# Neutrino oscillation bounds on quantum decoherence

Valentina De Romeri<EMAIL_ADDRESS>Instituto de Física Corpuscular
(CSIC-Universitat de València), Parc Científic UV C/ Catedrático José Beltrán,
2 E-46980 Paterna (Valencia) - Spain  Carlo Giunti<EMAIL_ADDRESS>Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Torino, Via P. Giuria
1, I–10125 Torino, Italy  Thomas Stuttard<EMAIL_ADDRESS>Niels
Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark
Christoph A. Ternes<EMAIL_ADDRESS>Istituto Nazionale di Fisica Nucleare
(INFN), Sezione di Torino, Via P. Giuria 1, I–10125 Torino, Italy
Dipartimento di Fisica, Università di Torino, via P. Giuria 1, I–10125 Torino,
Italy

###### Abstract

We consider quantum-decoherence effects in neutrino oscillation data. Working
in the open quantum system framework we adopt a phenomenological approach that
allows to parameterize the energy dependence of the decoherence effects. We
consider several phenomenological models. We analyze data from the reactor
experiments RENO, Daya Bay and KamLAND and from the accelerator experiments
NOvA, MINOS/MINOS+ and T2K. We obtain updated constraints on the decoherence
parameters quantifying the strength of damping effects, which can be as low as
$\Gamma_{ij}\lesssim 8\times 10^{-27}$ GeV at 90% confidence level in some
cases. We also present sensitivities for the future facilities DUNE and JUNO.

neutrinos, decoherence, reactors, accelerators

##  I Introduction

The outstanding experimental progress of the past years has allowed to
establish a clear picture of neutrino oscillations [1, 2] and to measure most
neutrino oscillation parameters with unprecedented accuracy [3, 4, 5]. While
few open questions remain to be answered, such as the determination of
possible leptonic CP violation, the neutrino mass ordering and the octant of
the atmospheric angle, the next generation of neutrino experiments will likely
allow to close up the neutrino physics picture and to determine all
oscillation parameters with good precision.

The established oscillation paradigm, and the consequent interpretation of
current and forthcoming neutrino oscillation data, relies on the three-
neutrino mass and mixing scheme. According to this scenario, neutrinos
propagate as a superposition of mass and flavor eigenstates. During
propagation, each component mass eigenstate evolves with a different frequency
thus resulting in the phenomenon of flavor conversion. Since neutrinos
interact only very weakly with other matter particles and since they are
stable, this quantum superposition effect is maintained over macroscopic
distances. In general, the study of neutrino oscillations considers the
neutrino to be isolated from its environment, and the oscillation effects to
be coherent. However, neutrino eigenstates may loose their quantum
superposition, for example if the neutrino system interacts with a stochastic
environment, thus resulting in a loss of coherence and in the damping of
neutrino oscillation probabilities – a phenomenon known as neutrino
decoherence.

Different physical origins can lead to similar, nonstandard decoherence
effects, see for instance Refs. [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19]. Let us note that the possible loss of flavor-coherence of the
neutrino beam can also occur during propagation in standard quantum mechanics,
depending on the neutrino wave-packet widths and separation [20, 9, 13, 21,
22, 23]. We will not consider decoherence from wave-packet separation in this
paper.

We will work in the framework of open quantum systems [24], commonly used in
quantum decoherence studies [25, 26, 27, 28, 29, 30, 31, 14, 32, 33, 34, 35,
36, 37, 38, 39, 40, 41, 42, 43]. We will assume that the oscillation
phenomenon can be changed by the interaction of the neutrino subsystem with
the environment. The fluctuating nature of space-time in a quantum theory of
gravity is a commonly cited potential source of a stochastic background that
might produce neutrino decoherence effects [44, 45, 46, 47]. In this work,
however, we will adopt a phenomenological approach and remain agnostic about
its origin. By analyzing several neutrino oscillation experiments, we will set
stringent limits on possible decoherence effects.

Experimental searches for neutrino quantum decoherence effects have been
performed with a vast range of facilities, and assuming different neutrino
sources. In general, the loss of coherence does not prevent neutrino flavor
conversion, but modifies the oscillation phenomenon through distance- and
energy-dependent damping terms, as the neutrinos propagate away from the
source. This suggests that long-baseline experiments are better suited for
probing neutrino quantum decoherence. Regarding neutrino sources,
astrophysical neutrinos are therefore excellent candidates. Solar neutrinos
have been used to place the strongest limits on energy-independent quantum
decoherence, as shown in [30, 48], in combination with data from the Kamioka
Liquid Scintillator Antineutrino Detector (KamLAND). It should be noted,
though, that depending on the model of decoherence, all sensitivity to measure
decoherence parameters might be lost if neutrino oscillations become averaged,
as in the case of solar neutrinos (or the diffuse high energy extragalactic
neutrino flux). We will comment on this important point later on. Earlier
works made use of atmospheric neutrinos using Super-Kamiokande data to obtain
bounds on the quantum decoherence parameters [26]. Currently, the most
stringent limits on energy-suppressed ($\propto E^{n},n>0$) decoherence
parameters have been obtained using atmospheric neutrinos observed at the
IceCube Neutrino Observatory [36]. Note that if the origin of the decoherence
effects is related to quantum gravity, one would naturally expect energy-
suppressed decoherence. Long-baseline accelerator and reactor experiments,
like the Main Injector Neutrino Oscillation Search (MINOS) and KamLAND have
been proven capable of setting more constraining bounds on energy-enhanced
($\propto E^{n},n<0$) decoherence parameters [49, 32]. Further analyses
include long-baseline accelerator data from MINOS and Tokai to Kamioka (T2K)
[40], as well as from NuMI Off-Axis $\nu_{e}$ Appearance (NOvA) and T2K [33].
Sensitivity studies have been presented for the Deep Underground Neutrino
Experiment (DUNE) in [50, 37] and for the Jiangmen Underground Neutrino
Observatory (JUNO) [51].

Building upon these earlier works, we work in the open quantum system
framework and analyze data from multiple experiments in order to set up-do-
date stringent constraints on the decoherence parameters. Keeping a
phenomenological approach, we will parameterize the decoherence effects and
their energy dependence in a model-independent way, so to allow for both
energy-enhanced and energy-suppressed scenarios. Since there are many possible
sources of neutrino decoherence, it is desirable to search for these effects
in as many experiments and energy regimes as possible. In this work we compare
data from a range of reactor and accelerator experiments to maximize the
sensitivity across a broad range of parameter space. As we will show,
depending on the energy dependence and on the distance travelled by neutrinos
from the source to the detector, different facilities will be more sensitive
to the damping effects than others. We analyze data from reactor experiments
such as the Reactor Experiment for Neutrino Oscillation (RENO) [52] and Daya
Bay [53], which allow for high-resolution, high-statistics measurements of the
antineutrino flux and consequently of possible damping features. We also
consider KamLAND [54, 55], which observes neutrinos from a large number of
nuclear reactor sites with longer baselines, thus providing complementary
information. We further analyze data from accelerator experiments, which are
sensitive to higher-energy neutrinos. We include the MINOS [56], NOvA [57, 58]
and T2K [59] experiments. The analysis of reactor and accelerator data yields
sensitivity to decoherence effects across the MeV to GeV regimes. Finally, we
provide sensitivity analyses for two future facilities: JUNO [60, 61], a
reactor experiment sensitive to the solar oscillation parameters and the
atmospheric mass splitting, and especially designed for measuring the neutrino
mass ordering, and for the DUNE long-baseline neutrino experiment [62, 63, 64,
65], also expected to determine the neutrino mass ordering, among other goals
in its extended physics program.

