aainstitutetext: Theoretical Division, Los Alamos National Laboratory,
Los Alamos, NM 87545, U.S.A.bbinstitutetext: Physics Division, Los Alamos
National Laboratory,
Los Alamos, NM 87545, U.S.A.

# Quasi-sterile neutrinos from dark sectors. Part I. BSM matter effects in
neutrino oscillations and the short-baseline anomalies.

Daniele S. M. Alves,111Corresponding author. b William C. Louis a and Patrick
G. deNiverville<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Quasi-sterile neutrinos are a natural consequence of dark sectors interacting
with the Standard Model (SM) sector via neutrino- and vector-portals.
Essentially, quasi-sterile neutrinos are light dark sector fermions with two
generic properties: (i) they mix with the active neutrinos of the SM, and (ii)
they are charged under a vector mediator that couples feebly to SM matter.
Various interesting phenomenological consequences result from this class of
particles. In this article, we investigate one such consequence: new, beyond
the SM matter effects that can alter in-medium neutrino oscillations. In
particular, for special windows of energy and matter densities, active
neutrinos can resonantly oscillate into sterile neutrinos. We take advantage
of this feature to build a quasi-sterile neutrino model that can explain the
MiniBooNE and LSND anomalies, while remaining compatible with observations
from long-baseline reactor- and accelerator-based neutrino experiments. This
model is also likely compatible with the recent results reported by the
MicroBooNE collaboration (albeit we cannot precisely quantify this claim due
to a lack of information in MicroBooNE’s public data releases to date).
Implications for solar neutrinos and $\nu_{e}$ disappearance searches are also
briefly discussed.

††preprint: LA-UR-22-20015

##  1 Introduction

Over the past decade, light dark sectors have emerged as a contender to the
WIMP paradigm in providing a possible explanation for the cosmological origin
and particle nature of dark matter Arkani-Hamed-ml-2008hhe ; Jaeckel-ml-2010ni
; Essig-ml-2013lka ; Alexander-ml-2016aln ; Lanfranchi-ml-2020crw . While
there is no consensus on a precise definition of dark sectors, a fairly
generic framework to characterize their phenomenology are the so-called
_portals_ : particle ‘mediators’ that couple to matter and/or forces both in
the Standard Model (SM) and in the dark sector, typically (but not
necessarily) by mixing with neutral SM particles — such as the photon, the
Higgs boson, or neutrinos — if their quantum numbers permit. Most dark sector
phenomenological studies, however, take the approach of ‘switching on’ only
one individual portal at a time. In this article, we investigate a broader
scenario of two simultaneous dark sector portals, namely, the _neutrino_ and
_vector_ portals, and focus on one generic phenomenological consequence:
_quasi-sterile neutrinos_. Specifically, the neutrino portal can be realized
by light (say, sub-keV) dark sector fermions mixing with the active neutrinos
of the Standard Model. Since these dark sector fermions are neutral under the
SM gauge interactions, they are, by definition, _sterile neutrinos_. However,
in the presence of a vector portal — a dark sector gauge interaction that
couples feebly to SM fermions — sterile neutrinos charged under the vector
portal might be more appropriately characterized as _quasi_ -sterile neutrinos
(see figure 1). From this, two phenomenological features naturally arise: (i)
the presence of beyond the Standard Model (BSM) neutrino oscillations due to
mixing of active and sterile neutrinos, and (ii) the possibility of BSM matter
effects, and therefore, of _resonant_ active-to-sterile neutrino oscillations
for specific regimes of matter density and neutrino energy.

Figure 1: A generic consequence of dark sectors with neutrino- and vector-
portals are quasi-sterile neutrinos. The BSM matter potential generated by the
dark vector interactions can alter neutrino oscillations in matter.

There are many interesting motivations to study quasi-sterile neutrinos. One
of them is the excess of electron recoil events recently reported by the
XENON1T collaboration XENON-ml-2020rca . Attempts to explain this excess as a
signal of solar neutrinos scattering off electrons via an anomalous neutrino
magnetic dipole moment are in tension with limits from stellar cooling
Corsico-ml-2014mpa ; bib2019arXiv191010568A ; Yusof-ml-2021fhu ; Studenikin-
ml-2021fmn . On the other hand, these astrophysical constraints could be
relaxed if the anomalous XENON1T electron recoil events were instead induced
by quasi-sterile solar neutrinos scattering via a light dark mediator (see
also Boehm-ml-2020ltd ; Bloch-ml-2020uzh ; Babu-ml-2020ivd ; Ge-ml-2021snv ).
This topic will be explored in a future publication future .

Another motivation for quasi-sterile neutrinos, which will be the focus of
this article, is the puzzle posed by the short-baseline neutrino oscillation
anomalies, which could be an indication of light sterile states mixing with
the active neutrinos of the SM. Two intriguing anomalies of this type are the
$\bar{\nu}_{\mu}\to\bar{\nu}_{e}$ and $\nu_{\mu}\to\nu_{e}$ appearance events
reported by the LSND LSND-ml-1996ubh ; LSND-ml-1997vun ; LSND-ml-1996vlr ;
LSND-ml-1997vqj ; LSND-ml-2001aii and MiniBooNE MiniBooNE-ml-2018esg ;
MiniBooNE-ml-2020pnu collaborations. The data in both of these experiments
show excesses of (anti-)electron-neutrino charged-current scattering events at
$L/E_{\nu}\sim 0.5-2$ m/MeV, at rates compatible with
$\stackon[-1.3pt]{$\bar{\nu}$}{\scalebox{0.25}{\hskip
2.0pt{{(}}}\scalebox{0.25}{\hskip 17.0pt {{)}}}}_{\\!e}$ appearance
probabilities of _O_(0.1–1%). These
$\stackon[-1.3pt]{$\bar{\nu}$}{\scalebox{0.25}{\hskip
2.0pt{{(}}}\scalebox{0.25}{\hskip 17.0pt {{)}}}}_{\\!e}$ appearance events
cannot be accommodated in the SM description of neutrino masses and mixings. A
simple BSM model including one sterile neutrino species in addition to the
three active species (the ‘vanilla’ 3+1 model) can provide a good fit to the
LSND excess, the MiniBooNE excess in _antineutrino_ mode, and the MiniBooNE
excess in _neutrino_ mode _for the energy range_ $E_{\nu}\gtrsim 400$ MeV.
However, an excess of $\nu_{e}$ appearance events with lower energies
$E_{\nu}\sim 150-300$ MeV still persists in the MiniBooNE data which cannot be
satisfactorily explained by any ‘3+$n$’ sterile neutrino model222Recent
results from the MicroBooNE collaboration MicroBooNE-ml-2021zai further
disfavor the interpretation of the MiniBooNE low energy excess as due to
single photon backgrounds from radiative $\Delta$ decays Hill-ml-2010zy ;
Zhang-ml-2012xn ; Wang-ml-2013wva ; Wang-ml-2014nat ; Alvarez-Ruso-ml-2021dna
. Maltoni-ml-2002xd ; Maltoni-ml-2007zf ; Conrad-ml-2012qt ; Kopp-ml-2013vaa ;
Giunti-ml-2015mwa ; Collin-ml-2016rao ; Boser-ml-2019rta ; Alvarez-Ruso-
ml-2021dna ; Brdar-ml-2021cgb . This challenge has led to numerous attempts to
explain the MiniBooNE low energy excess by invoking new physics of a dark
sector. In the ‘non-oscillatory’ approaches Gninenko-ml-2009ks ; Masip-
ml-2012ke ; Bai-ml-2015ztj ; Jordan-ml-2018qiy ; Ballett-ml-2018ynz ;
Bertuzzo-ml-2018itn ; Arguelles-ml-2018mtc ; Ballett-ml-2019pyw ; Fischer-
ml-2019fbw ; deGouvea-ml-2019qre ; Hostert-ml-2020oui ; Datta-ml-2020auq ;
Abdallah-ml-2020biq ; Abdullahi-ml-2020nyr ; Dutta-ml-2021cip ; Vergani-
ml-2021tgc , production and decays of dark sector fermions and/or light bosons
yield final states with $\nu_{e}$, $\gamma$, or collimated $e^{+}e^{-}$ pairs
in the MiniBooNE detector, all of which could potentially mimic a $\nu_{e}$
charged-current scattering event. In the ‘oscillatory’ approaches, BSM effects
alter neutrino propagation through a medium (either ordinary matter
Karagiorgi-ml-2012kw , the C$\nu$B Asaadi-ml-2017bhx , or extra dimensions
Pas-ml-2005rb ; Doring-ml-2018cob ), leading to resonant neutrino oscillations
within the window of energy where the MiniBooNE low energy excess is
concentrated. Previous attempts to implement oscillatory approaches have not
been successful, however, because new matter effects invariably alter the
effective in-medium mass splittings of active neutrinos Kopp-ml-2014fha ;
Denton-ml-2018dqq ; Barenboim-ml-2019hso ; Smirnov-ml-2021zgn , leading to
oscillation probabilities incompatible with observations at long-baseline
neutrino experiments.

In this article, we show that these difficulties can be overcome by a ‘triple-
resonance mechanism’ that is able to preserve the in-medium active neutrino
mass spectrum at high energies probed by accelerator-based neutrino
experiments, as well as the in-medium antineutrino mass spectrum at low
energies probed by reactor-based experiments.

At this point it is incumbent on us to comment on the recent $\nu_{e}$
searches by the MicroBooNE experiment based on data obtained from an exposure
of $7\times 10^{20}$ protons-on-target (POT) from the Booster Neutrino Beam
(BNB) MicroBooNE-ml-2021rmx . These analyses appear to strongly disfavor the
hypothesis that the MiniBooNE low energy excess is due to $\nu_{e}$ scattering
events. As we shall discuss in subsection 4.1, this conclusion is strongly
model dependent and likely does not hold at a high enough confidence level to
exclude the ‘oscillatory’ BSM explanation of the MiniBooNE low energy excess
proposed in this study.

This article is organized as follows: in section 2, we introduce and dissect
the main features of the resonant mechanism in a toy model, which will then be
used in section 3 to build a full model explaining the MiniBooNE excess.
Section 4 considers the full model’s consistency with the recent MicroBooNE
results, and provides a brief discussion of implications for solar neutrinos.
Section 5 summarizes this article’s main points and provides an outlook of
future experimental measurements that could falsify the quasi-sterile neutrino
explanation of short-baseline anomalies.

##  2 The minimal case as a toy model

We begin by considering the simplest case of a single quasi-sterile neutrino
flavor, which we shall call $S_{3}$, mixing with the active neutrinos of the
SM model. It turns out that this minimalistic model does not provide a
satisfactory explanation of the short-baseline neutrino anomalies. However, it
retains all the basic phenomenological features of the full model to be
considered in the next section, and therefore, for pedagogical purposes, we
will introduce it first in this section and highlight its key aspects and
shortcomings.

We adopt the PDG convention ParticleDataGroup-ml-2020ssz to relate the
neutrino mass eigenstates in the SM, $\nu_{i}$ ($i=1,2,3$), to the active
neutrino flavors, $\nu_{\ell}$ ($\ell=e,\mu,\tau$):

$\displaystyle|\nu_{1}\rangle~{}$ $\displaystyle=~{}$ $\displaystyle
U_{e1}\,|\nu_{e}\rangle~{}+~{}U_{\mu 1}\,|\nu_{\mu}\rangle~{}+~{}U_{\tau
1}\,|\nu_{\tau}\rangle$ (1a) $\displaystyle|\nu_{2}\rangle~{}$
$\displaystyle=~{}$ $\displaystyle U_{e2}\,|\nu_{e}\rangle~{}+~{}U_{\mu
2}\,|\nu_{\mu}\rangle~{}+~{}U_{\tau 2}\,|\nu_{\tau}\rangle$ (1b)
$\displaystyle|\nu_{3}\rangle~{}$ $\displaystyle=~{}$ $\displaystyle
U_{e3}\,|\nu_{e}\rangle~{}+~{}U_{\mu 3}\,|\nu_{\mu}\rangle~{}+~{}U_{\tau
3}\,|\nu_{\tau}\rangle$ (1c)

where the $U_{\ell i}$ coefficients above are the matrix elements of the PMNS
matrix $U$, which is parametrized by three mixing angles $\theta_{12}$,
$\theta_{23}$ and $\theta_{13}$, and a CP-violating phase
$\delta_{\text{CP}}$:

$U=\begin{pmatrix}1&~{}0&~{}0\\\ 0&~{}c_{23}&~{}s_{23}\\\
0&-s_{23}&~{}c_{23}\end{pmatrix}\cdot\begin{pmatrix}~{}c_{13}&~{}0&~{}~{}s_{13}\,e^{-i\delta_{\text{CP}}}\\\
0&~{}1&~{}0\\\
-s_{13}\,e^{i\delta_{\text{CP}}}&~{}0&~{}c_{13}\end{pmatrix}\cdot\begin{pmatrix}~{}c_{12}&s_{12}&~{}0\\\
-s_{12}&c_{12}&~{}0\\\ ~{}0&0&~{}1\end{pmatrix}$ (2)

with $s_{ij}\equiv\sin\theta_{ij}$ and $c_{ij}\equiv\cos\theta_{ij}$.

In this minimal model, we will introduce a small mixing between the sterile
neutrino $S_{3}$ and one of the active neutrino states, namely, $\nu_{3}$,
parametrized by a mixing angle $\theta_{{}_{\\!S_{3}}}\ll 1$. The mass
eigenstates _in vacuum_ are given by:

$\displaystyle|\nu_{1}\rangle~{}$ $\displaystyle=$
$\displaystyle~{}\sum_{~{}~{}~{}\ell\,=\,e,\,\mu,\,\tau}\\!\\!\\!U_{\ell
1}\,|\nu_{\ell}\rangle$ (3a) $\displaystyle|\nu_{2}\rangle~{}$
$\displaystyle=$
$\displaystyle~{}\sum_{~{}~{}~{}\ell\,=\,e,\,\mu,\,\tau}\\!\\!\\!U_{\ell
2}\,|\nu_{\ell}\rangle$ (3b) $\displaystyle|N_{3,\text{
vac}}^{{}_{(-)}}\rangle~{}$ $\displaystyle=$
$\displaystyle-e^{i\,\delta_{{}_{\\!S_{3}}}}\sin\theta_{{}_{\\!S_{3}}}\,|S_{3}\rangle\;+\;\cos\theta_{{}_{\\!S_{3}}}\,|\nu_{3}\rangle$
(3c) $\displaystyle=$
$\displaystyle-e^{i\,\delta_{{}_{\\!S_{3}}}}\sin\theta_{{}_{\\!S_{3}}}\,|S_{3}\rangle\;+\;\cos\theta_{{}_{\\!S_{3}}}\\!\\!\\!\\!\\!\sum_{~{}~{}~{}\ell\,=\,e,\,\mu,\,\tau}\\!\\!\\!\\!U_{\ell
3}\,|\nu_{\ell}\rangle$ $\displaystyle|N_{3,\text{ vac}}^{{}_{(+)}}\rangle~{}$
$\displaystyle=$
$\displaystyle~{}\cos\theta_{{}_{\\!S_{3}}}\,|S_{3}\rangle\;+\;e^{-i\,\delta_{{}_{\\!S_{3}}}}\sin\theta_{{}_{\\!S_{3}}}\,|\nu_{3}\rangle$
(3d) $\displaystyle=$
$\displaystyle~{}\cos\theta_{{}_{\\!S_{3}}}\,|S_{3}\rangle\;+\;e^{-i\,\delta_{{}_{\\!S_{3}}}}\sin\theta_{{}_{\\!S_{3}}}\\!\\!\\!\\!\\!\sum_{~{}~{}~{}\ell\,=\,e,\,\mu,\,\tau}\\!\\!\\!\\!U_{\ell
3}\,|\nu_{\ell}\rangle$

with mass eigenvalues $m_{1}$, $m_{2}$, $m_{3}$, and $M_{3}\gg m_{3}$,
respectively. Since $\sin\theta_{{}_{\\!S_{3}}}\ll 1$, the heavy eigenstate
$N_{3}^{{}_{(+)}}$ is mostly sterile with a small active component, whereas
$N_{3}^{{}_{(-)}}$ is composed mostly of the active neutrino state $\nu_{3}$,
with a small sterile component. In addition, note that because $S_{3}$ mixes
exclusively with $\nu_{3}$, the complex phase $\delta_{{}_{\\!S_{3}}}$ is
unphysical and can be fully reabsorbed by a unitary rotation $|S_{3}\rangle\to
e^{-i\,\delta_{{}_{\\!S_{3}}}}|S_{3}\rangle$. We will return to the issue of
additional CP-violating phases in subsection 3.2.

All of the above information can be captured in the neutrino Hamiltonian.
Working in the basis
$B=\\{|\nu_{1}\rangle,|\nu_{2}\rangle,|\nu_{3}\rangle,|S_{3}\rangle\\}$, we
can write the vacuum Hamiltonian in a familiar form:

$\displaystyle\widehat{H}_{B,\text{ vacuum}}$
$\displaystyle\simeq\begin{pmatrix}1&~{}~{}0~{}&0&~{}0\\\
0&~{}~{}1~{}&0&~{}0\\\
0&~{}~{}0~{}&~{}~{}\cos\theta_{{}_{\\!S_{3}}}&\sin\theta_{{}_{\\!S_{3}}}\\\
0&~{}~{}0&-\sin\theta_{{}_{\\!S_{3}}}&\cos\theta_{{}_{\\!S_{3}}}\end{pmatrix}\cdot\begin{pmatrix}\frac{m_{1}^{2}}{2\,E_{\nu}}&~{}0&~{}0&~{}0\\\
0~{}&\frac{m_{2}^{2}}{2\,E_{\nu}}&~{}0&~{}0\\\
0~{}&0~{}&\frac{m_{3}^{2}}{2\,E_{\nu}}&~{}0\\\
0~{}&0~{}&0~{}&\frac{M_{3}^{2}}{2\,E_{\nu}}\end{pmatrix}\cdot\begin{pmatrix}1&~{}~{}0~{}~{}&0&~{}0\\\
0&~{}~{}1~{}~{}&0&~{}0\\\
0&~{}~{}0~{}~{}&\cos\theta_{{}_{\\!S_{3}}}&-\sin\theta_{{}_{\\!S_{3}}}\\\
0&~{}~{}0~{}~{}&\sin\theta_{{}_{\\!S_{3}}}&~{}~{}\cos\theta_{{}_{\\!S_{3}}}\end{pmatrix}$
$\displaystyle=(\ldots)\times\mathds{1}_{3\times 3}~{}+~{}\frac{\delta
M_{3}^{2}}{4E_{\nu}}\begin{pmatrix}-1+2\frac{\delta m_{13}^{2}}{\delta
M_{3}^{2}}&0&0&0\\\ 0&-1+2\frac{\delta m_{23}^{2}}{\delta M_{3}^{2}}&0&0\\\
0&0&-\cos 2\theta_{{}_{\\!S_{3}}}~{}&\sin 2\theta_{{}_{\\!S_{3}}}\\\
0&0&~{}~{}\sin 2\theta_{{}_{\\!S_{3}}}~{}&\cos
2\theta_{{}_{\\!S_{3}}}\end{pmatrix}$ (4)

where we have defined the mass-squared splittings $\delta M_{3}^{2}\equiv
M_{3}^{2}-m_{3}^{2}$, $\delta m_{ij}^{2}\equiv m_{i}^{2}-m_{j}^{2}$, and we
did not bother to write the terms proportional to the identity operator in the
second line of (2), since they do not affect the difference between energy
eigenvalues and therefore do not contribute to neutrino oscillations.

