\leavevmode\nobreak\ ,\leavevmode\nobreak\ \,$ (55)

where the mass square matrix is given by,

$\left(M^{0}_{gauge}\right)^{2}_{3\times
3}=\frac{v^{2}}{4}\left(\begin{array}[]{ccc}{g^{\prime}}^{2}&-g^{\prime}g&g^{\prime}A_{2}\\\
-g^{\prime}g&g^{2}&-gA_{2}\\\
g^{\prime}A_{2}&-gA_{2}&A_{1}\end{array}\right)\leavevmode\nobreak\
.\leavevmode\nobreak\ \,$ (56)

where we have defined

$\displaystyle A_{2}$ $\displaystyle=$ $\displaystyle g^{\prime}_{X}+g_{X}A$
$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle{g^{\prime}_{X}}^{2}+g_{X}^{2}B+2g_{X}g^{\prime}_{X}A$ (57)

where the quantities $A$ and $B$ are defined as,

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{v_{2}^{2}}{v_{1}^{2}v_{2}^{2}}$ (58) $\displaystyle B$
$\displaystyle=$
$\displaystyle\frac{v_{2}^{2}+v_{\phi}^{2}}{v_{1}^{2}v_{2}^{2}}$ (59)

To obtain the masses of the physical gauge bosons, we aim to diagonalize the
mass-squared matrix into the form $\left(M^{0}_{diag}\right)^{2}_{3\times
3}=diag\left(m_{A}^{2},m_{Z}^{2},m_{Z^{\prime}}^{2}\right)$. Note that
$m_{A}^{2}=0$, which allows us to diagonalize this mass matrix using two
block-diagonal rotations: one with Weinberg’s angle $\theta_{W}$ and another
with an additional angle $\xi$. This process can be expressed as follows:

$\left(M^{0}_{diag}\right)^{2}_{3\times
3}=diag\left(0,m_{Z}^{2},m_{Z^{\prime}}^{2}\right)=R_{0,gauge}(\theta_{W},\xi)\left(M^{0}_{gauge}\right)^{2}R^{T}_{0,gauge}(\theta_{W},\xi)$
(60)

where the diagonalization matrix is given by,

$\displaystyle R_{0,gauge}(\theta_{W},\xi)$ $\displaystyle=$ $\displaystyle
R_{1}(\xi)R_{2}(\theta_{W})$ (67) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\ 0&\cos{\xi}&\sin{\xi}\\\
0&-\sin{\xi}&\cos{\xi}\end{array}\right)\left(\begin{array}[]{ccc}\cos{\theta_{W}}&\sin{\theta_{W}}&0\\\
-\sin{\theta_{W}}&\cos{\theta_{W}}&0\\\ 0&0&1\end{array}\right)$

The mixing angles can be expressed in terms of the gauge couplings:
$\theta_{W}$, the Weinberg’s angle, can be defined as
$\tan{\theta_{W}}=g^{\prime}/g$; and $\xi$ is the new mixing angle, which can
be expressed in terms of the new gauge coupling as,

$\tan{2\xi}=\frac{2g_{Z}A_{2}}{A_{1}-g_{Z}^{2}}$ (68)

where we have defined $g_{Z}=\sqrt{g^{2}+{g^{\prime}}^{2}}$. The mass of the
new gauge boson $Z^{\prime}$ depends on the choice of the couplings $g_{X}$
and $g_{X}^{\prime}$, as well as the vev $v_{\phi}$. It’s worth noting that
for $v_{\phi}=10$ GeV, with couplings $g_{X}\sim\mathcal{O}(10^{-3})$ and
$g_{X}^{\prime}\sim\mathcal{O}(10^{-4})$, we can generate a $Z^{\prime}$ with
a mass of around $\mathcal{O}(10)$ MeV, which can explain the $e^{+}e^{-}$
excess around 17 MeV reported by the ATOMKI Collaboration.

Note that we are primarily interested in the limit where $g_{X}$ and
$g^{\prime}_{X}\ll 1$. In this limit, the expression for the angle can be
simplified. First, note that $A$ and $B$ are of order $\mathcal{O}(1)$, so we
have $A_{1}-g_{Z}^{2}\simeq-g_{Z}^{2}$. Therefore, the right-hand side of Eq.
(68) is very small, allowing us to use the small angle approximation. In this
limit, Eq. (68) can be written as

$g_{Z}\xi=-g^{\prime}_{X}-g_{X}\sin^{2}{\beta}$ (69)

Define a vector
$\left(Z^{\prime\rho}_{0}\right)^{T}\equiv\left(B^{\rho},W^{3\rho},X^{\rho}\right)$
in interaction basis and a vector
$\left(Z^{\rho}_{0}\right)^{T}\equiv\left(A^{\rho},Z^{\rho},Z^{\prime\rho}\right)$
in mass basis. Consequently, Eq. (55) can be written as,

$\displaystyle\mathcal{L}^{0,gauge}_{mass}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(Z^{\prime\rho}_{0}\right)^{T}R^{T}_{0,gauge}(\theta_{W},\xi)\left(M^{0}_{diag}\right)^{2}R_{0,gauge}(\theta_{W},\xi)Z^{\prime}_{0\rho}$
(70) $\displaystyle=$
$\displaystyle\frac{1}{2}\left(Z^{\rho}_{0}\right)^{T}\left(M^{0}_{diag}\right)^{2}Z_{0\rho}$

where we have defined,

$Z_{0\rho}=R_{0,gauge}(\theta_{W},\xi)Z^{\prime}_{0\rho}$ (71)

that connects the interaction states and the mass states.

Therefore, we can explicitly express the states in the interaction basis in
terms of the mass basis as follows:

$\displaystyle B_{\mu}$ $\displaystyle=$
$\displaystyle\cos{\theta_{W}}A_{\mu}-\sin{\theta_{W}}\cos{\xi}Z_{\mu}+\sin{\theta_{W}}\sin{\xi}Z^{\prime}_{\mu}\leavevmode\nobreak\
,\leavevmode\nobreak\ \,$ $\displaystyle W_{3\mu}$ $\displaystyle=$
$\displaystyle\sin{\theta_{W}}A_{\mu}+\cos{\theta_{W}}\cos{\xi}Z_{\mu}-\cos{\theta_{W}}\sin{\xi}Z^{\prime}_{\mu}\leavevmode\nobreak\
,\leavevmode\nobreak\ \,$ $\displaystyle X_{\mu}$ $\displaystyle=$
$\displaystyle\sin{\xi}Z_{\mu}+\cos{\xi}Z^{\prime}_{\mu}\leavevmode\nobreak\
.\leavevmode\nobreak\ \,$ (72)

The terms of the model Lagrangian that give the interactions between the
fermions and the gauge bosons are,

$\displaystyle-\mathcal{L}_{kin,fermion}$ $\displaystyle=$
$\displaystyle\bar{q}^{\prime}_{L_{j}}i\gamma^{\rho}D_{\rho}{q}^{\prime}_{L_{j}}+\bar{u}^{\prime}_{R_{j}}i\gamma^{\rho}D_{\rho}u^{\prime}_{R_{j}}+\bar{d}^{\prime}_{R_{j}}i\gamma^{\rho}D_{\rho}d^{\prime}_{R_{j}}$
(73)
$\displaystyle+\bar{l}^{\prime}_{L_{j}}i\gamma^{\rho}D_{\rho}{l}^{\prime}_{L_{j}}+\bar{n}^{\prime}_{R_{j}}i\gamma^{\rho}D_{\rho}n^{\prime}_{R_{j}}+\bar{e}^{\prime}_{R_{j}}i\gamma^{\rho}D_{\rho}e^{\prime}_{R_{j}}$
$\displaystyle+\bar{\psi}^{\prime}_{L}i\gamma^{\rho}D_{\rho}\psi^{\prime}_{L}+\bar{\psi}^{\prime}_{R}i\gamma^{\rho}D_{\rho}\psi^{\prime}_{R}$

By expanding the terms in Eq. (73), we obtain the currents that couple to
charged and neutral gauge bosons. It is important to note that the charged
boson gauge current and the $A$ current remain the same as in the SM, while
the $Z$ gauge current will be modified due to the neutral gauge boson mixing.
In the limit of a small mixing angle, the SM $Z$ current can be recovered from
the modified current. The most crucial interaction current for our analysis is
the $Z^{\prime}$ current, which is given by:

$\mathcal{L}^{NC}_{Z^{\prime}}=-\sum_{f}\bar{f}\gamma^{\rho}\left(C_{f,V}+C_{f,A}\gamma^{5}\right)fZ^{\prime}_{\rho}$
(74)

where we have defined,

$\displaystyle C_{f,V}$ $\displaystyle=$ $\displaystyle
g_{Z}\sin{\xi}\left(-\frac{1}{2}T^{3}_{f}+\sin^{2}{\theta_{W}}Q_{f}\right)+g_{X}^{\prime}\cos{\xi}\left(Q_{f}-\frac{1}{2}T^{3}_{f}\right)$
$\displaystyle C_{f,A}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(g_{Z}\sin{\xi}+g_{X}^{\prime}\cos{\xi}\right)T^{3}_{f}$
(75)

In the limit, $g_{X},g_{X}^{\prime}\ll 1$ we can simplify the above
expressions and get,

$\displaystyle C_{f,V}$ $\displaystyle=$ $\displaystyle-
g_{X}\sin^{2}{\beta}\left(-\frac{1}{2}T^{3}_{f}+\sin^{2}{\theta_{W}}Q_{f}\right)+g_{X}^{\prime}\cos^{2}{\theta_{W}}Q_{f}$
$\displaystyle C_{f,A}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}g_{X}\sin^{2}{\beta}T^{3}_{f}$ (76)

Note that the gauge interaction currents of $Z^{\prime}$ depend solely on the
couplings $g_{X}$ and $g_{X}^{\prime}$. Additionally, the interactions of
$Z^{\prime}$ with the physical mass states of neutrinos will be further
suppressed by the mixing angles in the neutrino sector in comparison to the
electron interactions. Therefore, we can neglect the neutrino interactions by
appropriately choosing small mixing angles in the neutrino sector.

##  3 Beryllium Anomaly at ATOMKI Experiment

In 2015, the ATOMKI spectrometer detected an anomalous peak at an opening
angle of $140\degree$, accompanied by a corresponding bump in the $(m_{ee})$
distribution during the IPC decays of 8Be${}^{*}(18.15)$ Krasznahorkay:2015iga
; Krasznahorkay:2018snd . This behavior starkly contrasts with the expected SM
behavior, where both the $(\theta_{ee})$ and $(m_{ee})$ distributions are
anticipated to be monotonically falling curves. The bump observed in the data
exhibited a remarkable statistical significance of $6.8\sigma$ and could not
be attributed to experimental or nuclear physics effects. Subsequently, in
2018, the experiment was repeated with an improved setup, reaffirming the
presence of the anomaly and its consistency with the earlier measurements
Krasznahorkay:2018snd . Additionally, it was discovered that the anomaly
disappeared when the resonance peak was no longer present, leading to the
conclusion that it originated from the decay of the resonant excited states.

The observed bump in the data can be explained by the production of a massive,
short-lived neutral particle with low velocity during the decay of
8Be${}^{*}(18.15)$. This particle subsequently decays into an $e^{+}e^{-}$
pair, leading to the emergence of a peak at a large opening angle in the
experimental results.

${}^{8}\text{Be}^{*}(18.15)\rightarrow\text{${}^{8}$Be}+X(X\rightarrow
e^{+}e^{-})$ (77)

The best-fit mass and decay branching ratio for this hypothetical neutral
boson have been estimated with a goodness-of-fit statistic of
$\chi^{2}/dof=1.07$ Krasznahorkay:2018snd .

