# Neutrino CPT violation in the solar sector

G. Barenboim<EMAIL_ADDRESS>Instituto de Física Corpuscular, CSIC-
Universitat de València, C/Catedrático José Beltrán 2, Paterna 46980, Spain
Departament de Física Teòrica, Universitat de València, C/ Dr. Moliner 50,
Burjassot 46100, Spain  P. Martínez-Miravé<EMAIL_ADDRESS>Instituto de
Física Corpuscular, CSIC-Universitat de València, C/Catedrático José Beltrán
2, Paterna 46980, Spain  Departament de Física Teòrica, Universitat de
València, C/ Dr. Moliner 50, Burjassot 46100, Spain  C.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  M. Tórtola
<EMAIL_ADDRESS>Instituto de Física Corpuscular, CSIC-Universitat de
València, C/Catedrático José Beltrán 2, Paterna 46980, Spain  Departament de
Física Teòrica, Universitat de València, C/ Dr. Moliner 50, Burjassot 46100,
Spain

###### Abstract

In this paper we place new bounds on CPT violation in the solar neutrino
sector analyzing the results from solar experiments and KamLAND. We also
discuss the sensitivity of the next-generation experiments DUNE and Hyper-
Kamiokande, which will provide accurate measurements of the solar neutrino
oscillation parameters. The joint analysis of both experiments will further
improve the precision due to cancellations in the systematic uncertainties
regarding the solar neutrino flux. In combination with the next-generation
reactor experiment JUNO, the bound on CPT violation in the solar sector could
be improved by one order of magnitude in comparison with current constraints.
The distinguishability among CPT-violating neutrino oscillations and neutrino
non-standard interactions in the solar sector is also addressed.

##  I Introduction

Symmetries are a key ingredient of any area of science. They are present in
chemistry and biology, but it is indeed in particle physics where they play a
fundamental role. There, the structure of non-elementary particles themselves
and their interactions is deduced from symmetries and invariance properties.
The very existence of particle masses follows the same lines. Among all the
symmetries present in particle physics, CPT invariance played a defining role.
It surely is one of Nature’s most essential symmetries and its invariance has
been used as a guiding tool to construct models. This is why its experimental
validation is so crucial and the reason behind so many experimental efforts to
test it [1].

In particle physics, field theory is the mathematical tool to construct
models. Through the field theory lens, perhaps one of the most intriguing
properties of free antiparticles, courtesy of CPT invariance, is that they may
be mathematically regarded as if they were simply particles with the same mass
and opposite charge (as compared with their counterparts), but moving
backwards in time and space. Clearly, CPT invariance is embedded in the
founding pillars of our model since its construction. Shortly, it states that
the three independent operations of charge conjugation (C), parity (P), and
time reversal (T), if simultaneously performed, would not modify any
measurable property of the system. Furthermore, the CPT theorem [2] guarantees
that any local, relativistic quantum field theory that preserves Lorentz
invariance automatically conserves CPT. And precisely because of its anti-
unitary nature, the CPT operator connects the S-matrix of a process to the
inverse process’s S-matrix, where particles are replaced by their
antiparticles and all the spin components are reversed. Needless to say, this
does not imply that two CPT conjugated processes will have the same
probability. A particle will not decay at the same rate as its antiparticle to
the same final states. Instead, the sum of all partial decay rates, the
lifetime of this particle, would be the same as that of its antiparticle.

Precisely because it is so deeply buried in field theory, its evaluation using
elementary particles, and not composites, becomes essential, and the best
system where this can be done is in neutrino experiments. Certainly, a
potential discovery of CPT violation would imply that at least one of its key
axioms like Lorentz invariance, interaction locality, or unitarity must be
rejected. Neutrinos are not only the ideal system for CPT to be tested, they
offer also the best opportunity to do it from an experimental as well as a
theoretical point of view. The origin of neutrino mass and its smallness is
not satisfactorily addressed by the Standard Model and its incorporation
generally comes along with new particles, new interactions and new scales.
Some plausible explanations invoke the existence of a very high scale, not far
from where we expect gravity to be non-local, and where new physical laws can
appear or the old ones get modified to incorporate Lorentz and CPT violation.
Then, neutrinos can be thought of as our window to such high scales and our
chance to test fundamental symmetries to an unprecedented level.

