# Short-baseline oscillation scenarios at JUNO and TAO

V. S. Basto-Gonzalez<EMAIL_ADDRESS>Universidad de Pamplona,
km 1, vía salida a Bucaramanga, Campus Universitario, 543050 Pamplona,
Colombia  Departamento de Física, Pontifícia Universidade Católica do Rio de
Janeiro, C. P. 38071, 22452-970, Rio de Janeiro, RJ, Brazil  D. V. Forero
<EMAIL_ADDRESS>Universidad de Medellín, Carrera 87 N° 30 - 65
Medellín, Colombia  C. Giunti<EMAIL_ADDRESS>Istituto Nazionale di
Fisica Nucleare (INFN), Sezione di Torino, Via P. Giuria 1, I–10125 Torino,
Italy  A. A. Quiroga<EMAIL_ADDRESS>Departamento de Física, Pontifícia
Universidade Católica do Rio de Janeiro, C. P. 38071, 22452-970, Rio de
Janeiro, RJ, Brazil  C. A. Ternes<EMAIL_ADDRESS>Istituto Nazionale di
Fisica Nucleare (INFN), Sezione di Torino, Via P. Giuria 1, I–10125 Torino,
Italy

###### Abstract

We study the sensitivity of JUNO and TAO to the oscillations induced by two
well-motivated scenarios beyond the standard model: Large Extra Dimensions
(LED) and light sterile neutrinos in the context of 3+1 neutrino mixing. We
find that JUNO+TAO can set competitive bounds on the parameter space of each
scenario. In particular, we find that JUNO+TAO can be competitive with MINOS,
DUNE or KATRIN in the context of LED. If LED are present in nature, we show
that the parameters could be measured with a similar precision as the standard
oscillation parameters. We also show that JUNO+TAO can test nearly all of the
parameter space preferred by Gallium experiments in the context of 3+1 mixing.
Finally, we discuss the possibility to distinguish the two scenarios from each
other.

###### Contents

  1. I Introduction
  2. II Formalism
    1. II.1 Neutrino oscillations in presence of Large Extra Dimensions
    2. II.2 Neutrino oscillations in presence of a light sterile neutrino
  3. III Details of analysis
    1. III.1 Far detector
    2. III.2 Near detector
    3. III.3 Statistical analysis
  4. IV Results
    1. IV.1 JUNO and TAO sensitivity to large extra dimensions
    2. IV.2 Measuring LED parameters
    3. IV.3 Light sterile neutrinos at JUNO and TAO
    4. IV.4 Distinguishing LED oscillations from light sterile neutrinos
  5. V Conclusions

##  I Introduction

Neutrino oscillations is the leading mechanism that successfully explains
neutrino flavor conversion. Neutrino oscillations have been observed by
experiments measuring neutrinos from the Sun, Earth’s atmosphere, nuclear
reactors, and particle accelerators. The existence of neutrino oscillations
implies that neutrinos are massive and mixed and provides a direct evidence of
physics beyond the standard model.

Currently, the best scenario that fits most of the observed neutrino
oscillation data requires three active neutrinos, with at least two of them
having different non-zero masses [1, 2, 3]. It should be noted that neutrino
oscillation experiments are not sensitive to the absolute neutrino mass scale,
which is expected to be measured by different means, i.e. using decay
experiments or cosmological observations (see Refs. [4, 5] for recent reviews
on this topic). Some of the parameters describing neutrino oscillations are
fairly well measured ($\Delta m^{2}_{21},|\Delta
m^{2}_{31}|,\sin^{2}\theta_{12},\sin^{2}\theta_{13}$), while the measurement
of others needs further improvement ($\delta_{CP},\sin^{2}\theta_{23}$, the
sign of $\Delta m^{2}_{31}$). The phase that encodes the possibility of
leptons violating the Charge-Parity (CP) symmetry (the so call Dirac CP phase)
has been measured by T2K [6] and NOvA [7]. However, there is some disagreement
in the results, as can be seen in Fig. 6 of Ref. [7]. It is also not clear if
the true value of the atmospheric angle $\theta_{23}$ lies in the first or
second octant, or if it is exactly maximal. The last unknown is the neutrino
mass ordering. From the current data it is not clear if $\Delta m_{31}^{2}>0$
or $\Delta m_{31}^{2}<0$.

Future neutrino oscillation experiments such as DUNE [8], Hyper-Kamiokande
[9], ORCA [10], and JUNO [11, 12, 13], are expected to improve the current
precision of the neutrino oscillation parameters and address the remaining
unknowns discussed before. To fulfill these goals, large statistics and an
unprecedented control of systematic uncertainties have to be achieved. The
expected precision provides a unique avenue to probe different beyond the
standard model scenarios that might produce sub-leading effects on top of the
well established three-neutrino oscillation pattern.

Here, we focus on the next generation reactor experiment JUNO (the Jiangmen
Underground Neutrino Observatory) and its near detector TAO (the Taishan
Antineutrino Observatory). JUNO in combination with TAO aims to measure very
precisely several of the neutrino oscillation parameters
($\sin^{2}(\theta_{12}),\Delta m^{2}_{21},|\Delta m^{2}_{31}|$) and the
neutrino mass ordering (see Refs. [14, 15, 16, 17, 11, 18, 19, 20]). Apart
from this, a very rich program of new physics searches will be possible using
JUNO (see Refs. [21, 11, 22, 23, 24, 25, 26, 27, 28, 29, 30, 12, 31, 32, 33]).
In this paper we discuss the sensitivity of JUNO and TAO to oscillation
effects induced in two models of physics beyond standard three-neutrino
mixing: Large Extra Dimensions (LED) and a light sterile neutrino (3+1). In
particular, we discuss the sensitivity of JUNO and TAO to the parameters of
the two models.

The paper is structured as follows: in Section II we introduce the formalism
for LED and 3+1 oscillations. In Section III we discuss the details of our
simulations. Our main results are presented in Section IV. Finally, we
summarize and draw our conclusions in Section V.

##  II Formalism

In this section we discuss the relevant formalisms for LED and 3+1 neutrino
oscillations. The results of this section serve as physics input in our
numerical analyses discussed in the next sections.