$\displaystyle m_{X}=16.70\pm 0.35(stat)\pm 0.5(sys)\leavevmode\nobreak\
\text{MeV}$ (78)
$\displaystyle\frac{\Gamma\left({}^{8}\text{Be}^{*}(18.15)\rightarrow\text{${}^{8}$Be}\leavevmode\nobreak\
X\right)}{\Gamma\left({}^{8}\text{Be}^{*}(18.15)\rightarrow\text{${}^{8}$Be}\leavevmode\nobreak\
\gamma\right)}Br(X\rightarrow e^{+}e^{-})=5.8\times 10^{-6}$ (79)

In 2019 and more recently in 2022, the ATOMKI collaboration successfully
replicated its experiments using 4He Krasznahorkay:2019lyl ;
Krasznahorkay:2021joi  and 12C Krasznahorkay:2022pxs  nuclei. They observed
similar anomalous peaks, with 4He nuclei exhibiting an opening angle of
$115\degree$ and 12C nuclei showing peaks at $150\degree-160\degree$. These
observations are consistent with the presence of a new boson with a mass of 17
MeV. However, it is important to note that these results should be further
confirmed and validated with additional statistics and repeated measurements.
To establish the robustness of these findings and to test the hypothesis of
the 17 MeV new boson, new and independent measurements are essential. One
promising avenue is the use of MEG II at PSI MEGII:2018kmf ; Chiappini:2022egy
and the Montreal Tandem accelerator Azuelos:2022nbu , which have setups
similar to that of the ATOMKI experiment. These facilities have the potential
to replicate the measurements on 8Be nuclei and provide high-statistics data
for a more comprehensive analysis. Given the high statistical significance and
the potential for near future investigations, our primary focus remains on the
Beryllium anomaly. Various attempts have been made to explain the ATOMKI
anomaly by new physics involving a 17 MeV bosonic field Feng:2016jff ;
Feng:2016ysn ; DelleRose:2017xil ; DelleRose:2018eic ; DelleRose:2018pgm ;
Nam:2019osu ; Feng:2020mbt ; Barducci:2022lqd ; Denton:2023gat ;
Hostert:2023tkg .

In the following, we demonstrate that the gauge interaction current of the new
gauge boson $Z^{\prime}$ can explain the resonance bump in the Beryllium data
while also satisfying all existing bounds. We start our analysis by writing
the interactions of the fermions with the new gauge boson $Z^{\prime}$. The
interactions can be written using Eq. (2.4) as,

$\displaystyle-\mathcal{L}_{NC,Z^{\prime}}$ $\displaystyle=$
$\displaystyle\bar{u}\gamma^{\rho}\left[\left\\{g_{X}\sin^{2}{\beta}\left(\frac{1}{4}-\frac{2}{3}\sin^{2}{\theta_{W}}\right)+\frac{2}{3}g_{X}^{\prime}\cos^{2}{\theta_{W}}\right\\}+\left\\{-\frac{1}{4}g_{X}\sin^{2}{\beta}\right\\}\gamma_{5}\right]uZ^{\prime}_{\rho}$
(80) $\displaystyle+$
$\displaystyle\bar{d}\gamma^{\rho}\left[\left\\{g_{X}\sin^{2}{\beta}\left(-\frac{1}{4}+\frac{1}{3}\sin^{2}{\theta_{W}}\right)-\frac{1}{3}g_{X}^{\prime}\cos^{2}{\theta_{W}}\right\\}+\left\\{\frac{1}{4}g_{X}\sin^{2}{\beta}\right\\}\gamma_{5}\right]dZ^{\prime}_{\rho}$
$\displaystyle+$
$\displaystyle\bar{e}\gamma^{\rho}\left[\left\\{g_{X}\sin^{2}{\beta}\left(-\frac{1}{4}+\sin^{2}{\theta_{W}}\right)-g_{X}^{\prime}\cos^{2}{\theta_{W}}\right\\}+\left\\{\frac{1}{4}g_{X}\sin{\beta}^{2}\right\\}\gamma_{5}\right]eZ^{\prime}_{\rho}$

The interaction structure of the $Z^{\prime}$ with fermions described in Eq.
80 is capable of resolving the Beryllium anomaly. Before we identify the
parameter space that matches the data, let’s briefly outline a few important
points,

  1. 1.

First, note that the strengths of the vector and axial-vector interactions are
similar, i.e., $C_{V}^{f}\sim C_{A}^{f}$, for appropriate values of the
couplings: $g_{X}\sim\mathcal{O}(10^{-3})$ and
$g^{\prime}_{X}\sim\mathcal{O}(10^{-4})$.

  2. 2.

For pure vector coupling,
$\Gamma(^{8}\text{Be}^{*}\rightarrow^{8}\text{Be}Z^{\prime})\propto
k_{Z^{\prime}}^{3}$, while for pure axial-vector coupling,
$\Gamma(^{8}\text{Be}^{*}\rightarrow^{8}\text{Be}Z^{\prime})\propto
k_{Z^{\prime}}$, where $k_{Z^{\prime}}$ is the momentum of the $Z^{\prime}$.
Since we are considering a low-energy scenario, the momentum of the
$Z^{\prime}$ is small. As we saw in the last point, $C_{V}^{f}\sim C_{A}^{f}$,
therefore, we can neglect the effects of the vector coupling of $Z^{\prime}$
and its interference with the axial-vector component.

  3. 3.

A 17 MeV $Z^{\prime}$ can only decay to $e^{+}e^{-}$ and $\nu\bar{\nu}$. The
diphoton, $\gamma\gamma$, final state is prohibited by the Landau-Yang theorem
Landau:1948kw ; Yang:1950rg . A $3\gamma$ final state is possible but highly
suppressed. It’s important to note that the interaction of the $Z^{\prime}$
with neutrinos will be suppressed by the neutrino mixing angle compared to the
electron couplings. Therefore, we can consider the coupling of the
$Z^{\prime}$ to neutrinos as negligible. This helps circumvent the stringent
constraints from neutrino-electron scattering experiments and ensures that the
$Z^{\prime}$ mostly decays to an $e^{+}e^{-}$ pair with lifetime of the order
of $\mathcal{O}(10^{-10})$ sec.

A light boson coupled to both quarks and leptons is subject to a wide range of
experimental constraints. Below, we summarize the most significant constraints
that would impact any potential explanation of the ATOMKI anomaly involving a
BSM degree of freedom with a mass of approximately 17 MeV in the case of a
spin-1 particle.

  1. 1.

Rare neutral pion decay: The constraints on the quarks’ coupling to
$Z^{\prime}$ arise from the study of the rare pion decay process,
$\pi^{0}\rightarrow\gamma Z^{\prime}$( $Z^{\prime}\rightarrow e^{+}e^{-}$).
Currently, the NA48/2 experiment NA482:2015wmo ; Raggi:2015noa  provides the
strongest bounds on this interaction. These constraints are proportional to
the vector-vector anomaly factor $|C_{u,V}Q_{u}-C_{d,V}Q_{d}|^{2}$, which in
turn depends on the vector couplings of the $Z^{\prime}$. The bound can be
expressed as:$|2C_{u,V}+C_{d,V}|\lesssim 2.4\times
10^{-4}/\sqrt{\text{Br}(Z^{\prime}\rightarrow e^{+}e^{-})}$ for
$m_{Z^{\prime}}\simeq 17$ MeV Feng:2016jff ; Feng:2016ysn . It’s worth noting
that the contribution of the axial vector components is suppressed due to the
light quark masses, as these components are induced by chiral symmetry
breaking.

  2. 2.

Anomalous magnetic moment of charged leptons: The universal couplings of the
$Z^{\prime}$ boson to charged leptons affect both electron and muon anomalous
magnetic moments. As the $Z^{\prime}$ decays into $e^{+}e^{-}$, it possesses
couplings to both electron and muon, leading to new contributions to their
magnetic moments. Due to the $Z^{\prime}$ boson’s vector and axial-vector
couplings to charged leptons, the contribution to the magnetic moment is
negative for a $Z^{\prime}$ mass around 17 MeV. However, in our model, a
positive contribution arises from the light scalar particle, which has non-
universal couplings to charged leptons. By balancing these positive and
negative contributions, correct values for the anomalous magnetic moments can
be achieved. See section. 4 for details.

  3. 3.

Electron-positron scattering: An upper limit on the coupling of the
$Z^{\prime}$ boson to electrons can be determined through the investigation of
$Z^{\prime}$ production associated with initial-state radiation in
$e^{+}e^{-}$ collisions, specifically the process $e^{+}e^{-}\rightarrow\gamma
Z^{\prime}$, where the $Z^{\prime}$ subsequently decays into $e^{+}e^{-}$.
KLOE-2 has provided an upper bound on this coupling Anastasi:2015qla , given
by $\sqrt{C_{e,V}^{2}+C_{e,A}^{2}}\lesssim 6\times
10^{-4}/\sqrt{\text{Br}(Z^{\prime}\rightarrow e^{+}e^{-})}$.

  4. 4.

Electron beam dump experiments: A lower limit on the coupling of the
$Z^{\prime}$ boson to electrons can be derived from fixed-target electron beam
dump experiments such as NA64 NA64:2018lsq  and E141 Riordan:1987aw . These
experiments investigate the bremsstrahlung interaction of electrons with
target nuclei, leading to the process $e+N(A,Z)\rightarrow
e+N(A,Z)+Z^{\prime}$, followed by the subsequent decay of the newly produced
gauge boson, $Z^{\prime}\rightarrow e^{+}e^{-}$. In both cases, these
experiments search for decays-in-flight of the $Z^{\prime}$. The absence of
any such direct detection results in bounds, which are given by
$\sqrt{C_{e,V}^{2}+C_{e,A}^{2}}\gtrsim 3.6\times
10^{-5}/\sqrt{\text{Br}(Z^{\prime}\rightarrow e^{+}e^{-})}$ for NA64
NA64:2019auh ; Barducci:2022lqd  and $\sqrt{C_{e,V}^{2}+C_{e,A}^{2}}\gtrsim
6\times 10^{-5}/\sqrt{\text{Br}(Z^{\prime}\rightarrow e^{+}e^{-})}$ for E141
Andreas:2012mt , specifically for $m_{Z^{\prime}}\simeq 17$ MeV.

  5. 5.

Møller scattering: The SLAC E158 experiment SLACE158:2005uay  conducted
measurements of the parity-violating asymmetry in Møller scattering, a process
that is sensitive to the existence of a Z’ boson. Through a comparison between
the experimentally measured asymmetry and theoretical predictions, limitations
can be imposed on the product $C_{e,V}C_{e,A}$ for various Z’ boson masses. In
the case of $m_{Z^{\prime}}\simeq 17$ MeV, the experiment establishes a
constraint of $|C_{e,V}C_{e,A}|\lesssim 10^{-8}$ Kahn:2016vjr .

  6. 6.

Atomic parity violation:The measurement of the weak nuclear charge, denoted as
$Q_{W}^{\text{eff}}(Z,N)$ in the cesium (Cs) atom, imposes the most stringent
constraints in terms of atomic parity violation (APV) Davoudiasl:2012ag ;
Bouchiat:2004sp . This measurement necessitates that the contribution stemming
from new physics to the weak charge must satisfy $|\Delta Q_{W}|\lesssim 0.71$
at the $2\sigma$ confidence level Porsev:2009pr . Here, $\Delta
Q_{W}(Z,N)=Q_{W}^{\text{eff}}(Z,N)-Q_{W}^{\text{SM}}(Z,N)$ represents the
difference in weak charge between the actual measurement and the Standard
Model prediction. This bound translates into a constraint on the product of
$C_{e,A}$ and a combination of $C_{u,V}$ and $C_{d,V}$ based on the definition
of $\Delta Q_{W}$. Specifically, it is given by the equation:

$\Delta
Q_{W}(Z,N)=-\frac{2\sqrt{2}}{G_{F}}\frac{C_{e,A}}{m_{Z^{\prime}}^{2}}\left[(2C_{u,V}+C_{d,V})Z+(C_{u,V}+2C_{d,V})N\right].$
(81)

  7. 7.

Neutrino based experiments: Various experiments involving neutrinos in either
the initial and final states, such as neutrino-electron scattering
Denton:2023gat , coherent elastic neutrino-nucleus scattering (CE$\nu$NS)
Denton:2023gat , and decays of charged pions Hostert:2023tkg , kaons
Davoudiasl:2014kua , can impose constraints on combinations of couplings,
including those involving $C_{\nu}$. Since we assume very small neutrino
mixing angles, $\sim\mathcal{O}(10^{-6})$ and, consequently, small neutrino
couplings, our model automatically satisfies all of these constraints.