This paper is organized as follows. We introduce in Section II the open
quantum system formalism and the phenomenological scenarios that we analyze.
In Section III we present technical details of the experiments and of our
statistical analyses. We discuss the results of our analyses in Section IV,
presenting updated bounds on the decoherence parameters and sensitivities for
near-future experiments. We finally draw our conclusions in Section V.

##  II Theoretical framework

In this section we present the theoretical framework that we use to obtain the
neutrino oscillation probabilities in the presence of decoherence effects. We
adopt the open quantum system approach, treating propagating neutrinos as a
subsystem which interacts very weakly with a larger, unknown environment [66,
24]. The interaction of the neutrino system with the quantum environment is
the source of damping effects in the oscillations.

The evolution in time of neutrinos in the quantum (sub-)system can be
described through the Lindblad equation [67, 68]

$\frac{\partial\rho_{\nu}(t)}{\partial
t}=-i[H,\rho_{\nu}(t)]+\mathcal{D}[\rho_{\nu}(t)]\,,$ (1)

where $\rho_{\nu}(t)$ is a Hermitian density matrix describing the neutrino
states in the mass basis and $H$ is the Hamiltonian of the neutrino subsystem,
which encodes both the standard oscillation effects resulting from the unequal
neutrino masses, and any conventional matter effects. According to the density
matrix formalism, the density matrix of mixed quantum states is
$\rho_{\nu}=\sum_{j}\mathcal{P}_{j}|\psi_{j}\rangle\langle\psi_{j}|$, for a
system of $j$ states of probability $\mathcal{P}_{j}$. The dissipative term
$\mathcal{D}[\rho_{\nu}(t)]$ encodes the decoherence effects, such that when
$\mathcal{D}[\rho_{\nu}(t)]\to 0$ the standard neutrino oscillations without
damping effects are recovered. It is also a Hermitian matrix which can be
expressed as [69, 25, 26, 42]

$\mathcal{D}[\rho_{\nu}(t)]=\frac{1}{2}\sum_{j=1}^{N^{2}-1}\left([\mathcal{O}_{j},\rho_{\nu}(t)\mathcal{O}_{j}^{\dagger}]+[\mathcal{O}_{j}\rho_{\nu}(t),\mathcal{O}_{j}^{\dagger}]\right)\,,$
(2)

where $N$ refers to the dimension of the $SU(N)$ Hilbert space defining the
neutrino subsystem ($N=3$ when assuming three neutrino flavors) and
$\mathcal{O}_{j}$ are dissipative operators described as $N\times N$ complex
matrices which characterize the coupling of the neutrino subsystem to its
environment ($j_{\text{max}}=8$ for three neutrino flavors), for example in
quantum gravity space-time [70, 71, 27, 72]. The operators $\mathcal{O}_{j}$
act only on neutrino states in such a way that
$\sum_{j}\mathcal{O}^{\dagger}_{j}\mathcal{O}_{j}=\mathbb{I}$ and the trace of
$\rho_{\nu}(t)$ is unchanged. Let us notice that, while being Hermitian, the
matrix $\mathcal{D}[\rho_{\nu}(t)]$ can be nonunitary due to the interactions
of the neutrino subsystem with the environment including possible neutrino
loss. The general expression of $\mathcal{D}[\rho_{\nu}(t)]$ in Eq. (2) would
in principle allow for a model-independent analysis of neutrino decoherence
effects in oscillation data. However, given the large number of free
parameters that it contains, such an analysis is realistically not viable from
a practical point of view. To simplify the theoretical framework, focusing on
a $N$-neutrino system, we can expand the operators in Eq (2) in terms of the
Gell-Mann matrices from the $SU(N)$ group [25, 69, 38, 39, 42]

$\mathcal{D}[\rho_{\nu}(t)]=c_{k}\lambda^{k}~{}\left(\mathrm{with}~{}\rho_{\nu}=\sum\rho_{\nu}^{k}\lambda^{k}\mathrm{and}~{}\mathcal{O}_{j}=\sum\mathcal{O}^{j}_{k}\lambda^{k}\right)\,.$
(3)

In the previous expression, $c_{k}$ are the coefficients of the expansion, and
the index $k$ runs from 0 to $N^{2}-1$. In the specific case of three neutrino
flavors: $k=0\ldots 8$, $\lambda^{0}$ is the identity matrix and $\lambda^{k}$
are the Gell-Mann matrices satisfying
$[\lambda^{a},\lambda^{b}]=i\sum_{c}f_{abc}\lambda^{c}$, $f_{abc}$ being the
structure constants of $SU(3)$. We can then express the decoherence term as
$\mathcal{D}[\rho_{\nu}(t)]=(\mathbf{D}_{k\ell}~{}\rho_{\nu}^{\ell})\lambda^{k}$,
where $\rho_{\nu}^{\ell}$ are the coefficients of the neutrino density matrix,
expanded in the $SU(N)$ basis (see Eq. (3)), and
$\mathbf{D}_{k\ell}~{}\rho_{\nu}^{\ell}=c_{k}$ are the elements of a
$N^{2}\times N^{2}$ matrix representing the free parameters of the system.
Focusing again on a three-neutrino scenario, the matrix $\bf{D}$ can be
parameterized by 45 parameters as follows

$\bf{D}=\begin{pmatrix}-\Gamma_{0}&\beta_{01}&\beta_{02}&\beta_{03}&\beta_{04}&\beta_{05}&\beta_{06}&\beta_{07}&\beta_{08}\\\
\beta_{01}&-\Gamma_{1}&\beta_{12}&\beta_{13}&\beta_{14}&\beta_{15}&\beta_{16}&\beta_{17}&\beta_{18}\\\
\beta_{02}&\beta_{12}&-\Gamma_{2}&\beta_{23}&\beta_{24}&\beta_{25}&\beta_{26}&\beta_{27}&\beta_{28}\\\
\beta_{03}&\beta_{13}&\beta_{23}&-\Gamma_{3}&\beta_{34}&\beta_{35}&\beta_{36}&\beta_{37}&\beta_{38}\\\
\beta_{04}&\beta_{14}&\beta_{24}&\beta_{34}&-\Gamma_{4}&\beta_{45}&\beta_{46}&\beta_{47}&\beta_{48}\\\
\beta_{05}&\beta_{15}&\beta_{25}&\beta_{35}&\beta_{45}&-\Gamma_{5}&\beta_{56}&\beta_{57}&\beta_{58}\\\
\beta_{06}&\beta_{16}&\beta_{26}&\beta_{36}&\beta_{46}&\beta_{56}&-\Gamma_{6}&\beta_{67}&\beta_{68}\\\
\beta_{07}&\beta_{17}&\beta_{27}&\beta_{37}&\beta_{47}&\beta_{57}&\beta_{67}&-\Gamma_{7}&\beta_{78}\\\
\beta_{08}&\beta_{18}&\beta_{28}&\beta_{38}&\beta_{48}&\beta_{58}&\beta_{68}&\beta_{78}&-\Gamma_{8}\\\
\end{pmatrix}\,,$ (4)

where all entries are real scalars and the $\Gamma_{i}$ are positive in order
to satisfy the relation Tr$(\rho_{\nu}(t))=1$. As already mentioned, while a
general model-independent analysis might be contemplated, it proves convenient
to reduce the number of free parameters by imposing general physical
conditions on $\bf{D}$ [26, 25, 69, 73, 50, 42]. These include:

  * •

Unitarity of the system. Probability conservation implies
${\bf{D}}_{k0}={\bf{D}}_{0\ell}=0$ [39], given that $f_{ab0}=0$.

  * •

Complete positivity of the time-evolution $\rho_{\nu}$ place conditions on the
diagonal elements, thus making them not completely independent.

  * •

Entropy increase. The condition $\mathcal{O}_{j}=\mathcal{O}_{j}^{\dagger}$
implies that the Von Neumann entropy
$S=-\mathrm{Tr}(\rho_{\nu}\mathrm{ln}\rho_{\nu})$ increases with time [74].

  * •

Energy conservation of the neutrino subsystem is satisfied through the
commutation relation $[H,\mathcal{O}_{j}]=0$. This bound includes the
decoherence effect in the evolution.

Although in principle energy nonconservation in the neutrino subsystem could
be envisaged [25, 73], while of course always being satisfied for the global
system, in the following we work under the requirement of energy conservation
on the neutrino subsystem. These assumptions allow for a remarkable
simplification of the dissipator in Eq. (4), which eventually assumes the
diagonal form [32, 75, 50]

${\bf{D}}=-\mathrm{diag}(\Gamma_{21},\Gamma_{21},0,\Gamma_{31},\Gamma_{31},\Gamma_{32},\Gamma_{32},0)\,,$
(5)

in the case of three neutrino flavors. Comparing with Eq. (4), we see that
imposing the above-mentioned conditions in practice leads to $\beta_{ij}=0$
and $\Gamma_{1}=\Gamma_{2}\equiv\Gamma_{21}$,
$\Gamma_{4}=\Gamma_{5}\equiv\Gamma_{31}$ and
$\Gamma_{6}=\Gamma_{7}\equiv\Gamma_{32}$. In principle, also the so-called
relaxation parameters $\Gamma_{3}$ and $\Gamma_{8}$ could be present in Eq.
(5). However, since the terrestrial experiments that we consider here are not
expected to have competitive sensitivity with bounds from solar neutrinos
[48], we set $\Gamma_{3}=\Gamma_{8}=0$ throughout the paper.

As anticipated, the interaction of the neutrino subsystem with the
environment, through the matrix Eq. (5) will induce damping effects of the
form $e^{-\Gamma_{ij}L}$ in the oscillation probabilities, $L$ being the
experiment baseline and $\Gamma_{ij}$ quantifying the strength of the damping
effects, resulting in a coherence length $L_{\mathrm{coh}}=1/\Gamma_{ij}$.
While remaining agnostic of the (quantum-gravity) origin of such dissipating
terms, we follow a phenomenological approach and introduce a general form for
the energy-dependence of the dissipator matrix controlling the decoherence
effects. We assume a power-law energy- dependence following previous studies
in the literature [26, 30, 36, 76, 38, 40, 42]

$\Gamma_{ij}(E)=\Gamma_{ij}(E_{0})\left(\frac{E}{E_{0}}\right)^{n}\,,$ (6)

where $E_{0}$ is a pivot energy scale which we set to $E_{0}=1$ GeV, as
previously done in the literature, and $n$ is a power-law index to be tested
experimentally. Given the lack of a fundamental theory for quantum gravity,
the dependence of $\Gamma_{ij}$ on the neutrino energy is currently unknown.
We will make the hypothesis that it can assume the following integer values:
$n=[-2,-1,0,+1,+2]$. Note that this power-law dependence on the neutrino
energy breaks Lorentz invariance, except for the case with $n=-1$ which
imitates the oscillation energy dependence. Our choice for $n$ thus
accommodates: a case where decoherence might be induced by lightcone
fluctuations [43] ($n=-2$); a Lorentz-invariant case ($n=-1$) [26]; the
energy-independent case ($n=0$); $n=+1$ which mimics a cross section-like
energy-dependence, or as suggested in Ref. [77]; and the case ($n=+2$) that
can arise in quantum-gravity models, in which $\Gamma_{i}(E_{0})\sim O(M_{\rm
Planck}^{-1})\sim 10^{-19}$ GeV is expected [78, 79, 80, 81, 82]. For the sake
of completeness, let us also recall that the loss of coherence due to wave-
packet separation would correspond to the case $n=-4$. While we do not cover
this scenario here, we refer the reader to Refs. [22, 23] for recent analyses.
We also note that the $n=-1$ case is phenomenologically similar to invisible
neutrino decay [42]. Finally, it has recently been suggested that scenarios
with an extreme energy dependence ($n\leq-10$) could explain the Gallium
anomaly [83] without the need of light sterile neutrinos and hence without
entering in conflict with bounds from other short-baseline experiments [84].

###  II.1 Phenomenological models

Model A |  We vary $\Gamma_{21}=\Gamma_{31}=\Gamma_{32}$
---|---
Model B |  We vary $\Gamma_{21}=\Gamma_{31}$ and keep $\Gamma_{32}=0$
Model C |  We vary $\Gamma_{21}=\Gamma_{32}$ and keep $\Gamma_{31}=0$
Model D |  We vary $\Gamma_{31}=\Gamma_{32}$ and keep $\Gamma_{21}=0$
Model E |  We vary $\Gamma_{21}$ and keep $\Gamma_{31}=\Gamma_{32}=0$
Model F |  We vary $\Gamma_{31}$ and keep $\Gamma_{21}=\Gamma_{32}=0$
Model G |  We vary $\Gamma_{32}$ and keep $\Gamma_{21}=\Gamma_{31}=0$
Table 1: The phenomenological models considered in this paper.

The choice of D defines the flavor structure of the decoherence effects, and
ultimately the flavor ratio of a neutrino ensemble once coherence is
completely lost. In the analyses of this paper we consider several scenarios,
reflected in activation of different $\Gamma_{ij}$ simultaneously. The models
that we consider are summarized in Table 1, and can be grouped into three
categories; all three $\Gamma_{ij}$ activated (A), two $\Gamma_{ij}$ activated
(B, C, D), and only a single $\Gamma_{ij}$ activated (E, F, G). In each case
we test each of the above mentioned energy dependencies.