###  II.1 Neutrino oscillations in presence of Large Extra Dimensions

The Kaluza-Klein (KK) theory, in the contexts of Large Extra Dimensions, was
introduced in order to solve the hierarchy problem between the electroweak and
the Planck scale [34, 35, 36]. LED models incorporate neutrino masses [37, 38]
by allowing standard model gauge singlets, such as gravitons or right-handed
neutrinos, to propagate in a higher-dimensional bulk of extra dimensions while
restricting the Standard Model (SM) particles to propagate in our four-
dimensional brane. In these models, neutrinos acquire a Dirac mass term, which
is strongly suppressed by the volume of the Large Extra Dimensions. Thus, the
same mechanism which explains the gap between the electroweak and Planck
scales generates naturally small neutrino masses.

In the LED model, the SM is extended by adding three five-dimensional***It is
assumed that the compactification manifold is asymmetric, i.e., that one of
the extra dimensions is much larger than the sizes of the other dimensions,
and is compactified on a circle of radius $R_{ED}$ [39]. The effective
treatment is therefore five-dimensional. fermions
$\Psi^{\alpha}=(\psi_{L}^{\alpha},\psi_{R}^{\alpha})$, each associated to one
active neutrino with flavor $\alpha$. From our four-dimensional point of view,
these fields can be expressed as infinite towers of Kaluza-Klein modes,
$\psi^{\alpha(n)}_{L,R}$, for $n=0,\pm 1,\pm 2,\ldots$ Redefining the fields
as

$\displaystyle\nu_{R}^{\alpha(0)}$ $\displaystyle=$
$\displaystyle\psi_{R}^{\alpha(0)}\,,$ $\displaystyle\nu_{L,R}^{\alpha(n)}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\psi_{L,R}^{\alpha(n)}+\psi_{L,R}^{\alpha(-n)}\right),\quad\text{for}\quad
n=1,\ldots,\infty\,,$ (1)

the mass term in the Lagrangian can be written as [37, 40, 38, 39]

$\displaystyle\mathcal{L}=m^{D}_{\alpha\beta}\left(\overline{\nu}_{R}^{\alpha(0)}\nu_{L}^{\beta}+\sqrt{2}\sum_{n=0}^{\infty}\overline{\nu}_{R}^{\alpha(n)}\nu_{L}^{\beta}\right)+\sum_{n=1}^{\infty}\frac{n}{R_{ED}}\overline{\nu}_{R}^{\alpha(n)}\nu_{L}^{\alpha(n)}+h.c.\,,$
(2)

where $m^{D}_{\alpha\beta}$ are the entries of the Dirac mass matrix and
$R_{ED}$ is the compactification radius of the extra dimensions. Note that
repeated indices are summed over. The diagonalization of this mass term can be
performed in two steps [39]. First diagonalize the Dirac mass term using
$\operatorname{diag}(m_{1}^{D},m_{2}^{D},m_{3}^{D})=R^{\prime}{}^{\dagger}m^{D}U$
with the redefinitions

$\displaystyle\nu_{L}^{\alpha}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}U_{\alpha i}\nu_{L}^{i(0)}\,,$
$\displaystyle\nu_{R}^{\alpha(0)}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}R^{\prime}_{\alpha i}\nu_{R}^{i(0)}\,,$
$\displaystyle\nu_{L,R}^{\alpha(n)}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}R^{\prime}_{\alpha i}\nu_{L,R}^{i(n)},\quad
n=1,2,\ldots,\infty\,.$ (3)

The mass term can be written now as

$\mathcal{L}=\sum_{i=1}^{3}\overline{\nu}_{R}^{i}M_{i}\nu_{L}^{i}+h.c.\,,$ (4)

where

$\nu_{L,R}^{i}=(\nu_{L,R}^{i(0)},\nu_{L,R}^{i(1)},\ldots)^{T},\qquad
M_{i}=\begin{pmatrix}m_{i}^{D}&0&0&\ldots\\\
\sqrt{2}m_{i}^{D}&1/R_{ED}&0&\ldots\\\ \sqrt{2}m_{i}^{D}&0&2/R_{ED}&\ldots\\\
\vdots&\vdots&\vdots&\ddots\end{pmatrix}$ (5)

are infinite-dimensional vectors and matrices. In a last step we diagonalize
the $M_{i}$ using infinite-dimensional matrices $L_{i}$ and $R_{i}$. The mass
states are then given by $\nu_{L}^{i}{}^{\prime}=L_{i}^{\dagger}\nu_{L}^{i}$
and $\nu_{R}^{i}{}^{\prime}=R_{i}^{\dagger}\nu_{R}^{i}$. Here we are
interested in the decomposition of the standard flavor neutrinos
$\nu_{L}^{\alpha}$, which becomes now

$\nu_{L}^{\alpha}=\sum_{i=1}^{3}U_{\alpha
i}\sum_{n=0}^{\infty}L_{i}^{0n}\nu_{L}^{i(n)}{}^{\prime}\,,$ (6)

where the relevant entries of $L_{i}$ are given by [37, 41, 42]

$\left(L_{i}^{0n}\right)^{2}=\frac{2}{1+\pi^{2}\left(m_{i}^{D}R_{ED}\right)^{2}+\left(\lambda_{i}^{(n)}\right)^{2}/\left(m_{i}^{D}R_{ED}\right)^{2}}\,,$
(7)

with $n=0,1,2,\ldots,\infty$. In this equation, the $\lambda_{i}^{(n)}$ are
the eigenvalues of $R_{ED}^{2}M_{i}M_{i}^{\dagger}$ and can be obtained from
solving the equation

$\lambda_{i}^{(n)}-\pi\left(m_{i}^{D}R_{ED}\right)^{2}\cot\left(\pi\lambda_{i}^{(n)}\right)=0\,.$
(8)