Therefore, the primary constraints on the $Z^{\prime}$ couplings, under the
assumption that the $Z^{\prime}$ boson primarily decays to an $e^{+}e^{-}$
pair, can be summarized as follows.

$\displaystyle|2C_{u,V}+C_{d,V}|$ $\displaystyle\lesssim\leavevmode\nobreak\
2.4\times 10^{-4};$ $\displaystyle 3.6\times 10^{-5}\leavevmode\nobreak\
\lesssim\leavevmode\nobreak\ \sqrt{C_{e,V}^{2}+C_{e,A}^{2}}$
$\displaystyle\lesssim\leavevmode\nobreak\ 6\times 10^{-4};$
$\displaystyle|C_{e,V}C_{e,A}|$ $\displaystyle\lesssim\leavevmode\nobreak\
10^{-8};$ $\displaystyle|\Delta Q_{W}(Cs)|$
$\displaystyle\lesssim\leavevmode\nobreak\ 0.7;$ (82)

All these constraints can be projected into the $(g_{X},g_{X}^{\prime})$
plane, as all the coupling coefficients depend on the gauge coupling of the
$U(1)_{H}$ gauge group, denoted as $g_{X}$, and the gauge coupling mixing
parameter, $g_{X}^{\prime}$. The allowed parameter region in the
$(g_{X},g_{X}^{\prime})$ plane that satisfies all the above constraints is
depicted in figure. 1(a). Note that the couplings $g_{X}$ and $g_{X}^{\prime}$
are linearly dependent on each other within the allowed region. We find that
the allowed range of values for the parameter $g_{X}$ is $(1.8-6.6)\times
10^{-3}$, and for $g_{X}^{\prime}$, it is $(0.4-1.4)\times 10^{-4}$.

(a)

(b)

Figure 1:  (a) The allowed parameter range in the $(g_{X},g_{X}^{\prime})$
plane that satisfies all the relevant constraints summarized in Eq. (3).(b)The
blue points represent the values of axial coupling to protons/neutrons,
denoted as $C_{p/n}$, obtained using the approved range of
$(g_{X},g_{X}^{\prime})$ from the preceding figure. In this context, the dark
(light) shaded area corresponds to the $1\sigma$ ($2\sigma$) acceptable region
for $C_{p/n}$ in the case of a general axial vector coupling reproduced from
Ref. Barducci:2022lqd .

To perform the nuclear decay calculation, one must define the coupling of
$Z^{\prime}$ to nucleons. When considering only the axial vector coupling, it
can be expressed as:

$\mathcal{L}_{Z^{\prime}}^{N}=C_{p}\bar{p}\gamma^{\rho}\gamma^{5}pZ^{\prime}_{\rho}+C_{n}\bar{n}\gamma^{\rho}\gamma^{5}nZ^{\prime}_{\rho}$
(83)

where couplings $C_{p/n}$ can be represented in terms of the axial vector
couplings of up and down type quarks as follows:

$\displaystyle C_{p}$ $\displaystyle=$ $\displaystyle
g_{u,A}C_{u,A}+g_{d,A}C_{d,A};$ $\displaystyle C_{n}$ $\displaystyle=$
$\displaystyle g_{d,A}C_{u,A}+g_{u,A}C_{d,A};$ (84)

where the coefficients $g_{u/d,A}$ can be obtained from the results of neutron
$\beta$-decay Alexandrou:2021wzv  and Lattice QCD calculations
Alexandrou:2019brg  and are given as follows: $g_{u,A}=0.817$ and
$g_{d,A}=-0.450$. It’s worth noting that $C_{u,A}=-C_{d,A}\equiv C$.
Consequently, a simple relationship between $C_{p}$ and $C_{n}$ can be
established: $C_{p}=-C_{n}=1.267C$ where $C$ depends on $g_{X}$ only. By
employing the allowable values of $g_{X}$ from figure. 1(a), we can determine
the corresponding values of $C_{p}$ and $C_{n}$ , which are presented in
figure. 1(b) alongside the $1\sigma$ (dark shaded region) and $2\sigma$ (light
shaded region) allowed regions of effective nuclear couplings. In particular,
the values of $C_{p}$ are within the range of $-(0.297-1.050)\times 10^{-4}$,
and $C_{n}$ falls within the range of $(0.297-1.050)\times 10^{-4}$. These
values have the potential to explain the excess observed in the 8Be nuclear
decay at the ATOMKI experiment. Note that most of the values are within the
$1\sigma$ allowed region. Also note that, while the axial couplings $C_{p/n}$
depend solely on $g_{X}$, the coupling $g_{X}$ itself are related to
$g_{X}^{\prime}$ through the constraint conditions in Eq. (3).

##  4 Muon Anomalous Magnetic Moments

A persistent and significant discrepancy between experimental data and the
theoretical predictions of the SM lies in the anomalous magnetic moment of the
muon, denoted as $a_{\mu}=(g_{\mu}-2)/2$. Notably, the E989 experiments at
FNAL have recently released the most precise measurement of $a_{\mu}$
Muong-2:2023cdq , achieved through the combination of data from Runs 1, 2, and
3. This refined value is

$a_{\mu}^{FNAL}=116\leavevmode\nobreak\ 592\leavevmode\nobreak\ 055(24)\times
10^{-11},$ (85)

upon combining this result with those from BNL Muong-2:2006rrc , gives a
comprehensive experimental average

$a_{\mu}^{exp}=116\leavevmode\nobreak\ 592\leavevmode\nobreak\ 059(22)\times
10^{-11}.$ (86)

The FNAL result is entirely consistent with the most accurate previous
measurements Abi:2021gix  and, notably, enhances precision by a factor of two.
Additional analysis of the remaining data is anticipated to yield a further
twofold improvement in statistical accuracy. Simultaneously, the upcoming E34
experiment at J-PARC Saito:2012zz ; Lee:2019xkg  is dedicated to minimizing
statistical uncertainties and independently confirming the FNAL findings.

On the theoretical front, a global collaboration known as the Muon $g-2$
Theory Initiative has offered the most precise and comprehensive predictions
for the SM value Aoyama:2020ynm , which is calculated to be

$a_{\mu}^{th}=116\leavevmode\nobreak\ 591\leavevmode\nobreak\ 810(43)\times
10^{-11}.$ (87)

This SM prediction relies on the most contemporary and precise calculations
encompassing Quantum Electrodynamics (QED) Aoyama:2012wk ; Aoyama:2019ryr ,
Electroweak interactions Czarnecki:2002nt ; Gnendiger:2013pva , Hadronic
Vacuum Polarization (HVP) Davier:2017zfy ; Keshavarzi:2018mgv ;
Colangelo:2018mtw ; Hoferichter:2019mqg ; Davier:2019can ; Keshavarzi:2019abf
; Kurz:2014wya , and Hadronic Light-by-Light Scattering Melnikov:2003xd ;
Masjuan:2017tvw ; Colangelo:2017fiz ; Hoferichter:2018kwz ; Gerardin:2019vio ;
Bijnens:2019ghy ; Colangelo:2019uex ; Pauk:2014rta ; Danilkin:2016hnh ;
Jegerlehner:2017gek ; Knecht:2018sci ; Eichmann:2019bqf ; Roig:2019reh ;
Colangelo:2014qya ; Blum:2019ugy . Specifically, the HVP, which constitutes
the leading-order hadronic contribution, was derived from measurements of the
electron-positron to hadrons cross-section performed by various experiments.
However, the BMW collaboration recently introduced improvements in the
theoretical prediction for HVP through lattice QCD calculations, introducing a
noticeable tension with other data Borsanyi:2020mff . Furthermore, preliminary
measurements of the electron-positron to $\pi^{+}\pi^{-}$ cross-section from
the CMD-3 experiment CMD-3:2023alj  have exhibited substantial tension. It’s
important to note that these results are awaiting independent verification,
and multiple ongoing theoretical endeavors aim to clarify the current state of
theoretical understanding Colangelo:2022jxc .

(a)

(b)

Figure 2:  One-loop diagrams mediated by the light scalar $h_{1}$ and the
light vector $Z^{\prime}$ that contribute to the anomalous magnetic moments of
muon.

Hence, when we compare the experimental and theoretical values, a discrepancy
of 5.1 sigma emerges,

$\Delta a_{\mu}=a_{\mu}^{exp}-a_{\mu}^{th}=2.49(0.48)\times 10^{-9},$ (88)

with the uncertainties being combined by adding them in quadrature. Within the
framework of our model, the muon’s magnetic moment receives contributions from
both the light scalar $h_{1}$ and the light vector $Z^{\prime}$ mediated
through one-loop diagrams. The relevant diagrams are shown in figure. 2.

The Lagrangian in the physical mass basis, which leads to the expression for
the magnetic moment of muon depicted in figure. 2, can be written as follows:

$-\mathcal{L}\subset y_{\mu h_{1}}\bar{\mu}\mu
h_{1}+\bar{\mu}\gamma^{\rho}\left(C_{\mu,V}+C_{\mu,A}\gamma^{5}\right)\mu
Z^{\prime}_{\rho}$ (89)

We can derive the contribution to $\Delta a_{\mu}$ from figure. 2(a) as
follows:

$\Delta a_{\mu}^{(h_{1})}=y_{\mu
h_{1}}^{2}\frac{m_{\mu}^{2}}{8\pi^{2}}\int_{0}^{1}dx\frac{2x^{2}-x^{3}}{m_{\mu}^{2}x^{2}+m_{h_{1}}^{2}(1-x)}$
(90)

Similarly the vector boson contribution from figure. 2(b) is:

$\Delta
a_{\mu}^{(Z^{\prime})}=\frac{m_{\mu}^{2}}{4\pi^{2}}\int_{0}^{1}dx\frac{\left[C_{\mu,V}^{2}x^{2}(1-x)+C_{\mu,A}^{2}\\{x(1-x)(x-4)-\frac{4m_{\mu}^{2}x^{3}}{m_{Z^{\prime}}^{2}}\\}\right]}{m_{\mu}^{2}x^{2}+m_{Z^{\prime}}^{2}(1-x)}$
(91)

where

$\displaystyle C_{\mu,V}$ $\displaystyle=$ $\displaystyle
g_{X}\sin^{2}{\beta}\left(-\frac{1}{4}+\sin^{2}{\theta_{W}}\right)-g_{X}^{\prime}\cos^{2}{\theta_{W}}$
$\displaystyle C_{\mu,A}$ $\displaystyle=$
$\displaystyle\frac{1}{4}g_{X}\sin{\beta}^{2}$ (92)

Hence, the overall contribution to the anomalous magnetic moment is expressed
as:

$\Delta a_{\mu}=\Delta a_{\mu}^{(h_{1})}+\Delta a_{\mu}^{(Z^{\prime})}$ (93)

For a mediator mass around $\mathcal{O}(10)$ MeV, the $\Delta
a_{\mu}^{(Z^{\prime})}$ contribution consistently appears negative.
Conversely, $\Delta a_{\mu}^{(h_{1})}$ always yield a positive contribution.
To achieve the correct value for $\Delta a_{\mu}$, these contributions are
balanced against each other. A similar approach was taken in the case of low
scale $U(1)_{T3R}$ theory as well Dutta:2021afo . Note that $C_{\mu,V}$ and
$C_{\mu,A}$ rely on the gauge coupling of the $U(1)_{H}$ gauge group denoted
as $g_{X}$ and the gauge coupling mixing parameter $g_{X}^{\prime}$. The
relationship between $g_{X}$ and $g_{X}^{\prime}$ has been already determined
by addressing the ATOMKI anomaly in Sec. 3, given in figure. 1(a). The scalar
coupling to muon is bounded by muon mass. We examine four distinct benchmark
scenarios, each featuring a different light scalar mass, to calculate the
total contribution to $\Delta a_{\mu}$. Throughout these scenarios, we
maintain a fixed $Z^{\prime}$ mass of 17 MeV. A summary of the values is
provided in Table. 2. The primary constraints on the $(m_{h_{1}},y_{\mu
h_{1}})$ plane arise from fixed-target/beam dump experiments like E137
Dobrich:2015jyk ; Dolan:2017osp ; Bjorken:1988as ; Batell:2016ove  and Orsay
Batell:2016ove . We determine that couplings of the order of
$\mathcal{O}(10^{-4}-10^{-3})$ are permissible for
$m_{h_{1}}\sim\mathcal{O}(10-100)$ MeV Dutta:2020scq . Future experiment such
as FASER Feng:2017uoz ; Feng:2017vli ; Batell:2017kty , ShiP Alekhin:2015byh ;
Batell:2017kty , and Fermilab $\mu-$beam fixed target Chen:2017awl ;
Batell:2017kty  can probe this paramater space.