Activating differing $\Gamma_{ij}$ results in differing matrix elements of the
$[H,\rho_{\nu}(t)]$ term in Eq. (1) being damped. In particular, specific
active pairs of $\Gamma_{ij}$ damp differing oscillations frequencies
resulting from the solar ($\Delta m^{2}_{21}$) and atmospheric ($\Delta
m^{2}_{3j}$) mass splittings. Model D for instance results in the (faster)
atmospheric sector oscillations being damped whilst the (slower) solar sector
oscillations are not. Such a scenario could therefore represent new physics
that depends on the mass splittings (or equivalently oscillation wavelengths),
such as lightcone fluctuations [43] or a mass-dependent coupling. Model A
represents the case where all oscillation channels are damped together, and
can result from perturbations to the neutrino wavefunction phase [42]. In
general, combining experimental data from various oscillation
channels/experiments will be required to optimally test all models.

###  II.2 Neutrino oscillation probability and decoherence

The neutrino oscillation probability in presence of the dissipator under
consideration here, Eq. (5), has been derived many times in the literature,
see for example Ref. [50]. The result reads

$\displaystyle P(\nu_{\alpha}\to\nu_{\beta})=\delta_{\alpha\beta}$
$\displaystyle-2\sum_{k>j}\Re\left[\tilde{U}^{*}_{\alpha k}\tilde{U}_{\beta
k}\tilde{U}_{\beta j}\tilde{U}^{*}_{\beta
j}\right]~{}\left[1-\cos\left(\frac{\Delta\tilde{m}^{2}_{kj}L}{2E}\right)~{}e^{-\Gamma_{ij}(E)L}\right]$
$\displaystyle+2\sum_{k>j}\Im\left[\tilde{U}^{*}_{\alpha k}\tilde{U}_{\beta
k}\tilde{U}_{\beta j}\tilde{U}^{*}_{\beta
j}\right]~{}\sin\left(\frac{\Delta\tilde{m}^{2}_{kj}L}{2E}\right)~{}e^{-\Gamma_{ij}(E)L}\,,$
(7)

where $\Gamma_{ij}(E)$ is given in Eq. (6) and $\tilde{U}$ and
$\Delta\tilde{m}^{2}_{kj}$ are the mixing matrix and mass splittings in
matter. In our analyses we calculate these quantities using the expressions
provided in Ref. [85], which currently constitute the most precise
approximation for neutrino oscillations in constant matter [86]. In the
following, matter effects will be relevant in the analyses of KamLAND, NOvA
and DUNE. Note that the expression of the neutrino oscillation probability is
equivalent to the standard expression when $\Gamma_{ij}(E)\to 0$ (i.e. when no
decoherence effects are present).

Figure 1: Neutrino oscillation probabilities for the baselines, energies and
flavors relevant for several of the experiments considered in this paper, in
the case of Model A and for various $n$. The left panel is characteristic of
long-baseline accelerator neutrino experiments, whilst the right panel
represents reactor neutrino experiments.

Figure 2: Neutrino oscillation probabilities for the baselines, energies and
flavors relevant for several of the experiments considered in this paper, for
all phenomenological models (see Table 1). The left panel is characteristic of
long-baseline accelerator neutrino experiments, whilst the right panel
represents reactor neutrino experiments. We fix $n=0$ and
$\Gamma_{ij}=10^{-22}$ GeV (left panel), $\Gamma_{ij}=10^{-20}$ GeV (right
panel, models A-B-C-E) and $\Gamma_{ij}=10^{-19}$ GeV (right panel, models
D-F-G).

We show in Fig. 1 the $\nu_{\mu}$ (left panel) and $\bar{\nu}_{e}$ (right
panel) survival probabilities using baselines and energies relevant for the
experiments considered in this paper. As can be seen, different choices of $n$
can induce different forms of spectral distortions. We fix $\Gamma_{ij}(E)$ to
values in the ballpark of current bounds, which we will infer in Sec. IV.
Experiments using low-energy neutrinos will be more sensitive to scenarios
with negative $n$, while high-energy neutrinos will allow us to set stronger
bounds when $n>0$, as it is particularly clear from the left plot. In Fig. 2
we keep $n=0$ fixed and show the impact of the different models on the
oscillation probabilities. From the right panel (representing reactor
experiments) we see that the models can be separated into two groups: One
group, which has $\Gamma_{21}$ activated and one which has not. If
$\Gamma_{21}$ is active we obtain, in addition to possible spectral
distortions, also an overall shift of the oscillation probability. Note that a
simple shift might not result in an observable effect if one only takes ratios
of events at different detectors. Therefore we expect the reactor experiments
to have best sensitivity to models which have $\Gamma_{21}$ and at least one
other $\Gamma$ activated, while also using $n<0$. Also in the case of long-
baseline accelerator experiments (left panel) we find that there exist two
groups of models. One group which has $\Gamma_{32}$ active, and one without.
As can be seen from the left panel of Fig. 2, the deviation from the standard
three-neutrino oscillation probability is particularly large for the former
models (A-C-D-G). We can therefore expect accelerator experiments to have best
sensitivity to models with (at least) $\Gamma_{32}\neq 0$ and positive $n$. Of
course, this expectation based on oscillation plots is somewhat simplified,
since in the real analyses correlations with standard neutrino oscillation
parameters and systematic uncertainties of the experiments must be taken into
account. Nevertheless, as will be shown later, the strengths of the bounds for
the models under consideration will follow this simplified expectation in many
cases.

##  III Experiments included in the analysis

In this section we discuss the experimental data that have been considered in
this paper and give some details of the analyses performed. We have used data
from several reactor and long-baseline accelerator experiments, discussed in
Sections III.1 and III.2, respectively. We also detail in Section III.3 the
analysis procedure for the expected sensitivities at the next-generation
experiments JUNO and DUNE. It should be noted that the decoherence effects
discussed in this paper do not affect solar neutrino measurements. We will
therefore always impose a prior on the solar mixing angle, fixing
$\sin^{2}\theta_{12}=0.318\pm 0.016$ [3].