In the standard picture of the three-neutrino paradigm, the neutrino
oscillation parameters are fairly well measured, see Refs. [3, 4, 5]. However,
it is well known that new physics might affect these measurements. In this
paper, we assume a non-local field theory [6] where different mass terms for
neutrinos and antineutrinos can be present, allowing for different mixing
parameters. Hence, we consider that flavor oscillations of neutrinos are
described by a set of oscillation parameters, $\Delta m_{ji}^{2}$,
$\theta_{ij}$ and $\delta$, whereas antineutrinos oscillate with
$\Delta\overline{m}_{ji}^{2}$, $\overline{\theta}_{ij}$ and
$\overline{\delta}$. If CPT is conserved, the parameters must coincide (except
for CP-violating effects). This formalism has been used to compute bounds on
the differences between neutrino and antineutrino oscillation parameters [7,
8]. These works focused on analyzing data from accelerator and reactor
experiments [8], while also discussing future sensitivities at DUNE [7].
Sensitivity studies for other experiments have also been performed in Refs.
[9, 10, 11]. In this paper, we extend the discussion of Refs. [7, 8] to the
solar sector. The solar neutrino parameters have been measured using solar
neutrinos observed at many experiments and reactor antineutrinos using the
KamLAND detector. While the measurement of the mixing angles $\theta_{12}$ and
$\overline{\theta}_{12}$ showed good agreement***Note that KamLAND is
basically octant-blind and that if CPT was violated such that the neutrino
parameter was $\sin^{2}\theta_{12}\approx 0.3$ and its antineutrino
counterpart $\sin^{2}\overline{\theta}_{12}\approx 0.7$, we would have no
possibility to observe it., there was a small tension between the measurement
of $\Delta m_{21}^{2}$ and $\Delta\overline{m}_{21}^{2}$. This tension has
been at the 2$\sigma$ level for many years, while with the latest preliminary
Super-Kamiokande solar data it is reduced to the $\sim 1.4\sigma$ level [12].

In this paper, we discuss how this tension between the neutrino and
antineutrino solar mass splitting can evolve in the future. First, we
calculate a new bound on the differences between neutrino and antineutrino
oscillation parameters using the most recent data in Section II. We briefly
summarize the simulation details of the future experiments JUNO [13], Hyper-
Kamiokande [14] and DUNE [15, 16, 17, 18] in Section III, while in Section IV
we discuss the bound that can be obtained from the combined analysis of
reactor and solar experiments. In Section V we estimate the significance of
the tension if the best fit values of the measurements do not change. Next, in
Section VI we discuss the possibility of interpreting a CPT-violating signal
in terms of neutrino nonstandard interactions (NSI) in matter. Finally, we
conclude in Section VII.

##  II Updated bounds on CPT violation in the solar sector

As a first step, we update the bounds on CPT-violating neutrino and
antineutrino oscillation parameters in the solar sector. For a given neutrino
oscillation parameter $x$ and its antineutrino counterpart $\overline{x}$, the
limit on CPT violation is obtained by evaluating

$\chi^{2}(|\Delta
x|)=\chi^{2}(|x-\overline{x}|)=\chi_{\nu}^{2}(x)+\chi_{\overline{\nu}}^{2}(\overline{x})\,,$
(1)

where $\chi_{\nu}^{2}$ $(\chi_{\overline{\nu}}^{2})$ is the
$\chi^{2}$-function for the analysis of neutrino (antineutrino) data. Note
that this is not the standard procedure where one usually considers the same
value for neutrino and antineutrino oscillation parameters ($x$ =
$\overline{x}$). This approach, however, can lead to impostor solutions, as
shown in Ref. [7].