The neutrino oscillation probability is then given by

$P(\nu_{\alpha}\rightarrow\nu_{\beta})=\left|\sum_{i=1}^{3}\sum_{n=0}^{\infty}U_{\alpha
i}U_{\beta
i}^{*}\left(L_{i}^{0n}\right)^{2}\exp\left(-i\frac{\left(\lambda_{i}^{(n)}\right)^{2}L}{2ER_{ED}^{2}}\right)\right|^{2}\,,$
(9)

where we used $\nu_{\alpha}$ as abbreviation for the mostly active neutrino
$\nu_{L}^{\alpha}$. Here, $E$ and $L$ are the neutrino energy and the distance
traveled by the neutrino. From this formula it becomes clear that the standard
mass splittings, characterizing the solar and atmospheric oscillations are
given by [43, 44, 45, 46]

$\Delta
m_{j1}^{2}=\frac{\left(\lambda_{j}^{0}\right)^{2}-\left(\lambda_{1}^{0}\right)^{2}}{R_{ED}^{2}}\,.$
(10)

It is important to note that

$\Delta
m_{j1}^{2}\neq\left(m_{j}^{D}\right)^{2}-\left(m_{1}^{D}\right)^{2}\,.$ (11)

If this was the case, the LED parameters would also affect the main
oscillations. Instead, the LED parameters can only induce new oscillation
lengths on top of the regular $1-3$ and $1-2$ oscillations. Note that this
method limits the number of effective LED parameters to 2, the
compactification radius $R_{ED}$ and the lightest Dirac mass term
$m_{0}=m_{1}^{D}(m_{3}^{D})$ for normal (inverted) neutrino mass ordering. For
a given $R_{ED}$ and $m_{0}$ we use Eq. (8) to calculate
$\lambda_{1}^{(0)}(\lambda_{3}^{(0)})$ for normal (inverted) ordering. Next,
we use Eq. (10) to calculate the remaining two $\lambda_{i}^{(0)}$, and then,
substituting them into Eq. (8), we get the other two $m_{i}^{D}$. With all
three $m_{i}^{D}$ given, we obtain each $\lambda_{i}^{(n)}$ with $n>1$ by
solving Eq. (10) (see Ref. [43] for more details on this procedure). Note that
even though the sum is infinite, we have checked that taking $n$ up to
$n_{\text{max}}=4$ is enough for obtaining a convergent behavior for the
experiments considered in this analysis.

Neutrino oscillations with Large Extra Dimensions have been studied in the
context of solar [39], atmospheric [47], accelerator [48, 44, 49, 45, 46, 50]
and reactor neutrinos [51, 52, 53]. Interestingly, in Ref. [53] the authors
showed that the reactor and Gallium anomalies can be resolved with LED-induced
oscillations. We will come back to this in the next section.

###  II.2 Neutrino oscillations in presence of a light sterile neutrino

Light sterile neutrinos have been a hot topic in neutrino physics since many
years. Their presence has been motivated by several anomalies which appeared
at short-baseline neutrino oscillation experiments.

Light sterile neutrinos at the eV mass scale might explain the anomalous
appearance of electron neutrinos and antineutrinos in LSND [54] and MiniBooNE
[55, 56], although the recent results of the MicroBooNE experiment [57, 58,
59] disfavor the $\nu_{\mu}\to\nu_{e}$ interpretation of the MiniBooNE low-
energy anomaly (see, however, the caveats discussed in Ref. [60]). It is also
interesting that the MicroBooNE data may be interpreted as an indication of
short-baseline $\nu_{e}$ disappearance [61].

The deficit obtained in Gallium experiments [62, 63, 64, 65] might be due to
light sterile neutrinos. Note, however, that the values of the mixing
parameters required to explain this anomaly are in tension with several other
bounds (see the discussions in Refs. [66, 67, 68]).

The deficit observed in the rates of reactor experiments, the reactor
antineutrino anomaly [69], could be explained with light sterile neutrinos,
too. Note, however, that this anomaly might also be due to a misunderstanding
of the reactor neutrino fluxes [70, 67].

Recently, a strong indication in favor of such oscillations has been claimed
by the Neutrino-4 collaboration [71, 72, 73]. However, this result is
controversial and has been criticized in Refs. [74, 75, 76].

As one can see, there is a lot of interest in light sterile neutrinos. For
recent reviews we refer the reader to Refs. [77, 78, 79, 80]. In the 3+1
framework it is assumed that there is a non-standard massive neutrino
$\nu_{4}$ with mass $m_{4}\gtrsim 1\,\text{eV}$ such that
$m_{1},m_{2},m_{3}\ll m_{4}$, where $m_{1},m_{2},m_{3}$ are the masses of the
three standard massive neutrinos $\nu_{1},\nu_{2},\nu_{3}$. In the flavor
basis, besides the three standard active neutrinos
$\nu_{e},\nu_{\mu},\nu_{\tau}$ there is a sterile neutrino $\nu_{s}$, and the
mixing is given by

$\nu_{\alpha}=\sum_{j=1}^{4}U_{\alpha j}\nu_{j}\qquad(\alpha=e,\mu,\tau,s),$
(12)

where $U$ is the $4\times 4$ unitary mixing matrix. The effective short-
baseline (SBL) survival probability of electron neutrinos and antineutrinos is
given by

$P_{ee}^{\text{SBL}}=1-\sin^{2}\\!2\theta_{ee}\,\sin^{2}\\!\left(\dfrac{\Delta{m}^{2}_{41}L}{4E}\right),$
(13)

where $\Delta{m}^{2}_{41}=m_{4}^{2}-m_{1}^{2}$ and
$\sin^{2}\\!2\theta_{ee}=4|U_{e4}|^{2}(1-|U_{e4}|^{2})$, where $\theta_{ee}$
is an effective mixing angle that coincides with $\theta_{14}$ in the standard
parameterization of the mixing matrix. At longer baselines the oscillations
due to $\Delta m_{41}^{2}$ are washed out and the neutrino oscillation
probability is modified to

$P_{ee}^{\text{LBL}}=(1-|U_{e4}|^{2})^{2}P_{ee}^{3\times 3}+|U_{e4}|^{4},$
(14)

where $P_{ee}^{3\times 3}$ is the standard three-neutrino oscillation
probability.