When propagating through a medium, neutrinos receive additional contributions
to their energies due to the presence of matter potentials (which can be
understood as coherent forward scattering at zero momentum transfer). If this
effect is felt identically by all neutrino states — which is the case, in the
SM, of the matter potential generated by the neutral weak interactions
mediated by the $Z^{0}$ boson — the relative phases of the neutrino
propagation eigenstates are not changed, and therefore such matter potentials
do not affect neutrino oscillations in a medium. On the other hand, a matter
potential that affects different neutrino flavors non-identically will alter
the propagation eigenstates and their relative phases with respect to vacuum,
and can lead to significant distortions in neutrino oscillations for certain
energy and medium density regimes. The most well-known such effect is due to
the charged weak interactions mediated by the $W^{\pm}$ bosons, which generate
an effective matter potential to the electron-neutrino flavor, but not to
muon- nor tau-neutrino flavors, since the only charged leptons present in
ordinary matter are electrons. This is known as the standard
Mikheyev–Smirnov–Wolfenstein (MSW) effect PhysRevD.17.2369 ; Mikheyev-
ml-1985zog ; PhysRevD.22.2718 ; Smirnov-ml-2004zv .

In the framework of dark sectors, it is only natural to expect that dark gauge
bosons might mediate interactions between sterile neutrinos and SM fermions.
The interactions with stable SM fermions, specifically, will generate a new
matter potential source for neutrinos propagating through matter, and will
alter neutrino oscillations as long as the dark gauge boson does not couple
universally to all active and sterile neutrino species. As an example,
consider such a dark vector mediator $A_{\\!{}_{D}}$ coupling to the following
currents:

$\displaystyle\mathcal{L}_{\text{dark}}\supset-A_{{\\!{}_{D}}}^{\mu}$
$\displaystyle\Bigg{(}g_{{}_{S}}\,\bar{S}_{3}\;\frac{\\!\gamma_{\mu}(1-\gamma_{5})\\!\\!}{2}\;S_{3}+\sum_{i}g_{\nu}\;\bar{\nu}_{i}\;\frac{\\!\gamma_{\mu}(1-\gamma_{5})\\!\\!}{2}\;\nu_{i}$
$\displaystyle\;+g_{e}\,\bar{e}\,\gamma_{\mu}\,e+g_{p}\,\bar{p}\,\gamma_{\mu}\,p+g_{n}\,\bar{n}\,\gamma_{\mu}\,n+\cdots\Bigg{)}\,,$
(5)

where the dots denote unspecified mediator couplings to heavier fermions. The
resulting matter potential experienced by (anti-)neutrinos propagating through
matter is then given by (see, e.g., Linder-ml-2005fc ):

$\displaystyle V_{\nu_{i}}\big{|}_{\text{matter}}$
$\displaystyle=-V_{\overline{\nu}_{i}}\big{|}_{\text{matter}}$
$\displaystyle=-\frac{\;g_{\nu}}{\;2\,m_{A_{\\!{}_{D}}}^{2}\\!\\!}\;\\!\\!\\!\\!\\!\sum_{~{}~{}~{}f=e,\,p,\,n}\\!\\!\\!\\!g_{f}\,n_{f}\,,$
(6a) $\displaystyle V_{S_{3}}\big{|}_{\text{matter}}$
$\displaystyle=-V_{\overline{S}_{3}}\big{|}_{\text{matter}}$
$\displaystyle=-\frac{\;g_{{}_{S}}}{\;2\,m_{A_{\\!{}_{D}}}^{2}\\!\\!}\;\\!\\!\\!\\!\\!\sum_{~{}~{}~{}f=e,\,p,\,n}\\!\\!\\!\\!g_{f}\,n_{f}\,.$
(6b)

Above, $m_{A_{\\!{}_{D}}}$ is the dark mediator’s mass, and $n_{f}$ is the in-
medium number density of the SM fermion species $f$. It is clear from (6) that
these $A_{\\!{}_{D}}$-induced matter potentials are non-zero only if ordinary
matter has finite ‘dark charge’ density. The latter requirement rules out the
dark photon as a mediator candidate, since ordinary matter is, on average,
charge-neutral to a dark photon. In this specific example, the three active
neutrinos experience the same matter potential $V_{\nu_{i}}$, and therefore SM
neutrino oscillations are unaffected in regimes for which BSM effects can be
neglected. However, as long as $g_{{}_{S}}\neq g_{\nu}$, the matter potentials
for the sterile and active species will be distinct, and therefore neutrino
oscillations will be affected by the relative potential experienced by
$S_{3}$:

$\Delta
V\big{|}_{\text{matter}}=(V_{S_{3}}-V_{\nu_{i}})\big{|}_{\text{matter}}=-\frac{\;(g_{{}_{S}}-g_{\nu})}{\;2\,m_{A_{\\!{}_{D}}}^{2}\\!\\!}\;\\!\\!\\!\\!\\!\sum_{~{}~{}~{}f=e,\,p,\,n}\\!\\!\\!\\!g_{f}\,n_{f}\,.$
(7)

We note that a variety of phenomenological constraints restrict the mass and
couplings of the dark mediator. For instance, $S_{3}$ in-medium scattering
with finite momentum exchange must be sufficiently suppressed, otherwise this
would lead to decoherence of $S_{3}-\nu_{3}$ oscillations Nieves-ml-2020jjg .
Further, the combinations
$g_{p}(g_{\nu}-g_{{}_{S}}\sin\theta_{{}_{\\!S_{3}}})$,
$g_{n}(g_{\nu}-g_{{}_{S}}\sin\theta_{{}_{\\!S_{3}}})$, and
$g_{e}(g_{\nu}-g_{{}_{S}}\sin\theta_{{}_{\\!S_{3}}})$ are constrained by
neutrino-proton, neutrino-neutron, and neutrino-electron scattering,
respectively Kolb-ml-1987qy ; Bilmis-ml-2015lja ; COHERENT-ml-2017ipa ;
Coloma-ml-2020gfv . These constraints can be respected for sufficiently small
couplings, and, as evident from (7), it is possible to simultaneously keep
$\Delta V$ non-negligible if the mediator mass $m_{A_{\\!{}_{D}}}$ is
sufficiently light. Numerous other constraints from colliders, beam dumps,
fixed targets, meson decays, stellar cooling, anomalous magnetic dipole
moments, etc., also apply in different regions of parameter space of (2)
Ilten-ml-2018crw ; Fabbrichesi-ml-2020wbt ; Caputo-ml-2021eaa ; Agrawal-
ml-2021dbo . We will, however, defer these considerations to a future work.
For the remainder of this article, we will remain agnostic about the mass and
couplings of the dark mediator $A_{\\!{}_{D}}$, and instead treat the
effective matter potential $\Delta V$ as a free parameter in order to address
the MiniBooNE anomaly. A comprehensive study of experimentally viable UV
completions of $V_{\nu_{i}}$, $V_{S_{3}}$, and further signal predictions,
will be the focus of an upcoming publication future .

The effective matter potential (7) for $S_{3}$ modifies the neutrino
Hamiltonian in matter:

$\widehat{H}_{B,\text{ matter}}=\widehat{H}_{B,\text{
vacuum}}-\begin{pmatrix}~{}0&~{}~{}~{}0&~{}~{}~{}0&\,0\\\
~{}0&~{}~{}~{}0&~{}~{}~{}0&\,0\\\ ~{}0&~{}~{}~{}0&~{}~{}~{}0&\,0\\\
~{}0&~{}~{}~{}0&~{}~{}~{}0&|\Delta V|\\!\end{pmatrix}\,,$ (8)

where we chose the sign of $\Delta V$ to be negative (positive) for neutrinos
(antineutrinos) for reasons that will become clear shortly.

The neutrino propagation eigenstates of $\widehat{H}_{B,\text{matter}}$ in (8)
and their eigenvalues $\omega$ are easily obtained:

$\displaystyle\\!\\!\\!\\!\\!\\!|N_{1,\,m}\rangle~{}=~{}|\nu_{1}\rangle\,,$
$\displaystyle\omega_{1}~{}~{}$ $\displaystyle=~{}\frac{\delta
m^{2}_{13}}{2E_{\nu}}$ (9a)
$\displaystyle\\!\\!\\!\\!\\!\\!|N_{2,\,m}\rangle~{}=~{}|\nu_{2}\rangle\,,$
$\displaystyle\omega_{2}~{}~{}$ $\displaystyle=~{}\frac{\delta
m^{2}_{23}}{2E_{\nu}}$ (9b)
$\displaystyle\\!\\!\\!\\!\\!\\!|N_{3,\,m}^{{}_{(-)}}\rangle~{}=-\sin\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,|S_{3}\rangle\,+\,\cos\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,|\nu_{3}\rangle\,,\qquad$
$\displaystyle\omega_{3}^{{}_{(-)}}$ $\displaystyle=~{}\frac{\delta
M_{3}^{2}}{4E_{\nu}}(1-C_{3})-\frac{|\Delta V|}{2}$ (9c)
$\displaystyle\\!\\!\\!\\!\\!\\!|N_{3,\,m}^{{}_{(+)}}\rangle~{}=~{}~{}\cos\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,|S_{3}\rangle\,+\,\sin\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,|\nu_{3}\rangle\,,\qquad$
$\displaystyle\omega_{3}^{{}_{(+)}}$ $\displaystyle=~{}\frac{\delta
M_{3}^{2}}{4E_{\nu}}(1+C_{3})-\frac{|\Delta V|}{2}$ (9d)

where $C_{3}$ and the effective $\nu_{3}-S_{3}$ mixing angle in matter,
$\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}$, are defined through:

$\sin 2\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}=\frac{\sin
2\theta_{{}_{\\!S_{3}}}}{C_{3}}\equiv\frac{\sin
2\theta_{{}_{\\!S_{3}}}}{\;\sqrt{~{}\Big{(}\\!\sin
2\theta_{{}_{\\!S_{3}}\\!}\Big{)}^{2}+\left(\cos
2\theta_{{}_{\\!S_{3}}}-\frac{4E_{\nu}}{\delta M_{3}^{2}}\frac{|\Delta
V|}{2}\right)^{2}~{}}\;}\,.$ (10)

Expressions analogous to (9a)–(9d) and (10) hold for antineutrinos, with one
important difference: $-|\Delta V|$ should be replaced everywhere by $+|\Delta
V|$. Note, importantly, that because of our choice of sign for the matter
potential $\Delta V$, $\nu_{3}-S_{3}$ oscillations become _resonant_ (i.e.,
the mixing angle becomes maximal, $\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}=\pi/4$)
when $E_{\nu}$ is at the resonant energy value of

$E^{\text{res}}_{\nu_{3}}=\frac{\delta M_{3}^{2}\,\cos
2\theta_{{}_{\\!S_{3}}}}{2\,|\Delta V|}\,.$ (11)

Because the matter potential for antineutrinos has the opposite sign, no such
resonant effect is present for $\overline{\nu}_{3}-\overline{S}_{3}$
oscillations.

Figure 2 displays the effective active-sterile mixing angles in matter for
both neutrinos ($\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}$) and antineutrinos
($\bar{\theta}_{{}_{\\!S_{3}}}^{{}_{(m)}}$) as a function of (anti-)neutrino
energy, under three illustrative assumptions for the vacuum mixing angle
$\theta_{{}_{\\!S_{3}}}$. Note that the transition from
$\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\approx\theta_{{}_{\\!S_{3}}}$ to
$\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\to\pi/2$ as $E_{\nu}$ crosses the resonant
region becomes sharper for smaller values of the vacuum mixing angle
$\theta_{{}_{\\!S_{3}}}$. For antineutrinos, on the other hand, the active-
sterile mixing angle $\bar{\theta}_{{}_{\\!S_{3}}}^{{}_{(m)}}$ decreases
monotonically with increasing energy (see also subsection 2.3).

Figure 2: The in-medium effective mixing angles
$\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}$ (red curves) and
$\bar{\theta}_{{}_{\\!S_{3}}}^{{}_{(m)}}$ (blue curves) between
$\nu_{3}-S_{3}$ and $\bar{\nu}_{3}-\bar{S}_{3}$, respectively. The dependence
of these angles is shown as a function of (anti)neutrino energy $E_{\nu}$ in
units of the resonant energy $E^{\text{res}}_{\nu_{3}}$ for different choices
of the vacuum mixing angle $\theta_{{}_{\\!S_{3}}}$.

###  2.1 Neutrino propagation eigenstates in three energy regimes

With (9a)–(9d) and (10), we can now proceed to discuss the composition of the
propagation eigenstates at finite matter density, and their respective
oscillation phases, in three energy regimes relative to the resonant energy
$E^{\text{res}}_{\nu_{3}}$ defined in (11).

  * •

At very low energies, $E_{\nu}\ll E^{\text{res}}_{\nu_{3}}$, or alternatively,
at very low medium densities, $|\Delta V|\to 0$, we recover the vacuum
oscillation results, with the mixing angle between $\nu_{3}$ and $S_{3}$ being

$\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\simeq\theta_{{}_{\\!S_{3}}}$ (12)

and the propagation eigenstates and respective oscillation phases given by:

$\displaystyle\\!\\!\\!\\!\\!\\!|N_{1,\,m}\rangle~{}=~{}|\nu_{1}\rangle\,,$
$\displaystyle\omega_{1}~{}~{}$ $\displaystyle=\,~{}~{}\frac{\delta
m^{2}_{13}}{2E_{\nu}}$ (13a)
$\displaystyle\\!\\!\\!\\!\\!\\!|N_{2,\,m}\rangle~{}=~{}|\nu_{2}\rangle\,,$
$\displaystyle\omega_{2}~{}~{}$ $\displaystyle=\,~{}~{}\frac{\delta
m^{2}_{23}}{2E_{\nu}}$ (13b)
$\displaystyle\\!\\!\\!\\!\\!\\!|N_{3,\,m}^{{}_{(-)}}\rangle~{}=-\sin\theta_{{}_{\\!S_{3}}}\,|S_{3}\rangle\,+\,\cos\theta_{{}_{\\!S_{3}}}\,|\nu_{3}\rangle\,,~{}~{}$
$\displaystyle\omega_{3}^{{}_{(-)}}$
$\displaystyle\simeq-\,\frac{\,\sin^{2}\\!\theta_{{}_{\\!S_{3}}}\delta
M_{3}^{2}\,}{2E_{\nu}}\\!\left(\\!\frac{E_{\nu}}{\,E^{\text{res}}_{\nu_{3}}\,}\\!\cos
2\theta_{{}_{\\!S_{3}}}\\!\right)$ (13c)
$\displaystyle\\!\\!\\!\\!\\!\\!|N_{3,\,m}^{{}_{(+)}}\rangle~{}=~{}~{}\cos\theta_{{}_{\\!S_{3}}}\,|S_{3}\rangle\,+\,\sin\theta_{{}_{\\!S_{3}}}\,|\nu_{3}\rangle\,,~{}~{}$
$\displaystyle\omega_{3}^{{}_{(+)}}$ $\displaystyle\simeq\,~{}~{}\frac{\delta
M_{3}^{2}}{\,2E_{\nu}\,}\\!\left(\\!1-\frac{E_{\nu}}{\,E^{\text{res}}_{\nu_{3}}\,}\\!\cos
2\theta_{{}_{\\!S_{3}}}\\!\right)$ (13d)

  * •

At the resonance, $E_{\nu}=E^{\text{res}}_{\nu_{3}}$, the mixing between
$\nu_{3}$ and $S_{3}$ is maximal,

$\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}=\frac{\pi}{4}\,,$ (14)

and the propagation eigenstates and respective oscillation phases are given
by:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!|N_{1,\,m}\rangle~{}=~{}|\nu_{1}\rangle\,,$
$\displaystyle\omega_{1}~{}~{}$ $\displaystyle=\,~{}~{}\frac{\delta
m^{2}_{13}}{2E_{\nu}}$ (15a)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!|N_{2,\,m}\rangle~{}=~{}|\nu_{2}\rangle\,,$
$\displaystyle\omega_{2}~{}~{}$ $\displaystyle=\,~{}~{}\frac{\delta
m^{2}_{23}}{2E_{\nu}}$ (15b)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!|N_{3,\,m}^{{}_{(-)}}\rangle\,=~{}\frac{1}{\sqrt{2}}\,\big{(}\,|\nu_{3}\rangle\,-\,|S_{3}\rangle\,\big{)}\,,\qquad\qquad\qquad$
$\displaystyle\omega_{3}^{{}_{(-)}}$
$\displaystyle\simeq\,-\,\frac{\,\sin\theta_{{}_{\\!S_{3}}}\,\delta
M_{3}^{2}\,}{2E_{\nu}}\qquad$ (15c)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!|N_{3,\,m}^{{}_{(+)}}\rangle\,=~{}\frac{1}{\sqrt{2}}\,\big{(}\,|\nu_{3}\rangle\,+\,|S_{3}\rangle\,\big{)}\,,\qquad\qquad\qquad$
$\displaystyle\omega_{3}^{{}_{(+)}}$
$\displaystyle\simeq\,+\,\frac{\,\sin\theta_{{}_{\\!S_{3}}}\,\delta
M_{3}^{2}\,}{2E_{\nu}}$ (15d)

  * •

At very high energies, $E_{\nu}\gg E^{\text{res}}_{\nu_{3}}$, the mixing
between $\nu_{3}$ and $S_{3}$ is strongly suppressed, with

$\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\to\frac{\pi}{2}\,,$ (16)

so that their hierarchy is inverted, the active neutrino $\nu_{3}$ becomes the
heavier eigenstate, and the sterile neutrino $S_{3}$ becomes the lighter
eigenstate:

$\displaystyle\\!\\!\\!\\!\\!\\!|N_{1,\,m}\rangle~{}~{}=\,~{}~{}~{}|\nu_{1}\rangle\,,$
$\displaystyle\omega_{1}~{}~{}$ $\displaystyle=~{}\frac{\delta
m^{2}_{13}}{2E_{\nu}}$ (17a)
$\displaystyle\\!\\!\\!\\!\\!\\!|N_{2,\,m}\rangle~{}~{}=\,~{}~{}~{}|\nu_{2}\rangle\,,$
$\displaystyle\omega_{2}~{}~{}$ $\displaystyle=~{}\frac{\delta
m^{2}_{23}}{2E_{\nu}}$ (17b)
$\displaystyle\\!\\!\\!\\!\\!\\!|N_{3,\,m}^{{}_{(-)}}\rangle\,~{}\to-\,|S_{3}\rangle\,,\qquad\qquad\qquad$
$\displaystyle\omega_{3}^{{}_{(-)}}$ $\displaystyle\simeq~{}\frac{\delta
M_{3}^{2}}{2E_{\nu}}\\!\left(\\!1-\frac{E_{\nu}}{\,E^{\text{res}}_{\nu_{3}}\,}\\!\cos
2\theta_{{}_{\\!S_{3}}}\\!\right)$ (17c)
$\displaystyle\\!\\!\\!\\!\\!\\!|N_{3,\,m}^{{}_{(+)}}\rangle\,~{}\to\,~{}~{}|\nu_{3}\rangle\,,\qquad\qquad\qquad$
$\displaystyle\omega_{3}^{{}_{(+)}}$
$\displaystyle\simeq~{}\frac{\,\sin^{2}\\!\theta_{{}_{\\!S_{3}}}\,\delta
M_{3}^{2}\,}{2E_{\nu}}\\!\left(1+\frac{E^{\text{res}}_{\nu_{3}}}{E_{\nu}}\right)$
(17d)

Figure 3: [_For illustration purposes only — not to scale_] Effective mass-
squared eigenvalues of neutrino propagation eigenstates (left panel) and
antineutrino propagation eigenstates (right panel) as a function of
(anti-)neutrino energy in the quasi-sterile neutrino toy model. The
eigenstates’ flavor compositions are denoted by their color shading.