The constraints reported in Refs. [7, 8] relied on previous measurements of
solar neutrinos [19, 20, 21, 22, 23, 24, 25, 26, 27]. Here we update those
limits using the most recent preliminary results from the Super-Kamiokande
Collaboration [12]. In Fig. 1 we compare the bound that is obtained using
solar data as of 2022 and the one obtained from the older solar data set
discussed in Ref. [3]. From the left panel of Fig. 1, one sees that the bound
on $|\Delta\sin^{2}\theta_{12}|$ is a bit weaker now. This is due to the fact
that the current best fit value for the solar mixing angle
($\sin^{2}\theta_{12}=0.305$) is in slightly less agreement with the KamLAND
best fit point ($\sin^{2}\theta_{12}=0.316$) than the previous solar best fit
value ($\sin^{2}\theta_{12}=0.320$). It is only a tiny difference, but it
propagates to the derived limit on CPT violation. The results for
$|\Delta(\Delta m_{21}^{2})|$ are shown in the right panel of Fig. 1. In this
case, the most striking feature is the better agreement between the solar and
KamLAND measurements, compared to the results obtained in 2019. Now the
significance of the tension between both determinations, i.e. the value of
$\Delta\chi 2$ for $|\Delta(\Delta m_{21}^{2})|$= 0, is lowered from $\sim
2.2\sigma$ to $\sim 1.2\sigma$. The new bounds on CPT violation in the
neutrino sector at 3$\sigma$ are therefore

$\displaystyle|\Delta(\Delta m_{21}^{2})|=$ $\displaystyle|\Delta
m_{21}^{2}-\Delta\overline{m}_{21}^{2}|$ $\displaystyle<3.7\times
10^{-5}~{}\text{eV}^{2},$ $\displaystyle|\Delta(\Delta m_{31}^{2})|=$
$\displaystyle|\Delta m_{31}^{2}-\Delta\overline{m}_{31}^{2}|$
$\displaystyle<2.5\times 10^{-4}~{}\text{eV}^{2},$
$\displaystyle|\Delta\sin^{2}\theta_{12}|=$
$\displaystyle|\sin^{2}\theta_{12}-\sin^{2}\overline{\theta}_{12}|$
$\displaystyle<0.187,$ (2) $\displaystyle|\Delta\sin^{2}\theta_{13}|=$
$\displaystyle|\sin^{2}\theta_{13}-\sin^{2}\overline{\theta}_{13}|$
$\displaystyle<0.029,$ $\displaystyle|\Delta\sin^{2}\theta_{23}|=$
$\displaystyle|\sin^{2}\theta_{23}-\sin^{2}\overline{\theta}_{23}|$
$\displaystyle<0.19,$

where the constraints not related to the solar sector are taken from Ref. [8].
Note that the updates in the long-baseline sector, Refs. [28, 29], which
appeared after Ref. [8] are not expected to change these bounds significantly.

Figure 1:  Limits on CPT-violating neutrino oscillation parameters in the
solar sector using KamLAND and solar neutrino data. In both panels, solid
(dashed) lines correspond to the analysis with the 2022 (2019) Super-
Kamiokande solar data set.

##  III Next generation experiments

Although the current bounds on neutrino properties we have presented above
represent the world’s best absolute bound on CPT violation†††Relative
precision bounds, as the bound from the kaon system [30], depend on the choice
of the scale used in the denominator, the averaged mass of the kaons, which
clearly is not associated to CPT in any way., next-generation experiments have
the potential to improve them to an impressive level. In this section, we
discuss the simulation details of the next generation of most CPT-wise
relevant experiments. Regarding antineutrinos, we consider JUNO, while for the
neutrino analysis we focus on the solar analyses of Hyper-Kamiokande and DUNE.

###  III.1 Reactor antineutrinos at JUNO

The Jiangmen Underground Neutrino Observatory (JUNO) [13] is a next-generation
medium-baseline reactor experiment. Its main scientific goals include the
precise measurement of the solar neutrino oscillation parameters, the
atmospheric mass splitting, and the neutrino mass ordering, apart from many
other physics opportunities. JUNO will consist of eight reactors [31, 32]
located at around 52 km distance from the main detector. The main signal at
the JUNO detector 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. Our
simulation of the experiment follows the descriptions of Refs. [13, 31, 32],
and in particular the one of Ref. [33]. Regarding the reactor flux, we use the
parameterization of Huber-Mueller [34, 35]. The fission fractions are assumed
to be the same for all reactors and are kept fixed at $f_{235}=0.561$,
$f_{238}=0.076$, $f_{239}=0.307$, $f_{241}=0.056$ [36]. We assume a detector
with a fiducial mass of 20 kt, a selection efficiency of 82.2% [32] and an
energy resolution of 2.9% [37]. The cross-section is taken from Ref. [38]. An
important source of background in JUNO is the contribution of antineutrinos
from the nuclear power plants at Daya Bay and Huizhou, which depends on the
oscillation parameters. This signal has been included in our simulation as two
independent reactors with 17.4 GW thermal power at 215 and 265 km distance,
respectively. We also include accidental, fast neutron, 9Li/8He, $\alpha-$n
and geo-neutrino background components, as extracted from Refs. [13, 32]. Our
statistical analysis considers an overall 2% flux uncertainty, an 0.8%
uncertainty on the power of each core, and a 1% uncorrelated shape
uncertainty. The other relevant oscillation parameters
($\Delta\overline{m}_{31}^{2}$ and $\sin^{2}\overline{\theta}_{13}$) have been
marginalized over in the analysis, although their effect on the precision of
the solar neutrino parameters is negligible.