###  III.1 Reactor experiments: KamLAND, Daya Bay and RENO

KamLAND was a reactor experiment which was placed at the site of the former
Kamiokande experiment and was used to measure electron antineutrinos from more
than 50 reactors, located at distances ranging from $\sim 100$ km to $\sim
1000$ km. Due to the long-baselines, matter effects are not negligible at
KamLAND, and must be included in the calculation of the neutrino oscillation
probability. For the present analysis we include the data presented in Refs.
[54, 55]. The $\chi^{2}$ function used for KamLAND reads

$\chi^{2}_{\text{KL}}(\vec{p})=\min_{\vec{\alpha}}\left\\{\sum_{i=1}^{N_{\text{KL}}}\left(\frac{N_{\text{dat},i}-N_{\text{exp},i}(\vec{p},\vec{\alpha})}{\sigma^{\text{KL}}_{i}}\right)^{2}+\frac{(N^{\text{tot}}_{\text{dat}}-N^{\text{tot}}_{\text{exp}}(\vec{p},\vec{\alpha}))^{2}}{N^{\text{tot}}_{\text{dat}}}+\sum_{k}\left(\frac{\alpha_{k}-\mu_{k}}{\sigma_{k}}\right)^{2}\right\\}+\chi^{2}_{\theta_{12}}\,,$
(8)

where the index $i$ runs over the energy bins. In the previous expression,
$N_{\text{dat},i}$ denote the observed event numbers per energy bin, while
$N_{\text{exp},i}(\vec{p},\vec{\alpha})$ indicate the expected event numbers
for a given set of oscillation parameters $\vec{p}$ and systematic
uncertainties $\vec{\alpha}$. The second term is a penalty term on the total
number of events, while the third term contains penalty factors for all of the
systematic uncertainties $\alpha_{k}$, with expectation value $\mu_{k}$ and
standard deviation $\sigma_{k}$. The systematic uncertainties that are
included in our analysis account for reactor uncertainties (normalizations
related to the different reactors and an uncorrelated shape error) and
detector uncertainties (detection efficiency and energy scale). The last term
contains the above mentioned penalty for $\sin^{2}\theta_{12}$. We use GLoBES
[87, 88] to compute the event numbers and to perform the statistical analysis.
The reactor fluxes are parameterized as in Ref. [89] and the inverse beta-
decay cross section is taken from Ref. [90]. We have found that the
measurement of $\Delta m_{21}^{2}$ obtained from the analysis of KamLAND data
is robust under the decoherence effects discussed here. We will therefore
include another penalty when marginalizing over $\Delta m_{21}^{2}$ in all
experiments discussed in the following subsections, corresponding to $\Delta
m_{21}^{2}=(7.53\pm 0.22)\times 10^{-5}$ eV2.

Apart from KamLAND, we also include data from the reactor experiments Daya Bay
and RENO. We analyze the data sets corresponding to 2900 days of running time
at RENO [52] and 1958 days of running time at Daya Bay [53]. In the
statistical analyses, we include uncertainties related to the thermal power
for each core, to the detection efficiencies, uncertainties on the fission
fractions, a shape uncertainty for each energy bin, and an uncertainty on the
energy scale. The $\chi^{2}$ function for RENO is

$\chi^{2}_{\text{RENO}}(\vec{p})=\min_{\vec{\alpha}}\left\\{\sum_{i=1}^{N_{\text{RENO}}}\left(\frac{R^{F/N}_{\text{dat},i}-R^{F/N}_{\text{exp},i}(\vec{p},\vec{\alpha})}{\sigma^{\text{RENO}}_{i}}\right)^{2}+\sum_{k}\left(\frac{\alpha_{k}-\mu_{k}}{\sigma_{k}}\right)^{2}\right\\}+\chi^{2}_{\text{solar}}\,.$
(9)

Here, $R^{F/N}_{i}=F_{i}/N_{i}$, is the ratio of events at the far ($F_{i}$)
and near ($N_{i}$) detectors in the $i$th energy bin. In particular,
$R_{\text{dat},i}$ are the background-subtracted observed event ratios, while
$R_{\text{exp},i}(\vec{p},\vec{\alpha})$ are the expected event ratios for a
given set of oscillation parameters $\vec{p}$. The uncertainty for each bin is
given by $\sigma^{\text{RENO}}_{i}$. The second term contains penalty factors
for all the systematic uncertainties $\alpha_{k}$, again with expectation
value $\mu_{k}$ and standard deviation $\sigma_{k}$. The last term
$\chi^{2}_{\text{solar}}$ contains the solar penalty for $\sin^{2}\theta_{12}$
and the KamLAND penalty for $\Delta m_{21}^{2}$. Similarly, for Daya Bay, we
define

$\displaystyle\chi^{2}_{\text{DB}}(\vec{p})=$
$\displaystyle\min_{\vec{\alpha}}\left\\{\sum_{i=1}^{N_{\text{DB}}}\left(\frac{R^{F/N_{1}}_{\text{dat},i}-R^{F/N_{1}}_{\text{exp},i}(\vec{p},\vec{\alpha})}{\sigma_{i}^{F/N_{1}}}\right)^{2}+\sum_{i=1}^{N_{\text{DB}}}\left(\frac{R^{N_{2}/N_{1}}_{\text{dat},i}-R^{N_{2}/N_{1}}_{\text{exp},i}(\vec{p},\vec{\alpha})}{\sigma_{i}^{N_{2}/N_{1}}}\right)^{2}\right.$
$\displaystyle\left.+\sum_{k}\left(\frac{\alpha_{k}-\mu_{k}}{\sigma_{k}}\right)^{2}\right\\}+\chi^{2}_{\text{solar}}\,.$
(10)

Here, we take the ratios between the far and the first near detector and
between the two near detectors. We use the same reactor flux model and inverse
beta-decay cross section as in the analysis of KamLAND.

###  III.2 Accelerator experiments: MINOS/MINOS+, T2K and NOvA

MINOS was an accelerator-based neutrino oscillation experiment studying muon
neutrinos produced at Fermilab at the NuMI beam facility and detected at two
detectors located at 1.04 km and 735 km away from the source. First, during
the MINOS phase, the neutrino beam peaked at an energy of 3 GeV. Later, during
the MINOS+ phase, the energy of the beam was increased, peaking at 7 GeV. Here
we consider data corresponding to an exposure of $10.56\times 10^{20}$ proton-
on-target (POT) in MINOS and $5.80\times 10^{20}$ POT in MINOS+, collected in
the same detectors [56]. We adopted the analysis procedure followed by the
experimental collaboration for the search of active-sterile neutrino
oscillations in Ref. [56], by modifying the public MINOS/MINOS+ code to
account for decoherence effects instead of active-sterile oscillations. The
statistical analysis is performed by using the following $\chi^{2}$
definition:

$\chi^{2}_{\mathrm{MINOS}}(\vec{p})=\sum_{\text{Det.}}\sum_{i,j}(N_{\text{dat},i}-N_{\text{exp},i}(\vec{p}))~{}[V_{\text{cov.}}]^{-1}_{ij}~{}(N_{\text{dat},j}-N_{\text{exp},j}(\vec{p}))+\chi^{2}_{\text{solar}}\,,$
(11)

where $N_{\text{dat},i}$ and $N_{\text{exp},i}$ are the observed and the
predicted event rates, in the energy bin $i$. The covariance matrix
$V_{\text{cov.}}$ accounts for the contributions from several sources of
systematic uncertainties [56], and the first sum runs over the two detectors.
The last term, $\chi^{2}_{\text{solar}}$, is the same as in Eqs. (9) and
(III.1).