Importantly, at high very neutrino energies $E_{\nu}\to\infty$, the active
neutrino $\nu_{3}$ receives an effective contribution to its mass of $\delta
m_{3}^{2}=\sin^{2}\\!\theta_{{}_{\\!S_{3}}}\,\delta M_{3}^{2}$ (note (17d)),
an effect that had been previously noted in Kopp-ml-2014fha ; Denton-
ml-2018dqq ; Barenboim-ml-2019hso ; Smirnov-ml-2021zgn . Compatibility with
observations from high energy atmospheric and accelerator neutrinos then
enforces only two possibilities:

    1. a)

The effective mass contribution to $\nu_{3}$ should be extremely suppressed so
as to not make any difference, i.e., it should be of order of the current
experimental uncertainty on the atmospheric mass-squared splitting.
Quantitatively, this would imply

$\sin^{2}\\!\theta_{{}_{\\!S_{3}}\,}\delta
M_{3}^{2}~{}\lesssim~{}\mathcal{O}(10^{-4})~{}\text{eV}^{2}\,.$ (18)

    2. b)

In vacuum, the active neutrino mass hierarchy is inverted, and at high
neutrino energies $E_{\nu}\gg E^{\text{res}}_{\nu_{3}}$ the effective mass
contribution to $\nu_{3}$ due to matter effects flips the neutrino mass
ordering to a normal hierarchy. This would require

$\displaystyle\sin^{2}\\!\theta_{{}_{\\!S_{3}}\,}\delta M_{3}^{2}~{}$
$\displaystyle\simeq~{}2\,\left|\delta m^{2}_{32}\right|$ (19)
$\displaystyle\sim~{}(5\;\pm\;0.5)\times 10^{-3}~{}\text{eV}^{2}\,.$

Option (a) above would preclude any detectable effect at MiniBooNE, and
therefore, while perfectly compatible with experimental constraints, it is not
relevant for this minimal model case study. Which leaves us with (b) as the
only option of interest.

Figures 2 and 3 provide a clear illustration of the three energy regimes
discussed, as well as the transitions between them. In particular, figure 3
(left panel) depicts how the composition and eigenvalues of the propagation
eigenstates change as the neutrino energy crosses the resonant region set by
$E^{\text{res}}_{\nu_{3}}$.

###  2.2 Neutrino oscillation probabilities

We can now proceed to derive various appearance and disappearance oscillation
probabilities. The first step is to obtain the flavor eigenstates as a
function of the propagation eigenstates in (9).

Inverting the relations in (9c) and (9d), we have:

$\displaystyle|\nu_{3}\rangle~{}$
$\displaystyle=\,~{}~{}\cos\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,|N_{3,\,m}^{{}_{(-)}}\rangle~{}+~{}\sin\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,|N_{3,\,m}^{{}_{(+)}}\rangle$
(20a) $\displaystyle|S_{3}\rangle~{}$
$\displaystyle=\,-\sin\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,|N_{3,\,m}^{{}_{(-)}}\rangle~{}+~{}\cos\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,|N_{3,\,m}^{{}_{(+)}}\rangle$
(20b)

Using (3) and (20a), the muon-neutrino flavor is then given by:

$\displaystyle|\nu_{\mu}\rangle$ $\displaystyle=U^{*}_{\mu
1}\,|\nu_{1}\rangle+U^{*}_{\mu 2}\,|\nu_{2}\rangle+U^{*}_{\mu
3}\,|\nu_{3}\rangle$ (21) $\displaystyle=U^{*}_{\mu
1}\,|\nu_{1}\rangle+U^{*}_{\mu 2}\,|\nu_{2}\rangle+U^{*}_{\mu
3\,}\cos\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,|N_{3,\,m}^{{}_{(-)}}\rangle+U^{*}_{\mu
3\,}\sin\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,|N_{3,\,m}^{{}_{(+)}}\rangle\,,\qquad$

which, after propagating through a distance $L$, is time-evolved to:

$\displaystyle|\nu_{\mu}(L)\rangle$ $\displaystyle=U^{*}_{\mu
1}\,e^{-i\,\omega_{1}L}\;|\nu_{1}\rangle+U^{*}_{\mu
2}\,e^{-i\,\omega_{2}L}\;|\nu_{2}\rangle$ (22) $\displaystyle\quad+U^{*}_{\mu
3\,}\cos\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,e^{-i\,\omega_{3}^{{}_{(-)}}L}\;|N_{3,\,m}^{{}_{(-)}}\rangle+U^{*}_{\mu
3\,}\sin\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,e^{-i\,\omega_{3}^{{}_{(+)}}L}\;|N_{3,\,m}^{{}_{(+)}}\rangle\,.\qquad$

Likewise, the electron-neutrino flavor is given by:

$\displaystyle|\nu_{e}\rangle$
$\displaystyle=U^{*}_{e1}\,|\nu_{1}\rangle+U^{*}_{e2}\,|\nu_{2}\rangle+U^{*}_{e3}\,|\nu_{3}\rangle$
(23)
$\displaystyle=U^{*}_{e1}\,|\nu_{1}\rangle+U^{*}_{e2}\,|\nu_{2}\rangle+U^{*}_{e3\,}\cos\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,|N_{3,\,m}^{{}_{(-)}}\rangle+U^{*}_{e3\,}\sin\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,|N_{3,\,m}^{{}_{(+)}}\rangle\,,\qquad$

and through propagation evolves to:

$\displaystyle|\nu_{e}(L)\rangle$
$\displaystyle=U^{*}_{e1}\,e^{-i\,\omega_{1}L}\;|\nu_{1}\rangle+U^{*}_{e2}\,e^{-i\,\omega_{2}L}\;|\nu_{2}\rangle$
(24)
$\displaystyle\quad+U^{*}_{e3\,}\cos\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,e^{-i\,\omega_{3}^{{}_{(-)}}L}\;|N_{3,\,m}^{{}_{(-)}}\rangle+U^{*}_{e3\,}\sin\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,e^{-i\,\omega_{3}^{{}_{(+)}}L}\;|N_{3,\,m}^{{}_{(+)}}\rangle\,.\qquad$

We note that the expressions (22) and (24) above are valid for neutrino
propagation through a medium with constant density and composition. In cases
where the medium density/composition changes as the neutrino propagates, so
does the Hamiltonian change. In such cases, (22) and (24) have to replaced
with the appropriate states propagated under time-ordered Hamiltonian
evolution.

The $\nu_{\mu}\to S_{3}$ oscillation probability is then given by:

$\displaystyle\text{P}(\nu_{\mu}\rightarrow S_{3})$
$\displaystyle\equiv|\langle S_{3}|\nu_{\mu}(L)\rangle|^{2}$
$\displaystyle=|U_{\mu
3}|^{2}~{}\sin^{2}2\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}~{}\sin^{2}\\!\left(\frac{C_{3}\,\delta
M_{3}^{2}}{4E_{\nu}}L\right)\,,$ (25)

and the $\nu_{e}\to S_{3}$ oscillation probability by:

$\displaystyle\text{P}(\nu_{e}\rightarrow S_{3})$ $\displaystyle\equiv|\langle
S_{3}|\nu_{e}(L)\rangle|^{2}$
$\displaystyle=|U_{e3}|^{2}~{}\sin^{2}2\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}~{}\sin^{2}\\!\left(\frac{C_{3}\,\delta
M_{3}^{2}}{4E_{\nu}}L\right).$ (26)

Finally, the $\nu_{\mu}\rightarrow\nu_{e}$ appearance probability is given by:

$\displaystyle\text{P}(\nu_{\mu}\rightarrow\nu_{e})$
$\displaystyle=|\langle\nu_{e}|\nu_{\mu}(L)\rangle|^{2}$ (27)
$\displaystyle=\Big{|}\,U_{e1}\,U^{*}_{\mu
1}\,e^{-i\,\omega_{1}L}+U_{e2}\,U^{*}_{\mu 2}\,e^{-i\,\omega_{2}L}$
$\displaystyle\qquad+U_{e3}\,U^{*}_{\mu
3}\,\cos^{2}\\!\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,e^{-i\,\omega_{3}^{{}_{(-)}}L}+U_{e3}\,U^{*}_{\mu
3}\,\sin^{2}\\!\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,e^{-i\,\omega_{3}^{{}_{(+)}}L}\,\Big{|}^{2}\,.$

In MiniBooNE, the baseline-to-energy ratio is in the range of
$L/E_{\nu}\sim\mathcal{O}(1-10)\text{ eV}^{-2}$. Therefore, when computing the
$\nu_{e}$ appearance probabilities at MiniBooNE, we can ignore the difference
in oscillation phases between the two ‘solar’ neutrino states $\nu_{1}$ and
$\nu_{2}$, since $(\omega_{2}L-\omega_{1}L)\,|_{\text{\tiny
MB}}\sim\mathcal{O}(10^{-5}-10^{-4})$. In this case we have

$\text{P}(\nu_{\mu}\rightarrow\nu_{e})\Big{|}_{\text{MB}}\simeq|U_{e3}|^{2}\,|U_{\mu
3}|^{2}\,\left|-e^{-i\,\omega_{2}\,L}+\cos^{2}\\!\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,e^{-i\,\omega_{3}^{{}_{(-)}}L}+\sin^{2}\\!\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,e^{-i\,\omega_{3}^{{}_{(+)}}L}\,\right|^{2}.$
(28)

We can look at three energy regimes of MiniBooNE’s appearance probability in
(28):

For low energy neutrinos, $E_{\nu}\ll E^{\text{res}}_{\nu_{3}}$, or
alternatively, at negligible medium densities, $V_{m}\to 0$, the appearance
probability (28) reduces to the familiar form:

$\text{P}(\nu_{\mu}\rightarrow\nu_{e})\,\Big{|}_{E_{\nu}\to 0}\approx
4\;|U_{e3}|^{2}\,|U_{\mu
3}|^{2}\,\sin^{4}\\!\theta_{{}_{\\!S_{3}}}~{}\sin^{2}\\!\left(\frac{\delta
M_{3}^{2}}{4E_{\nu}}L\right).$ (29)

At the resonance, $E_{\nu}=E^{\text{res}}_{\nu_{3}}$,

$\displaystyle\text{P}(\nu_{\mu}\rightarrow\nu_{e})\,\Big{|}_{E^{\text{res}}_{\nu_{3}}}\approx
4\,|U_{e3}|^{2}|U_{\mu 3}|^{2}\left[\sin^{2}\\!\left(\frac{\,2\,\delta
m_{32}^{2}/\\!\sin\theta_{{}_{\\!S_{3}}}\,}{4\,E^{\text{res}}_{\nu_{3}}}L\right)-\frac{1}{4}\sin^{2}\\!\left(\frac{\,4\,\delta
m_{32}^{2}/\\!\sin\theta_{{}_{\\!S_{3}}}\,}{4\,E^{\text{res}}_{\nu_{3}}}L\right)\right],$

where we have used (19). Note that the $\nu_{e}$ appearance probability is
maximized at the resonance, and, in the case of (2.2), it is upper-bounded by
$4\,|U_{e3}|^{2}|U_{\mu 3}|^{2}\sim(0.04-0.06)$. It peaks at:

$\frac{\delta
m_{32}^{2}}{\,\sin\theta_{{}_{\\!S_{3}}}\,}\frac{L}{\,E^{\text{res}}_{\nu_{3}}\,}=(2n+1)\,\pi,\qquad
n\in\mathbb{Z}\,.$ (31)

If we set $L$ in (31) to MiniBooNE’s baseline333The center of the MiniBooNE
detector is located 541 m from the production target; however, the baseline’s
first $\sim$50 m constitute the decay region MiniBooNE-ml-2008paa . Hence, the
beam neutrinos propagate in matter through a distance of $\sim$490 m. In this
specific case, the evolution of the initial $\nu_{\mu}$ state through the
decay region can be neglected, and the MiniBooNE baseline can be set to
$L|_{\text{\tiny MB}}=490$ m. of $L|_{\text{\tiny MB}}=490$ m, and also set
$n=0$ or $n=-1$, we can obtain the maximum value of
$\big{|}{\sin\theta_{{}_{\\!S_{3}}}\\!}\big{|}$ that is able to saturate the
appearance probability at the resonance. Choosing
$E^{\text{res}}_{\nu_{3}}\big{|}_{\text{\tiny MB}}=300$ MeV for concreteness,
we have

$\big{|}{\sin\theta_{{}_{\\!S_{3}}}^{\text{(max)}}}\big{|}=\frac{\,|\delta
m_{32}^{2}|\,}{\pi}~{}\frac{L}{\,E^{\text{res}}_{\nu_{3}}\,}\Bigg{|}_{\text{MB}}\qquad\Rightarrow\qquad\big{|}{\theta_{{}_{\\!S_{3}}}^{\text{(max)}}}\big{|}\simeq
6.6\times 10^{-3}\,.$ (32)

Figure 4: The $\nu_{\mu}\to\nu_{e}$ appearance probability at MiniBooNE, as a
function of true neutrino energy, predicted by the toy model (28) assuming
$E^{\text{res}}_{\nu_{3}}\big{|}_{\text{\tiny MB}}=295$ MeV. Three different
choices for the active-sterile vacuum mixing angle $\theta_{{}_{\\!S_{3}}}$
are shown: below (left panel), at (middle panel), and above (right panel) the
appearance probability saturation condition; see (31) and (32). The red data
points with error bars were taken from figure 21 of MiniBooNE-ml-2020pnu ;
they assume $E_{\nu}^{\text{true}}\sim E_{\nu}^{\text{\tiny QE}}$, which is a
very coarse approximation. As such, they should be interpreted as a
_qualitative_ measure of MiniBooNE’s anomalous appearance probabilities.

Figure 4 shows the toy model’s $\nu_{e}$ appearance probability at MiniBooNE
for $E^{\text{res}}_{\nu_{3}}\big{|}_{\text{\tiny MB}}=295$ MeV and three
different choices of $\big{|}\theta_{{}_{\\!S_{3}}\\!}\big{|}$. In the left
panel
$\big{|}\theta_{{}_{\\!S_{3}}\\!}\big{|}>\big{|}{\theta_{{}_{\\!S_{3}}}^{\text{(max)}}}\big{|}$,
and therefore the $\nu_{e}$ appearance probability is below the saturation
condition in (31). In the middle panel
$\big{|}\theta_{{}_{\\!S_{3}}\\!}\big{|}\approx\big{|}{\theta_{{}_{\\!S_{3}}}^{\text{(max)}}}\big{|}$;
the appearance probability is saturated and displays a single sharp peak at
the resonant energy. Finally, in the right panel
$\big{|}\theta_{{}_{\\!S_{3}}\\!}\big{|}<\big{|}{\theta_{{}_{\\!S_{3}}}^{\text{(max)}}}\big{|}$;
the appearance probability is saturated but oscillates rapidly within the
resonant region.

The span of the resonant region is nearly-independent of
$\theta_{{}_{\\!S_{3}}}$ and $E^{\text{res}}_{\nu_{3}}\big{|}_{\text{\tiny
MB}}$. As a rough approximation, we can consider the width of the resonant
oscillations’ envelope, $\Delta E^{\text{res}}$, whose magnitude is given by:

$\displaystyle\Delta E^{\text{res}}\big{|}_{{}_{\text{\tiny MB}}}$
$\displaystyle\sim 2~{}\big{|}{\tan
2\theta_{{}_{\\!S_{3}}}^{\text{(max)}}}\big{|}~{}E^{\text{res}}_{\nu_{3}}\big{|}_{{}_{\text{\tiny
MB}}}$ $\displaystyle\sim\frac{4}{\pi}~{}|\delta
m_{32}^{2}|~{}L\big{|}_{{}_{\text{\tiny MB}}}$ $\displaystyle\sim
8\;\text{MeV}\,,$ (33)

which is narrow for MiniBooNE’s detector resolution. Averaging the oscillation
probability within the resonant range $\Delta E^{\text{res}}$ yields about
$(40-45)\%$ of the peak value of $4\,|U_{e3}|^{2}|U_{\mu 3}|^{2}$; i.e.,
P$(\nu_{\mu}\rightarrow\nu_{e})\sim(1.6-2.7)\%$ in the resonant region.
Nuclear effects and energy resolution preclude the MiniBooNE detector from
resolving such a narrow peak in the energy distribution of $\nu_{e}$ events.
Smearing this rate over the average bin size of 100 MeV, we obtain an average
appearance probability of roughly $~{}(0.3-0.5)\%$ within the $200-300$ MeV
range, which is still too small to account for the reported excess in
appearance probability of $1.4-1.8$% in this energy bin MiniBooNE-ml-2020pnu .
This is compounded by the fact that this model predicts a negligible $\nu_{e}$
appearance probability in the regions outside the resonant region, including
MiniBooNE’s higher energy bins with reported excesses (see figure 4). This
constitutes one of three important reasons why this minimal model falls short
of explaining the MiniBooNE excess. We will come back to this issue shortly.