###  III.2 Solar neutrinos at Hyper-Kamiokande

Hyper-Kamiokande [14] will be the successor of the water Cherenkov detector
Super-Kamiokande. With a fiducial volume 8.3 times larger than its predecessor
and improved photomultiplier detectors, it will provide great capabilities for
neutrino detection. Hyper-Kamokande’s rich physics program will include the
study of solar, atmospheric and long-baseline accelerator neutrinos produced
at the Japan Proton Accelerator Research Complex (J-PARC). The relevant
process for the study of solar neutrinos is elastic scattering on electrons,
i.e. $\nu_{\alpha}+e^{-}\rightarrow\nu_{\alpha}+e^{-}$. Note that, although
this channel is sensitive to the three neutrino flavors, the cross-section for
electron neutrinos, $\nu_{e}$, is larger than the one for $\nu_{\mu}$ and
$\nu_{\tau}$. We perform two different sensitivity analyses. In the first one,
our simulation follows the approach taken in Ref. [39], where we assume the
same energy resolution as in Super-Kamiokande IV [40]. Likewise, we consider
an energy threshold of 5 MeV‡‡‡Although Super-Kamiokande IV reached an energy
threshold of 3.5 MeV, here we take a more conservative approach. In any case,
the impact of the threshold is actually small, given the large energy
uncorrelated uncertainties in the lowest-energy bins.. We perform an analysis
of the day and night spectrum, taking into account the contributions from 8B
and $hep$ neutrinos. The analysis considers the ratio between the oscillated
and non-oscillated solar neutrino flux, as in [27, 39]. We include a 4% and
30% uncertainty in the normalization of the 8B and $hep$ flux, respectively.
Following [39, 27], we incorporate in the analysis the uncertainty in the
energy scale and uncorrelated systematic uncertainties of the same order of
magnitude as those in Super-Kamiokande IV [40]. We also assume ten years of
data-taking. We refer to the simulation following these considerations as
conservative. Our second analysis takes into account several improvements
expected for Hyper-Kamiokande and hence, it will be referred to as optimal. In
the first place, and according to Ref. [41], we consider an improvement by a
factor of two in the efficiency. Secondly, we assume an energy threshold of
4.5 MeV [14]. Finally, we consider that both the energy resolution and the
uncorrelated systematic uncertainties are improved by a factor of two with
respect to Super-Kamiokande IV, resulting from the improvement of the
photomultipliers. The remaining aspects of the simulation are identical to the
conservative scenario. These considerations seem rather reasonable in the
light of the preliminary sensitivity shown in Ref. [42].