$m_{h_{1}}$ | $g_{X}$ | $g_{X}^{\prime}$ | $y_{\mu h_{1}}$ | $\Delta a_{\mu}$
---|---|---|---|---
12 MeV | $1.8\times 10^{-3}$ | $0.4\times 10^{-4}$ | $0.4\times 10^{-3}$ | $1.98\times 10^{-9}$
15 MeV | $1.8\times 10^{-3}$ | $0.4\times 10^{-4}$ | $0.4\times 10^{-3}$ | $1.89\times 10^{-9}$
20 MeV | $1.8\times 10^{-3}$ | $0.4\times 10^{-4}$ | $0.4\times 10^{-3}$ | $1.78\times 10^{-9}$
50 MeV | $1.6\times 10^{-3}$ | $0.39\times 10^{-4}$ | $0.4\times 10^{-3}$ | $1.08\times 10^{-9}$
Table 2:  We present four benchmark scenarios, each with a different light
scalar mass, to obtain values of $\Delta a_{\mu}$. Here, we have kept the
vector boson mass fixed at $m_{Z^{\prime}}=17$ MeV.

One-loop diagrams, similar to those illustrated in figure. 2 but involving
electrons as the charged leptons, can impact the magnetic moment of the
electron. Despite both the muon and electron having the same coupling strength
to the vector boson $Z^{\prime}$, their Yukawa couplings differ. Specifically,
the electron Yukawa coupling, denoted as $y_{eh_{1}}$, is of the order of
$\mathcal{O}(10^{-6})$ in our model. However, it’s essential to note that the
contribution from the scalar-mediated diagram is three orders of magnitude
smaller than that from the vector-mediated diagram within the relevant
parameter space. Consequently, the total contribution to the electron’s
magnetic moment amounts to approximately $\mathcal{O}(10^{-14})$, which is one
order of magnitude smaller than the predicted value and predominantly comes
from the $Z^{\prime}$ contribution.

##  5 MiniBooNE Excess Events

In 2020, after collecting data for 17 years from 2002 to 2019, MiniBooNE
updated its results Aguilar-Arevalo:2020nvw . They reported an excess of
single shower ($1-sh$) events, with $560.6\pm 119.6$ events in the neutrino
mode and $77.4\pm 28.5$ events in the antineutrino mode, over the estimated
background within the energy range of 200 to 1250 MeV. These observations were
based on $11.27\times 10^{20}$protons-on-target (POT) in the antineutrino mode
and $18.75\times 10^{20}$ POT in the neutrino mode. The combined excess events
totaled $638.0\pm 132.8$, leading to a $4.8\sigma$ tension with the
predictions of the two-neutrino oscillation scenario within the Standard
Model.

The initial explanation for the excess observed in MiniBooNE data was the
$3+1$ oscillation scenario Sorel:2003hf ; Karagiorgi:2009nb ; Giunti:2011cp ;
Kopp:2011qd ; Abazajian:2012ys ; Conrad:2012qt ; Kopp:2013vaa ; Collin:2016aqd
; Gariazzo:2017fdh ; Asaadi:2017bhx ; Dentler:2018sju ; Boser:2019rta ;
Diaz:2019fwt , which introduced an eV-scale sterile neutrino responsible for
inducing short-distance oscillations: $\nu_{\mu}\rightarrow\nu_{e}$. However,
this hypothesis encounters tension with other neutrino data Collin:2016rao ;
MINOS:2020iqj ; IceCube:2020phf ; IceCube:2020tka  and conflicts with
cosmological observations Hamann:2011ge ; Archidiacono:2013xxa ;
Hagstotz:2020ukm . Recent findings from the MicroBooNE experiment
MicroBooNE:2016pwy  at Fermilab (FNAL) have added complexity to the situation.
MicroBooNE results have cast doubt on the notion that neutral current
$\Delta\rightarrow N\gamma$ backgrounds alone are responsible for the observed
excess, with a confidence level of $94.8\%$ in disfavor of this explanation
MicroBooNE:2021zai . Furthermore, the idea that generic $\nu_{e}$ interactions
are the primary cause of the excess is also challenged by MicroBooNE data
MicroBooNE:2021tya . These observations strongly advocate the necessity for
new physics to explain the excess. It’s worth noting that MiniBooNE cannot
distinguish between photons and electrons, whereas MicroBooNE’s advanced
calorimetry and precise resolution enable a clear separation between electrons
and photons Contin:2017mck ; MicroBooNE:2021nxr . This capability allows
MicroBooNE to explore BSM physics as a potential solution to this low-energy
excess.

Numerous new physics scenarios have been proposed to account for this anomaly.
Among them, solutions based on neutrinos Bertuzzo:2018itn ; Ballett:2018ynz ;
Ballett:2019cqp ; Ballett:2019pyw ; Fischer:2019fbw ; Dentler:2019dhz ;
deGouvea:2019qre ; Abdallah:2020biq ; Datta:2020auq ; Dutta:2020scq ;
Abdullahi:2020nyr ; Abdallah:2020vgg ; Abdullahi:2023ejc  hold particular
promise, as they can explain the absence of excess events in the MiniBooNE
beam dump mode MiniBooNEDM:2018cxm . Within the realm of neutrino-based
solutions, several scenarios are conceivable Brdar:2020tle . One particularly
intriguing possibility involves the coherent upscattering of light neutrinos
to heavy neutrinos, followed by the decay of the heavy neutrino into an
electron-positron pair through the mediation of a light particle
Bertuzzo:2018itn ; Dutta:2020scq . The neutrino’s energy can be reconstructed
by analyzing the energy and angular distributions of the mediator arising from
the heavy neutrino’s decay Martini:2012fa . However, many of these neutrino-
based solutions, especially those involving vector mediators, face potential
conflicts with the absence of signal observations in other neutrino
experiments like CHARM-II CHARM-II:1989nic ; CHARM-II:1992vao ; CHARM-
II:1994dzw ; Arguelles:2018mtc , MINER$\nu$A MINERvA:2015jih ; MINERvA:2015nqi
; MINERvA:2019hhc , and T2K ND280 Kudenko:2008ia ; T2K:2011qtm ;
Assylbekov:2011sh ; T2KND280FGD:2012umz ; T2K:2012bge ; T2K:2019jwa ;
T2K:2020lrr . This discrepancy arises because the coherent scattering cross-
section for the vector mediator becomes more pronounced with increasing
neutrino energy Arguelles:2018mtc . Conversely, when mediated by a scalar
particle, the cross-section diminishes with rising neutrino energy and
therefore does not lead to any excess events in other neutrino experiments
Dutta:2020scq .

Figure 3:  The Feynman diagrams represent the upscattering-double decay
scenario, involving both light mediators, $h_{1}$ and $Z^{\prime}$,
contributing to the MiniBooNE excess events in our model.

In the context of our model, the heavy sterile neutrino, denoted as
$\nu_{s_{2}}$, is generated through the upscattering process:
$\nu_{\mu}A\rightarrow\nu_{s_{2}}A$. Importantly, in our model, this
upscattering process can be mediated by both of the two light mediators, 17
MeV $Z^{\prime}$ for the ATOMKI anomaly and light $h_{1}$ for the muon $g-2$.
Note that, since this upscattering process is coherent, it experiences an
enhancement by a factor of $A^{2}$. Subsequently, the produced heavy neutrino
promptly undergoes decay, leading to either the lightest sterile neutrino or
active neutrinos, along with an on-shell mediator, either $Z^{\prime}$ of
$h_{1}$. This mediator, in turn, decays into a pair of $e^{+}e^{-}$. This
specific scenario can be categorized as an upscattering-double decay scenario,
as identified in Ref. Brdar:2020tle . In this scenario, there are four
possible diagrams, as illustrated in figure. 3, that can contribute to the
observed excess of events. Our plan is

  1. 1.

First, we perform calculations for the contribution of all four diagrams
individually and then sum their results. Let’s denote the predicted number of
events from our model as $N_{th}$, and the corresponding experimental value as
$N_{exp}$. In this context, we anticipate that $N_{th}^{MB}=N_{exp}^{MB}$,
within some permissible deviation. Here $MB$ stands for MiniBooNE.

  2. 2.

Next, our objective is to evaluate whether the parameter space of this model
yields ambiguous predictions for the number of events in other neutrino
experiments. We can express the predicted number of events in these
experiments as $N_{exp}^{i}=\left(N_{th}^{i}/N_{th}^{MB}\right)N_{exp}^{MB}$,
essentially normalizing the events in the i-experiment to match the MiniBooNE
excess.

  3. 3.

To simultaneously satisfy both of the above conditions, we need to select the
model’s parameter space in a manner that the upscattering cross section is
predominantly governed by the scalar-mediated contribution, and this cross
section should decrease as the neutrino energy increases.

Under the assumption that the heavy neutrino, as well as both the scalar and
vector mediators, have short lifetimes, we can express the total number of
events as follows:

$N^{MB}_{th}=\xi_{exp}\int_{{E_{\nu}}_{min}}^{{E_{\nu}}_{max}}dE_{\nu}\Phi(E_{\nu})\int_{{E_{R}}_{min}}^{{E_{R}}_{max}}dE_{R}\frac{d\sigma(E_{R},E_{\nu})}{dE_{R}}\times
Br\left(mediator\rightarrow e^{+}e^{-}\right),$ (94)

where $\xi_{exp}$ accounts for the exposure time of the detector to the beam,
the number of protons on target, and the effective area of the detector, and
it depends on the specific experiment; $E_{\nu}$ is the energy of the incoming
$\nu_{\mu}$ beam; $\Phi(E_{\nu})$ is the corresponding flux; $E_{R}$ is the
recoil energy of the target nuclei in the detector. To compare with the
previous MiniBooNE solutions, we just need to calculate
$\xi_{model}=N^{MB}_{th}/\xi_{exp}$.

The differential scattering cross-section with respect to the recoil energy
$E_{R}$ for $\nu A\rightarrow\nu_{s}A$ mediated through the light scalar
$h_{1}$ shown in upper panel of figure. 3 can be expressed as follows:

$\frac{d\sigma}{dE_{R}}=\frac{y_{\nu}^{2}}{16\pi
E_{\nu}^{2}}\times\left[Zf_{p}+(A-Z)f_{n}\right]^{2}\times\frac{\left({m^{2}_{\nu_{s_{2}}}}+2M_{A}E_{R}\right)\left(2M_{A}+E_{R}\right)}{\left(m_{h_{1}}^{2}+2M_{A}E_{R}\right)^{2}}\times
F^{2}(E_{R})$ (95)

where $m_{A}$ is the mass of the target nucleus; $Z$ is the number of protons
in the target nucleus; $A-Z$ represents the number of neutrons in the target
nucleus, where $A$ is the total number of nucleons; $F({E_{R}})$ is the
nuclear form factor Helm:1956zz ; Engel:1991wq , which characterizes the
spatial distribution of nucleons within the target nucleus and depends on the
momentum transfer. The coupling factors $f_{p,n}$ are expressed as Falk:1999mq

$\frac{f_{p,n}}{m_{N}}=\sum_{q=u,d,s}f^{(p,n)}_{T_{q}}\frac{f_{q}}{m_{q}}+\frac{2}{27}\left(1-\sum_{q=u,d,s}f^{(p,n)}_{T_{q}}\right)\sum_{q=c,b,t}\frac{f_{q}}{m_{q}}$
(96)

where we set $f_{(u,d)}=y_{(u,d)h_{1}}$ and $f_{c,s,t,b}=0$; the parameters
$f_{T_{q}}^{(N)}$ are defined as
$m_{N}f_{T_{q}}^{(N)}=\braket{N}{m_{q}\bar{q}q}{N}$ for $N=p,n$ Shifman:1978zn
; our results used the values: $f_{T_{u}}^{(p)}=0.020$,
$f_{T_{d}}^{(p)}=0.041$,$f_{T_{u}}^{(n)}=0.0189$, and $f_{T_{d}}^{(n)}=0.045$
Alarcon:2011zs ; Alarcon:2012nr ; Crivellin:2013ipa ; Hoferichter:2015dsa ;
Junnarkar:2013ac .