##  III Details of analysis

In this section we describe the simulation and analysis details used in this
paper. We basically follow the descriptions of Refs. [81, 82, 13] and [12, 13]
for the far and near detector, respectively. Throughout the paper we assume an
exposure time of eight effective years (2400 days) at both detectors†††An
exposure of eight years with eight reactors is approximately equivalent to the
exposure of six years with ten reactors at the far detector, previously used
in many JUNO sensitivity analyses..

###  III.1 Far detector

We assume the eight reactors configuration from Ref. [82], since two of the
originally considered Taishan cores might only become operative in a later
stage of the experiment [12]. Therefore, the main signal at JUNO will come
from the two 4.6 GW reactor cores at the Taishan power plant and the six 2.9
GW reactors at the Yangjiang power plant. We also include contributions from
Daya Bay and Huizhou, at 215 km and 265 km distance. We simplify both
contributions as single reactors with 17.4 GW thermal power [81].

The fluxes for each reactor core are parameterized as in Refs. [83, 84]. The
fission fractions [12] are assumed to be the same for all reactors and are
kept fixed at: $f_{235}=0.561$, $f_{238}=0.076$, $f_{239}=0.307$,
$f_{241}=0.056$.

Regarding the detector, we assume a fiducial mass of 20 kt and 82% selection
efficiency [13]. The cross section is taken from Ref. [85]. Using this
criteria we expect 49 (5) inverse beta decay (IBD) events per day from the
main reactors (Daya Bay and Huizhou). For the analysis we also include an
energy resolution of 3%. Apart from the background contributions from Daya Bay
and Huizhou which depend on the oscillation parameters, we also include
accidental, fast neutron 9Li/8He, $\alpha-$n and geo-neutrino background
components, which we have extracted from Refs. [81, 13] and normalized to an
exposure of 8 years. Hence, the number of signal-events in the far detector is
computed in the following way. The probability-weighted flux is given by

$\Phi_{\text{osc}}(E)=\sum_{\text{cores}}\frac{W_{\text{th,core}}}{L_{\text{core}}^{2}}P_{ee}(E,L_{\text{core}})\Phi_{0}(E)\,,$
(15)

where $E$ is the neutrino energy, $W_{\text{th}}$ is the thermal power of the
core, $L_{\text{core}}$ is the core-detector distance and $\Phi_{0}(E)$ is the
flux from the reactor, where the contribution from each isotope is weighted
with the above stated fission fractions and $P_{ee}(E,L)$ is the
$\bar{\nu}_{e}$ survival probability. The number of signal events in the
energy bin $k$ is then

$N_{k}=\mathcal{N}\int
dE\int_{E_{k,\text{min}}}^{E_{k,\text{max}}}dE^{\prime}\Phi_{\text{osc}}(E)\sigma_{\text{IBD}}(E)R(E,E^{\prime})\,$
(16)

where the normalization constant $\mathcal{N}$ contains efficiencies, detector
size and exposure time, $\sigma_{\text{IBD}}(E)$ is the cross section,
$E^{\prime}$ the reconstructed energy and

$R(E,E^{\prime})=\frac{1}{\sqrt{2\pi}\sigma_{E}}\exp\left(-\frac{\left(E-E^{\prime}\right)^{2}}{2\sigma_{E}^{2}}\right)$
(17)

is used for energy smearing with the resolution given by

$\sigma_{E}=\alpha\sqrt{E/\text{MeV}-0.78}~{}\text{MeV}$ (18)

with $\alpha=3\%$. The total number of events is obtained by adding signal and
background contributions.

###  III.2 Near detector

The computation of events at the near detector is similar, but with a few
important differences. We assume the signal from all reactors except the
nearest one to be negligible. This is a valid simplification, because the
contribution from the second Taishan reactor is supposed to be at the
$\mathcal{O}(1\%)$ level [12], while the contribution from other backgrounds
is at the $\mathcal{O}(10\%)$ level. Due to the short baseline, we cannot
assume pointlike source and detector anymore, but we must average over the
dimensions of the reactor (which is larger than the detector). We assume a
fiducial mass of 1 t and an energy resolution of 1.5% following Refs. [12,
13]. We assume the near detector to be positioned at 30 m from the core and
expect to observe 2700 IBD events per day in the absence of short baseline
oscillations. We include accidental, fast neutron and 9Li/8He background
components. The spectral forms are extracted from Ref. [81], but normalized to
produce 190, 200 and 54 events per day [12] for the three background
components. The background/signal ratio that we obtain with these parameters
is $\sim 16\%$ in the absence of short baseline oscillations. Due to the
vertexing capabilities of TAO, and for the purpose of this analysis we
consider the near detector to be divided in two segments with fiducial masses
of 0.5 t. The spherical detector is split in half along the direction of
propagation with the two average baselines given by $(30\pm 3/8\times 0.65)$ m
[86].

###  III.3 Statistical analysis

In our analyses, we use a standard $\chi^{2}$ test hypothesis to quantify our
results. For a given set of oscillation parameters $\vec{p}$ it is given by

$\chi^{2}_{D}(\vec{p},\vec{\alpha})=\sum_{i=1}^{200}\frac{(N^{D}_{i}-T^{D}_{i}(\vec{p},\vec{\alpha}))^{2}}{N^{D}_{i}}\,,$
(19)

where $D=N_{1},N_{2}$ for the near detector segments and $D=F$ for the far
detector. Here, $N^{D}_{i}$ is the fake data spectrum and $T^{D}_{i}$ the
prediction for the oscillation parameters $\vec{p}$. It also depends on the
systematic uncertainties $\vec{\alpha}$. In both quantities the background
components are already included. In our analyses, we include several sources
of systematic uncertainties which are penalized by

$\chi^{2}_{\text{sys.}}(\vec{\alpha})=\sum_{k}\frac{(\alpha_{k}-\mu_{k})^{2}}{\sigma_{k}^{2}}\,,$
(20)

where $\mu_{k}$ and $\sigma_{k}$ are the expectation values and standard
deviations for each systematic uncertainty $\alpha_{k}$. The uncertainties
that we consider are:

  * •

A 0.8% normalization of the thermal power of each reactor.

  * •

A 2% overall normalization of the flux.

  * •

A 0.13% normalization of the selection efficiency.

  * •

A shape uncertainty in each bin. We will vary the penalty on this quantity.