For very high energy neutrinos, $E_{\nu}\gg E^{\text{res}}_{\nu_{3}}$, or
alternatively, in very dense matter, $|\Delta V|\gg\delta M_{3}^{2}/E_{\nu}$,
(28) reduces to:

$\text{P}(\nu_{\mu}\rightarrow\nu_{e})\,\Big{|}_{E_{\nu}\gg
E^{\text{res}}_{\nu_{3}}}~{}\approx~{}4\,|U_{e3}|^{2}|U_{\mu
3}|^{2}\,\sin^{2}\\!\left[\,\frac{\,\delta
m_{32}^{2}\,(1+2\,E^{\text{res}}_{\nu_{3}}/E_{\nu})\,}{4E_{\nu}}\,L\,\right].$
(34)

Note that (34) recovers the SM limit of $\nu_{\mu}\to\nu_{e}$ oscillation
probabilities at very high energies, since the effective mass-squared
splitting controlling these oscillations, namely, $\delta
m_{32}^{2}\,(1+2\,E^{\text{res}}_{\nu_{3}}/E_{\nu})$, approaches the SM value
of $\delta m_{32}^{2}$ as $E^{\text{res}}_{\nu_{3}}/E_{\nu}\to 0$. (As
discussed in option (b) of subsection 2.1, this is because the condition in
(19) effectively flips the active neutrino in-medium mass hierarchy at high
energies.)

Unfortunately, however, the active neutrino mass ordering does not relax to
the SM expectation quickly enough at high energies, leading to strong tensions
with measurements from long-baseline neutrino experiments. Consider, for
instance, T2K’s measurement of $\delta m_{32}^{2}$ in neutrino mode. The
average matter density along the Tokai-to-Kamioka baseline is
$\rho|_{\text{\tiny T2K}}=2.6$ g/cm3 Hagiwara-ml-2011kw ; therefore, the
resonant energy at T2K is related to the resonant energy at MiniBooNE through:

$\displaystyle E^{\text{res}}_{\nu_{3}}\big{|}_{{}_{\text{\tiny T2K}}}$
$\displaystyle\approx E^{\text{res}}_{\nu_{3}}\big{|}_{{}_{\text{\tiny
MB}}}~{}~{}\frac{\;\rho|_{{}_{\text{\tiny MB}}}}{\;\rho|_{{}_{\text{\tiny
T2K}}}}$ $\displaystyle\sim E^{\text{res}}_{\nu_{3}}\big{|}_{{}_{\text{\tiny
MB}}}~{}~{}\frac{[1.6-2.2]\text{ g/cm}^{\\!3}}{\;2.6\text{ g/cm}^{\\!3}\;}\,.$
(35)

Above, the estimated range of soil densities along MiniBooNE’s baseline,
$\rho|_{{}_{\text{\tiny MB}}}\approx[1.6-2.2]\text{ g/cm}^{\\!3}$, is based on
the assumption that the soil’s composition is dominated by silty clay MBsoil .
Then, for the benchmark choice of
$E^{\text{res}}_{\nu_{3}}\big{|}_{\text{\tiny MB}}=300$ MeV, (2.2) gives
$E^{\text{res}}_{\nu_{3}}\big{|}_{\text{\tiny T2K}}\sim[185-254]$ MeV. For the
remainder of this paper, we will benchmark the soil density along MiniBooNE’s
baseline to $\rho|_{{}_{\text{\tiny MB}}}\approx 1.6\text{ g/cm}^{\\!3}$, and
therefore we will use $E^{\text{res}}_{\nu_{3}}\big{|}_{\text{\tiny T2K}}\sim
185$ MeV. With this information, we can consider T2K’s determination of
$\delta m_{32}^{2}$, which is dominated by its sensitivity to the dip in the
$\nu_{\mu}$ survival probability at around
$E_{\nu_{\mu}}^{\text{\,dip}}\big{|}_{\text{\tiny T2K}}\sim 600$ MeV. Using
(34), the effective $\delta m_{32\text{,\,eff}}^{2}$ predicted at T2K in this
energy range would be:

$\displaystyle\delta m_{32\text{,\,eff\,}}^{2}\big{|}_{{}_{\text{T2K}}}$
$\displaystyle\sim|\delta
m_{32}^{2}|\big{|}_{{}_{\text{SM}}}\times\left(\\!1+2\,\frac{\,E^{\text{res}}_{\nu_{3}}\,}{\;E_{\nu_{\mu}}^{\text{\,dip}}\;}\right)\\!\Bigg{|}_{{}_{\text{T2K}}}$
$\displaystyle\sim 1.62\times|\delta m_{32}^{2}|\big{|}_{{}_{\text{SM}}}\,,$
(36)

which is in significant tension with neutrino data from T2K bibT2K-ml-2021xwb
, whose error on its most recent determination of $\delta m_{32}^{2}$ is only
about 4%. This constitutes the second of three main reasons why this minimal
model is phenomenologically unviable.

Figure 5: The effective ‘atmospheric’ mass-squared splitting predicted by the
toy model for both neutrinos ($\delta m_{32\text{,\,eff}}^{2}$) and
antineutrinos ($\delta\overline{m}_{32\text{,\,eff}}^{2}$) as a function of
(anti-)neutrino energy, assuming $\theta_{{}_{\\!S_{3}}}=0.003$,
$E^{\text{res}}_{\nu_{3}}\big{|}_{\text{\tiny MB}}\sim 300$ MeV, and
$E^{\text{res}}_{\nu_{3}}\big{|}_{\text{\tiny T2K}}\simeq
E^{\text{res}}_{\nu_{3}}\big{|}_{\text{\tiny NO$\nu$A}}\simeq
E^{\text{res}}_{\nu_{3}}\big{|}_{\text{\tiny MINOS}}\sim 185$ MeV. The data
points with error bars denote recent combined measurements of $\delta
m_{32}^{2}$ by long-baseline accelerator experiments bibT2K-ml-2018rhz ;
MINOS-ml-2020llm ; NOvA-ml-2021nfi .

Figure 5 shows this toy model’s prediction for the effective $\delta
m_{32\text{,\,eff}}^{2}$ as a function of neutrino energy in long-baseline
neutrino experiments such as T2K bibT2K-ml-2018rhz , MINOS MINOS-ml-2020llm ,
and NO$\nu$A NOvA-ml-2021nfi . Here, we take $\theta_{{}_{\\!S_{3}}}=0.003$
and make the same simplified assumption that the average matter density
through which neutrinos propagate in these long-baseline experiments results
in a long-baseline resonant energy of
$E^{\text{res}}_{\nu_{3}}\big{|}_{\text{\tiny T2K}}\sim 185$ MeV (see
discussion around (2.2)). It is clear from figure 5 that severe tensions
between data and the toy model prediction for $\delta m_{32\text{,\,eff}}^{2}$
persist even at much higher neutrino energies than the range probed by T2K.

###  2.3 Matter effects on antineutrinos

We now briefly discuss the consequences of the minimal model for
antineutrinos. The sign of the matter potential, $\Delta V$, was chosen such
that antineutrinos do not experience resonant oscillations. Furthermore, the
maximum allowed strength of the $\bar{\nu}_{3}-\bar{S}_{3}$ mixing angle,
given in (32), is too small to yield a significant amplitude for non-resonant
$\bar{\nu}_{3}-\bar{S}_{3}$ oscillations (see also figure 2). Consequently,
the $\bar{\nu}_{\mu}\to\bar{\nu}_{e}$ appearance probability predicted by this
minimal model is negligible, and therefore cannot account for the excesses of
$\bar{\nu}_{e}$-like events reported by LSND and MiniBooNE. This is the third
main reason why this minimal model is phenomenologically unviable.

Additionally, there is another shortcoming of the minimal model which is
related to the second reason discussed at the end of subsection 2.2. Before we
state it, we need to highlight how these BSM matter effects alter the in-
medium oscillations of antineutrinos. Their propagation eigenstates are given
by analogous expressions to (9) and (10) with $|\Delta V|\to-|\Delta V|$:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!|\bar{N}_{1,\,m}\rangle~{}=~{}|\bar{\nu}_{1}\rangle\,,$
$\displaystyle\bar{\omega}_{1}~{}~{}$ $\displaystyle=~{}\frac{\delta
m^{2}_{13}}{2E_{\bar{\nu}}}$ (37a)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!|\bar{N}_{2,\,m}\rangle~{}=~{}|\bar{\nu}_{2}\rangle\,,$
$\displaystyle\bar{\omega}_{2}~{}~{}$ $\displaystyle=~{}\frac{\delta
m^{2}_{23}}{2E_{\bar{\nu}}}$ (37b)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!|\bar{N}_{3,\,m}^{{}_{(-)}}\rangle~{}=-\sin\bar{\theta}_{{}_{\\!S_{3}}}^{{}_{\,(m)}}|\bar{S}_{3}\rangle\,+\,\cos\bar{\theta}_{{}_{\\!S_{3}}}^{{}_{\,(m)}}|\bar{\nu}_{3}\rangle\,,~{}~{}~{}~{}$
$\displaystyle\bar{\omega}_{3}^{{}_{(-)}}$ $\displaystyle=~{}\frac{\delta
M_{3}^{2}}{4E_{\bar{\nu}}}\left(\\!1-\overline{C}_{3}+\frac{E_{\bar{\nu}}}{E^{\text{res}}_{\nu_{3}}}\cos
2\theta_{{}_{\\!S_{3}}}\\!\right)\\!\\!\\!$ (37c)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!|\bar{N}_{3,\,m}^{{}_{(+)}}\rangle~{}=~{}~{}\cos\bar{\theta}_{{}_{\\!S_{3}}}^{{}_{\,(m)}}|\bar{S}_{3}\rangle\,+\,\sin\bar{\theta}_{{}_{\\!S_{3}}}^{{}_{\,(m)}}|\bar{\nu}_{3}\rangle\,,~{}~{}~{}~{}$
$\displaystyle\bar{\omega}_{3}^{{}_{(+)}}$ $\displaystyle=~{}\frac{\delta
M_{3}^{2}}{4E_{\bar{\nu}}}\left(\\!1+\overline{C}_{3}+\frac{E_{\bar{\nu}}}{E^{\text{res}}_{\nu_{3}}}\cos
2\theta_{{}_{\\!S_{3}}}\\!\right)\\!\\!\\!$ (37d)

where we have recast $|\Delta V|$ in terms of $E^{\text{res}}_{\nu_{3}}$ using
(11). Furthermore, $\overline{C}_{3}$ and the effective
$\bar{\nu}_{3}-\bar{S}_{3}$ mixing angle in matter,
$\bar{\theta}_{{}_{\\!S_{3}}}^{{}_{\,(m)}}$, are defined through:

$\sin 2\bar{\theta}_{{}_{\\!S_{3}}}^{{}_{\,(m)}}=\frac{\sin
2\theta_{{}_{\\!S_{3}}}}{\overline{C}_{3}}\equiv\frac{\sin
2\theta_{{}_{\\!S_{3}}}}{\;\sqrt{~{}\big{(}\\!\sin
2\theta_{{}_{\\!S_{3\\!}}}\big{)}^{2}+\big{(}\\!\cos
2\theta_{{}_{\\!S_{3\\!}}}\big{)}^{2}\left(1+\frac{E_{\bar{\nu}}}{\,E^{\text{res}}_{\nu_{3}}\,}\right)^{2}\;}}\,.$
(38)

Figures 2 and 3 (right panel) illustrate, respectively, the in-medium
$\bar{\nu}_{3}-\bar{S}_{3}$ mixing angle as a function of antineutrino energy,
and the flavor composition and eigenvalues of the in-medium propagation
eigenstates as a function of antineutrino energy.

We can proceed to examine two energy regimes of experimental interest: (i) the
low energy region $E_{\bar{\nu}}\sim\mathcal{O}(1-10)\,$MeV, relevant for
reactor experiments; and (ii) the high energy region
$E_{\bar{\nu}}\;\gtrsim\;(0.2-10)\,$GeV, relevant for long-baseline
accelerator experiments. Firstly, we note that in this minimal model
$\bar{\nu}_{3}-\bar{S}_{3}$ mixing is always suppressed, and the active
neutrino state $\bar{\nu}_{3}$ is mostly composed of $\bar{N}_{3}^{{}_{(-)}}$.
Therefore, the effective antineutrino mass splitting
$\delta\overline{m}_{3i\text{,\,eff\,}}^{2}$ ($i=1,2$) can be approximated by:

$\displaystyle\delta\overline{m}_{3i\text{,\,eff\,}}^{2}\Big{|}_{\bar{\nu}}~{}$
$\displaystyle\simeq$
$\displaystyle~{}~{}~{}2\,E_{\bar{\nu}}\,\left(\bar{\omega}_{3}^{{}_{(-)}}-\bar{\omega}_{i}\right)$
(39) $\displaystyle~{}\rightarrow$ $\displaystyle~{}\begin{dcases}-\,|\delta
m_{3i}^{2}|\,\left(1-2\cos
2\theta_{{}_{\\!S_{3}}}\frac{E_{\bar{\nu}}}{E^{\text{res}}_{\nu_{3}}}\right)&\text{for
}E_{\bar{\nu}}\,\ll\,E^{\text{res}}_{\nu_{3}}\,,\\\ ~{}~{}~{}|\delta
m_{3i}^{2}|\,\left(1-2\,\frac{E^{\text{res}}_{\nu_{3}}}{E_{\bar{\nu}}}\right)&\text{for
}E_{\bar{\nu}}\,\gg\,E^{\text{res}}_{\nu_{3}}\,.\end{dcases}$

Interestingly, as it is clear from (39), the antineutrino mass ordering in
this minimal model displays the same ‘hierarchy-flipping’ pattern of the
active neutrinos, with an inverted mass hierarchy in the asymptotically low
energy limit and a normal in-medium mass hierarchy in the asymptotically high
energy limit. This effect is illustrated in figure 3 (right panel).

Reactor-based antineutrino experiments such as Daya Bay DayaBay-ml-2018yms and
RENO RENO-ml-2018dro have measured $|\delta m^{2}_{31}|$ with $\sim\\!5\%$
precision at $2\sigma$ C.L. within the energy range
$E_{\bar{\nu}}\sim\mathcal{O}(1-10)\,$MeV. From (39), we can check that the
minimal model is within $2\sigma$-compatibility with these measurements,
predicting

$\delta\overline{m}_{31\text{,\,eff\,}}^{2}\big{|}_{\text{reactor
}\bar{\nu}}~{}\simeq~{}-|\delta
m_{31}^{2}|\big{|}_{{}_{\text{SM}}}\,\left[1-0.02\times\\!\bigg{(}\\!\frac{E_{\bar{\nu}}}{\,3\,\text{MeV}\,}\\!\bigg{)}\right].$
(40)

The minimal model’s compatibility with $\bar{\nu}$-data is not as good in the
high energy limit, however. Several long-baseline accelerator experiments,
such as K2K, T2K, MINOS, and NO$\nu$A have also measured the active mass-
squared splittings and mixing angles in antineutrino mode, albeit with less
precision that in neutrino mode. Taking T2K again as an example, we can check
the minimal model’s prediction for the effective
$\delta\overline{m}_{32\text{,\,eff}}^{2}$ that would be observed at T2K in
antineutrino mode:

$\displaystyle\delta\overline{m}_{32\text{,\,eff\,}}^{2}\big{|}_{{}_{\text{T2K,}\,\bar{\nu}}}$
$\displaystyle~{}\sim~{}|\delta
m_{32}^{2}|\big{|}_{{}_{\text{SM}}}\times\left(\\!1-2\,\frac{\,E^{\text{res}}_{\nu_{3}}\,}{\;E_{\bar{\nu}_{\mu}}^{\text{\,dip}}\;}\right)\\!\Bigg{|}_{{}_{\text{T2K}}}$
$\displaystyle~{}\sim~{}0.38\times|\delta
m_{32}^{2}|\big{|}_{{}_{\text{SM}}}\,,$ (41)

which is in significant tension with T2K’s most recent determination of
$\delta m_{32}^{2}$ in antineutrino mode Berns-ml-2021iss , which had an error
of $\sim$8%. This tension is analogous to the one (2.2) for neutrino mode,
with the main difference being that the latter overpredicts $|\delta
m_{32}^{2}|$, whereas the former underpredicts it. This is clearly illustrated
in figure 5.

##  3 Beyond the minimal model: a viable explanation for the MiniBooNE excess

We begin this section by recapping the phenomenological shortcomings of the
minimal model just discussed:

  1. 1.

_The strength of the resonant $\nu_{\mu}\to\nu_{e}$ oscillations in the
minimal model is insufficient to account for the full low energy
$\nu_{e}$-like excess at MiniBooNE._

There are two reasons for this. First, the resonant energy window is very
narrow. Second, the active neutrino experiencing resonant oscillations,
$\nu_{3}$, has a suppressed $\nu_{e}$ component, thereby suppressing the
resonant amplitude by a factor of $|U_{e3}|^{2}|U_{\mu
3}|^{2}\sim(0.01-0.015)$, as shown in (28) and figure 4.

  2. 2.