Figure 4:  This figure illustrates the cross sections’ dependency on the
incoming neutrino energy for various scalar masses in conjunction with a 17
MeV vector gauge boson. The lines represent the combined cross section for
both scalar and vector interactions for the benchmark points given in table.
3.

The expression for the differential scattering cross-section mediated by the
light vector $Z^{\prime}$ given by the two figures in the bottom panel of
figure. 3 is given by,

$\displaystyle\frac{d\sigma}{dE_{R}}$ $\displaystyle=$
$\displaystyle\frac{g_{\nu}^{2}}{4\pi
E_{\nu}^{2}}\times\left[(A+Z)C_{u,V}+(2A-Z)C_{d,V}\right]^{2}\times
F^{2}(E_{R})$ (97) $\displaystyle\times$
$\displaystyle\frac{\left(2M_{A}E_{\nu}^{2}+M_{A}E_{R}^{2}-2M_{A}E_{\nu}E_{R}-{m^{2}_{\nu_{s_{2}}}}E_{\nu}+\frac{{m^{2}_{\nu_{s_{2}}}}}{2}E_{R}-\frac{{m^{2}_{\nu_{s_{2}}}}}{2}M_{A}-M_{A}^{2}E_{R}\right)}{\left(m_{Z^{\prime}}^{2}+2M_{A}E_{R}\right)^{2}}$

where $g_{\nu}$ represents the effective coupling between the active and
sterile neutrinos with the vector gauge boson. This coupling is small due to
the small mixing angle in the neutrino sector. It is worth noting that the
mixing angle is essentially a free parameter in our model since we are not
making any other adjustments that would affect it.

Parameters | BP1 | BP2 | BP3 | BP4
---|---|---|---|---
$m_{h_{1}}$ | 12 MeV | 15 MeV | 20 MeV | 50 MeV
$m_{\nu_{s_{2}}}$ | 420 MeV | 420 MeV | 420 MeV | 420 MeV
$y_{\nu}$ | 0.0096 | 0.0182 | 0.0190 | 0.0220
$y_{uh_{1}}$ | $5\times 10^{-6}$ | $5\times 10^{-6}$ | $5\times 10^{-6}$ | $5\times 10^{-6}$
$y_{dh_{1}}$ | $10^{-5}$ | $10^{-5}$ | $10^{-5}$ | $10^{-5}$
$N_{MB}^{th}$ | 636 | 647 | 681 | 658
Table 3:  We present four benchmark scenarios with varying masses of the
scalar boson that successfully account for the observed number of excess
events at the MB detector. Importantly, these scenarios do not lead to an
excessive production of events at the ND-T2K and MINERvA detectors. We have
set the vector boson mass to $m_{Z^{\prime}}=17$ MeV, while the gauge
couplings are held constant at $g_{X}=1.8\times 10^{-3}$ and
$g_{X}^{\prime}=0.41\times 10^{-4}$.

To estimate the number of events predicted by our model, denoted as
$N^{th}_{MB}$, we consider four benchmark points detailed in Table. 3.
Specifically, we explore four different scalar masses ranging from 10 MeV to
50 MeV, while keeping the masses of $\nu_{s_{2}}$ and $Z^{\prime}$ fixed at
420 MeV and 17 MeV, respectively. For the $Z^{\prime}$-mediated diagram, we
choose the couplings to be $g_{X}=1.8\times 10^{-3}$ and
$g_{X}^{\prime}=4.1\times 10^{-5}$. It is important to note that the neutrino
coupling with $Z^{\prime}$ is further suppressed by mixing angle in the
neutrino sector. Consequently, the contribution to the cross-section from the
$Z^{\prime}$-mediated diagram is orders of magnitude smaller compared to the
scalar-mediated diagram. Thus, the total cross-section is predominantly
determined by the scalar-mediated process. Concerning the scalar-mediated
diagram, the quark couplings are constrained by the quark masses. However, the
neutrino coupling $y_{\nu}$ is a free parameter and can be relatively large.
Assuming typical neutrino and scalar energies of $E_{\nu_{s_{2}}}$ and
$E_{h_{1}}$ around 1 GeV, we estimate that the length of the path they travel
before decaying is less than or equal to $10^{-4}$ m, indicating prompt
decays. In a related study Brdar:2020tle , it was demonstrated that a process
dominated by scalar interactions does not lead to an overproduction of events
at the T2K ND280 and MINER$\nu$A experiments, provided that $\nu_{S_{2}}$
decays promptly with a decay length of less than or equal to $10^{-4}$ meter.

##  6 Summary and discussions

In light of the absence of evidence for new physics at the TeV scale in LHC
experiments, attention is turning towards low-energy experiments that may
provide insight into new physics phenomena at the MeV scale. Several
intriguing findings have emerged recently from such experiments, including the
Beryllium anomaly at the ATOMKI experiment, the anomalous magnetic moment of
the muon, and the excess low-energy events observed at MiniBooNE. To address
these anomalies simultaneously, we propose a straightforward model extension
of the gauge and scalar sectors within the SM. This extension introduces new
scalar fields $H_{2}$ and $\phi_{H}$ that carry charges under a new gauge
group, denoted as $U(1)_{H}$. The model is $U(1)_{H}$ extensions of the Type-I
2HDM plus a singlet scalar $\phi_{H}$. The mixing within the scalar sector
yields a new light scalar field, while mixing within the gauge sector results
in a light vector boson. Additionally, our model incorporates
$U(1)_{H}$-charged sterile neutrinos at the MeV scale.

The resolution of the Beryllium anomaly primarily relies on the presence of
the vector boson mass around 17 MeV, whereas the excess events observed at
MiniBooNE can be predominantly attributed to the scalar particle. Notably,
both the scalar and vector particles contribute to the muon $(g-2)$, albeit
with opposite signs. Careful fine-tuning between these contributions allows us
to arrive at the correct value for the muon’s magnetic moment. Hence, by
utilizing the capabilities of the two light mediators within our model, we
simultaneously provide explanations for these recent significant anomalies.

These results will soon undergo verification through upcoming experiments.
Notably, MEG II at PSI is set to replicate the Beryllium experiment conducted
at ATOMKI, E34 at J-PARC aims to validate our model’s predictions regarding
the muon’s magnetic moment, and MicroBooNE is actively scrutinizing the
MiniBooNE results for further insights

###### Acknowledgements.

This work is supported in part by KIAS Individual Grants under Grant No.
PG021403 (PK), and by National Research Foundation of Korea (NRF) Grant No.
NRF-2019R1A2C3005009 (SG PK). We acknowledge that we have used the TikZ-
Feynman Ellis:2016jkw  package to generate the Feynman diagram of Figs. 2 and
3.