Flux-related uncertainties are correlated among near and far detector, while
the selection efficiencies are not. Therefore, the whole $\chi^{2}$ function
is given by

$\chi^{2}(\vec{p})=\min_{\vec{\alpha}}\left\\{\chi^{2}_{N_{1}}(\vec{p},\vec{\alpha})+\chi^{2}_{N_{2}}(\vec{p},\vec{\alpha})+\chi^{2}_{F}(\vec{p},\vec{\alpha})+\chi^{2}_{\text{sys.}}(\vec{\alpha})\right\\}\,.$
(21)

It has been shown [51] that the $\theta_{13}$ measurement is fairly robust
under LED. Therefore, when we marginalize over the reactor angle, we include
the external penalty $\theta_{13}=(8.53\pm 0.13)\degree$ [1].

In the context of 3+1 neutrinos, the oscillations due to $\Delta m_{41}^{2}$
are averaged out for the baselines relevant for Daya Bay, RENO and Double
Chooz, if $\Delta m_{41}^{2}\gtrsim 0.1$ eV2. We focus on these values of
$\Delta m_{41}^{2}$, and hence we can impose a prior on $\theta_{13}$ using
the value which has been obtained taking ratios between the near and far
detectors of these experiments.

##  IV Results

In this section we present the main results of this paper on the sensitivity
of JUNO + TAO to LED and light sterile neutrinos. In Subsection IV.1 we
examine the sensitivity to the LED parameters that can be reached at JUNO+TAO.
In Subsection IV.2 we select a point in the LED parameter space that is
consistent with the reactor and Gallium anomalies (in the context of LED) and
we show that JUNO+TAO can measure these values of the LED parameters with very
good precision. In Subsection IV.3 we present the combined JUNO+TAO
sensitivity to 3+1 neutrino oscillation parameters. Finally, in Subsection
IV.4 we discuss if JUNO+TAO can distinguish the LED scenario from 3+1 neutrino
oscillations.

###  IV.1 JUNO and TAO sensitivity to large extra dimensions

We first estimate the sensitivity of JUNO+TAO to the LED parameters. For this
purpose, we create a fake set of data points without LED oscillations using
the following values of the standard neutrino oscillation parameters for
normal (NO) and inverted (IO) neutrino mass ordering [1]:

$\displaystyle\sin^{2}\theta_{12}$ $\displaystyle=$ $\displaystyle
0.318,\quad\sin^{2}\theta_{13}=0.022,\quad\Delta m_{21}^{2}=7.5\times
10^{-5}~{}\text{eV}^{2},$ $\displaystyle\Delta m_{31}^{2,\text{NO}}$
$\displaystyle=$ $\displaystyle 2.55\times
10^{-3}~{}\text{eV}^{2}\,,\quad\Delta m_{31}^{2,\text{IO}}=-2.45\times
10^{-3}~{}\text{eV}^{2}\,.$ (22)

The analysis of the simulated data is performed by considering the values of
the standard oscillation parameters as variables determined by the fit. The
sensitivity to the LED parameters is obtained by marginalizing over the
standard oscillation parameters.

Let us comment on the different implications of the normal and inverted
orderings for the short-baseline oscillations of reactor neutrinos in the LED
framework. For normal ordering, we have that
$\lambda_{1}^{(0)},\lambda_{2}^{(0)}\ll\lambda_{3}^{(0)}$, and also
$m_{1}^{D},m_{2}^{D}\ll m_{3}^{D}$. Therefore, since for $m_{i}^{D}R_{ED}\ll
1$ we have $L_{i}^{0k}\simeq\sqrt{2}m_{i}^{D}R_{ED}/k$ for $k\geq 1$ [39], the
dominant LED contribution to the neutrino oscillation probability comes from
the term with $i=3$ in Eq. (9), which is suppressed by the smallness of
$|U_{e3}|^{2}$. If one assumes inverted ordering, however, the dominant LED
contributions appear in the terms with $i=1$ and $i=2$, which are not
suppressed. Therefore, the LED oscillation effects for fixed $R_{ED}$ and
$m_{0}$ can induce a much larger deviation from standard 3-neutrino
oscillations for inverted ordering than for normal ordering. This translates
to better sensitivities to the LED parameters for inverted ordering compared
to the ones for normal ordering.

The results for several sets of systematic uncertainties are shown in the left
panel of Fig. 1 for normal (solid lines) and inverted ordering (dashed lines).
As anticipated, the results for inverted ordering are very strong. Moreover,
the result hardly depends on the flux uncertainty. We find that in the case of
inverted ordering, JUNO can provide the strongest bound on $R_{ED}$. The bound
is stronger than in any current experiment‡‡‡Note that experiments measuring
$\nu_{\mu}\to\nu_{\mu}$ oscillations depend on $|U_{\mu i}|^{2}$, where all
elements are large, and hence the bounds for these experiments are very
similar for both orderings., and also better than expectations from other
future experiments, see right panel of Fig. 1. The only experiment which can
set competitive bounds for inverted ordering is KATRIN. From here on we turn
our focus to the discussion of normal ordering. The blue line in the left
panel of Fig. 1 is obtained including the first three systematic uncertainties
in the list in Section III.3 and is the minimal set of uncertainties that we
consider in this paper. We also performed several analyses with different flux
shape uncertainties. This systematic has a very important effect on the
sensitivity, as can be seen in the figure. The green, magenta and orange lines
are obtained using a bin-to-bin shape uncertainty of 1%, 1.5% and 3%,
respectively. The black line is obtained leaving this systematic uncertainty
completely free.

One can see from the left panel of Fig. 1 that $m_{0}$ is unbounded for very
small values of $R_{ED}$. This is expected, since in the limit $R_{ED}\to 0$
the Dirac mass becomes equivalent to the physical neutrino mass and we recover
standard neutrino oscillations which do not depend on the overall neutrino
mass scale.