###  III.3 Solar neutrinos at DUNE

The Deep Underground Neutrino Experiment (DUNE) [15, 16, 17, 18] is a next-
generation multipurpose neutrino experiment whose capabilities include the
measurement of MeV neutrinos, among which, are solar neutrinos. Using the
charged-current reaction $\nu_{e}+{}^{40}\text{Ar}\to e^{-}+{}^{40}\text{K}$,
it will be sensitive to electron neutrinos from the Sun, with energies above 9
MeV. A lower energy threshold could be possible if a significant background
reduction is achieved. Our sensitivity analysis considers 10 years of data and
the projected full size of DUNE’s far detector, consisting of 40 kton of
liquid argon. For the cross-section, we consider the baseline configuration
implemented in SNOwGLoBES [43] and we consider an energy resolution
$\sigma(E)/E=0.2$ [16, 44]. We also include backgrounds from 222Rn and neutron
capture [45] with a 10% uncertainty each, and we take into account the
efficiency linearly increasing from 30% at 9 MeV to 60% at 21 MeV [46].
Likewise, the uncertainty in the flux normalization for 8B and $hep$ neutrinos
is the same as in the analysis carried out for Hyper-Kamiokande. The
assumptions in this scenario are rather conservative and it might be possible
that the background contribution could be reduced [47, 48, 49]. Moreover,
existing studies suggest that a much better detection efficiency can be
achieved in the MeV range [50, 51, 52]. We perform two analyses in this paper:
the first one uses only the spectral information for two angular bins, labeled
as day and night, and the conservative choice of backgrounds and efficiencies,
while in the second one, we separate the angular information for the night
events into 10 bins of the same width and, in addition, consider the more
favorable backgrounds and efficiencies. In particular, we consider a reduction
of one order of magnitude in the neutron background rate and perfect
efficiency. The first case can be considered as the conservative scenario,
while the latter is an optimal scenario.

##  IV Future sensitivity to CPT violation in the solar sector

In this section, we discuss how well future experiments can improve the bound
on CPT violation in the solar sector. We will consider the combinations of
JUNO with Hyper-Kamiokande, with DUNE, and with a combined analysis of Hyper-
Kamiokande and DUNE. As explained above, for both solar experiments we
consider a conservative and an optimal experimental configuration. When
discussing the combined analysis we consider always either both conservative
or both optimal configurations.

In order to derive the projected sensitivities, we generate a CPT-conserving
data set using $\sin^{2}\theta_{12}=\sin^{2}\overline{\theta}_{12}=0.32$ and
$\Delta m_{21}^{2}=\Delta\overline{m}_{21}^{2}=7.53\times 10^{-5}$ eV2. The
expected sensitivity of the different experiments to measure the solar
parameters is shown in the left panel of Figure 2. It is clear that in the
CPT-conserving scenario, the measurement will be dominated by JUNO, as
indicated by the green contour. The individual analyses of Hyper-Kamiokande
and DUNE are shown in orange and blue (using the solid lines for the
conservative cases and the dashed ones for the optimal cases), respectively.
One can see that the solar sensitivity at DUNE is better than at Hyper-
Kamiokande for the measurements of both solar oscillation parameters. We
obtain an interesting result for the combined analysis of both Hyper-
Kamiokande and DUNE, see the red contours, where again the solid line is
obtained from the conservative analyses, while the dashed lines are obtained
for the optimal analyses. Note that the experiments share the systematic
uncertainties related to the solar flux and, therefore, the combined
sensitivity is much better than the simple sum of $\chi^{2}$ functions (which
would be very similar to the DUNE-alone analysis). The reason for this
improvement, particularly in the determination of the solar mixing angle, is
that the use of two different detection channels allows breaking the
degeneracy between the 8B flux normalization and $\sin^{2}\theta_{12}$.

Figure 2:  Expected 2$\sigma$ regions from Hyper-Kamiokande (HK), DUNE,
HK+DUNE using conservative and optimal configurations, and JUNO, assuming the
same oscillation parameters for neutrinos and antineutrinos.

Using the different solar analyses of neutrino data, and the antineutrino
results from JUNO we can estimate the future sensitivity to CPT violation in
the solar sector. The $\chi^{2}$ profiles for the differences of the neutrino
and antineutrino oscillation parameters are shown in Fig. 3. The left panel
contains the sensitivity to constrain the difference in the solar mixing
angle. Note that, even though the measurement of the mixing angle at Hyper-
Kamiokande is rather weak, the potential limit from JUNO and Hyper-Kamiokande
data will substantially improve the current 3$\sigma$ bound. This comes from
JUNO’s much more accurate determination of solar oscillation parameters
compared to KamLAND. As expected, the combination of JUNO and DUNE (see the
solid and dashed blue lines in the left panel of Fig. 3) shows a clear
improvement in the sensitivity to $|\Delta\sin^{2}\theta_{12}|$. The joint
analysis of JUNO, Hyper-Kamiokande and DUNE (red lines) is expected to improve
the current limit by one order of magnitude. Notice here that the impact of
considering the conservative or the optimal configurations of DUNE and Hyper-
Kamiokande in the combined analysis results in a difference of a factor two
approximately in the limits on $|\Delta\sin^{2}\theta_{12}|$. The expected
$3\sigma$ sensitivities from the different analyses are summarized in the
second column of Table 1.