## References

  * (1) Super-Kamiokande Collaboration, Y. Fukuda et al., Evidence for oscillation of atmospheric neutrinos, Phys. Rev. Lett. 81 (1998) 1562–1567, [hep-ex/9807003].
  * (2) SNO Collaboration, Q. R. Ahmad et al., Direct evidence for neutrino flavor transformation from neutral current interactions in the Sudbury Neutrino Observatory, Phys. Rev. Lett. 89 (2002) 011301, [nucl-ex/0204008].
  * (3) Planck Collaboration, N. Aghanim et al., Planck 2018 results. VI. Cosmological parameters, arXiv:1807.06209.
  * (4) J. Gulyás, T. J. Ketel, A. J. Krasznahorkay, M. Csatlós, L. Csige, Z. Gácsi, M. Hunyadi, A. Krasznahorkay, A. Vitéz, and T. G. Tornyi, A pair spectrometer for measuring multipolarities of energetic nuclear transitions, Nucl. Instrum. Meth. A 808 (2016) 21–28, [arXiv:1504.00489].
  * (5) A. J. Krasznahorkay et al., Observation of Anomalous Internal Pair Creation in Be8 : A Possible Indication of a Light, Neutral Boson, Phys. Rev. Lett. 116 (2016), no. 4 042501, [arXiv:1504.01527].
  * (6) A. J. Krasznahorkay et al., New results on the 8Be anomaly, J. Phys. Conf. Ser. 1056 (2018), no. 1 012028.
  * (7) A. J. Krasznahorkay et al., New evidence supporting the existence of the hypothetic X17 particle, arXiv:1910.10459.
  * (8) A. J. Krasznahorkay, M. Csatlós, L. Csige, J. Gulyás, A. Krasznahorkay, B. M. Nyakó, I. Rajta, J. Timár, I. Vajda, and N. J. Sas, New anomaly observed in He4 supports the existence of the hypothetical X17 particle, Phys. Rev. C 104 (2021), no. 4 044003, [arXiv:2104.10075].
  * (9) A. J. Krasznahorkay et al., New anomaly observed in C12 supports the existence and the vector character of the hypothetical X17 boson, Phys. Rev. C 106 (2022), no. 6 L061601, [arXiv:2209.10795].
  * (10) A. Keshavarzi, K. S. Khaw, and T. Yoshioka, Muon g-2: A review, Nucl. Phys. B 975 (2022) 115675, [arXiv:2106.06723].
  * (11) Muon g-2 Collaboration, J. Grange et al., Muon (g-2) Technical Design Report, arXiv:1501.06858.
  * (12) Muon g-2 Collaboration, A. T. Fienberg, The Status and Prospects of the Muon $g-2$ Experiment at Fermilab, in 54th Rencontres de Moriond on QCD and High Energy Interactions, pp. 163–166, ARISF, 5, 2019.  arXiv:1905.05318.
  * (13) Muon g-2 Collaboration, D. P. Aguillard et al., Measurement of the Positive Muon Anomalous Magnetic Moment to 0.20 ppm, arXiv:2308.06230.
  * (14) Muon g-2 Collaboration, G. W. Bennett et al., Final Report of the Muon E821 Anomalous Magnetic Moment Measurement at BNL, Phys. Rev. D 73 (2006) 072003, [hep-ex/0602035].
  * (15) G. Colangelo et al., Prospects for precise predictions of $a_{\mu}$ in the Standard Model, arXiv:2203.15810.
  * (16) J-PARC g-’2/EDM Collaboration, N. Saito, A novel precision measurement of muon g-2 and EDM at J-PARC, AIP Conf. Proc. 1467 (2012) 45–56.
  * (17) J. Jaeckel, M. Lamont, and C. Vallée, The quest for new physics with the Physics Beyond Colliders programme, Nature Phys. 16 (2020), no. 4 393–401.
  * (18) MiniBooNE Collaboration, A. A. Aguilar-Arevalo et al., The MiniBooNE Detector, Nucl. Instrum. Meth. A 599 (2009) 28–46, [arXiv:0806.4201].
  * (19) MiniBooNE Collaboration, A. Aguilar-Arevalo et al., Updated MiniBooNE Neutrino Oscillation Results with Increased Data and New Background Studies, arXiv:2006.16883.
  * (20) B. Dutta, S. Ghosh, and J. Kumar, A sub-GeV dark matter model, Phys. Rev. D 100 (2019) 075028, [arXiv:1905.02692].
  * (21) B. Dutta, S. Ghosh, and J. Kumar, $U(1)_{T3R}$ Extension of Standard Model: A Sub-GeV Dark Matter Model, in Snowmass 2021, 3, 2022.  arXiv:2203.07786.
  * (22) R. Foot, New Physics From Electric Charge Quantization?, Mod. Phys. Lett. A6 (1991) 527–530.
  * (23) X. G. He, G. C. Joshi, H. Lew, and R. R. Volkas, NEW Z-prime PHENOMENOLOGY, Phys. Rev. D43 (1991) 22–24.
  * (24) X.-G. He, G. C. Joshi, H. Lew, and R. R. Volkas, Simplest Z-prime model, Phys. Rev. D44 (1991) 2118–2132.
  * (25) P. Ko, Y. Omura, and C. Yu, A Resolution of the Flavor Problem of Two Higgs Doublet Models with an Extra $U(1)_{H}$ Symmetry for Higgs Flavor, Phys. Lett. B 717 (2012) 202–206, [arXiv:1204.4588].
  * (26) S. L. Glashow and S. Weinberg, Natural Conservation Laws for Neutral Currents, Phys. Rev. D 15 (1977) 1958.
  * (27) G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher, and J. P. Silva, Theory and phenomenology of two-Higgs-doublet models, Phys. Rept. 516 (2012) 1–102, [arXiv:1106.0034].
  * (28) P. Ko, Y. Omura, and C. Yu, Higgs phenomenology in Type-I 2HDM with $U(1)_{H}$ Higgs gauge symmetry, JHEP 01 (2014) 016, [arXiv:1309.7156].
  * (29) P. Ko, Y. Omura, and C. Yu, Dark matter and dark force in the type-I inert 2HDM with local U(1)H gauge symmetry, JHEP 11 (2014) 054, [arXiv:1405.2138].
  * (30) P. Ko, Y. Omura, and C. Yu, Higgs and dark matter physics in the type-II two-Higgs-doublet model inspired by $E_{6}$ GUT, JHEP 06 (2015) 034, [arXiv:1502.00262].
  * (31) P. Ko, Y. Omura, and C. Yu, Diphoton Excess at 750 GeV in leptophobic U(1)′ model inspired by $E_{6}$ GUT, JHEP 04 (2016) 098, [arXiv:1601.00586].
  * (32) L. Delle Rose, S. Khalil, and S. Moretti, Explanation of the 17 MeV Atomki anomaly in a U(1)’ -extended two Higgs doublet model, Phys. Rev. D 96 (2017), no. 11 115024, [arXiv:1704.03436].
  * (33) E. Bertuzzo, S. Jana, P. A. N. Machado, and R. Zukanovich Funchal, Dark Neutrino Portal to Explain MiniBooNE excess, Phys. Rev. Lett. 121 (2018), no. 24 241801, [arXiv:1807.09877].
  * (34) A. Datta, S. Kamali, and D. Marfatia, Dark sector origin of the KOTO and MiniBooNE anomalies, Phys. Lett. B 807 (2020) 135579, [arXiv:2005.08920].
  * (35) B. Dutta, S. Ghosh, and T. Li, Explaining $(g-2)_{\mu,e}$, the KOTO anomaly and the MiniBooNE excess in an extended Higgs model with sterile neutrinos, Phys. Rev. D 102 (2020), no. 5 055017, [arXiv:2006.01319].
  * (36) P. Ballett, M. Hostert, and S. Pascoli, Dark Neutrinos and a Three Portal Connection to the Standard Model, Phys. Rev. D 101 (2020), no. 11 115025, [arXiv:1903.07589].
  * (37) B. Dutta, S. Ghosh, T. Li, A. Thompson, and A. Verma, Non-standard neutrino interactions in light mediator models at reactor experiments, JHEP 03 (2023) 163, [arXiv:2209.13566].
  * (38) CMS Collaboration, A search for decays of the Higgs boson to invisible particles in events with a top-antitop quark pair or a vector boson in proton-proton collisions at $\sqrt{s}$ = 13 TeV, arXiv:2303.01214.
  * (39) ATLAS Collaboration, Combination of searches for invisible decays of the Higgs boson using 139 fb$-$1 of proton-proton collision data at s=13 TeV collected with the ATLAS experiment, Phys. Lett. B 842 (2023) 137963, [arXiv:2301.10731].
  * (40) P. F. de Salas, D. V. Forero, S. Gariazzo, P. Martínez-Miravé, O. Mena, C. A. Ternes, M. Tórtola, and J. W. F. Valle, 2020 global reassessment of the neutrino oscillation picture, JHEP 02 (2021) 071, [arXiv:2006.11237].
  * (41) R. Foot and X.-G. He, Comment on Z Z-prime mixing in extended gauge theories, Phys. Lett. B 267 (1991) 509–512.
  * (42) MEG II Collaboration, A. M. Baldini et al., The design of the MEG II experiment, Eur. Phys. J. C 78 (2018), no. 5 380, [arXiv:1801.04688].
  * (43) M. Chiappini et al., Commissioning and preliminary performance of the MEG II drift chamber, Nucl. Instrum. Meth. A 1041 (2022) 167314.
  * (44) G. Azuelos et al., Status of the X17 search in Montreal, J. Phys. Conf. Ser. 2391 (2022), no. 1 012008, [arXiv:2211.11900].
  * (45) J. L. Feng, B. Fornal, I. Galon, S. Gardner, J. Smolinsky, T. M. P. Tait, and P. Tanedo, Protophobic Fifth-Force Interpretation of the Observed Anomaly in 8Be Nuclear Transitions, Phys. Rev. Lett. 117 (2016), no. 7 071803, [arXiv:1604.07411].
  * (46) J. L. Feng, B. Fornal, I. Galon, S. Gardner, J. Smolinsky, T. M. P. Tait, and P. Tanedo, Particle physics models for the 17 MeV anomaly in beryllium nuclear decays, Phys. Rev. D 95 (2017), no. 3 035017, [arXiv:1608.03591].
  * (47) L. Delle Rose, S. Khalil, S. J. D. King, S. Moretti, and A. M. Thabt, Atomki Anomaly in Family-Dependent $U(1)^{\prime}$ Extension of the Standard Model, Phys. Rev. D 99 (2019), no. 5 055022, [arXiv:1811.07953].
  * (48) L. Delle Rose, S. Khalil, S. J. D. King, and S. Moretti, New Physics Suggested by Atomki Anomaly, Front. in Phys. 7 (2019) 73, [arXiv:1812.05497].
  * (49) C. H. Nam, 17 MeV Atomki anomaly from short-distance structure of spacetime, Eur. Phys. J. C 80 (2020), no. 3 231, [arXiv:1907.09819].
  * (50) J. L. Feng, T. M. P. Tait, and C. B. Verhaaren, Dynamical Evidence For a Fifth Force Explanation of the ATOMKI Nuclear Anomalies, Phys. Rev. D 102 (2020), no. 3 036016, [arXiv:2006.01151].
  * (51) D. Barducci and C. Toni, An updated view on the ATOMKI nuclear anomalies, JHEP 02 (2023) 154, [arXiv:2212.06453]. [Erratum: JHEP 07, 168 (2023)].
  * (52) P. B. Denton and J. Gehrlein, Neutrino constraints and the ATOMKI X17 anomaly, Phys. Rev. D 108 (2023), no. 1 015009, [arXiv:2304.09877].
  * (53) M. Hostert and M. Pospelov, Pion decay constraints on exotic 17 MeV vector bosons, Phys. Rev. D 108 (2023), no. 5 055011, [arXiv:2306.15077].
  * (54) L. Landau, On the angular momentum of a system of two photons, Dokl. Akad. Nauk SSSR 60 (1948), no. 2 207–209.
  * (55) C.-N. Yang, Selection Rules for the Dematerialization of a Particle Into Two Photons, Phys. Rev. 77 (1950) 242–245.
  * (56) NA48/2 Collaboration, J. R. Batley et al., Search for the dark photon in $\pi^{0}$ decays, Phys. Lett. B 746 (2015) 178–185, [arXiv:1504.00607].
  * (57) NA48/2 Collaboration, M. Raggi, NA48/2 studies of rare decays, Nuovo Cim. C 38 (2016), no. 4 132, [arXiv:1508.01307].
  * (58) A. Anastasi et al., Limit on the production of a low-mass vector boson in $\mathrm{e}^{+}\mathrm{e}^{-}\to\mathrm{U}\gamma$, $\mathrm{U}\to\mathrm{e}^{+}\mathrm{e}^{-}$ with the KLOE experiment, Phys. Lett. B 750 (2015) 633–637, [arXiv:1509.00740].
  * (59) NA64 Collaboration, D. Banerjee et al., Search for a Hypothetical 16.7 MeV Gauge Boson and Dark Photons in the NA64 Experiment at CERN, Phys. Rev. Lett. 120 (2018), no. 23 231802, [arXiv:1803.07748].
  * (60) E. M. Riordan et al., A Search for Short Lived Axions in an Electron Beam Dump Experiment, Phys. Rev. Lett. 59 (1987) 755.
  * (61) NA64 Collaboration, D. Banerjee et al., Improved limits on a hypothetical X(16.7) boson and a dark photon decaying into $e^{+}e^{-}$ pairs, Phys. Rev. D 101 (2020), no. 7 071101, [arXiv:1912.11389].
  * (62) S. Andreas, C. Niebuhr, and A. Ringwald, New Limits on Hidden Photons from Past Electron Beam Dumps, Phys. Rev. D 86 (2012) 095019, [arXiv:1209.6083].