Considering the case with free (no) shape uncertainty, at 90% C.L. for 1
degree of freedom ($\Delta\chi^{2}=2.71$), the LED compactification radius
$R_{ED}$ has a sensitivity upper bound of $0.87~{}\mu\text{m}$
($0.35~{}\mu\text{m}$). The bounds for the different systematic uncertainties
considered in this section are summarized in Tab. 1 for normal and inverted
mass ordering. These results show again the importance of the systematic
uncertainties for probing the LED parameters in the case of normal ordering.
Notice however that, for the intermediate cases, the sensitivity to $R_{ED}$
does not significantly change when the shape uncertainty is larger than 3%.
Within this range lies a realistic estimate for the flux-related uncertainties
that is of the order of $\sim 5\%$. In what follows, and for comparison
purposes, we take as a reference the case with 3% bin-to-bin uncorrelated
uncertainty.

Case | no shape unc. | 1.0% shape unc. | 1.5% shape unc. | 3.0% shape unc. | free shape unc.
---|---|---|---|---|---
Bound (90% C.L.), NO | 0.35 $\mu$m | 0.57 $\mu$m | 0.65 $\mu$m | 0.81 $\mu$m | 0.87 $\mu$m
Bound (90% C.L.), IO | 0.115 $\mu$m | 0.128 $\mu$m | 0.134 $\mu$m | 0.142 $\mu$m | 0.151 $\mu$m
Table 1: JUNO+TAO 90% C.L. sensitivity upper bound (1 degree of freedom) on
the compatification radius $R_{ED}$ for each set of systematic uncertainties
considered in Section IV.1.

In the right panel of Fig. 1, we compare the JUNO+TAO sensitivity
corresponding to $3\%$ shape uncertainty (orange lines) with the estimated
sensitivities of KATRIN [43] (blue lines), DUNE [87, 50] (purple line; see
also Ref. [44]), and with the sensitivity of MINOS [48] (red line). Note that
the bound obtained by the MINOS collaboration from the data is much stronger
than the sensitivity [48]. This is due to a fluctuation of the data. Since we
are discussing experimental sensitivities, it is appropriate to compare the
JUNO+TAO sensitivity with that of MINOS, not with the MINOS bound obtained
from the data. When the JUNO+TAO data will appear, the resulting bound on the
LED parameters will depend on the fluctuation of the data and it will be
appropriate to compare it with the MINOS bound obtained from the data.

As can be seen in Fig. 1, JUNO+TAO can set competitive bounds on the LED
parameters, which, depending on the treatment of systematic uncertainties, can
be more stringent than those from the other experiments. For the optimistic
case of $1\%$ flux-shape systematic uncertainty, the JUNO+TAO sensitivity is
compatible with the DUNE sensitivity showing a synergy of future reactor and
accelerator neutrino experiments.

Figure 1: Left: JUNO+TAO 90% C.L. sensitivity in the $R_{ED}$-$m_{0}$ plane
for several choices of systematic uncertainties. The solid and dashed lines
correspond, respectively, to normal and inverted ordering. Right: Comparison
of the JUNO+TAO sensitivity with those of KATRIN [43], DUNE [87, 50], and
MINOS [48].

###  IV.2 Measuring LED parameters

In this subsection we discuss how well JUNO+TAO can measure values of the LED
parameters that are consistent with a LED explanation of the reactor and
Gallium anomalies [53]. For this purpose, we generate a simulated data set
with the same values of the standard oscillation parameters (focusing here
only on normal ordering) considered in the previous subsection (see Eq. (22))
and the following specific values of the LED parameters:

$\displaystyle R_{ED}$ $\displaystyle=$ $\displaystyle 0.5~{}\mu\text{m}\,,$
(23) $\displaystyle m_{0}$ $\displaystyle=$ $\displaystyle
0.03~{}\text{eV}\,.$ (24)

We assume the neutrino mass ordering to be normal. Using this benchmark point
one can explain the reactor antineutrino anomaly [69] and the Gallium neutrino
anomaly [62, 63, 64, 65] as a result of LED oscillations [53].

As can be seen in Fig. 2, relatively small closed contours at 2$\sigma$ can be
obtained with this benchmark point. Indeed, the expected sensitivity to
measure the LED parameters is at a similar level ($\lesssim 1\%$) as the
expected sensitivity to the standard neutrino oscillation parameters [13]. The
JUNO+TAO sensitivity with $3\%$ shape uncertainty (full green line) is also
shown in Fig. 2 in comparison with the allowed region from the reactor and
Gallium anomalies, extracted from Ref. [53] (blue line).

We considered for illustration only one point in the region of the LED
parameters inside of the blue contour in Fig. 2, that is motivated by the LED
explanation of the reactor and Gallium anomalies [53]. However, it should be
noted that basically all of the region inside of the blue contour lies within
the sensitivity reach of JUNO+TAO. Therefore, it is plausible that a similar
precision of the JUNO+TAO determination of the LED parameters can be obtained
for any benchmark point compatible with the LED explanation of the reactor and
Gallium anomalies.

Figure 2: 2$\sigma$ regions obtained from the analysis of fake data in
presence of LED oscillations for a several assumptions on the flux shape
uncertainty (red, orange and black lines), in comparison with the region
allowed from reactor and Gallium anomalies (blue). We also show the JUNO+TAO
sensitivity (green line) to emphasize that JUNO+TAO can test all of the values
preferred by the reactor and Gallium anomalies.

###  IV.3 Light sterile neutrinos at JUNO and TAO

The JUNO and TAO detectors can also be used to measure or bound active-sterile
neutrino oscillation parameters [13, 86]. In this subsection we explore the
sensitivity to $\sin^{2}2\theta_{14}$ and $\Delta m_{41}^{2}$ in the 3+1
neutrino mixing framework. The analysis follows the description given above,
with the only difference that now the LED neutrino oscillation probability is
replaced by the oscillation probability describing 3+1 neutrino mixing.

The results of our analyses are shown in the left panel of Fig. 3. We show the
sensitivity expected at 2$\sigma$ (dashed) and 3$\sigma$ (solid) confidence
level for different choices of the flux shape uncertainty, as we have done in
Section IV.1 for LED-induced oscillations.