| $~{}~{}|\Delta\sin^{2}\theta_{12}|~{}~{}$ |  $|\Delta(\Delta m_{21}^{2})|$ [$10^{-5}$eV2]
---|---|---
current bound |  0.187 |  3.7
JUNO + HK conservative |  0.092 |  7.2
JUNO + HK optimal |  0.073 |  4.7
JUNO + DUNE conservative |  0.043 |  2.9
JUNO + DUNE optimal |  0.029 |  1.1
JUNO + HK + DUNE conservative |  0.018 |  2.4
JUNO + HK + DUNE optimal |  0.011 |  0.8
Table 1: 3$\sigma$ bounds on $|\Delta\sin^{2}\theta_{12}|$ and $|\Delta(\Delta
m_{21}^{2})|$ from current data in comparison with the future sensitivity
expected from the combination of JUNO, Hyper-Kamiokande (HK) and DUNE for the
conservative and optimal configurations.

The sensitivity to $|\Delta(\Delta m_{21}^{2})|$ is shown in the right panel
of Figure 3, and in the third column of Table 1. As in the case of the mixing
angles, the sensitivity of the JUNO+Hyper-Kamiokande analysis is weaker than
for JUNO+DUNE, and it is further improved when combining DUNE and Hyper-
Kamiokande. Note, however, that here the improvement is not as strong as in
the last case, since, as it was shown in Fig. 2, the combination of solar
experiments is more effective for the determination of the mixing angle. On
the other hand, in this case, the assumed configuration of DUNE has a strong
impact on the result, as can be seen by comparing the blue and red solid lines
with the dashed ones. Indeed, for the optimal setup of DUNE, the sensitivity
to $|\Delta(\Delta m_{21}^{2})|$ is improved by more than a factor of two.

Summarizing, we find that the next generation of solar neutrino experiments in
combination with the reactor experiment JUNO will be able to improve the
current bounds on CPT-violating oscillation parameters for neutrinos and
antineutrinos significantly, even by more than one order of magnitude in the
case of the mixing angles.

Figure 3:  Future sensitivity to CPT-violating neutrino and antineutrino
oscillation parameters for different configurations and combinations of
experiments in comparison with the current bounds.

##  V Measuring CPT violation

In this section, we assume that CPT is violated in Nature. Then, we generate
the antineutrino oscillation data using
$\Delta\overline{m}_{21}^{2}=7.53\times 10^{-5}$ eV2, whereas neutrino
oscillation data are generated using $\Delta m_{21}^{2}=6.10\times 10^{-5}$
eV2. We still assume
$\sin^{2}\theta_{12}=\sin^{2}\overline{\theta}_{12}=0.32$, since the very
small differences in the best fit values of the measurements of the mixing
angle would most likely not be visible, not even with the next generation of
neutrino experiments. In the left panel of Figure 4 we show the expected
region in the solar plane obtained from the different parameter settings. Note
that the overall sensitivity of the solar experiments improves for smaller
values of $\Delta m_{21}^{2}$, as can be seen by comparing the regions of
Figure 4 with those of Figure 2. We observe the same behavior as before,
namely the combination of DUNE and Hyper-Kamiokande improves mildly
(significantly) the determination of the mass splitting (mixing angle) when
comparing with the analysis of DUNE alone. Likewise, the optimal DUNE
configuration has a significant impact on the sensitivity to the solar mass
splitting $\Delta m^{2}_{21}$.

In the right panel of Figure 4 we show the $\Delta\chi^{2}$ profiles for the
CPT-violating scenario. This quantifies how the current mild tension in the
measurements of KamLAND and solar data could evolve in the future, in case the
best-fit values of the measurements do not change significantly in either
sector. Note that the tension from the combination of JUNO and Hyper-
Kamiokande would be similar to the current one due essentially to KamLAND and
Super-Kamiokande. However, once we consider DUNE (DUNE+Hyper-Kamiokande), the
tension is pushed to 2.0$\sigma$ (2.3$\sigma$) for the conservative
configuration of both DUNE and Hyper-Kamiokande. The huge improvement of
sensitivity seen in the left panel of Figure 4 when using the optimal
configuration is also evident in the right panel. Actually, DUNE’s optimal
configuration could push the significance of the tension to 5.1$\sigma$
(5.9$\sigma$) for the analysis of DUNE alone (DUNE+Hyper-Kamiokande).