  * (63) SLAC E158 Collaboration, P. L. Anthony et al., Precision measurement of the weak mixing angle in Moller scattering, Phys. Rev. Lett. 95 (2005) 081601, [hep-ex/0504049].
  * (64) Y. Kahn, G. Krnjaic, S. Mishra-Sharma, and T. M. P. Tait, Light Weakly Coupled Axial Forces: Models, Constraints, and Projections, JHEP 05 (2017) 002, [arXiv:1609.09072].
  * (65) H. Davoudiasl, H.-S. Lee, and W. J. Marciano, ’Dark’ Z implications for Parity Violation, Rare Meson Decays, and Higgs Physics, Phys. Rev. D 85 (2012) 115019, [arXiv:1203.2947].
  * (66) C. Bouchiat and P. Fayet, Constraints on the parity-violating couplings of a new gauge boson, Phys. Lett. B 608 (2005) 87–94, [hep-ph/0410260].
  * (67) S. G. Porsev, K. Beloy, and A. Derevianko, Precision determination of electroweak coupling from atomic parity violation and implications for particle physics, Phys. Rev. Lett. 102 (2009) 181601, [arXiv:0902.0335].
  * (68) H. Davoudiasl, H.-S. Lee, and W. J. Marciano, Muon g-2, rare kaon decays, and parity violation from dark bosons, Phys. Rev. D 89 (2014), no. 9 095006, [arXiv:1402.3620].
  * (69) C. Alexandrou, S. Bacchio, M. Constantinou, K. Hadjiyiannakou, K. Jansen, and G. Koutsou, Quark flavor decomposition of the nucleon axial form factors, Phys. Rev. D 104 (2021) 074503, [arXiv:2106.13468].
  * (70) C. Alexandrou, S. Bacchio, M. Constantinou, J. Finkenrath, K. Hadjiyiannakou, K. Jansen, G. Koutsou, and A. Vaquero Aviles-Casco, Nucleon axial, tensor, and scalar charges and $\sigma$-terms in lattice QCD, Phys. Rev. D 102 (2020), no. 5 054517, [arXiv:1909.00485].
  * (71) Muon g-2 Collaboration, B. Abi et al., Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm, Phys. Rev. Lett. 126 (2021), no. 14 141801, [arXiv:2104.03281].
  * (72) J-PARC Muon g-2/EDM Collaboration, S. Lee, Muon $g-2$/EDM Experiment at J-PARC, PoS NuFact2019 (2019) 074.
  * (73) T. Aoyama et al., The anomalous magnetic moment of the muon in the Standard Model, Phys. Rept. 887 (2020) 1–166, [arXiv:2006.04822].
  * (74) T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio, Complete Tenth-Order QED Contribution to the Muon g-2, Phys. Rev. Lett. 109 (2012) 111808, [arXiv:1205.5370].
  * (75) T. Aoyama, T. Kinoshita, and M. Nio, Theory of the Anomalous Magnetic Moment of the Electron, Atoms 7 (2019), no. 1 28.
  * (76) A. Czarnecki, W. J. Marciano, and A. Vainshtein, Refinements in electroweak contributions to the muon anomalous magnetic moment, Phys. Rev. D 67 (2003) 073006, [hep-ph/0212229]. [Erratum: Phys.Rev.D 73, 119901 (2006)].
  * (77) C. Gnendiger, D. Stöckinger, and H. Stöckinger-Kim, The electroweak contributions to $(g-2)_{\mu}$ after the Higgs boson mass measurement, Phys. Rev. D 88 (2013) 053005, [arXiv:1306.5546].
  * (78) M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, Reevaluation of the hadronic vacuum polarisation contributions to the Standard Model predictions of the muon $g-2$ and ${\alpha(m_{Z}^{2})}$ using newest hadronic cross-section data, Eur. Phys. J. C 77 (2017), no. 12 827, [arXiv:1706.09436].
  * (79) A. Keshavarzi, D. Nomura, and T. Teubner, Muon $g-2$ and $\alpha(M_{Z}^{2})$: a new data-based analysis, Phys. Rev. D 97 (2018), no. 11 114025, [arXiv:1802.02995].
  * (80) G. Colangelo, M. Hoferichter, and P. Stoffer, Two-pion contribution to hadronic vacuum polarization, JHEP 02 (2019) 006, [arXiv:1810.00007].
  * (81) M. Hoferichter, B.-L. Hoid, and B. Kubis, Three-pion contribution to hadronic vacuum polarization, JHEP 08 (2019) 137, [arXiv:1907.01556].
  * (82) M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, A new evaluation of the hadronic vacuum polarisation contributions to the muon anomalous magnetic moment and to $\mathbf{\boldsymbol{\alpha}(m_{Z}^{2})}$, Eur. Phys. J. C 80 (2020), no. 3 241, [arXiv:1908.00921]. [Erratum: Eur.Phys.J.C 80, 410 (2020)].
  * (83) A. Keshavarzi, D. Nomura, and T. Teubner, $g-2$ of charged leptons, $\alpha(M^{2}_{Z})$ , and the hyperfine splitting of muonium, Phys. Rev. D 101 (2020), no. 1 014029, [arXiv:1911.00367].
  * (84) A. Kurz, T. Liu, P. Marquard, and M. Steinhauser, Hadronic contribution to the muon anomalous magnetic moment to next-to-next-to-leading order, Phys. Lett. B 734 (2014) 144–147, [arXiv:1403.6400].
  * (85) K. Melnikov and A. Vainshtein, Hadronic light-by-light scattering contribution to the muon anomalous magnetic moment revisited, Phys. Rev. D 70 (2004) 113006, [hep-ph/0312226].
  * (86) P. Masjuan and P. Sanchez-Puertas, Pseudoscalar-pole contribution to the $(g_{\mu}-2)$: a rational approach, Phys. Rev. D 95 (2017), no. 5 054026, [arXiv:1701.05829].
  * (87) G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, Dispersion relation for hadronic light-by-light scattering: two-pion contributions, JHEP 04 (2017) 161, [arXiv:1702.07347].
  * (88) M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold, and S. P. Schneider, Dispersion relation for hadronic light-by-light scattering: pion pole, JHEP 10 (2018) 141, [arXiv:1808.04823].
  * (89) A. Gérardin, H. B. Meyer, and A. Nyffeler, Lattice calculation of the pion transition form factor with $N_{f}=2+1$ Wilson quarks, Phys. Rev. D 100 (2019), no. 3 034520, [arXiv:1903.09471].
  * (90) J. Bijnens, N. Hermansson-Truedsson, and A. Rodríguez-Sánchez, Short-distance constraints for the HLbL contribution to the muon anomalous magnetic moment, Phys. Lett. B 798 (2019) 134994, [arXiv:1908.03331].
  * (91) G. Colangelo, F. Hagelstein, M. Hoferichter, L. Laub, and P. Stoffer, Longitudinal short-distance constraints for the hadronic light-by-light contribution to $(g-2)_{\mu}$ with large-$N_{c}$ Regge models, JHEP 03 (2020) 101, [arXiv:1910.13432].
  * (92) V. Pauk and M. Vanderhaeghen, Single meson contributions to the muon‘s anomalous magnetic moment, Eur. Phys. J. C 74 (2014), no. 8 3008, [arXiv:1401.0832].
  * (93) I. Danilkin and M. Vanderhaeghen, Light-by-light scattering sum rules in light of new data, Phys. Rev. D 95 (2017), no. 1 014019, [arXiv:1611.04646].
  * (94) F. Jegerlehner, The Anomalous Magnetic Moment of the Muon, vol. 274.  Springer, Cham, 2017.
  * (95) M. Knecht, S. Narison, A. Rabemananjara, and D. Rabetiarivony, Scalar meson contributions to a $\mu$ from hadronic light-by-light scattering, Phys. Lett. B 787 (2018) 111–123, [arXiv:1808.03848].
  * (96) G. Eichmann, C. S. Fischer, and R. Williams, Kaon-box contribution to the anomalous magnetic moment of the muon, Phys. Rev. D 101 (2020), no. 5 054015, [arXiv:1910.06795].
  * (97) P. Roig and P. Sanchez-Puertas, Axial-vector exchange contribution to the hadronic light-by-light piece of the muon anomalous magnetic moment, Phys. Rev. D 101 (2020), no. 7 074019, [arXiv:1910.02881].
  * (98) G. Colangelo, M. Hoferichter, A. Nyffeler, M. Passera, and P. Stoffer, Remarks on higher-order hadronic corrections to the muon g$-$2, Phys. Lett. B 735 (2014) 90–91, [arXiv:1403.7512].
  * (99) T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, L. Jin, C. Jung, and C. Lehner, Hadronic Light-by-Light Scattering Contribution to the Muon Anomalous Magnetic Moment from Lattice QCD, Phys. Rev. Lett. 124 (2020), no. 13 132002, [arXiv:1911.08123].
  * (100) S. Borsanyi et al., Leading hadronic contribution to the muon magnetic moment from lattice QCD, Nature 593 (2021), no. 7857 51–55, [arXiv:2002.12347].
  * (101) CMD-3 Collaboration, F. V. Ignatov et al., Measurement of the $e^{+}e^{-}\to\pi^{+}\pi^{-}$ cross section from threshold to 1.2 GeV with the CMD-3 detector, arXiv:2302.08834.
  * (102) B. Dutta, S. Ghosh, P. Huang, and J. Kumar, Explaining g$\mu$-2 and RK(*) using the light mediators of U(1)T3R, Phys. Rev. D 105 (2022), no. 1 015011, [arXiv:2105.07655].
  * (103) B. Döbrich, J. Jaeckel, F. Kahlhoefer, A. Ringwald, and K. Schmidt-Hoberg, ALPtraum: ALP production in proton beam dump experiments, JHEP 02 (2016) 018, [arXiv:1512.03069].
  * (104) M. J. Dolan, T. Ferber, C. Hearty, F. Kahlhoefer, and K. Schmidt-Hoberg, Revised constraints and Belle II sensitivity for visible and invisible axion-like particles, JHEP 12 (2017) 094, [arXiv:1709.00009].
  * (105) J. Bjorken, S. Ecklund, W. Nelson, A. Abashian, C. Church, B. Lu, L. Mo, T. Nunamaker, and P. Rassmann, Search for Neutral Metastable Penetrating Particles Produced in the SLAC Beam Dump, Phys. Rev. D 38 (1988) 3375.
  * (106) B. Batell, N. Lange, D. McKeen, M. Pospelov, and A. Ritz, Muon anomalous magnetic moment through the leptonic Higgs portal, Phys. Rev. D 95 (2017), no. 7 075003, [arXiv:1606.04943].
  * (107) J. L. Feng, I. Galon, F. Kling, and S. Trojanowski, ForwArd Search ExpeRiment at the LHC, Phys. Rev. D 97 (2018), no. 3 035001, [arXiv:1708.09389].
  * (108) J. L. Feng, I. Galon, F. Kling, and S. Trojanowski, Dark Higgs bosons at the ForwArd Search ExpeRiment, Phys. Rev. D 97 (2018), no. 5 055034, [arXiv:1710.09387].
  * (109) B. Batell, A. Freitas, A. Ismail, and D. Mckeen, Flavor-specific scalar mediators, Phys. Rev. D 98 (2018), no. 5 055026, [arXiv:1712.10022].
  * (110) S. Alekhin et al., A facility to Search for Hidden Particles at the CERN SPS: the SHiP physics case, Rept. Prog. Phys. 79 (2016), no. 12 124201, [arXiv:1504.04855].
  * (111) C.-Y. Chen, M. Pospelov, and Y.-M. Zhong, Muon Beam Experiments to Probe the Dark Sector, Phys. Rev. D 95 (2017), no. 11 115005, [arXiv:1701.07437].
  * (112) M. Sorel, J. M. Conrad, and M. Shaevitz, A Combined analysis of short baseline neutrino experiments in the (3+1) and (3+2) sterile neutrino oscillation hypotheses, Phys. Rev. D 70 (2004) 073004, [hep-ph/0305255].
  * (113) G. Karagiorgi, Z. Djurcic, J. M. Conrad, M. H. Shaevitz, and M. Sorel, Viability of Delta m**2 ~ 1- eV**2 sterile neutrino mixing models in light of MiniBooNE electron neutrino and antineutrino data from the Booster and NuMI beamlines, Phys. Rev. D 80 (2009) 073001, [arXiv:0906.1997]. [Erratum: Phys.Rev.D 81, 039902 (2010)].
  * (114) C. Giunti and M. Laveder, Implications of 3+1 Short-Baseline Neutrino Oscillations, Phys. Lett. B 706 (2011) 200–207, [arXiv:1111.1069].
  * (115) J. Kopp, M. Maltoni, and T. Schwetz, Are There Sterile Neutrinos at the eV Scale?, Phys. Rev. Lett. 107 (2011) 091801, [arXiv:1103.4570].
  * (116) K. N. Abazajian et al., Light Sterile Neutrinos: A White Paper, arXiv:1204.5379.
  * (117) J. M. Conrad, C. M. Ignarra, G. Karagiorgi, M. H. Shaevitz, and J. Spitz, Sterile Neutrino Fits to Short Baseline Neutrino Oscillation Measurements, Adv. High Energy Phys. 2013 (2013) 163897, [arXiv:1207.4765].