Note that our results on the sensitivity differ from those obtained in Ref.
[86], particularly for very small and large values of $\Delta m_{41}^{2}$.
This is due to the fact that there are crucial differences in the analyses.
The total exposure assumed in this paper is larger than that assumed in Ref.
[86]. On the other hand, our treatment of backgrounds is more conservative.
Most importantly, the authors of Ref. [86] studied the TAO sensitivity alone,
without taking into account the contribution from the far detector. The same
argument applies when comparing with the results obtained by the TAO
collaboration in Ref. [12]. The addition of the far detector allows to bound
the mixing also for small and large values of $\Delta m_{41}^{2}$. It should
be also noted, that the treatment of the shape uncertainty in Ref. [12]
slightly differs from ours.

In the left panel of Fig. 3, we also show the preferred region obtained from
the combined analysis of the Gallium experiments [66], where of particular
importance are the recent results of the BEST experiment [66]. We find that
JUNO+TAO can exclude nearly all of the parameter space preferred by the
Gallium experiments. Indeed, only for the shape-free analysis there is some
overlap at 3$\sigma$ between the Gallium and JUNO+TAO regions for $\Delta
m_{41}^{2}\gtrsim 10\,\text{eV}^{2}$. Note that most of the parameter space
preferred from the analysis of Neutrino-4 data [73, 76], which is in agreement
with the BEST results, could be therefore also ruled out by JUNO+TAO. Also, a
large part of the parameter space allowed by the analysis in Ref. [61] of the
MicroBooNE data [57, 58, 59] in terms of $\nu_{e}$ disappearance is within the
sensitivity reach of JUNO+TAO.

It must be mentioned that the bound on $\sin^{2}2\theta_{14}$ at large values
of $\Delta m_{41}^{2}$ is due to the fact that we assume a 2% total
normalization uncertainty on the flux. If we relax this assumption and leave
the normalization free, we obtain the solid lines in the right panel of Fig.
3, where we also show (dashed lines) the bounds using our nominal 2% penalty
(these are the same as in the left panel). Note that even with a free
normalization and free shape uncertainty JUNO+TAO can test the Gallium region
for $\Delta m_{41}^{2}\lesssim 10-20$ eV2.

Figure 3: Left: The 2$\sigma$ (dashed) and 3$\sigma$ (solid) sensitivity to
sterile oscillation parameters obtained in the analysis of JUNO+TAO. We also
show the region preferred by Gallium experiments in blue [66]. Right: Impact
of the overall flux normalization on the sensitivity. The dashed lines
correspond to the 2$\sigma$ sensitivity curves in the left panel, while the
solid lines are obtained leaving the flux normalization free.

###  IV.4 Distinguishing LED oscillations from light sterile neutrinos

Finally, in this subsection we investigate if JUNO+TAO can distinguish LED
oscillations from 3+1 oscillations. For this purpose, we generate fake data
sets consistent with LED oscillations. We consider several benchmark points in
the LED parameter space, summarized in the first column of Table 2 and
graphically indicated in the left panel of Fig. 4. We take points inside and
outside of the sensitivity regions of JUNO+TAO and some points inside the
region preferred by the LED explanation of the reactor and Gallium anomalies
[53]. In the right panel of Fig. 4, we plot the minimal $\Delta\chi^{2}$ value
obtained for each analysis as a function of the shape uncertainty. As one can
see, the effect of the shape uncertainty saturates for values between
approximately 3% and 8%, depending on the benchmark. We have checked that even
for larger values of the shape uncertainty no significant change of
$\Delta\chi^{2}$ can be observed. This is a result of our two detector
analysis. Since the flux uncertainties are correlated among detectors (and
detector segments) the effect of the flux uncertainty saturates at some point,
and increasing the uncertainty has no further effect anymore. In the last two
columns of Table 2 we report the values of the 3+1 oscillation parameters for
which the minimal $\Delta\chi^{2}$ has been found in the case of 10% shape
uncertainty. Interestingly, we find that the minimal $\Delta\chi^{2}$ values
span several orders of magnitude, as shown by the second column of Table 2 and
the right panel of Fig. 4.

We find that $\Delta\chi^{2}_{\text{min}}$ is very small in the cases of the
red and green benchmark points. The red point lies outside of the sensitivity
region of JUNO+TAO, indicated in the left panel of Fig. 4 by the red, green,
and black lines for negligible, 1%, and free flux shape uncertainties,
respectively (see the discussion in Section IV.1). Therefore, in this case the
simulated data are not very different from those given by standard 3-neutrino
oscillations. Indeed, they are best fitted within the 3+1 scenario with small
values of $\sin^{2}2\theta_{14}$. The green point lies within the sensitivity
region of JUNO+TAO if the shape uncertainties are negligible or small (1%),
but it is outside of the sensitivity region for free shape uncertainties.
Therefore, in this case the data are not fitted well with 3-neutrino
oscillations. However, we find that 3+1 oscillations provide a decent fit with
$\Delta\chi^{2}_{\text{min}}<5$. As a result, for these benchmark points a
distinction of the two models is not possible.

For the blue benchmark point (note that it corresponds to the benchmark point
of Section IV.2), we find that $\Delta\chi^{2}_{\text{min}}$ depends
importantly on the value of the shape uncertainty. If the shape is mostly
unknown we can fit the LED data rather well using 3+1 oscillations and the
$\Delta\chi^{2}_{\text{min}}$ is small. However, for shape uncertainties
smaller than about 2% we obtain $\Delta\chi_{\text{min}}^{2}\gtrsim 15$ and a
distinction between the two models should not be possible.

For the other benchmark points the fit of the simulated LED data in terms of
3+1 oscillations is rather poor, even for a conservative choice of the shape
uncertainty. This means that in these cases a distinction between the two
models is feasible with JUNO+TAO. In general, we find that if the benchmark
point is far away from the sensitivity curves a better distinction is
possible. The reason for this behavior is due to the fact that the amplitudes
$L_{i}^{0n}$ increase when increasing $m_{i}^{D}$ and $R_{ED}$. Therefore, if
we generate the fake data with points which are far away from the sensitivity
curve, more oscillation lengths become relevant, and the data can not be
fitted well anymore within the 3+1 scenario. A better fit could be performed
considering $N>1$ sterile neutrinos, because of the equivalence of the LED and
$3+N$ sterile neutrino models discussed in Ref. [47].