Figure 4:  Left: Expected $2\sigma$ regions from Hyper-Kamiokande (HK), DUNE,
HK+DUNE using conservative and optimal configurations, and JUNO, assuming
different oscillation parameters for neutrinos and antineutrinos. Right:
$\Delta\chi^{2}$ profiles of the difference of mass splittings, quantifying
the tension for the different combinations.

##  VI Disentangling CPT violation and NSI

In this section we discuss the possibility of confusing CPT violation with
neutrino nonstandard interactions. In order to do so we generate a CPT
violating data set but analyze it in a CPT-conserving way including NSI. The
same approach was followed in Ref. [53] in the context of accelerator
neutrinos. Since the solar parameters are by far best measured by JUNO (and
since JUNO is not as much affected by NSI), this reduces to generating a solar
data set considering only solar neutrino oscillations with $\Delta
m_{21}^{2}=6.10\times 10^{-5}$ eV2 and then trying to reconstruct it assuming
$\Delta m_{21}^{2}=7.53\times 10^{-5}$ eV2 (the value measured by KamLAND)
together with NSI.

The evolution of neutrinos in the Sun can be described in an effective two-
neutrino approach, as long as NSI are smaller or of the same order as neutrino
interactions in the Standard Model. In that case, the survival probability is
given by [54, 55]

$\displaystyle P_{ee}=\cos^{4}\theta_{13}P^{2\nu}_{ee}+\sin^{4}\theta_{13}\,,$
(3)

where $P^{2\nu}_{ee}$ is the electron neutrino survival probability derived in
the effective two-neutrino framework from the Hamiltonian

$\displaystyle H^{2\nu}=\frac{\Delta m^{2}_{21}}{4E}\begin{pmatrix}-\cos
2\theta_{12}&\sin 2\theta_{12}\\\ \sin 2\theta_{12}&\cos
2\theta_{12}\end{pmatrix}+\sqrt{2}G_{F}\Bigg{[}\cos^{2}\theta_{13}N_{e}\begin{pmatrix}1&0\\\
0&0\end{pmatrix}+\sum_{f=e,u,d}N_{f}\begin{pmatrix}0&\varepsilon_{f}\\\
\varepsilon^{*}_{f}&\varepsilon^{\prime}_{f}\end{pmatrix}\Bigg{]}\,.$ (4)

This effective Hamiltonian describes the evolution of the states $\nu$ =
$(\nu_{e},\,\nu_{x})$, where $\nu_{x}$ is an admixture of the $\nu_{\mu}$ and
$\nu_{\tau}$ states. It depends on the number density of matter fields,
$N_{f}$, with which neutrinos interact while propagating, i.e. electrons,
$u$-quarks and $d$-quarks. We have introduced the effective NSI parameters
$\varepsilon_{f}$ and $\varepsilon^{\prime}_{f}$, which are related to the NSI
couplings in the neutral-current NSI Lagrangian [56, 57, 58, 59, 39]. For
simplicity, we only allow for NSI between neutrinos and $d$-quarks, denoting
the corresponding NSI parameters, $\varepsilon_{d}$ and
$\varepsilon^{\prime}_{d}$ as $\varepsilon$ and $\varepsilon^{\prime}$.