_The effective ‘atmospheric’ splitting $\delta m_{32}^{2}$ predicted at long-
baseline experiments is in strong tension with data._

This is because, in the high energy regime $E_{\nu}\gg
E^{\text{res}}_{\nu_{3}}$, the active state that mixes with the quasi-sterile
state (which is $\stackon[-1.3pt]{$\bar{\nu}$}{\scalebox{0.25}{\hskip
2.0pt{{(}}}\scalebox{0.25}{\hskip 17.0pt {{)}}}}_{\\!3}$ in the minimal model)
receives a residual contribution to its in-medium effective mass given by
(17d) for $\nu_{3}$ and (37c) for $\bar{\nu}_{3}$. Hence, in order to preserve
the expected active neutrino mass-squared splittings at high energies, the
parameters of the minimal model have to be adjusted to effectively ‘flip’ the
mass hierarchy at $E_{\nu}\gg E^{\text{res}}_{\nu_{3}}$ according to (19).
However, this hierarchy-flip does not happen ‘instantaneously’ as $E_{\nu}$
crosses the resonant region — the effective mass-squared splitting $\delta
m_{32\text{,\,eff\,}}^{2}$ does not fully relax to the SM value until high
enough energies, well above the energy range probed by, e.g., T2K, NO$\nu$A,
and MINOS. As a result, the predicted $\delta m_{32\text{,\,eff\,}}^{2}$ is in
tension with accelerator-based measurements of $\delta m_{32}^{2}$, as shown
in (2.2), (2.3), and figure 5.

  3. 3.

_The minimal model cannot address the $\bar{\nu}_{e}$-like excesses at LSND
and MiniBooNE, nor the $\nu_{e}$-like excess at MiniBooNE above
$E_{\nu}\gtrsim 400$_ MeV.

This issue is related to 2 above. Since $\nu_{3}-S_{3}$ mixing has been
constrained by the ‘hierarchy-flip’ relation in (19), this model predicts
negligible appearance probabilities for $\bar{\nu}_{\nu}\to\bar{\nu}_{e}$
oscillations over all energy ranges, as well as for $\nu_{\nu}\to\nu_{e}$
oscillations outside the resonant region (see figure 4).

The reasons behind these shortcomings, as summarized above, offer clues as to
how the minimal model could be augmented and modified to overcome
phenomenological tensions.

In particular, shortcoming 1 above could be addressed if $\nu_{1}$ and/or
$\nu_{2}$ — which have larger $\nu_{e}$ components — were involved in the
resonant oscillation mechanism. However, this needs to be done in such a way
that the effective mass spectrum of active neutrinos is not significantly
altered at high energies. In other words, the incorporation of resonance
effects for $\nu_{1,2}$ in order to address shortcoming 1 must simultaneously
address shortcoming 2.

The most straightforward way to achieve this goal is to augment the minimal
model by introducing two additional quasi-sterile species, $S_{1}$ and
$S_{2}$, which will mix with $\nu_{1}$ and $\nu_{2}$, respectively. This will
result in two new resonant energies, $E^{\text{res}}_{\nu_{1}}$ and
$E^{\text{res}}_{\nu_{2}}$, which must fall within the low energy window of
interest for MiniBooNE. These additional resonances will be weighted by
factors of $|U_{e1}|^{2}|U_{\mu 1}|^{2}$ and $|U_{e2}|^{2}|U_{\mu 2}|^{2}$,
respectively, and therefore will allow for much larger resonant appearance
probabilities than the minimal model. Furthermore, the parameters of this
augmented model must be arranged such that the effective shift in the active
neutrino masses is the same for $\nu_{1}$, $\nu_{2}$, and $\nu_{3}$, so that
the in-medium active mass hierarchy is preserved at high energies $E_{\nu}\gg
E^{\text{res}}_{\nu_{1,2,3}}$. It turns out that this latter requirement will
be sufficient to address shortcoming 2 as well, as we will detail shortly.

Unfortunately, the model’s augmentation by two new quasi-sterile states
$S_{1}$ and $S_{2}$ is still insufficient to address shortcoming 3. As we
shall see, this is because the three resonant regions will remain narrow, and
$\stackon[-1.3pt]{$\bar{\nu}$}{\scalebox{0.25}{\hskip
2.0pt{{(}}}\scalebox{0.25}{\hskip 17.0pt
{{)}}}}_{\\!i}-\stackon[-1.2pt]{$\bar{S}\,$}{\scalebox{0.25}{\hskip
2.0pt{{(}}}\scalebox{0.25}{\hskip 17.0pt {{)}}}}_{\\!i}$ oscillations will
remain suppressed away from the resonant region. Therefore, a fully viable
model would still need an additional source of non-resonant active-to-sterile
oscillations. The simplest option is to introduce a new ‘vanilla’ sterile
neutrino that is neutral under the dark sector gauge interactions, and
therefore does not experience a matter potential. This vanilla sterile state
will mix with a linear combination of active neutrinos to generate an off-
resonance appearance probability, and, importantly, it will not qualitatively
interfere with the resonant mechanism. This ordinary ‘3+1’-like mechanism is
well-known for providing an excellent fit to the LSND oscillation data, as
well as to the MiniBooNE oscillation data for antineutrinos and mid-energy
neutrinos. If one wishes, the vanilla ‘3+1’ parameters could also be adjusted
to fit $\stackon[-1.3pt]{$\bar{\nu}$}{\scalebox{0.25}{\hskip
2.0pt{{(}}}\scalebox{0.25}{\hskip 17.0pt {{)}}}}_{\\!e}$ disappearance data,
such as the Gallium and/or reactor anomalies, in the spirit of global fits.

Our goal in this study, however, is not to perform global fits of this model,
much less re-fit the vanilla ‘3+1’ parameters to all existing neutrino data
(something that has already been done _ad nauseam_ in the literature; see,
e.g., Kopp-ml-2013vaa ; Gariazzo-ml-2017fdh ; Dentler-ml-2018sju ; Diaz-
ml-2019fwt ; Moulai-ml-2019gpi ; Berryman-ml-2021yan ; Dasgupta-ml-2021ies ;
Gonzalez-Garcia-ml-2021dve ). Instead, we will benchmark the vanilla ‘3+1’
parameters to yield a good fit to the LSND, MiniBooNE $\bar{\nu}$-mode, and
Gallium anomalies, and focus our attention on deconstructing the parameter
space of the resonant mechanism.

###  3.1 Full model — part I: the triple-resonance mechanism

For pedagogical purposes, we begin by describing the resonant mechanism of the
full model with the ‘3+1’ non-resonant effects tuned off. In the next
subsection, we will then introduce the ‘vanilla’ sterile state to complete the
description of the full model.

As stated in the previous subsection, the full model will contain three quasi-
sterile neutrinos, $S_{1}$, $S_{2}$, and $S_{3}$, all of which experience the
same effective matter potential in (7). Each of the $S_{i}$ will mix
exclusively with $\nu_{i}$ with a vacuum mixing angle
$\theta_{{}_{\\!S_{i}}}$, in a manner analogous to the $\nu_{3}-S_{3}$ mixing
discussed in section 2.

The effective matter Hamiltonian in the basis
$B=\\{|\nu_{1}\rangle,|S_{1}\rangle,|\nu_{2}\rangle,|S_{2}\rangle,|\nu_{3}\rangle,|S_{3}\rangle\\}$
can then be written in block-diagonal form as:

$\\!\\!\\!\\!\widehat{H}_{B,\text{
matter}}=\begin{pmatrix}\widehat{H}_{1}^{\,(2\times 2)}&&\\\
&\widehat{H}_{2}^{\,(2\times 2)}&\\\ &&\widehat{H}_{3}^{\,(2\times
2)}\end{pmatrix}$ (42)

where each block $\widehat{H}_{i}^{\,(2\times 2)}$ acts on the subspace
$\\{|\nu_{i}\rangle,|S_{i}\rangle\\}$ and is defined by:

$\displaystyle\widehat{H}_{i}=\widehat{H}\\!\left[\nu_{i},S_{i}\right]$
$\displaystyle\equiv\begin{pmatrix}~{}~{}\cos\theta_{{}_{\\!S_{i}}}&\sin\theta_{{}_{\\!S_{i}}}\\\
-\sin\theta_{{}_{\\!S_{i}}}&\cos\theta_{{}_{\\!S_{i}}}\end{pmatrix}\cdot\begin{pmatrix}\frac{m_{i}^{2}}{2E_{\nu}}&~{}0\\\
0~{}&\frac{M_{i}^{2}}{2E_{\nu}}\end{pmatrix}\cdot\begin{pmatrix}\,\cos\theta_{{}_{\\!S_{i}}}&-\sin\theta_{{}_{\\!S_{i}}}\\\
\,\sin\theta_{{}_{\\!S_{i}}}&~{}~{}\cos\theta_{{}_{\\!S_{i}}}\end{pmatrix}-\begin{pmatrix}~{}0~{}&0\\\
~{}0~{}&|\Delta V|\\!\end{pmatrix}$
$\displaystyle=\frac{m_{i}^{2}}{2E_{\nu}}\begin{pmatrix}\,1~{}&~{}0\,\,\\\
\,0~{}&~{}0\,\,\end{pmatrix}\,+~{}\frac{\delta
M_{i}^{2}}{4E_{\nu}}\left(1-\frac{E_{\nu}}{\,E^{\text{res}}_{\nu_{i}}\,}\cos
2\theta_{{}_{\\!S_{i}}}\\!\right)\begin{pmatrix}\,1~{}&~{}0\,\,\\\
\,0~{}&~{}1\,\,\end{pmatrix}$ (43) $\displaystyle\quad+\frac{\delta
M_{i}^{2}}{4E_{\nu}}\begin{pmatrix}-\cos
2\theta_{{}_{\\!S_{i}}}\\!\Big{(}\\!1-\frac{E_{\nu}}{\,E^{\text{res}}_{\nu_{i}}\,}\\!\Big{)}~{}&~{}\sin
2\theta_{{}_{\\!S_{i}}}\\\ ~{}~{}\sin 2\theta_{{}_{\\!S_{i}}}~{}&~{}\cos
2\theta_{{}_{\\!S_{i}}}\\!\Big{(}1-\frac{E_{\nu}}{\,E^{\text{res}}_{\nu_{i}}\,}\Big{)}\end{pmatrix}.$

Above, $\delta M_{i}^{2}\equiv M_{i}^{2}-m_{i}^{2}$, and $M_{i}$ is the
(vacuum) mass of the heavier eigenstate, which is composed predominantly of
the sterile state $S_{i}$. Note that in the last equality of (3.1) we have
traded the matter potential $|\Delta V|$ for $E^{\text{res}}_{\nu_{i}}$ using:

$|\Delta V|~{}=~{}\frac{\delta M_{1}^{2}}{\,2\,E^{\text{res}}_{\nu_{1}}\,}\cos
2\theta_{{}_{\\!S_{1}}}=\frac{\delta
M_{2}^{2}}{\,2\,E^{\text{res}}_{\nu_{2}}\,}\cos
2\theta_{{}_{\\!S_{2}}}=\frac{\delta
M_{3}^{2}}{\,2\,E^{\text{res}}_{\nu_{3}}\,}\cos 2\theta_{{}_{\\!S_{3}}}\,.$
(44)

In analogy with (9), we can proceed to diagonalize (3.1) to obtain the
propagation eigenstates for $i=1,2,3$:

$\displaystyle|{N}_{i,\,m}^{{}_{(-)}}\rangle$
$\displaystyle=-\sin\theta_{{}_{\\!S_{i}}}^{{}_{(m)}}|{S}_{i}\rangle+\cos\theta_{{}_{\\!S_{i}}}^{{}_{(m)}}|{\nu}_{i}\rangle\,,~{}~{}~{}~{}~{}~{}$
$\displaystyle{\omega}_{i}^{{}_{(-)}}$
$\displaystyle=\frac{m_{i}^{2}}{2E_{\nu}}+\frac{\delta
M_{i}^{2}}{4E_{\nu}}\left(\\!1-{C}_{i}-\frac{E_{\nu}}{E^{\text{res}}_{\nu_{i}}}\cos
2\theta_{{}_{\\!S_{i}}}\\!\right)\\!\\!\\!$ (45a)
$\displaystyle|{N}_{i,\,m}^{{}_{(+)}}\rangle$
$\displaystyle=~{}~{}\cos\theta_{{}_{\\!S_{i}}}^{{}_{(m)}}|{S}_{i}\rangle+\sin\theta_{{}_{\\!S_{i}}}^{{}_{(m)}}|{\nu}_{i}\rangle\,,~{}~{}~{}~{}~{}~{}$
$\displaystyle{\omega}_{i}^{{}_{(+)}}$
$\displaystyle=\frac{m_{i}^{2}}{2E_{\nu}}+\frac{\delta
M_{i}^{2}}{4E_{\nu}}\left(\\!1+{C}_{i}-\frac{E_{\nu}}{E^{\text{res}}_{\nu_{i}}}\cos
2\theta_{{}_{\\!S_{i}}}\\!\right)\\!\\!\\!$ (45b)

where, as usual, ${C}_{i}$ and the effective ${\nu}_{i}-{S}_{i}$ mixing angle
in matter, $\theta_{{}_{\\!S_{i}}}^{{}_{(m)}}$, are defined through:

$\sin 2\theta_{{}_{\\!S_{i}}}^{{}_{(m)}}=\frac{\sin
2\theta_{{}_{\\!S_{i}}}}{{C}_{i}}\equiv\frac{\sin
2\theta_{{}_{\\!S_{i}}}}{\;\sqrt{~{}\big{(}\\!\sin
2\theta_{{}_{\\!S_{i\\!}}}\big{)}^{2}+\big{(}\\!\cos
2\theta_{{}_{\\!S_{i\\!}}}\big{)}^{2}\left(1-\frac{E_{\nu}}{\,E^{\text{res}}_{\nu_{i}}\,}\right)^{2}\;}}\,.$
(46)

####  3.1.1 Accelerator constraints on the triple-resonance mechanism

We begin by discussing how this triple-resonance mechanism addresses
shortcoming 2 of the minimal model. In the high energy limit of $E_{\nu}\gg
E^{\text{res}}_{\nu_{1,2,3}}$, each active neutrino $\nu_{i}$ is composed
predominantly of the propagation eigenstate ${N}_{i,\,m}^{{}_{(+)}}$ in (45b),
with an effective mass-squared given by:

$m_{i\text{,\,eff\,}}^{2}~{}=~{}2\,E_{\nu}\;\omega_{i}^{{}_{(+)}}~{}\simeq~{}m_{i}^{2}+\sin^{2}\\!\theta_{{}_{\\!S_{i}}}\,\delta
M_{i}^{2}\left(1+\frac{E^{\text{res}}_{\nu_{i}}}{E_{\nu}}\right)$ (47)

From (47), we see that in the asymptotically high energy limit of
$E^{\text{res}}_{\nu_{i}}/E_{\nu}\to 0$, the active mass hierarchy is
preserved, i.e., $\delta m_{ij\text{,\,eff\,}}^{2}\to\delta m_{ij}^{2}$
($i,j=1,2,3$), if:

$\sin^{2}\\!\theta_{{}_{\\!S_{1}}}\,\delta
M_{1}^{2}~{}\approx~{}\sin^{2}\\!\theta_{{}_{\\!S_{2}}}\,\delta
M_{2}^{2}~{}\approx~{}\sin^{2}\\!\theta_{{}_{\\!S_{3}}}\,\delta M_{3}^{2}\,.$
(48)

While (48) may appear like an extremely fine-tuned relation, note that if we
write explicitly the $\nu_{i}-S_{i}$ mixing terms in the Lagrangian,

$\mathcal{L}\supset\sum_{i=1,\,2,\,3}~{}\Big{[}\;m^{\\!\text{\tiny(mix)}\\!}_{i}\,\big{(}\nu_{i}\,S_{i}+\nu_{i}^{c}\,S_{i}^{c}\big{)}+M_{i}\,S_{i}S_{i}^{c}\;\Big{]}+\cdots\,,$
(49)

the mass-mixing term $m^{\\!\text{\tiny(mix)}\\!}_{i}$ is essentially
$m^{\\!\text{\tiny(mix)}\\!}_{i}\approx\sin\\!\theta_{{}_{\\!S_{i}}}M_{i}$.
Hence, when considering this model from a UV completion perspective, one could
envision model-building the relations in (48) by invoking global symmetries,
instead of enforcing (48) via _ad hoc_ fine-tuning. These same global
symmetries would also provide a natural mechanism for the alignment of
$S_{i}-\nu_{i}$ mixing with the SM neutrino mass basis.

We can also check the triple-resonance mechanism’s prediction for $\delta
m_{32\text{,\,eff\,}}^{2}$ at T2K in neutrino mode. Following the discussion
at the end of subsection 2.2, and using (47) and (48), we have:

$\displaystyle\delta m_{32\text{,\,eff\,}}^{2}\big{|}_{{}_{\text{T2K}}}$
$\displaystyle\sim\delta
m_{32}^{2}\big{|}_{{}_{\text{SM}}}\left[\;1+\frac{\,\sin^{2}\\!\theta_{{}_{\\!S_{i}}}\,\delta
M_{i}^{2}\,}{~{}~{}~{}\delta m_{32}^{2}\big{|}_{{}_{\text{\tiny
SM}}}}\,\left(\frac{\,E^{\text{res}}_{\nu_{3}}-E^{\text{res}}_{\nu_{2}}\,}{\;E_{\nu_{\mu}}^{\text{\,dip}}\;}\right)\\!\Bigg{|}_{{}_{\text{T2K}}}\right]$
(50) $\displaystyle\sim\delta
m_{32}^{2}\big{|}_{{}_{\text{SM}}}\times(1\;\pm\;0.08)~{}~{}~{}\text{(from
T2K's $\nu$ data),}$

where the second line in (50) is a $2\sigma$-constraint on this prediction to
be compatible with T2K’s determination of $\delta m_{32}^{2}$ in neutrino mode
bibT2K-ml-2021xwb . Defining

$R_{m_{\nu}^{2}}\equiv\frac{\,\sin^{2}\\!\theta_{{}_{\\!S_{i}}}\,\delta
M_{i}^{2}\,}{~{}~{}~{}|\delta m_{32}^{2}|\big{|}_{{}_{\text{\tiny SM}}}}\,,$
(51)

remembering that $E_{\nu_{\mu}}^{\text{\,dip}}\big{|}_{\text{\tiny T2K}}\sim
600$ MeV, and assuming the relation between
$E^{\text{res}}_{\nu_{i}}\big{|}_{\text{\tiny T2K}}$ and
$E^{\text{res}}_{\nu_{i}}\big{|}_{\text{\tiny MB}}$ in (2.2), we can recast
the constraint in (50) as:

$R_{m_{\nu}^{2}}\,\left[\frac{~{}|E^{\text{res}}_{\nu_{3}}-E^{\text{res}}_{\nu_{2}}|\big{|}_{\text{\tiny
MB}}}{\text{MeV}}\right]~{}\lesssim~{}0.8\times 10^{2}\,.$ (52)

An analogous calculation for antineutrinos yields

$\displaystyle\delta\overline{m}_{32\text{,\,eff\,}}^{2}\big{|}_{{}_{\text{T2K,}\,\bar{\nu}}}$
$\displaystyle\sim\delta
m_{32}^{2}\big{|}_{{}_{\text{SM}}}\left[\;1-\frac{\,\sin^{2}\\!\theta_{{}_{\\!S_{i}}}\,\delta
M_{i}^{2}\,}{~{}~{}~{}\delta m_{32}^{2}\big{|}_{{}_{\text{\tiny
SM}}}}\,\left(\frac{\,E^{\text{res}}_{\nu_{3}}-E^{\text{res}}_{\nu_{2}}\,}{\;E_{\bar{\nu}_{\mu}}^{\text{\,dip}}\;}\right)\\!\Bigg{|}_{{}_{\text{T2K}}}\right]$
(53) $\displaystyle\sim\delta
m_{32}^{2}\big{|}_{{}_{\text{SM}}}\times(1\;\pm\;0.16)~{}~{}~{}\text{(from
T2K's $\bar{\nu}$ data),}$

which provides a weaker constraint than in (50), (52) because T2K’s
determination of $\delta m_{32}^{2}$ in antineutrino mode is less precise
Berns-ml-2021iss .