We further consider data collected by T2K [59] and NOvA [58]. The T2K
collaboration observed events induced by neutrinos and antineutrinos,
corresponding to an exposure at Super-Kamiokande of 1.97$\times 10^{21}$ POT
in neutrino mode and 1.63$\times 10^{21}$ POT in antineutrino mode. NOvA has
instead reached 13.6$\times 10^{20}$ POT in neutrino mode [57] and 12.5$\times
10^{20}$ POT in antineutrino mode. For the energy reconstruction in both
experiments we assume Gaussian smearing adding bin-to-bin efficiencies, which
are adjusted to reproduce the best-fit spectra reported by the experimental
collaborations. Our statistical analysis includes several sources of
systematic uncertainties, related to the signal and background predictions.
The $\chi^{2}$ function for the analysis of T2K and NOvA data (and DUNE, as we
will see later) is given by

$\chi^{2}_{\mathrm{T/N/D}}(\vec{p})=\min_{\vec{\alpha}}\sum_{\text{channels}}2\sum_{i}\left[N_{\text{exp},i}(\vec{p},\vec{\alpha})-N_{\text{dat},i}+N_{\text{dat},i}\log\left(\frac{N_{\text{dat},i}}{N_{\text{exp},i}(\vec{p},\vec{\alpha})}\right)\right]+\sum_{i}\left(\frac{\alpha_{i}}{\sigma_{i}}\right)^{2}+\chi^{2}_{\text{solar}}\,,$
(12)

where again $N_{\text{dat},i}$ and $N_{\text{exp},i}$ are the data and
prediction in bin $i$ and the first sum is taken over the different
oscillation channels: $\nu_{\mu}\to\nu_{\mu}$,
$\overline{\nu}_{\mu}\to\overline{\nu}_{\mu}$, $\nu_{\mu}\to\nu_{e}$ and
$\overline{\nu}_{\mu}\to\overline{\nu}_{e}$. The last two terms are equivalent
to all other cases.

###  III.3 Future experiments: JUNO and DUNE

In this subsection we discuss the details for the sensitivity analyses at JUNO
and DUNE. When performing these sensitivity analyses we generate a mock data
set for both experiments using the neutrino oscillation parameters from Ref.
[3] and without decoherence effects. In the statistical analysis we vary the
decoherence parameters and marginalize over the standard parameters,
equivalently to the other experiments.

JUNO [60] is a next-generation reactor experiment, which aims to precisely
measure the solar neutrino oscillation parameters, the atmospheric mass
splitting, and the neutrino mass ordering. JUNO will consist of eight reactors
[91, 61] located at around 52 km distance from the main detector. Our
simulation of the experiment follows the descriptions in Refs. [60, 91, 61,
92] assuming 8 years of running time. Our statistical analyses are performed
with

$\chi^{2}_{\mathrm{JUNO}}(\vec{p})=\min_{\vec{\alpha}}\left\\{\sum_{i=1}^{N_{\text{JUNO}}}\left(\frac{N_{\text{dat},i}-N_{\text{exp},i}(\vec{p},\vec{\alpha})}{\sigma_{i}}\right)^{2}+\sum_{k}\left(\frac{\alpha_{k}-\mu_{k}}{\sigma_{k}}\right)^{2}\right\\}+\chi^{2}_{\text{solar}}\,,$
(13)

where the definition of the quantities is equivalent to the other cases,
barring that this time $N_{\text{dat},i}$ consists of the mock data set.

DUNE [62, 63, 64, 65] is going to be the successor experiment of NOvA. It will
also consist of two detectors, which are going to be exposed to a megawatt-
scale neutrino beam produced at Fermilab, composed of (nearly) only muon
neutrinos or antineutrinos. The near detector will be placed approximately 600
meters away from the source of the beam, while the far detector, divided into
four modules, each using 10 kton of argon as detection material, will be
installed 1285 kilometres away, deep underground at the Sanford Underground
Research Facility in South Dakota. To simulate the neutrino signal at DUNE we
use the latest configuration file for GLoBES provided by the DUNE
collaboration [93], which assumes 6.5 years of running time in both neutrino
and antineutrino modes. Our analysis includes disappearance and appearance
channels, simulating both signals and backgrounds. The simulated backgrounds
include contamination of antineutrinos (neutrinos) in the neutrino
(antineutrino) mode, and also misinterpretation of flavors. In addition, we
also consider the DUNE-High Energy (HE) flux configuration, which has been
proposed to optimize the study of tau neutrinos at DUNE [94]. In this option
the beam will peak at higher energies in comparison to the nominal
configuration. When considering the sensitivity of DUNE we will compare the
sensitivity using the standard DUNE beam with the high energy beam [94]. The
$\chi^{2}$ function for DUNE is the same as for T2K and NOvA, see Eq. (12).

We conclude by summarizing in Table 2 some of the experimental details of the
experiments considered in this paper.

Experiment | Baseline | Energy range | Main oscillation channel
---|---|---|---
KamLAND [54] |  $\mathcal{O}(100)-\mathcal{O}(1000)$ km |  $1.8-8.0$ MeV |  $\overline{\nu}_{e}\to\overline{\nu}_{e}$
Daya Bay [53] and RENO [52] |  $\mathcal{O}(100)-\mathcal{O}(1000)$ m |  $1.8-8.0$ MeV |  $\overline{\nu}_{e}\to\overline{\nu}_{e}$
T2K [59] | 295 km |  $0.2-2.0$ GeV |  $\nu_{\mu}(\overline{\nu}_{\mu})\to\nu_{\mu}(\overline{\nu}_{\mu})$ and $\nu_{\mu}(\overline{\nu}_{\mu})\to\nu_{e}(\overline{\nu}_{e})$
NOvA [58] | 812 km |  $0.8-5.0$ GeV |  $\nu_{\mu}(\overline{\nu}_{\mu})\to\nu_{\mu}(\overline{\nu}_{\mu})$ and $\nu_{\mu}(\overline{\nu}_{\mu})\to\nu_{e}(\overline{\nu}_{e})$
MINOS/MINOS+ [56] | 735 km |  $0-40.0$ GeV |  $\nu_{\mu}(\overline{\nu}_{\mu})\to\nu_{\mu}(\overline{\nu}_{\mu})$
JUNO [61] |  $\sim 53$ km |  $1.8-8.0$ MeV |  $\overline{\nu}_{e}\to\overline{\nu}_{e}$
DUNE [93] | 1285 km |  $0.5-20$ GeV |  $\nu_{\mu}(\overline{\nu}_{\mu})\to\nu_{\mu}(\overline{\nu}_{\mu})$ and $\nu_{\mu}(\overline{\nu}_{\mu})\to\nu_{e}(\overline{\nu}_{e})$
DUNE HE [94] | 1285 km |  $0.5-20$ GeV |  $\nu_{\mu}(\overline{\nu}_{\mu})\to\nu_{\mu}(\overline{\nu}_{\mu})$ and $\nu_{\mu}(\overline{\nu}_{\mu})\to\nu_{e}(\overline{\nu}_{e})$

Table 2: Details of the experiments considered in this paper. Note that even
though the range for DUNE and DUNE-HE is the same, the high-energy beam is
much broader and peaks at larger energies resulting in different
sensitivities.