The results from this analysis are shown in Fig. 5. There, we plot the regions
in the plane of the NSI parameters ($\varepsilon$, $\varepsilon^{\prime}$)
that can mimic the effect of CPT-violating neutrino mass splittings in the
solar neutrino signal. It should be noted that, for the separate analyses of
Hyper-Kamiokande and the conservative setup of DUNE, the point
$\varepsilon=\varepsilon^{\prime}=0$ is within the 2$\sigma$ region. This is
due to the fact that the tension between the two mass splittings,
corresponding to $|\Delta(\Delta m^{2}_{21})|\simeq 1.4\times
10^{-5}\mathrm{eV}$, is quite small for these three data sets, as shown in the
right panel of Fig. 4. Combining the conservative analyses of DUNE and Hyper-
Kamiokande restricts a bit more the allowed region, but
$\varepsilon=\varepsilon^{\prime}=0$ still lies at the border of the $2\sigma$
contour. Only when assuming the optimal DUNE configuration the standard
scenario can be excluded at large confidence level as shown by the dashed
contours in Fig. 5. However, even though the
$\varepsilon=\varepsilon^{\prime}=0$ value would be excluded from the
analysis, the CPT-violating scenario could still be fit very well with NSI.
Indeed, using the combination of optimal Hyper-Kamiokande and optimal DUNE the
best fit value is found at $\varepsilon=-0.01$ and
$\varepsilon^{\prime}=-0.09$ and $\chi^{2}\approx 4.7$. We therefore conclude
that this scenario of CPT violation could not be distinguished from NSI,
unless independent constraints on $\varepsilon^{\prime}$ from another type of
experiment rule out the values preferred here.

Figure 5:  Regions in the NSI plane ($\varepsilon$, $\varepsilon^{\prime}$)
that can mimic the signal of CPT-violating solar neutrino mass splittings at
the $2\sigma$ level in different experiments using conservative and optimal
configurations.

##  VII Conclusions

The Standard Model of particle physics describes to an amazing precision all
the experimental results obtained so far, with the exception of the neutrino
sector. Neutrino masses were absent in the original model, and its
incorporation necessarily brings along new physics. Not surprisingly,
neutrinos themselves offer the opportunity not only to test the model itself,
but also to test the paradigm where the model is immersed: local relativistic
quantum field theory.

If Nature can be described by local relativistic quantum field theory, CPT
conservation is one of the few solid predictions of the paradigm, and then
particles and antiparticles are bounded to have the same mass and, if
unstable, the same lifetime. It is therefore extremely important to know
whether the language we use to describe Nature is the correct one, and test
it.

We do not know the origin of the neutrino mass, let alone its scale. However,
we have measured neutrino mass squared differences to an admirable precision.
In fact, neutrino mass squared differences are better known than some of the
absolute masses of charged fermions and offer us a unique opportunity to test
CPT conservation in an elementary particle system without charge contamination
to an unprecedented level. If CPT violation is related or originated from
quantum gravity, and therefore suppressed by the Planck scale, naively we
would expect its magnitude to be given by $\langle
v^{2}\rangle/M_{\mbox{Planck}}\sim 10^{-5}$ eV, exactly in the ballpark the
next generation of experiments will explore.

In this work, we have shown that CPT violation can be bounded or discovered by
comparing antineutrino measurements at the reactor experiment JUNO with solar
neutrino observations in DUNE and Hyper-Kamiokande. It is important to stress
that DUNE improvements regarding angular bining and efficiencies have promoted
the experiment to be a mayor player in the solar neutrino arena, as our
results for the optimal setup of DUNE show. Improvements in energy resolution
and a reduction of the energy-uncorrelated systematics would also improve
significantly the solar neutrino results of Hyper-Kamiokande.

Had a positive CPT-violating signal been found, we have explored if it could
be explained with NSI. Our results show that both sources of new physics can
actually be confused in solar neutrino experiments. This highlights the
relevance of combining different types of experiments, such as solar and
neutrino scattering experiments to search for physics beyond the Standard
Model [57, 60].

## Acknowledgments

This work has been supported by a STSM Grant from COST Action “Quantum gravity
phenomenology in the multi-messenger approach”, CA18108. C.A.T. is thankful
for the hospitality at IFIC, Valencia where this work has been initiated.
C.A.T. is 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), by a Departments of Excellence grant awarded by MIUR and the research
grant TAsP (Theoretical Astroparticle Physics) funded by Istituto Nazionale di
Fisica Nucleare (INFN). P.M.M. is supported by the Spanish fellowship
FPU18/04571. This work has also been supported by the Spanish grants
PID2020-113775GB-I00 (AEI/10.13039/501100011033), PID2020-113334GB-I00 / AEI /
10.13039/501100011033 and CIPROM/2021/054 (Generalitat Valenciana). G.B. has
received support from the European Union’s Horizon 2020 research and
innovation program under the Marie Skłodowska-Curie grant agreement
860881-HIDDeN.

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