####  3.1.2 Reactor constraints on the triple-resonance mechanism

Independent constraints on the parameters of the triple-resonance mechanism
come from lower energy electron antineutrinos at short- and long-baselines
reactor-based experiments.

We will not rehash the derivation of the antineutrino propagation eigenstates,
which can be easily obtained by reversing the signs of $|\Delta V|$ and
$E_{\nu}/E^{\text{res}}_{\nu_{i}}$ in (3.1), (45), and (46). In the low energy
limit of $E_{\bar{\nu}}\ll E^{\text{res}}_{\nu_{1,2,3}}$, each active
antineutrino $\bar{\nu}_{i}$ is composed predominantly of the propagation
eigenstate $\bar{N}_{i,\,m}^{{}_{(-)}}$, with an effective mass-squared given
by:

$\overline{m}_{i\text{,\,eff\,}}^{2}\big{|}_{{\bar{\nu}}}~{}=~{}2\,E_{\bar{\nu}}\;\bar{\omega}_{i}^{{}_{(-)}}~{}\simeq~{}m_{i}^{2}+\sin^{2}\\!\theta_{{}_{\\!S_{i}}}\delta
M_{i}^{2}\left(\frac{E_{\bar{\nu}}}{E^{\text{res}}_{\nu_{i}}}\,\cos
2\theta_{{}_{\\!S_{i}}}\\!\right)\,.$ (54)

With (54), we can revisit the prediction for $\delta m_{31}^{2}$ from short-
baseline reactor measurements (by, e.g., Daya Bay and RENO), discussed at the
end of subsection 2.3. Using Daya Bay for concreteness, we have,

$\displaystyle\delta\overline{m}_{31\text{,\,eff\,}}^{2}\big{|}_{\text{reactor
}{\bar{\nu}}_{e}}$ $\displaystyle\simeq\delta
m_{31}^{2}\big{|}_{{}_{\text{SM}}}\,\left[\;1+R_{m_{\nu}^{2}}\;\frac{|\delta
m_{32}^{2}|}{\,\delta
m_{31}^{2}}\Bigg{|}_{{}_{\text{SM}}}\\!\left(\\!\frac{E_{\bar{\nu}_{e}}^{\text{\,dip}}}{\;E^{\text{res}}_{\nu_{3}}\;}-\frac{E_{\bar{\nu}_{e}}^{\text{\,dip}}}{\;E^{\text{res}}_{\nu_{1}}\;}\\!\right)\\!\Bigg{|}_{{}_{\text{DayaB}}}\right]\qquad$
(55) $\displaystyle\sim\delta
m_{31}^{2}\big{|}_{{}_{\text{SM}}}\times(1\;\pm\;0.05)~{}~{}~{}\text{(from
Daya Bay's $\bar{\nu}_{e}$ data at $2\sigma$).}$

Above, we made the approximation $\cos 2\theta_{{}_{\\!S_{i}}}\approx 1$ and
used (51). Using $E_{\bar{\nu}_{e}}^{\text{\,dip}}\big{|}_{\text{\tiny
DayaB}}\sim 3$ MeV, we can recast the constraint in the second line of (55)
as:

$R_{m_{\nu}^{2}}\,\left[\frac{~{}|E^{\text{res}}_{\nu_{3}}-E^{\text{res}}_{\nu_{1}}|\big{|}_{\text{\tiny
MB}}}{\text{MeV}}\right]~{}\lesssim~{}1.5\times 10^{3}\,,$ (56)

which is about an order-of-magnitude weaker than (52).

As for reactor $\bar{\nu}_{e}$’s at long-baselines, of interest to us is
KamLAND’s determination of the ‘solar’ mass-squared splitting $\delta
m_{21}^{2}$ with an error of $\sim$2.5% KamLAND-ml-2013rgu . Using (54), we
can obtain the triple-resonance mechanism’s prediction for
$\delta\overline{m}_{21\text{,\,eff\,}}^{2}$ at KamLAND:

$\displaystyle\delta\overline{m}_{21\text{,\,eff\,}}^{2}\big{|}_{{}_{\text{KamLAND}}}$
$\displaystyle\simeq\delta
m_{21}^{2}\big{|}_{{}_{\text{SM}}}\left[\;1+R_{m_{\nu}^{2}}\;\frac{|\delta
m_{32}^{2}|}{\,\delta
m_{21}^{2}}\Bigg{|}_{{}_{\text{SM}}}\\!\left(\\!\frac{E_{\bar{\nu}_{e}}}{\;E^{\text{res}}_{\nu_{2}}\;}-\frac{E_{\bar{\nu}_{e}}}{\;E^{\text{res}}_{\nu_{1}}\;}\\!\right)\\!\Bigg{|}_{{}_{\text{KamL}}}\right]\qquad$
(57) $\displaystyle\sim\delta
m_{21}^{2}\big{|}_{{}_{\text{SM}}}\times(1\;\pm\;0.05)~{}~{}~{}\text{(from
KamLAND's $\bar{\nu}_{e}$ data at $2\sigma$).}$

KamLAND observes $\bar{\nu}_{e}$’s with energies ranging from
$E_{\bar{\nu}_{e}}\sim(0.9-7)$ MeV. As a simplification, we will use the
energy of maximal $\bar{\nu}_{e}$ disappearance at KamLAND as an input in
(57), namely, $E_{\bar{\nu}_{e}}^{\text{\,dip}}\big{|}_{\text{\tiny KamL}}\sim
3.6$ MeV. Making the additional assumption that the resonant energies
$E^{\text{res}}_{\nu_{i}}$ for KamLAND and T2K are the same, and using (2.2),
we can recast the constraint in (57) as:

$R_{m_{\nu}^{2}}\,\left[\frac{~{}|E^{\text{res}}_{\nu_{2}}-E^{\text{res}}_{\nu_{1}}|\big{|}_{\text{\tiny
MB}}}{\text{MeV}}\right]~{}\lesssim~{}23\,.$ (58)

###  3.2 Full model — part II: the ‘vanilla’ off-resonance component

We now complete the description of the full model by introducing its last
component, namely, a sterile neutrino that does not couple to dark vector
mediators and therefore does not experience the matter potential in (7). We
will denote this sterile species as $S_{0}$, and will often refer to it as the
_vanilla_ sterile neutrino.

In a generic UV realization of the off-resonance mechanism, $S_{0}$ will mix
with a linear combination of the active neutrinos, $\nu_{1}$, $\nu_{2}$, and
$\nu_{3}$, with a mixing angle $\theta_{{}_{\\!S_{0}}}$. Its purpose is to
induce non-resonant oscillations that could explain the $\bar{\nu}_{e}$-like
event excess at LSND and MiniBooNE, as well as the mid-energy $\nu_{e}$-like
event excess at MiniBooNE, through
$\stackon[-1.3pt]{$\bar{\nu}$}{\scalebox{0.25}{\hskip
2.0pt{{(}}}\scalebox{0.25}{\hskip 17.0pt
{{)}}}}_{\\!\mu}\to\stackon[-1.2pt]{$\bar{S}\,$}{\scalebox{0.25}{\hskip
2.0pt{{(}}}\scalebox{0.25}{\hskip 17.0pt
{{)}}}}_{\\!0}\to\stackon[-1.3pt]{$\bar{\nu}$}{\scalebox{0.25}{\hskip
2.0pt{{(}}}\scalebox{0.25}{\hskip 17.0pt {{)}}}}_{\\!e}$ oscillations. This
vanilla off-resonance mechanism will also lead to predictions for
$\stackon[-1.3pt]{$\bar{\nu}$}{\scalebox{0.25}{\hskip
2.0pt{{(}}}\scalebox{0.25}{\hskip 17.0pt {{)}}}}_{\\!\mu}$ and
$\stackon[-1.3pt]{$\bar{\nu}$}{\scalebox{0.25}{\hskip
2.0pt{{(}}}\scalebox{0.25}{\hskip 17.0pt {{)}}}}_{\\!e}$ disappearance in a
variety of other experiments, such as MINOS, IceCube, BEST, RENO/NEOS, etc.
Since the resonant and non-resonant mechanisms do not interfere in any
significant way, one could in principle fit the parameters of the vanilla
sterile neutrino independently of the resonant part of the model. We will not
perform this exercise in this paper, however, since our focus is on the novel
part of the resonance mechanism.

Instead, we will fix the parameters of the off-resonance component to a
simplified benchmark that will allow us to solve for all the propagation
eigenstates _analytically_. In this simplified benchmark, the active state
mixing with $S_{0}$ will consist exclusively of $\nu_{1}$. Consider initially
the limit in which we turn off the mixing of $\nu_{1}$ with $S_{1}$ (i.e.,
$\theta_{{}_{\\!S_{1}}}\\!\to 0$). The two mass eigenstates would then be:

$\displaystyle|\nu^{\prime}_{1\,}\rangle$
$\displaystyle=\cos\theta_{{}_{\\!S_{0}}}\,|\nu_{1}\rangle-\sin\theta_{{}_{\\!S_{0}}}\,|S_{0}\rangle\,,\qquad$
$\displaystyle{\omega}^{\prime}_{1}$
$\displaystyle~{}~{}\xrightarrow[\,\theta_{{}_{\\!S_{1}}}\\!\to
0\;]{}~{}~{}\frac{m_{1}^{2}}{\,2E_{\nu}\,}$ (59a)
$\displaystyle|\nu_{S\,}\rangle$
$\displaystyle=\sin\theta_{{}_{\\!S_{0}}}\,|\nu_{1}\rangle+\cos\theta_{{}_{\\!S_{0}}}\,|S_{0}\rangle\,,\qquad$
$\displaystyle{\omega}_{0}$
$\displaystyle~{}~{}\xrightarrow[\,\theta_{{}_{\\!S_{1}}}\\!\to
0\;]{}~{}~{}\frac{M_{0}^{2}}{\,2E_{\nu}\,}$ (59b)

The choice of $\nu_{1}-S_{0}$ mixing is motivated by a combination of
anomalies and experimental bounds. First, since $\nu_{1}$ has a significant
$\nu_{e}$-component, $\nu_{1}-S_{0}$ mixing can easily accommodate a large
(off-resonance) $\nu_{e}$ disappearance rate,
$\overline{P}(\nu_{e}\not\to\nu_{e})\approx\overline{P}(\nu_{e}\to
S_{0})=1/2\times|U_{e1}|^{2}\sin^{2}2\theta_{{}_{\\!S_{0}}}$, which is
convenient to address the Gallium anomalies, for instance GALLEX-ml-1997lja ;
SAGE-ml-1998fvr ; Abdurashitov-ml-2005tb ; Kaether-ml-2010ag ; Giunti-
ml-2012tn ; Barinov-ml-2021mjj . Secondly, $\nu_{1}$ has a smaller
$\nu_{\mu}$-component than $\nu_{2}$ and $\nu_{3}$, and therefore
$\nu_{1}-S_{0}$ mixing is subject to weaker
$\stackon[-1.3pt]{$\bar{\nu}$}{\scalebox{0.25}{\hskip
2.0pt{{(}}}\scalebox{0.25}{\hskip 17.0pt {{)}}}}_{\\!\mu}$ disappearance
constraints from, e.g., MINOS MINOS-ml-2017cae and IceCube IceCube-ml-2016rnb
; IceCube-ml-2020tka .

Figure 6: [_For illustration purposes only — not to scale_] Effective mass-
squared eigenvalues of neutrino propagation eigenstates (left panel) and
antineutrino propagation eigenstates (right panel) as a function of
(anti-)neutrino energy in the full model. The eigenstates’ flavor compositions
are denoted by their color shading.

Given (59), we could now straightforwardly turn on $\nu_{1}-S_{1}$ mixing.
However, this would then require numerically diagonalizing the Hamiltonian for
the $\\{\nu_{1},S_{0},S_{1}\\}$ subsystem in order to obtain the propagation
eigenstates. In order to avoid loosing analytical control over our treatment,
we will introduce a second simplification to our benchmark: we will mix
$S_{1}$ with $\nu^{\prime}_{1}$ in (59a) instead. This will allow us to retain
our analytical expressions for the propagation eigenstates at arbitrary matter
densities by simply making the replacement $\nu_{1}\to\nu^{\prime}_{1}$ in
(42), (3.1), and (45). This small deformation will not qualitatively affect
any of the important features of the resonance mechanism. Quantitatively, we
have checked that if $S_{1}$ were to mix with $\nu_{1}$ instead of
$\nu_{1}^{\prime}$, this would shift best fit parameters by
$\mathcal{O}(10\%)$, and would also require an $\mathcal{O}(10\%)$ adjustment
of relation (48) in order to preserve the in-medium active neutrino mass
hierarchy at high energies.

Finally, we briefly return to the issue of potential CP violating phases
introduced by mixings of the active neutrinos with the sterile and quasi-
sterile states. In analogy with the toy model case, complex phases
$\delta_{{}_{\\!S_{i}}}$ from $\nu_{i}-S_{i}$ mixing are unphysical and can be
absorbed by unitary rotations of the $|S_{i}\rangle$ states. This is possible
because, by construction, $\nu_{i}-S_{i}$ mixing is aligned with the active
Yukawa basis, i.e., there is no “cross-mixing” of the type $\nu_{j}-S_{k}$
with $j\neq k$. Similarly, the unphysical complex phase from $\nu_{1}-S_{0}$
mixing in (59) can be absorbed by a unitary rotation of $|S_{0}\rangle$.
Therefore, in the benchmark models we will be considering in the next section,
no additional sources of CP-violation exist beyond the SM CP-violating phase
$\delta_{\text{CP}}$. However, more generally, a more anarchical mixing
pattern between $S_{0}$ and a linear combination of $\nu_{i}$ and $S_{i}$
states will introduce additional physical CP-violating phases. The complex
phases from $S_{0}-S_{i}$ mixing, in particular, will only be relevant in the
very narrow resonance window, and while they may introduce quantitative
modifications to the resonant oscillation pattern, they will not qualitatively
affect the overall mechanism’s ability to address the MiniBooNE excess.
Outside the resonance window, the complex phases from $S_{0}-S_{i}$ mixing
will have a negligible effect in active neutrino appearance and disappearance
probabilities due to the suppressed $S_{0}-S_{i}$ and $\nu_{i}-S_{i}$ vacuum
mixing angles. In contrast, the complex phases from $S_{0}-\nu_{i}$ mixing
will be relevant outside the resonance window, where neutrino oscillations are
well described by the standard ‘3+1’ model. Such effects have been considered
elsewhere in the literature (see, e.g., Donini-ml-1999jc ; Klop-ml-2014ima ;
Cianci-ml-2017okw ; Palazzo-ml-2020apc ), and are not the focus of the present
study.

Figure 6 illustrates the components of the full model, including the neutrino
and antineutrino propagation eigenstates, their eigenvalues, and their flavor
composition. The relation (48), combined with the fact that all active
neutrinos cross the resonance within a narrow energy range, results in a much
smaller deviation of the in-medium active mass spectrum relative to the SM
expectation. Also, note that unlike the toy model discussed in section 2, the
in-medium mass hierarchy in this full model realization does not ‘flip’ at
high energies.

###  3.3 Full model — part III: fitting the MiniBooNE low energy excess

With the full model ingredients introduced, we can recast the active neutrino
flavor eigenstates in terms of the propagation eigenstates in order to derive
$\nu_{\mu}\to\nu_{e}$ oscillation probabilities for MiniBooNE. With $\ell$
denoting active neutrino flavors ($\ell=e,\mu,\tau$), we have:

$\displaystyle|\nu_{\ell}(L)\rangle~{}=~{}~{}~{}\sin\theta_{{}_{\\!S_{0}}}\\!\\!\\!$
$\displaystyle U^{*}_{\ell
1\,}\,\Big{[}\;e^{-i\,\omega_{0}\,(L\,+\,\overline{L}_{\text{\tiny
decay}})}\;|\nu_{S\,}\rangle~{}\Big{]}$ (60)
$\displaystyle~{}~{}\,+\,\cos\theta_{{}_{\\!S_{0}}}\\!\\!\\!$ $\displaystyle
U^{*}_{\ell
1\,}\,\Big{[}\,\cos\theta_{{}_{\\!S_{1}}}^{{}_{(m)}}\,e^{-i\,\omega_{1}^{{}_{(-)}}L}\;|N_{1,\,m}^{{}_{(-)}}\rangle~{}+~{}\sin\theta_{{}_{\\!S_{1}}}^{{}_{(m)}}\,e^{-i\,\omega_{1}^{{}_{(+)}}L}\;|N_{1,\,m}^{{}_{(+)}}\rangle~{}\Big{]}\qquad$
$\displaystyle+$ $\displaystyle U^{*}_{\ell
2\,}\,\Big{[}\,\cos\theta_{{}_{\\!S_{2}}}^{{}_{(m)}}\,e^{-i\,\omega_{2}^{{}_{(-)}}L}\;|N_{2,\,m}^{{}_{(-)}}\rangle~{}+~{}\sin\theta_{{}_{\\!S_{2}}}^{{}_{(m)}}\,e^{-i\,\omega_{2}^{{}_{(+)}}L}\;|N_{2,\,m}^{{}_{(+)}}\rangle~{}\Big{]}\qquad$
$\displaystyle+$ $\displaystyle U^{*}_{\ell
3\,}\,\Big{[}\,\cos\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,e^{-i\,\omega_{3}^{{}_{(-)}}L}\;|N_{3,\,m}^{{}_{(-)}}\rangle~{}+~{}\sin\theta_{{}_{\\!S_{3}}}^{{}_{(m)}}\,e^{-i\,\omega_{3}^{{}_{(+)}}L}\;|N_{3,\,m}^{{}_{(+)}}\rangle~{}\Big{]}\,.\qquad$

Above, the matter mixing angles $\theta_{{}_{\\!S_{i}}}^{{}_{(m)}}$ are
defined in (46), and the propagation eigenphases $\omega_{i}^{{}_{(\pm)}}$ and
$\omega_{0}$ are defined in (45) and (59b), respectively. As before, we take
MiniBooNE’s propagation length $L$ to be the average distance through which
neutrinos travel in matter, $L\approx 490\,$m, and we assume constant matter
density and earth composition throughout.444In particular, we neglect possible
effects of other structures in the neutrino’s propagation path, such as the
steel and concrete beam absorber, vault and detector walls, the mineral oil
target before interaction, etc. The additional ‘averaged decay baseline’
$\overline{L}_{\text{\tiny decay}}$ appearing in the first line of (60) is an
approximation to account for the state’s evolution within the decay region of
the BNB beamline, which for simplicity we take to be
$\overline{L}_{\text{\tiny decay}}\sim 30\,$m. Only the phase of the $\nu_{S}$
component of $\nu_{\ell}$ changes non-negligibly within the decay region (see
(59)); therefore, the phase change of the $\nu_{1}^{\prime}$ component within
the decay region is ignored in (60).