##  IV Results

In this section we discuss the results from the analyses performed. Given the
form of the neutrino oscillation probability provided in Section II.2 we
expect a long-baseline experiment to be especially suited to bound the
decoherence effects discussed in this paper. Note, however, that if the
baseline is too long, oscillations can become fully averaged and any
sensitivity to decoherence is lost. This explains why solar neutrino
experiments have no sensitivity to the decoherence parameters of interest in
our analyses111On the other hand, solar neutrinos can still be used to bound
the relaxation parameters [48] although we do not consider them here.. Due to
our choice of $E_{0}$ in Eq. (6) experiments with low (high) energy neutrinos
will be sensitive to models with negative (positive) energy dependence
(encoded in the parameter $n$).

The results of our analyses for all the models under consideration are
presented in Figs. 3-9. In the left panels we show the 90% C.L. upper limits
that can be obtained with current data from the different experiments
described in Sec. III, for each $n$. In the right panels we show the
sensitivities for JUNO and DUNE (using the standard and the high-energy flux
configuration), always in comparison with the current strongest bound,
obtained from the different experiments at each $n$ and represented by the
grey-shaded region.

Figure 3: Current 90% C.L. upper limits (left) and future sensitivities for
Model A ($\Gamma\equiv\Gamma_{21}=\Gamma_{31}=\Gamma_{32}$), as a function of
$n$ (see Eq. (6)).

In Fig. 3 we consider Model A
($\Gamma\equiv\Gamma_{21}=\Gamma_{31}=\Gamma_{32}$). As can be seen, for
negative $n$ the strongest bound is obtained from the analysis of KamLAND
data. For $n=0$ all accelerators produce similar results, while for positive
$n$ the bound obtained from MINOS/MINOS+ is the strongest by far. The bounds
obtained from the analysis of Daya Bay and RENO data are very similar and
always about one order of magnitude weaker than the bound from KamLAND. Note
that the slope of the corresponding limits, as a function of $n$, is different
for each experiment. This is due to the pivot point $E_{0}=1$ GeV chosen in
Eq. (6). Notice that the choice of a different reference value would change
the bounds obtained in this work by a factor
$\left(\frac{E_{0(\mathrm{old})}}{E_{0(\mathrm{new})}}\right)^{n}$. Reactors
use energies much smaller and hence they are mostly sensitive to negative $n$.
The energies relevant for T2K are close to $E_{0}$ and therefore the curve is
nearly horizontal. NOvA and MINOS use larger energies and hence they obtain
stronger bounds for positive $n$. This feature is particularly striking in the
case of MINOS/MINOS+. In this case the overall bound is dominated by the
analysis of KamLAND and MINOS data. In the right panel of Fig. 3 we compare
the current best bound for each $n$ with the sensitivities for JUNO and
DUNE/DUNE-HE. We find that JUNO will be able to improve the bound for negative
$n$, while DUNE will improve the bound for $n\geq 0$. Note that the
sensitivity for DUNE using the high-energy beam can improve or worsen when
comparing with the standard beam depending on $n$. Since the standard energy
beam peaks at smaller energies, stronger bounds can be obtained for negative
energy dependencies than with the high-energy beam.

Figure 4: Current 90% C.L. upper limits (left) and future sensitivities for
Model B ($\Gamma\equiv\Gamma_{21}=\Gamma_{31}$ and $\Gamma_{32}=0$), as a
function of $n$ (see Eq. (6)).

Figure 5: Current 90% C.L. upper limits (left) and future sensitivities for
Model C ($\Gamma\equiv\Gamma_{21}=\Gamma_{32}$ and $\Gamma_{31}=0$), as a
function of $n$ (see Eq. (6)).

Figure 6: Current 90% C.L. upper limits (left) and future sensitivities for
Model D ($\Gamma\equiv\Gamma_{31}=\Gamma_{32}$ and $\Gamma_{21}=0$), as a
function of $n$ (see Eq. (6)).

The bounds and sensitivities for Models B and C (Figs. 4 and 5) are very
similar to the ones obtained for Model A, for $n<0$. As anticipated in Section
II.2 we see that as long as $\Gamma_{21}$ and either $\Gamma_{31}$ or
$\Gamma_{32}$ are active, the effect on the oscillation probability at
reactors is similar to the case when all $\Gamma$ are switched on, and hence
the results are nearly the same. For positive $n$ the bound is dominated by
the analysis of data from accelerators. In this case we find that the
sensitivity is better in Model C than in Model B, since Model C uses
$\Gamma_{32}$ and Model B $\Gamma_{31}$, which has a smaller effect on the
oscillation probability as we saw in Section II.2. The situation changes for
model D ($\Gamma_{31}=\Gamma_{32}$ and $\Gamma_{21}=0$), shown in Fig. 6.
Since the fast oscillations due to $\Delta m_{31}^{2}$ and $\Delta m_{32}^{2}$
are averaged at KamLAND, KamLAND has no sensitivity at all to bound this
model. For the other experiments the bounds on this model are again very
similar to the ones obtained for Model A. Note that since JUNO will be able to
resolve the fast oscillations, the JUNO sensitivity does not disappear as
shown in the right panel of the figure.

For the remaining models only one parameter is active, $\Gamma_{21}$ (see Fig.
7), $\Gamma_{31}$ (see Fig. 8), and $\Gamma_{32}$ (see Fig. 9).

Figure 7: Current 90% C.L. upper limits (left) and future sensitivities for
Model E ($\Gamma\equiv\Gamma_{21}$ and $\Gamma_{31}=\Gamma_{32}=0$), as a
function of $n$ (see Eq. (6)).

In the first case we find that all experiments except NOvA are capable of
bounding $\Gamma_{21}$. It should be noted that for all experiments under
consideration, except KamLAND and JUNO, the $\Delta m_{21}^{2}$-driven
oscillation is not developed yet, and the effect of activating $\Gamma_{21}$
can be cancelled by modifying $\sin^{2}\theta_{12}$ accordingly. These
experiments are capable of bounding $\Gamma_{21}$ because of the prior from
the analysis of solar data [3] imposed in our analyses. If we left
$\sin^{2}\theta_{12}$ free in the analysis the bounds in this model would
become weaker or would eventually disappear completely as in the case of NOvA.
In the case of KamLAND and JUNO the main oscillation channel is directly
affected and therefore we could always set a bound. The other two cases (where
we switch on only $\Gamma_{31}$ or $\Gamma_{32}$) are very similar. The
differences in the bounds come from the fact that these models are either
effecting the oscillation term which goes with $\Delta m_{31}^{2}$ or the one
with $\Delta m_{32}^{2}$, which have slightly different amplitudes, as already
discussed. In the case of the reactor neutrinos only in a few cases a bound
can be obtained, because the data can be fit well using only one oscillating
term with a modified amplitude. However, experiments of next generation will
be capable of probing all models for every energy dependence.

Figure 8: Current 90% C.L. upper limits (left) and future sensitivities for
Model F ($\Gamma\equiv\Gamma_{31}$ and $\Gamma_{21}=\Gamma_{32}=0$), as a
function of $n$ (see Eq. (6)).