  * (118) J. Kopp, P. A. N. Machado, M. Maltoni, and T. Schwetz, Sterile Neutrino Oscillations: The Global Picture, JHEP 05 (2013) 050, [arXiv:1303.3011].
  * (119) G. H. Collin, C. A. Argüelles, J. M. Conrad, and M. H. Shaevitz, First Constraints on the Complete Neutrino Mixing Matrix with a Sterile Neutrino, Phys. Rev. Lett. 117 (2016), no. 22 221801, [arXiv:1607.00011].
  * (120) S. Gariazzo, C. Giunti, M. Laveder, and Y. F. Li, Updated Global 3+1 Analysis of Short-BaseLine Neutrino Oscillations, JHEP 06 (2017) 135, [arXiv:1703.00860].
  * (121) J. Asaadi, E. Church, R. Guenette, B. J. P. Jones, and A. M. Szelc, New light Higgs boson and short-baseline neutrino anomalies, Phys. Rev. D 97 (2018), no. 7 075021, [arXiv:1712.08019].
  * (122) M. Dentler, A. Hernández-Cabezudo, J. Kopp, P. A. N. Machado, M. Maltoni, I. Martinez-Soler, and T. Schwetz, Updated Global Analysis of Neutrino Oscillations in the Presence of eV-Scale Sterile Neutrinos, JHEP 08 (2018) 010, [arXiv:1803.10661].
  * (123) S. Böser, C. Buck, C. Giunti, J. Lesgourgues, L. Ludhova, S. Mertens, A. Schukraft, and M. Wurm, Status of Light Sterile Neutrino Searches, Prog. Part. Nucl. Phys. 111 (2020) 103736, [arXiv:1906.01739].
  * (124) A. Diaz, C. A. Argüelles, G. H. Collin, J. M. Conrad, and M. H. Shaevitz, Where Are We With Light Sterile Neutrinos?, Phys. Rept. 884 (2020) 1–59, [arXiv:1906.00045].
  * (125) G. H. Collin, C. A. Argüelles, J. M. Conrad, and M. H. Shaevitz, Sterile Neutrino Fits to Short Baseline Data, Nucl. Phys. B 908 (2016) 354–365, [arXiv:1602.00671].
  * (126) MINOS+, Daya Bay Collaboration, P. Adamson et al., Improved Constraints on Sterile Neutrino Mixing from Disappearance Searches in the MINOS, MINOS+, Daya Bay, and Bugey-3 Experiments, Phys. Rev. Lett. 125 (2020), no. 7 071801, [arXiv:2002.00301].
  * (127) IceCube Collaboration, M. G. Aartsen et al., eV-Scale Sterile Neutrino Search Using Eight Years of Atmospheric Muon Neutrino Data from the IceCube Neutrino Observatory, Phys. Rev. Lett. 125 (2020), no. 14 141801, [arXiv:2005.12942].
  * (128) IceCube Collaboration, M. G. Aartsen et al., Searching for eV-scale sterile neutrinos with eight years of atmospheric neutrinos at the IceCube Neutrino Telescope, Phys. Rev. D 102 (2020), no. 5 052009, [arXiv:2005.12943].
  * (129) J. Hamann, S. Hannestad, G. G. Raffelt, and Y. Y. Y. Wong, Sterile neutrinos with eV masses in cosmology: How disfavoured exactly?, JCAP 09 (2011) 034, [arXiv:1108.4136].
  * (130) M. Archidiacono, N. Fornengo, C. Giunti, S. Hannestad, and A. Melchiorri, Sterile neutrinos: Cosmology versus short-baseline experiments, Phys. Rev. D 87 (2013), no. 12 125034, [arXiv:1302.6720].
  * (131) S. Hagstotz, P. F. de Salas, S. Gariazzo, M. Gerbino, M. Lattanzi, S. Vagnozzi, K. Freese, and S. Pastor, Bounds on light sterile neutrino mass and mixing from cosmology and laboratory searches, Phys. Rev. D 104 (2021), no. 12 123524, [arXiv:2003.02289].
  * (132) MicroBooNE Collaboration, R. Acciarri et al., Design and Construction of the MicroBooNE Detector, JINST 12 (2017), no. 02 P02017, [arXiv:1612.05824].
  * (133) MicroBooNE Collaboration, P. Abratenko et al., Search for Neutrino-Induced Neutral-Current $\Delta$ Radiative Decay in MicroBooNE and a First Test of the MiniBooNE Low Energy Excess under a Single-Photon Hypothesis, Phys. Rev. Lett. 128 (2022) 111801, [arXiv:2110.00409].
  * (134) MicroBooNE Collaboration, P. Abratenko et al., Search for an Excess of Electron Neutrino Interactions in MicroBooNE Using Multiple Final-State Topologies, Phys. Rev. Lett. 128 (2022), no. 24 241801, [arXiv:2110.14054].
  * (135) G. Contin et al., The STAR MAPS-based PiXeL detector, Nucl. Instrum. Meth. A 907 (2018) 60–80, [arXiv:1710.02176].
  * (136) MicroBooNE Collaboration, P. Abratenko et al., Search for an anomalous excess of inclusive charged-current $\nu_{e}$ interactions in the MicroBooNE experiment using Wire-Cell reconstruction, Phys. Rev. D 105 (2022), no. 11 112005, [arXiv:2110.13978].
  * (137) P. Ballett, S. Pascoli, and M. Ross-Lonergan, U(1)’ mediated decays of heavy sterile neutrinos in MiniBooNE, Phys. Rev. D 99 (2019) 071701, [arXiv:1808.02915].
  * (138) P. Ballett, M. Hostert, and S. Pascoli, Neutrino Masses from a Dark Neutrino Sector below the Electroweak Scale, Phys. Rev. D 99 (2019), no. 9 091701, [arXiv:1903.07590].
  * (139) O. Fischer, A. Hernández-Cabezudo, and T. Schwetz, Explaining the MiniBooNE excess by a decaying sterile neutrino with mass in the 250 MeV range, Phys. Rev. D 101 (2020), no. 7 075045, [arXiv:1909.09561].
  * (140) M. Dentler, I. Esteban, J. Kopp, and P. Machado, Decaying Sterile Neutrinos and the Short Baseline Oscillation Anomalies, Phys. Rev. D 101 (2020), no. 11 115013, [arXiv:1911.01427].
  * (141) A. de Gouvêa, O. L. G. Peres, S. Prakash, and G. V. Stenico, On The Decaying-Sterile Neutrino Solution to the Electron (Anti)Neutrino Appearance Anomalies, JHEP 07 (2020) 141, [arXiv:1911.01447].
  * (142) W. Abdallah, R. Gandhi, and S. Roy, Understanding the MiniBooNE and the muon $g-2$ anomalies with a light $Z^{\prime}$ and a second Higgs doublet, arXiv:2006.01948.
  * (143) A. Abdullahi, M. Hostert, and S. Pascoli, A dark seesaw solution to low energy anomalies: MiniBooNE, the muon (g-2), and BaBar, Phys. Lett. B 820 (2021) 136531, [arXiv:2007.11813].
  * (144) W. Abdallah, R. Gandhi, and S. Roy, Two-Higgs doublet solution to the LSND, MiniBooNE and muon g-2 anomalies, Phys. Rev. D 104 (2021), no. 5 055028, [arXiv:2010.06159].
  * (145) A. M. Abdullahi, J. Hoefken Zink, M. Hostert, D. Massaro, and S. Pascoli, A panorama of new-physics explanations to the MiniBooNE excess, arXiv:2308.02543.
  * (146) MiniBooNE DM Collaboration, A. A. Aguilar-Arevalo et al., Dark Matter Search in Nucleon, Pion, and Electron Channels from a Proton Beam Dump with MiniBooNE, Phys. Rev. D 98 (2018), no. 11 112004, [arXiv:1807.06137].
  * (147) V. Brdar, O. Fischer, and A. Y. Smirnov, Model-independent bounds on the nonoscillatory explanations of the MiniBooNE excess, Phys. Rev. D 103 (2021), no. 7 075008, [arXiv:2007.14411].
  * (148) M. Martini, M. Ericson, and G. Chanfray, Neutrino energy reconstruction problems and neutrino oscillations, Phys. Rev. D 85 (2012) 093012, [arXiv:1202.4745].
  * (149) CHARM-II Collaboration, K. De Winter et al., A Detector for the Study of Neutrino - Electron Scattering, Nucl. Instrum. Meth. A 278 (1989) 670.
  * (150) CHARM-II Collaboration, D. Geiregat et al., Calibration and performance of the CHARM-II detector, Nucl. Instrum. Meth. A 325 (1993) 92–108.
  * (151) CHARM-II Collaboration, P. Vilain et al., Precision measurement of electroweak parameters from the scattering of muon-neutrinos on electrons, Phys. Lett. B 335 (1994) 246–252.
  * (152) C. A. Argüelles, M. Hostert, and Y.-D. Tsai, Testing New Physics Explanations of the MiniBooNE Anomaly at Neutrino Scattering Experiments, Phys. Rev. Lett. 123 (2019), no. 26 261801, [arXiv:1812.08768].
  * (153) MINERvA Collaboration, J. Wolcott et al., Measurement of electron neutrino quasielastic and quasielasticlike scattering on hydrocarbon at $\langle E_{\nu}\rangle$ = 3.6 GeV, Phys. Rev. Lett. 116 (2016), no. 8 081802, [arXiv:1509.05729].
  * (154) MINERvA Collaboration, J. Park et al., Measurement of Neutrino Flux from Neutrino-Electron Elastic Scattering, Phys. Rev. D 93 (2016), no. 11 112007, [arXiv:1512.07699].
  * (155) MINERvA Collaboration, E. Valencia et al., Constraint of the MINER$\nu$A medium energy neutrino flux using neutrino-electron elastic scattering, Phys. Rev. D 100 (2019), no. 9 092001, [arXiv:1906.00111].
  * (156) T2K Collaboration, Y. Kudenko, The Near neutrino detector for the T2K experiment, Nucl. Instrum. Meth. A 598 (2009) 289–295, [arXiv:0805.0411].
  * (157) T2K Collaboration, K. Abe et al., The T2K Experiment, Nucl. Instrum. Meth. A 659 (2011) 106–135, [arXiv:1106.1238].
  * (158) S. Assylbekov et al., The T2K ND280 Off-Axis Pi-Zero Detector, Nucl. Instrum. Meth. A 686 (2012) 48–63, [arXiv:1111.5030].
  * (159) T2K ND280 FGD Collaboration, P. A. Amaudruz et al., The T2K Fine-Grained Detectors, Nucl. Instrum. Meth. A 696 (2012) 1–31, [arXiv:1204.3666].
  * (160) T2K Collaboration, K. Abe et al., T2K neutrino flux prediction, Phys. Rev. D 87 (2013), no. 1 012001, [arXiv:1211.0469]. [Addendum: Phys.Rev.D 87, 019902 (2013)].
  * (161) T2K Collaboration, K. Abe et al., Search for heavy neutrinos with the T2K near detector ND280, Phys. Rev. D 100 (2019), no. 5 052006, [arXiv:1902.07598].
  * (162) T2K Collaboration, K. Abe et al., Measurement of the charged-current electron (anti-)neutrino inclusive cross-sections at the T2K off-axis near detector ND280, JHEP 10 (2020) 114, [arXiv:2002.11986].
  * (163) R. H. Helm, Inelastic and Elastic Scattering of 187-Mev Electrons from Selected Even-Even Nuclei, Phys. Rev. 104 (1956) 1466–1475.
  * (164) J. Engel, Nuclear form-factors for the scattering of weakly interacting massive particles, Phys. Lett. B 264 (1991) 114–119.
  * (165) T. Falk, A. Ferstl, and K. A. Olive, Variations of the neutralino elastic cross-section with CP violating phases, Astropart. Phys. 13 (2000) 301–316, [hep-ph/9908311].
  * (166) M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Remarks on Higgs Boson Interactions with Nucleons, Phys. Lett. B 78 (1978) 443–446.
  * (167) J. M. Alarcon, J. Martin Camalich, and J. A. Oller, The chiral representation of the $\pi N$ scattering amplitude and the pion-nucleon sigma term, Phys. Rev. D 85 (2012) 051503, [arXiv:1110.3797].
  * (168) J. M. Alarcon, L. S. Geng, J. Martin Camalich, and J. A. Oller, The strangeness content of the nucleon from effective field theory and phenomenology, Phys. Lett. B 730 (2014) 342–346, [arXiv:1209.2870].
  * (169) A. Crivellin, M. Hoferichter, and M. Procura, Accurate evaluation of hadronic uncertainties in spin-independent WIMP-nucleon scattering: Disentangling two- and three-flavor effects, Phys. Rev. D 89 (2014) 054021, [arXiv:1312.4951].
  * (170) M. Hoferichter, J. Ruiz de Elvira, B. Kubis, and U.-G. Meißner, High-Precision Determination of the Pion-Nucleon $\sigma$ Term from Roy-Steiner Equations, Phys. Rev. Lett. 115 (2015) 092301, [arXiv:1506.04142].
  * (171) P. Junnarkar and A. Walker-Loud, Scalar strange content of the nucleon from lattice QCD, Phys. Rev. D 87 (2013) 114510, [arXiv:1301.1114].
  * (172) J. Ellis, TikZ-Feynman: Feynman diagrams with TikZ, Comput. Phys. Commun. 210 (2017) 103–123, [arXiv:1601.05437].