Given the relations (44), (48), and (51), we can fully specify all the
parameters of the full model by a set of 7 independent parameters: 5 related
to the triple-resonance mechanism, which we choose to be
$\theta_{{}_{\\!S_{2}}}$, $R_{m_{\nu}^{2}}$, $E^{\text{res}}_{\nu_{1}}$,
$E^{\text{res}}_{\nu_{2}}$, $E^{\text{res}}_{\nu_{3}}$; and 2 related to the
off-resonance oscillations, $\theta_{{}_{\\!S_{0}}}$ and $M^{2}_{0}$.

In fitting MiniBooNE’s $\nu_{e}$ appearance data, we will fix the off-
resonance parameters to the following (non-optimized) values:555This choice of
off-resonance parameters is motivated by the Gallium $\nu_{e}$ disappearance
anomalies GALLEX-ml-1997lja ; SAGE-ml-1998fvr ; Abdurashitov-ml-2005tb ;
Kaether-ml-2010ag ; Giunti-ml-2012tn ; Barinov-ml-2021mjj , including the most
recent results from the BEST collaboration Barinov-ml-2021asz . However, we
note that these parameters are in strong tension with more recent oscillation
bounds from reactor-based antineutrino experiments Giunti-ml-2021kab , such as
the combined results from RENO and NEOS RENO-ml-2020hva . Notwithstanding, the
$\nu_{e}$ spectral shape at RENO, NEOS, and other reactor experiments (Daya
Bay, Double Chooz) shows a significant departure from flux models’
expectations at energies around $\sim 5-6\;$MeV, whose origin is not
understood NEOS-ml-2016wee ; RENO-ml-2016ujo ; DayaBay-ml-2016ssb ; Huber-
ml-2016xis ; Zacek-ml-2018bij ; Qian-ml-2018wid ; DoubleChooz-ml-2019qbj . We
speculate that there might be additional systematic uncertainties in reactor
antineutrino measurements that have not been accounted for Hayes-ml-2013wra ;
Dwyer-ml-2014eka ; Novella-ml-2015eaw ; Hayes-ml-2016qnu ; therefore, we
choose to ignore reactor-based constraints on ‘3+1’ sterile neutrino models
Dentler-ml-2017tkw ; Giunti-ml-2017yid ; Giunti-ml-2020uhv ; Serebrov-
ml-2021ndf .

$\theta_{{}_{\\!S_{0}}}=0.30\,,\qquad M^{2}_{0}=1.0\;\text{eV}^{2}\,.$ (61)

Figure 7: Left panel: the $\nu_{\mu}\to\nu_{e}$ appearance probability at
MiniBooNE predicted by the benchmark model B1 specified in (61), (63a). The
non-resonant component (yellow region) is the expectation from this model if
the triple-resonance mechanism were ‘turned off’ (i.e., if the active-quasi-
sterile mixing angles
$\theta_{{}_{\\!S_{1}}},\theta_{{}_{\\!S_{2}}},\theta_{{}_{\\!S_{3}}}$ were
set to zero). The red data points with error bars were taken from figure 21 of
MiniBooNE-ml-2020pnu ; they assume $E_{\nu}^{\text{true}}\sim
E_{\nu}^{\text{\tiny QE}}$, which is a very coarse approximation. As such,
they should be interpreted as a _qualitative_ measure of MiniBooNE’s anomalous
appearance probabilities. Right panel: the expected $\nu_{e}$ appearance event
rate at MiniBooNE predicted by the full model benchmark B1. The intrinsic
(beam) $\nu_{e}$ component of the background takes $\nu_{e}\not\to\nu_{e}$
oscillations into account. The red data points correspond to the measurements
reported by MiniBooNE in MiniBooNE-ml-2020pnu , and the error bars include
both statistical and systematic uncertainties.

As for the parameters of the triple-resonance mechanism, we have found that if
either $E^{\text{res}}_{\nu_{1}}$ or $E^{\text{res}}_{\nu_{2}}$ is within
$\mathcal{O}(20\%)$ of $\sim$300 MeV, the full model can provide excellent
fits to MiniBooNE’s data while simultaneously satisfying the constraints in
(52), (56), and (58). These excellent fits (i.e., within $1\sigma$ of the
best-fit point) are achieved if the mixing angle $\theta_{{}_{\\!S_{j}}}$
associated with the resonant energy $E^{\text{res}}_{\nu_{j}}$ closest to
$\sim$300 MeV satisfies the condition of appearance probability saturation at
the resonance, analogous to (32):

$\displaystyle\sin\theta_{{}_{\\!S_{j}}}$
$\displaystyle~{}\lesssim~{}\big{|}{\sin\theta_{{}_{\\!S_{j}}}^{\text{(max)}}}\big{|}~{}=~{}\frac{\,R_{m_{\nu}^{2}}\,|\delta
m_{32}^{2}|\,}{2\,\pi}~{}\frac{L}{\,E^{\text{res}}_{\nu_{j}}\,}\Bigg{|}_{\text{MB}}$
(62) $\displaystyle~{}\lesssim~{}3.3\times
10^{-3}\;R_{m_{\nu}^{2}}\;\frac{\,(300\;\text{MeV})\,}{~{}~{}~{}~{}E^{\text{res}}_{\nu_{j}}\big{|}_{{}_{\text{MB}}}\,}\,.$

Figures 7 and 8 (left panels) show the $\nu_{\mu}\to\nu_{e}$ appearance
probability at MiniBooNE as a function of true neutrino energy for two
illustrative benchmark points:

B1 $\displaystyle:\quad$ $\displaystyle\theta_{{}_{\\!S_{2}}}$
$\displaystyle={\bf{0.003}},$ $\displaystyle\quad R_{m_{\nu}^{2}}$
$\displaystyle=0.9,$ $\displaystyle\quad E^{\text{res}}_{\nu_{1}}$
$\displaystyle=290\;\text{MeV},$ $\displaystyle\quad E^{\text{res}}_{\nu_{2}}$
$\displaystyle=300\;\text{MeV},$ $\displaystyle\quad E^{\text{res}}_{\nu_{3}}$
$\displaystyle=310\;\text{MeV},$ (63a) B2 $\displaystyle:\quad$
$\displaystyle\theta_{{}_{\\!S_{2}}}$ $\displaystyle={\bf{0.006}},$
$\displaystyle\quad R_{m_{\nu}^{2}}$ $\displaystyle=0.9,$ $\displaystyle\quad
E^{\text{res}}_{\nu_{1}}$ $\displaystyle=290\;\text{MeV},$ $\displaystyle\quad
E^{\text{res}}_{\nu_{2}}$ $\displaystyle=300\;\text{MeV},$ $\displaystyle\quad
E^{\text{res}}_{\nu_{3}}$ $\displaystyle=310\;\text{MeV},$ (63b)

with the off-resonance parameters set to the values in (61). The right panels
in figures 7 and 8 show the expected contribution of these two benchmarks to
the MiniBooNE $\nu_{e}$ event spectrum. These rates were obtained using the
information provided by the MiniBooNE collaboration in its 2021 public data
release hepdata.114365.v1 , and include the effects of oscillations of the
intrinsic $\nu_{e}$ background from the Booster Neutrino Beam (BNB). Note that
even for benchmark B2 in (63b), which lies outside the appearance probability
saturation range in (62), the triple-resonance mechanism provides an adequate
fit to the MiniBooNE $\nu_{e}$-like excess, as can be seen from figure 8 (left
panel).

Figure 8: Same as in figure 7 but for the benchmark model B2 specified in
(61), (63b).

At this point, it is germane to comment on the $\cos\theta_{e}$ distribution
of MiniBooNE’s excess events ($\theta_{e}$ is defined as the angle of the
reconstructed electron relative to the neutrino beam). The $\cos\theta_{e}$
spectral shape of the MiniBooNE excess appears to be more forward-peaked in
the region $\cos\theta_{e}>0.9$ than what is expected from the kinematics of
$\nu_{e}$ charged-current quasi-elastic (CCQE) scattering, but not forward
enough to be consistent with $\nu-e^{-}$ elastic scattering. In fact, the
forward excess in $\cos\theta_{e}>0.9$ has motivated many non-oscillatory
interpretations of the MiniBooNE anomaly. Indeed, if statistically
significant, this spectral feature could in principle rule out explanations of
the $\nu_{e}$-like excess based on _bona fide_ $\nu_{e}$ appearance. However,
we note that the MiniBooNE collaboration has not performed any studies to
quantify the experimental systematics associated with the $\cos\theta_{e}$
distribution (and, in fact, MiniBooNE’s modeling of the $\nu_{e}$ backgrounds
in the forward low energy kinematic region has limited statistics). Hence, at
present one cannot rule out the $\nu_{e}$ CCQE scattering hypothesis for the
MiniBooNE excess based on its $\cos\theta_{e}$ event distribution.

Figure 9: Effective mass-squared splittings $\delta m_{ij\text{,\,eff}}^{2}$
predicted by the full model benchmarks B1 and B2 in (63) as a function of
(anti-)neutrino energy (both benchmarks’ predictions are indistinguishable as
displayed in this plot). All effective mass-squared splittings shown have been
normalized to the appropriate central value $\delta
m_{ij,\text{\tiny\,PDG}}^{2}$ reported in the PDG ParticleDataGroup-ml-2020ssz
. Solid purple curve: the effective ‘solar’ mass-squared splitting for
antineutrinos, $\delta\overline{m}_{21\text{,\,eff}}^{2}$, expected at
KamLAND. Solid orange (green) curve: the effective ‘atmospheric’ mass-squared
splitting for neutrinos (antineutrinos), $\delta m_{32\text{,\,eff}}^{2}$
($\delta\overline{m}_{32\text{,\,eff}}^{2}$), expected at long-baseline
accelerator neutrino experiments running in neutrino (antineutrino) mode. The
dotted parts of these curves indicate energy ranges that have not been
experimentally probed. The data points with error bars denote recent
measurements of $\delta\overline{m}_{21\text{,\,eff}}^{2}$, $\delta
m_{32\text{,\,eff}}^{2}$, and $\delta\overline{m}_{32\text{,\,eff}}^{2}$ by
long-baseline experiments KamLAND-ml-2013rgu ; bibT2K-ml-2018rhz ; MINOS-
ml-2020llm ; NOvA-ml-2021nfi .

Finally, we look at how neutrino oscillations in the full model can be
satisfactorily reconciled with data from long-baseline experiments. In figure
9 we show the effective active mass-squared splittings $\delta
m_{32\text{,\,eff\,}}^{2}$, $\delta\overline{m}_{32\text{,\,eff\,}}^{2}$, and
$\delta\overline{m}_{21\text{,\,eff\,}}^{2}$ expected in long-baseline
experiments as a function of (anti-)neutrino energy for the benchmarks in
(63). The rapid relaxation of these active mass-squared splittings towards
their SM expectation indicates that the triple-resonance mechanism, unlike the
models discussed in Karagiorgi-ml-2012kw ; Asaadi-ml-2017bhx ; Doring-
ml-2018cob ; Barenboim-ml-2019hso , can explain the MiniBooNE excess while
remaining compatible with observations from long-baseline neutrino
experiments.

Figure 10: Similar to figure 9, but for atmospheric (anti-)neutrinos and solar
neutrinos propagating through different media within the Earth, and ignoring
the SM matter effects (i.e., setting $G_{F}=0$). The left-most data point
corresponds to the global fit to solar neutrino data, including day-night
asymmetries, obtained in Esteban-ml-2020cvm . The other data points correspond
to atmospheric neutrino measurements from Super-Kamiokande Super-Kamiokande-
ml-2005mbp ; Super-Kamiokande-ml-2017yvm and IceCube DeepCore IceCube-
ml-2017lak .

In figure 10 we show the effective active mass-squared splittings $\delta
m_{32\text{,\,eff\,}}^{2}$ ($\delta\overline{m}_{32\text{,\,eff\,}}^{2}$) for
atmospheric (anti-)neutrinos and $\delta{m}_{21\text{,\,eff\,}}^{2}$ for solar
neutrinos when propagating through different media within the Earth. These
effective mass-squared splittings are obtained in the limit $G_{F}\to 0$, and
should replace the corresponding vacuum values in the calculation of
terrestrial MSW matter effects. Specifically, we show the effective mass-
squared splittings $\delta m_{ij\text{,\,eff\,}}^{2}\big{|}_{G_{F}\to 0}$ in
the Earth’s inner core ($\rho=13.0\,\text{g/cm}^{3}$), outer core
($\rho=11.3\,\text{g/cm}^{3}$), mantle ($\rho=5.0\,\text{g/cm}^{3}$), and
crust ($\rho=3.3\,\text{g/cm}^{3}$). The assumed densities for these various
media follow the PREM model Dziewonski used in the atmospheric and solar
neutrino analyses by the Super-Kamiokande collaboration. Figure 10 also shows
the global solar neutrino data determination of $\delta m_{21}^{2}$ Esteban-
ml-2020cvm , and the determination of $\delta m_{32}^{2}$ by Super-Kamiokande
runs I–IV Super-Kamiokande-ml-2017yvm and by IceCube DeepCore IceCube-
ml-2017lak . Note that despite the deformation of the active neutrino spectrum
near the triple-resonance region, the effective active mass-squared splittings
$\delta m_{ij\text{,\,eff\,}}^{2}\big{|}_{G_{F}\to 0}$ remain compatible with
experimental determinations of $\delta m_{ij}^{2}$ within the energy ranges
probed. In particular, deviations from the SM expectation are strongly
suppressed for atmospheric antineutrinos, as well as for atmospheric neutrinos
in the mid-to-high energy range $E_{\nu}\gtrsim 400$ MeV. For sub-GeV
atmospheric neutrinos in the low energy range $E_{\nu}\lesssim 200$ MeV, the
active mass spectrum departs significantly from the SM expectation; however,
in this energy range $\delta m_{32}^{2}$ is very poorly constrained by the
zenith-angle dependence of the atmospheric neutrino flux Super-Kamiokande-
ml-2005mbp . In the much lower energy regime relevant for solar neutrinos, the
range of $\delta{m}_{21\text{,\,eff\,}}^{2}\big{|}_{G_{F}\to 0}$ in the
Earth’s media (inner/outer core, mantle, and crust) is in moderate tension
($\lesssim 2\sigma$) with the global solar neutrino data; this can be traced
back to the slight discrepancy in the favored range for $\delta m_{21}^{2}$
between solar data _vs_ KamLAND KamLAND-ml-2013rgu . Note, however, that the
global solar fit to $\delta m_{21}^{2}$ assumes that the flux of neutrinos
emitted by the Sun is the one predicted in the SM. This flux might be modified
due to the new matter effects induced in the solar medium by the dark sector
vector mediator; in fact, some UV completions of the dark sector matter
potential could alleviate this mild tension in the determination of $\delta
m_{21}^{2}$. This topic will be considered in further detail in future work
future .

##  4 Further implications of quasi-sterile neutrinos and BSM matter effects

There are other experimental observations that could be significantly impacted
by the resonant mechanism of quasi-sterile neutrinos, and in particular by the
specific realization considered in section 3 to explain the MiniBooNE and LSND
anomalies. Two obvious issues are (i) the implications for solar neutrinos,
and (ii) the expected BSM $\nu_{e}$ flux at MicroBooNE, which is of
significant relevance at present in light of the recent results reported by
the MicroBooNE collaboration in MicroBooNE-ml-2021rmx . In this section we
briefly comment on generic experimental implications of the triple-resonance
mechanism for solar neutrinos and MicroBooNE, while deferring a comprehensive
investigation of these issues to a future publication future .

###  4.1 Implications for MicroBooNE

The MicroBooNE collaboration recently released the results of three searches
for $\nu_{e}$ scattering events based on data obtained with Fermilab’s BNB in
neutrino-mode from an exposure of $7\times 10^{20}$ POT MicroBooNE-ml-2021nxr
; MicroBooNE-ml-2021jwr ; MicroBooNE-ml-2021sne . These analyses targeted
independent final states of $\nu_{e}$-nuclear scattering: the ‘Wire-Cell’
search in MicroBooNE-ml-2021nxr was fully inclusive in all possible hadronic
final states; the ‘Deep-Learning’ search in MicroBooNE-ml-2021jwr selected
exclusive two-body $e^{-}p$ final states to obtain increased sensitivity to
$\nu_{e}$ CCQE scattering; and the semi-inclusive ‘Pandora’ search in
MicroBooNE-ml-2021sne applied a pion veto on its selection of $e^{-}+Np$ final
states ($N\geq 0$). The non-observation of an excess of events in these
searches (and, in fact, the observation of a _deficit_ of events in the
inclusive Wire-Cell and CCQE Deep-Learning searches) allowed MicroBooNE to
exclude an _ad hoc_ parametrization of the MiniBooNE low energy excess, coined
the ‘empirical LEE’ model, at a confidence level greater than $\sim 97\%$.