Figure 9: Current 90% C.L. upper limits (left) and future sensitivities for
Model G ($\Gamma\equiv\Gamma_{32}$ and $\Gamma_{21}=\Gamma_{31}=0$), as a
function of $n$ (see Eq. (6)).

We summarize all bounds that we have obtained for each model in Table 3. As
can be seen, the most constraining limit is often obtained in KamLAND and
MINOS, although other experiments also contribute in a few cases.

Decoherence Model | $\mathbf{n=-2}$ | $\mathbf{n=-1}$ | $\mathbf{n=0}$ | $\mathbf{n=+1}$ | $\mathbf{n=+2}$
---|---|---|---|---|---
A: $\Gamma_{21}=\Gamma_{31}=\Gamma_{32}$ |  $7.8\times 10^{-27}$ (KL) |  $1.8\times 10^{-24}$ (KL) |  $5.1\times 10^{-24}$ (M) |  $3.0\times 10^{-25}$ (M) |  $1.3\times 10^{-26}$ (M)
B: $\Gamma_{21}=\Gamma_{31}$, $\Gamma_{32}=0$ |  $7.9\times 10^{-27}$ (KL) |  $1.8\times 10^{-24}$ (KL) |  $2.4\times 10^{-23}$ (N) |  $2.3\times 10^{-24}$ (M) |  $1.0\times 10^{-25}$ (M)
C: $\Gamma_{21}=\Gamma_{32}$, $\Gamma_{31}=0$ |  $7.9\times 10^{-27}$ (KL) |  $1.8\times 10^{-24}$ (KL) |  $9.4\times 10^{-24}$ (M) |  $5.7\times 10^{-25}$ (M) |  $2.5\times 10^{-26}$ (M)
D: $\Gamma_{31}=\Gamma_{32}$, $\Gamma_{21}=0$ |  $6.9\times 10^{-25}$ (R) |  $2.1\times 10^{-23}$ (T2K) |  $5.6\times 10^{-24}$ (M) |  $3.3\times 10^{-25}$ (M) |  $1.5\times 10^{-26}$ (M)
E: $\Gamma_{21}$, $\Gamma_{31}=\Gamma_{32}=0$ |  $7.9\times 10^{-27}$ (KL) |  $1.8\times 10^{-24}$ (KL) |  $3.2\times 10^{-23}$ (M) |  $2.2\times 10^{-24}$ (M) |  $1.0\times 10^{-25}$ (M)
F: $\Gamma_{31}$, $\Gamma_{21}=\Gamma_{32}=0$ |  $1.0\times 10^{-24}$ (R) |  $1.9\times 10^{-23}$ (T2K) |  $2.3\times 10^{-23}$ (N) |  $2.2\times 10^{-24}$ (M) |  $1.0\times 10^{-25}$ (M)
G: $\Gamma_{32}$, $\Gamma_{21}=\Gamma_{31}=0$ |  $4.0\times 10^{-23}$ (T2K) |  $6.5\times 10^{-23}$ (T2K) |  $1.1\times 10^{-23}$ (M) |  $6.6\times 10^{-25}$ (M) |  $3.0\times 10^{-26}$ (M)

Table 3: Summary of results: each column shows the most constraining upper
limit on $\Gamma_{ij}$, in GeV, for each model (A to G) and value of $n$. We
also clarify, within parenthesis, which experiment sets the bound (KL =
KamLAND, R = RENO, M = MINOS/MINOS+, N = NOvA).

##  V Conclusions

Working in the open quantum system framework, we have analyzed a set of
neutrino oscillation experiments and we have obtained bounds on quantum
decoherence for several phenomenological models. We parametrize the energy
dependence of such quantum decoherence effects as $\propto E^{n}$, and
consider both energy-suppressed ($n>0$), energy-independent ($n=0$) and
energy-enhanced ($n<0$) models. Our analysis of MINOS/MINOS+ and T2K data
updates the results that were obtained in Ref. [40], while providing the first
analysis of NOvA data in this context. The data of KamLAND had already been
analyzed in Ref. [32] using $n=-1,0,1$. For these cases we find good agreement
on the bounds with the authors of Ref. [32] and in addition we provide the
bounds for $n=\pm 2$. Our analyses provide the strongest bounds to date for
negative values of $n$. It should be further noted that very strong bounds
were obtained from a combined analyses of solar and KamLAND data in Ref. [30].
However, these limits are model-dependent and in particular do not apply to a
scenario in which only decoherence effects are included (as it is always the
case in this paper), as was already argued in Refs. [32, 76]. We find that the
bounds for negative $n$ will be improved by about one or two orders of
magnitude, depending on the model under consideration, at JUNO. Regarding
$n\geq 0$ the bounds obtained from MINOS/MINOS+, T2K, and NOvA, can be
improved using DUNE and DUNE-HE. The capabilities of DUNE where already
discussed in Ref. [50] focusing only on $n=0$. For completeness, we note that
very strong bounds on quantum decoherence can also obtained from atmospheric
neutrinos [36].

As a final note, in this paper we have performed analyses of each experiment
individually. We only imposed a prior on $\sin^{2}\theta_{12}$ from solar
neutrino measurements, and after finding that KamLAND’s $\Delta m_{21}^{2}$
measurement is robust under decoherence we added another prior on this
parameter in the analysis of the remaining experiments. The measurements of
the standard parameters at the other experiments might not be robust anymore.
We have found, for example, that the measurement of $\sin^{2}\theta_{13}$ at
Daya Bay and RENO is affected, and therefore this parameter must be left to
vary freely in the analyses of the remaining experiments. A truly combined
analysis might improve a bit the bounds that we have obtained here. The
measurements of the standard parameters might be more precise from a
combination of experiments and might therefore break the degeneracies with the
$\Gamma_{ij}$ leading to slightly stronger bounds. Since the improvement from
such a complicated analysis is not expected to be very large, we decided to
focus on the experiments individually.

## Acknowledgments

This work has been supported by a STSM Grant from the COST Action “Quantum
gravity phenomenology in the multi-messenger approach”, CA18108. C.A.T. is
very thankful for the hospitality at the Niels Bohr Institute in Copenhagen
where part of this work has been performed. V.D.R. is grateful for the kind
hospitality received at the INFN and Physics Department of the University of
Torino, where part of the work was done. V.D.R. acknowledges financial support
by the CIDEGX/2022/20 grant (project “D’AMAGAT”) funded by Generalitat
Valenciana. C.G. and C.A.T. are supported by the research grant “The Dark
Universe: A Synergic Multimessenger Approach” number 2017X7X85K under the
program “PRIN 2017” funded by the Italian Ministero dell’Istruzione,
Università e della Ricerca (MIUR). C.A.T. also acknowledges support from
Departments of Excellence grant awarded by MIUR and the research grant TAsP
(Theoretical Astroparticle Physics) funded by Istituto Nazionale di Fisica
Nucleare (INFN). T.S. is supported by research grant CF19-0652 funded by
Carlsbergfondet.

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