Note that we did not assign a precise statistical meaning to the
$\Delta\chi^{2}$’s that we obtained, because a precise quantitative
statistical comparison of the LED and 3+1 models would require a complex
calculation of the distribution of the $\Delta\chi^{2}$ in the frequentist
approach, or a complex evaluation of the Bayes factor in the Bayesian
approach. As an indicative value of the $\Delta\chi^{2}$ that may allow to
distinguish the two models, one can consider $\Delta\chi^{2}=9$, that
corresponds to $3\sigma$ for a $\chi^{2}$ distribution with one degree of
freedom. However, in the case of a real observation of an anomalous result at
JUNO+TAO a more sophisticated statistical estimate should be done.

Benchmark point | $\Delta\chi_{\text{min}}^{2}$ | $\sin^{2}2\theta_{14}$ | $\Delta m_{41}^{2}$
---|---|---|---
$R_{ED}=0.1~{}\mu$m, $m_{0}=0.5$ eV | 1050 | 0.252 | 0.137 eV2
$R_{ED}=0.3~{}\mu$m, $m_{0}=0.07$ eV | 107 | 0.092 | 0.036 eV2
$R_{ED}=0.3~{}\mu$m, $m_{0}=0.05$ eV | 31 | 0.047 | 0.034 eV2
$R_{ED}=1.5~{}\mu$m, $m_{0}=0.005$ eV | 19 | 0.025 | 0.116 eV2
$R_{ED}=0.5~{}\mu$m, $m_{0}=0.03$ eV | 5.8 | 0.059 | 0.155 eV2
$R_{ED}=1.0~{}\mu$m, $m_{0}=0.005$ eV | 2.5 | 0.010 | 0.115 eV2
$R_{ED}=0.2~{}\mu$m, $m_{0}=0.01$ eV | 1.6 | 0.001 | 0.208 eV2
Table 2: The benchmark points used to generate the fake data sets and the
corresponding minimal $\Delta\chi^{2}$ values and values of the 3+1
oscillation parameters at the minimum for a shape uncertainty of 10%.

Figure 4: Left: Positions of the benchmark points (the stars) chosen to
generate the fake data, in comparison with the JUNO+TAO sensitivity and the
preferred region from Gallium and reactor experiments. Right: The minimum
$\Delta\chi^{2}$ value obtained when analyzing the LED generated fake data
using 3+1 neutrino oscillation parameters for several LED data sets (as
indicated in the legend).

##  V Conclusions

We have discussed the JUNO and TAO sensitivity to neutrino oscillations
induced by Large Extra Dimensions and by 3+1 active-sterile neutrino mixing.
We have shown that the experiments can set competitive bounds on the
parameters for both scenarios.

In the LED scenario, and for the realistic case of $3\%$ flux-shape systematic
uncertainty, we found that JUNO+TAO is sensitive to values of the
compactification radius $R_{ED}>0.81\,\mu\text{m}$ at $90\%$ C.L. (for NO),
which is in the sensitivity range of LBL experiments such as MINOS/MINOS+ and
DUNE.

We have shown that, in the case of light sterile neutrinos, JUNO+TAO can test
most of the region preferred by the analysis of the data of the Gallium
experiments that includes the recent results of the BEST experiment [66].

Considering the LED framework, since there is no updated analysis of the
Gallium data that includes the BEST data, we considered the results in Ref.
[53] of the LED explanation of the reactor anomaly and the GALLEX+SAGE Gallium
anomaly, that is anyhow consistent with the BEST data. We have shown that
JUNO+TAO can test all the $2\sigma$ region of the LED parameter space that is
allowed by the analysis in Ref. [53]. Moreover, we have shown that for a
benchmark point inside of this allowed region, JUNO+TAO can measure the LED
parameters with a precision that is similar to that of the standard neutrino
oscillation parameters. It is plausible that a similar precision can be
obtained for any value of the LED parameters that lies in the $2\sigma$ region
allowed by the analysis in Ref. [53], since it is within the sensitivity reach
of JUNO+TAO.

We have discussed also the possibility to distinguish LED oscillations from
3+1 active-sterile oscillations in case of a deviation of the JUNO+TAO data
from the prediction of three-neutrino mixing. We have shown that the
possibility of a distinction of the two models depends on the knowledge of the
reactor flux shape that will be reached when JUNO and TAO will operate and on
the true values of the LED parameters. In general, it is likely that the two
models can be distinguished if the true values of the LED parameters lie in
the sensitivity region of JUNO+TAO.

It is worth mentioning that also other types of experiments and analyses have
been performed in order to test the existence of Large Extra Dimensions, as
for instance tabletop gravitational experiments [88, 89, 90, 91], collider
experiments [92, 93, 94, 95, 96], analyses of astrophysical [97, 98, 99, 100,
101] and cosmological data [102, 103, 104]. In the case of astrophysical data,
strong constraints ranging in $R_{ED}<0.16-916\,\text{nm}$ can be obtained.
However, these limits depend on the technique and some assumptions [105].
Interestingly, neutrino oscillation experiments provide a stronger constraint
than tabletop experiments, which test the gravitational force at sub-
millimeter distances producing a limit of $R_{ED}<37~{}\mu\text{m}$ at 95% of
C.L [105].

Also in the case of light sterile neutrinos other searches exist, for example
using cosmological probes or $\beta$-decay (see the recent reviews in Ref.
[79, 106, 80]). In all cases, neutrino oscillation experiments provide
complementary tests of these scenarios and an observation in one sector and
non-observation in another could have interesting impacts on our understanding
of these fields.

## Acknowledgments

CG and CAT are supported by the research grant “The Dark Universe: A Synergic
Multimessenger Approach” number 2017X7X85K under the program “PRIN 2017”
funded by the Ministero dell’Istruzione, Università e della Ricerca (MIUR).
AAQ thanks Arman Esmaili and Hiroshi Nunokawa for useful discussions regarding
the LED model. VSBG’s work was partly supported by CNPQ under grant No.
151087/2014-8 and Universidad de Pamplona through Proyectos Convocatoria
Interna 2018.

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