While at face value these results appear to strongly disfavor $\nu_{e}$
appearance interpretations of the MiniBooNE low energy excess, we contend that
such a general conclusion is premature. In particular, we highlight several
caveats that could invalidate the generality of MicroBooNE’s exclusion of
$\nu_{\mu}\to\nu_{e}$ oscillation hypotheses addressing the MiniBooNE anomaly:

  1. 1.

The observed $\nu_{e}$ rates in the MicroBooNE BNB data were significantly
lower than the background expectation for the three most sensitive channels —
the CCQE Deep-Learning search, for example, found a $p$-value of $p=0.014$ for
the background-only hypothesis. This led to stronger than expected exclusions
of the empirical LEE model signal strength. Assuming this discrepancy between
data and background expectation was due to a downward statistical fluctuation,
it is likely that more data will lead to a weakening of MicroBooNE’s
constraints. On the other hand, if tensions persist with more data, this would
point to a possible mismodelling of expected backgrounds, which would require
a revision of the MicroBooNE bounds. Either way, it is unlikely that upcoming
MicroBooNE analyses with twice as much beam luminosity will lead to tighter
bounds on models predicting BSM $\nu_{e}$ appearance at MiniBooNE and
MicroBooNE.

  2. 2.

The MicroBooNE analyses tested only the very specific empirical LEE model,
which was an _ad hoc_ $\nu_{e}$ excess flux obtained by ‘reverse-engineering’
the MiniBooNE $\nu_{e}$-like excess. The MicroBooNE collaboration chose to
normalize the empirical LEE flux relative to the _intrinsic_ $\nu_{e}$ flux
(i.e., the background $\nu_{e}$ component in the BNB originating from decays
of muons and kaons produced at the source). Figure 11 shows the expected
empirical LEE $\nu_{e}$ spectrum at MicroBooNE taken from figure 2 of
MicroBooNE-ml-2021jwr ; for each energy bin, the empirical LEE flux is
weighted relative to the intrinsic $\nu_{e}$ flux integrated over that same
bin. The fact that the reported MicroBooNE exclusions were cast in terms of
the empirical LEE signal strength constitutes perhaps the most important
caveat concerning these results: _the mapping of the MiniBooNE excess to a
truth-level $\nu_{e}$ spectrum is not injective_, and in fact other $\nu_{e}$
spectra departing considerably from the empirical LEE model can also provide
good fits to the MiniBooNE excess (see also Arguelles-ml-2021meu ). We will
come back to this point shortly.

  3. 3.

The MicroBooNE analyses assumed that the empirical LEE $\nu_{e}$ flux, when
normalized to the intrinsic $\nu_{e}$ flux, was identical at MiniBooNE and
MicroBooNE. This simplified assumption fails to capture important features of
BSM oscillation models. For instance, the fact that MicroBooNE’s baseline is
about 70 m shorter than MiniBooNE’s could have a non-negligible effect on
$\nu_{\mu}\to\nu_{e}$ oscillation probabilities. Furthermore, intrinsic
$\nu_{e}$ disappearance could be non-negligible at MicroBooNE (see also
Denton-ml-2021czb ), especially if the Gallium anomalies are indeed an
indication of large $\nu_{e}$ disappearance at
$L/E_{\nu_{e}}\sim\mathcal{O}(\text{m}/\text{MeV})$. In this case,
MicroBooNE’s constraints on a realistic oscillation model would be less severe
than those implied from the empirical LEE signal strength due to a reduction
of the intrinsic $\nu_{e}$ backgrounds. (Note that a non-negligible $\nu_{e}$
disappearance probability would not be as consequential for the MiniBooNE
excess because the backgrounds from intrinsic $\nu_{e}$ CCQE scattering events
at MiniBooNE are sub-dominant in the low energy region
$E_{\nu}\big{|}_{\text{\tiny MB}}\lesssim 400$ MeV.)

Figure 11: The _effective_ BSM $\nu_{e}$ fluxes at the MicroBooNE (left) and
MiniBooNE (right) detectors from the BNB normalized by the expected intrinsic
$\nu_{e}$ background flux in the SM, as predicted by three different models:
the _ad hoc_ empirical LEE model used in the MicroBooNE analyses (red hatched
histogram); the quasi-sterile benchmark model B1 defined in (63a) and
displayed in figure 7 (light green histogram); and the quasi-sterile benchmark
model B2 defined in (63b) and displayed in figure 8 (dark green histogram).
For B1 and B2, the ‘effective’ BSM $\nu_{e}$ fluxes include
$\nu_{\mu}\to\nu_{e}$ oscillations as well as intrinsic $\nu_{e}$
disappearance.

The triple-resonance model presented in section 3 provides a clear example of
caveats 2 and 3 above. Figure 11 (left panel) contrasts the empirical LEE
$\nu_{e}$ flux at MicroBooNE with the _effective_ $\nu_{e}$ flux predicted by
the quasi-sterile benchmarks B1 and B2 specified in (61) and (63a), (63b). In
order to provide an ‘apples-to-apples’ comparison with the empirical LEE
model, the effective $\nu_{e}$ fluxes for B1 and B2 shown in figure 11 include
not only $\nu_{\mu}\to\nu_{e}$ appearance, but also intrinsic $\nu_{e}$
disappearance (which is predominantly due to $\nu_{e}$-to-sterile oscillations
with an averaged disappearance probability of $\overline{P}_{\nu_{e}\to
S_{0}}\approx 1/2\times|U_{e1}|^{2}\sin^{2}2\theta_{{}_{\\!S_{0}}}\approx
11\%$ away from the resonant region). The effects of $\nu_{\mu}$
disappearance, which would affect the normalization of the $\nu_{\mu}$ flux
and consequently the constrained $\nu_{e}$ predictions, have been ignored.
These effects are small away from the resonance, where they are dominated by
$\nu_{\mu}$-to-sterile oscillations with an averaged $\nu_{\mu}$ disappearance
probability of $\overline{P}_{\nu_{\mu}\to S_{0}}\approx 1/2\times|U_{\mu
1}|^{2}\sin^{2}2\theta_{{}_{\\!S_{0}}}\approx 2\%$. Around the resonant
region, $\nu_{\mu}$ disappearance is more significant — averaging to
$\mathcal{O}(10\%)$ for the benchmarks B1 and B2 — but still within the
$\nu_{\mu}$ flux uncertainty at MicroBooNE. A more comprehensive study
including the effects of $\nu_{\mu}$ disappearance are deferred to a future
study.

Figure 11 (right panel) contrasts the empirical LEE, B1, and B2 predictions
for the BSM $\nu_{e}$ flux at MiniBooNE normalized to the intrinsic $\nu_{e}$
flux. We can compare these predictions with the ones for MicroBooNE shown in
the left panel: while the empirical LEE prediction is, by definition,
identical at MicroBooNE and MiniBooNE, the predictions from the B1 and B2
quasi-sterile benchmarks are quantitatively different. This is due to the
difference in baselines between MicroBooNE and MiniBooNE mentioned in caveat 3
above. In particular, this difference is more pronounced for B2 because this
benchmark does not saturate the appearance probability at the resonance, and
MicroBooNE’s shorter baseline takes the $\nu_{\mu}\to\nu_{e}$ oscillation
probability further away from the saturation condition relative to MiniBooNE
(see (62) and discussion around it).

Figure 12: Expected $\nu_{e}$ event rates for two MicroBooNE searches: the
CCQE Deep-Learning MicroBooNE-ml-2021jwr (left) and the inclusive Wire-Cell
MicroBooNE-ml-2021nxr (right) signal regions. The gray histograms denote the
expected backgrounds obtained by the MicroBooNE collaboration in each
analysis. In particular, the intrinsic $\nu_{e}$ background rate shown in gray
is the one expected in the SM and does not assume BSM $\nu_{e}$ oscillations.
The hatched histograms (blue on left panel and red on right panel) are the
$\nu_{e}$ event rate prediction from the empirical LEE model used in the
MicroBooNE analyses. The light and dark green histograms denote approximate
$\nu_{e}$ event rate predictions from the B1 and B2 quasi-sterile benchmarks
(see text for details). Since intrinsic $\nu_{e}$ disappearance has not been
included in the background histograms, this effect has been absorbed in the
green histograms for the B1 and B2 predictions. The data error bars include
statistical and systematic uncertainties.

It is difficult, however, to draw any firm conclusion from the comparisons of
truth-level $\nu_{e}$ rate predictions shown in figure 11. Smearing and bias
effects in energy reconstruction, and the rapid reduction of event selection
efficiency towards lower energies, make it non-trivial to infer which part of
the $\nu_{e}$ signal spectrum dominates the expected $\nu_{e}$ event rates in
the various MicroBooNE analyses. Unlike MiniBooNE’s public data releases, the
MicroBooNE collaboration has not released sufficient information to allow one
to perform an end-to-end statistical analysis of the consistency of an
_arbitrary_ oscillation model with the MicroBooNE data.

The public data release for the CCQE Deep-Learning analysis hepdata.114859.v1
, for instance, provided a Monte Carlo $\nu_{e}$ event sample
hepdata.114859.v1-ml-t7 that can be re-weighted to obtain _unconstrained_
$\nu_{e}$ event rate predictions from an arbitrary model. Using these tools,
we have obtained the unconstrained $\nu_{e}$ event rate predicted by the B1
and B2 benchmarks in the CCQE Deep-Learning signal region. However, with the
information in the public data release alone, we cannot proceed beyond this
point. Specifically, no tools were provided to construct the full covariance
matrix for an arbitrary model; without it, one cannot obtain the _constrained_
$\nu_{e}$ event rate prediction, nor its associated $p$-value. Given these
limitations, we instead took a heuristic approach: we inferred a bin-by-bin
re-weighting procedure that reproduced the constrained $\nu_{e}$ rates given
in hepdata.114859.v1-ml-t1 . We applied this same re-weighting procedure to
our B1 and B2 benchmarks to obtain their expected constrained $\nu_{e}$ rates,
shown in figure 12 (left panel). While we still could not obtain their
corresponding $p$-values without a covariance matrix, inspection of figure 12
(left) allows us to at least draw a qualitative conclusion from it: the
predictions from the quasi-sterile benchmark models are in significantly less
tension with MicroBooNE’s data than the empirical LEE model’s prediction.
Indeed, it is conceivable that these benchmarks are not incompatible with data
at a high enough confidence level to be excluded.

As for the inclusive Wire-Cell analysis, its public data release to date
hepdata.114862.v1 provided smearing tables to map true neutrino energy into
reconstructed neutrino energy hepdata.114862.v1-ml-t5 , as well as event
selection efficiencies as a function of true neutrino energy
hepdata.114862.v1-ml-t3 , which in principle would allow one to obtain the
unconstrained $\nu_{e}$ event rate for an arbitrary model. However, similarly
to the public data release for the CCQE Deep-Learning analysis, no tools were
provided to allow one to construct the covariance matrix for such an arbitrary
model, without which one cannot obtain the constrained $\nu_{e}$ event rate.
Furthermore, the event selection efficiencies provided in the public data
release were binned too coarsely in the low energy region to be useful.
Therefore, here too we resorted to a semi-heuristic bin re-weighting procedure
that reproduced the binned ratio of fully contained, constrained event rate
predictions hepdata.114862.v1-ml-t1 for the empirical LEE model relative to
the intrinsic $\nu_{e}$ backgrounds. In figure 12 (right panel) we show the
expected events rates in the inclusive Wire-Cell signal region obtained from
applying this semi-heuristic procedure to the B1 and B2 benchmarks. Again,
without the means to obtain the covariance matrices for these benchmark
models, we could not estimate a $p$-value for the B1 and B2 predictions shown
in figure 12 (right). However, our prior qualitative conclusion from the CCQE
Deep-Learning analysis is not contradicted here: the quasi-sterile oscillation
model can provide a good explanation for the MiniBooNE excess while being
consistent with the recent constraints from MicroBooNE. Further data will
hopefully make this situation less ambiguous.

###  4.2 Implications for solar neutrinos

The triple-resonance mechanism that explains the MiniBooNE excess should
considerably alter neutrino propagation and flavor conversion in the Sun.
Several compounding effects come into play. First, the quasi-sterile resonant
energies in the solar core are expected to fall in the range
$E^{\text{res}}_{\nu_{i}}\big{|}_{\odot}\sim\mathcal{O}(\rho_{\text{\tiny
MB}}/\rho_{\odot})\,E^{\text{res}}_{\nu_{i}}\big{|}_{\text{\tiny
MB}}\sim\mathcal{O}(\text{MeV})$, which is of the same magnitude as the
resonant energy of the standard MSW effect. Second, in the vicinity of the
resonant energies $E^{\text{res}}_{\nu_{1,2}}\big{|}_{\odot}$, the effective
active mass-squared splitting $\delta m_{21\text{,\,eff\,}}^{2}$ in the solar
medium will deviate by $\mathcal{O}(1-10)$ from the SM expectation, hence
altering the MSW effect itself. Third, the quasi-sterile neutrinos $S_{i}$
($i=1,2,3$) could have short mean-free-paths in the solar medium due to their
vector portal interactions with ordinary matter; this would cause decoherence
of superpositions of active and quasi-sterile neutrino states.

However, precise predictions can only be made once a concrete model is
specified for the dark vector mediator responsible for the BSM matter
potential. In particular, the quasi-sterile resonant energies in the solar
core will depend not only on the ratio of dirt-to-solar core densities,
$\rho_{\text{\tiny MB}}/\rho_{\odot}$, but also on the specific couplings of
the dark mediator to electrons, neutrons, and protons, since the relative
neutron-to-proton and nucleon-to-electron compositions in the solar core
differ by $\mathcal{O}(1)$ from the shallow Earth crust composition through
which neutrinos propagate in accelerator experiments. In addition, in order to
compute the quasi-sterile neutrinos’ mean-free-path in the solar medium, the
mediator’s mass and couplings are needed. Finally, the model-dependent solar
emission of quasi-sterile neutrinos could lead to scattering signals in
neutrino and dark matter experiments such as SK, SNO, Borexino, and XENON1T
(see also Harnik-ml-2012ni ; Cerdeno-ml-2016sfi ; Gann-ml-2021ndb ; Goldhagen-
ml-2021kxe ).

Our preliminary investigations indicate that is it possible to build a dark
sector UV completion that is experimentally viable and leads to predictions
compatible with existing solar neutrino data. This is however beyond the scope
of this work; a complete study of implications of this model to active and
quasi-sterile solar neutrinos will appear in a follow-up paper future .

##  5 Summary and outlook

An inevitable consequence of dark sector models with quasi-sterile neutrinos
are new matter effects in neutrino propagation through a medium, which can
lead to two generic effects. The first is resonant active-to-sterile neutrino
oscillations around a resonant energy $E^{\text{res}}_{\nu}$ determined by the
medium’s forward scattering potential. This effect occurs for either neutrinos
or antineutrinos, but not both, and the choice is determined by the sign of
the effective matter potential. The second effect, which affects both
neutrinos and antineutrinos, is an energy- and medium-dependent modification
of the active neutrino mass spectrum. With suitable parameter choices, this
modification may disappear in the asymptotic limits $E_{\nu}\ll
E^{\text{res}}_{\nu}$ and $E_{\nu}\gg E^{\text{res}}_{\nu}$, but it is
inevitable and increasingly pronounced as $E_{\nu}$ approaches the resonant
energy. Generic realizations of these effects are obviously not
phenomenologically viable given the swath of experimental measurements
targeting different neutrino flavors, energies, and baselines.

In this paper, we have studied a specific realization of a model with quasi-
sterile neutrinos and resonant oscillations that can address the short-
baseline $\nu_{\mu}\to\nu_{e}$ anomalies while preserving the active neutrino
mass spectrum in regimes probed by long-baseline experiments. We have also
argued that the recent $\nu_{e}$ measurements by MicroBooNE likely do not
exclude this model. Additional data from MicroBooNE expected in the near
future is unlikely to unambiguously resolve the initial tensions between
prediction and observations; however, in the longer term, the SBN program,
including MicroBooNE, SBND, and ICARUS, will be able to fully test our
proposed explanation of the MiniBooNE and LSND anomalies.

Further, while this model can accommodate non-negligible
$\stackon[-1.3pt]{$\bar{\nu}$}{\scalebox{0.25}{\hskip
2.0pt{{(}}}\scalebox{0.25}{\hskip 17.0pt {{)}}}}_{\\!e}$ disappearance
probabilities, it does not address the significant tensions between the
existing $\stackon[-1.3pt]{$\bar{\nu}$}{\scalebox{0.25}{\hskip
2.0pt{{(}}}\scalebox{0.25}{\hskip 17.0pt {{)}}}}_{\\!e}$ disappearance
measurements. Proposed experiments such as IsoDAR@Yemilab Alonso-ml-2021jxx ;
Alonso-ml-2021kyu will be rid of large systematic flux uncertainties, and will
be able to reconstruct the antineutrinos energy and propagated distance with
high precision on an event-by-event basis. As such, it should be able to
resolve the ambiguities and inconsistencies that currently plague the
$\stackon[-1.3pt]{$\bar{\nu}$}{\scalebox{0.25}{\hskip
2.0pt{{(}}}\scalebox{0.25}{\hskip 17.0pt {{)}}}}_{\\!e}$ disappearance
experimental landscape. Being an _antineutrino_ experiment, IsoDAR@Yemilab
will not be able to probe the resonant mechanism within the model proposed in
this study; nonetheless, it will be able to test this model through its non-
resonant $\bar{\nu}_{e}$ oscillation predictions.

The resonant mechanism of this quasi-sterile neutrino model should also have
significant consequences for solar neutrinos, and offer testable predictions
for active solar neutrino emission, as well as quasi-sterile solar neutrino
emission which could lead to distinct signals in dark matter and solar
neutrino experiments. A comprehensive study of the BSM solar neutrino
phenomenology in this context is deferred to a future publication future .

###### Acknowledgements.

DSMA benefited from conversations with the late Ann Nelson on model building
aspects of this study; Ann is sorely missed. DSMA also acknowledges the Aspen
Center for Physics (ACP) where this work was partially developed — the ACP is
supported by National Science Foundation grant PHY-1607611. We are grateful to
En-Chuan Huang for useful explanations regarding the MiniBooNE data, and Janet
Conrad and Mark Ross-Lonergan for discussions regarding the recent MicroBooNE
analyses. This work was supported by the Los Alamos National Laboratory LDRD
program, and by the DOE Office of Science High Energy Physics under contract
number DE-AC52-06NA25396.

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