11institutetext: High Energy Theory Group, Physics Department, Brookhaven
National Laboratory, Upton, NY 11973, USA22institutetext: Dipartimento di
Matematica e Fisica, Università di Roma Tre Via della Vasca Navale 84, 00146
Rome, Italy

# How to Identify Different New Neutrino Oscillation Physics Scenarios at DUNE

Peter B. Denton1110000-0002-5209-872X 2 Alessio
Giarnetti2220000-0001-8487-8045 2 and Davide Meloni3330000-0001-7680-6957
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Next generation neutrino oscillation experiments are expected to measure the
remaining oscillation parameters with very good precision. They will have
unprecedented capabilities to search for new physics that modify oscillations.
DUNE, with its broad band beam, good particle identification, and relatively
high energies will provide an excellent environment to search for new physics.
If deviations from the standard three-flavor oscillation picture are seen
however, it is crucial to know which new physics scenario is found so that it
can be verified elsewhere and theoretically understood. We investigate several
benchmark new physics scenarios by looking at existing long-baseline
accelerator neutrino data from NOvA and T2K and determine at what sensitivity
DUNE can differentiate among them. We consider sterile neutrinos and both
vector and scalar non-standard neutrino interactions, all with new complex
phases, the latter of which could conceivably provide absolute neutrino mass
scale information. We find that, in many interesting cases, DUNE will have
good model discrimination. We also perform a new fit to NOvA and T2K data with
scalar NSI.

##  1 Introduction

Neutrino oscillation physics is on the advent of reaching the precision era.
Current long-baseline accelerator experiments NOvA Ayres:2007tu  and T2K
Abe:2011ks  are making the first appearance measurements in accelerator
neutrinos and the next-generation accelerator experiments HK Abe:2015zbg  and
DUNE Acciarri:2015uup  will measure the appearance probabilities precisely
providing a wealth of information about the three remaining oscillation
unknowns: the octant of $\theta_{23}$, the atmospheric mass ordering, and the
value of the complex phase $\delta$.

These powerful experiments will provide the strongest tests yet of the
standard three-flavor oscillation hypothesis. In the event there is new
physics, however, it is important to check if that new physics can be robustly
identified compared to alternatives. To address this question, we use several
benchmark new physics points, motivated by the slight tension in the NOvA and
T2K data Denton:2020uda , see also Chatterjee:2020kkm ; Chatterjee:2020yak ;
Miranda:2019ynh ; Forero:2021azc ; deGouvea:2022kma ; Majhi:2022wyp  and first
test the level at which DUNE can identify them, and then the level at which
they can be differentiated. As scalar NSI has not yet been tested with T2K and
NOvA data, we derive the first constraints on scalar NSI with existing long-
baseline data.

Testing these benchmarks against no new physics and against each other will
provide a fairly comprehensive overview of the capability of DUNE to correctly
identify the new physics scenarios and the parameters within the scenario that
are preferred, although partial degeneracies among different new physics
scenarios for precise values of the parameters may weakened this sensitivity
in some cases. HK will also have sensitivity to many of these scenarios and
their combination will be particularly powerful for disentangling things.
Nonetheless, as we will show in section 6, DUNE alone will provide very good
model discrimination capabilities.

##  2 New Physics Scenarios

There are numerous new physics scenarios affecting neutrino oscillations
Arguelles:2019xgp ; Arguelles:2022xxa  including neutrino decay
Berryman:2014qha ; Picoreti:2015ika ; SNO:2018pvg ; Gonzalez-Garcia:2008mgl ;
Gomes:2014yua ; Abrahao:2015rba ; Pagliaroli:2016zab ; Coloma:2017zpg ;
Choubey:2017dyu ; Gago:2017zzy ; Choubey:2017eyg ; Denton:2018aml ;
deSalas:2018kri ; Ascencio-Sosa:2018lbk ; Choubey:2018cfz ; Funcke:2019grs ;
Abdullahi:2020rge ; Ghoshal:2020hyo ; Porto-Silva:2020gma ; Choubey:2020dhw ;
Picoreti:2021yct , Lorentz invariance violation and CPT violation
Kostelecky:2003cr ; LSND:2005oop ; MINOS:2008fnv ; MINOS:2010kat ;
IceCube:2010fyu ; Kostelecky:2011gq ; MiniBooNE:2011pix ; DoubleChooz:2012eiq
; MINOS:2012ozn ; Diaz:2013iba ; Rebel:2013vc ; Super-Kamiokande:2014exs ;
EXO-200:2016hbz ; IceCube:2017qyp ; T2K:2017ega ; DayaBay:2018fsh , background
dark matter fields Reynoso:2016hjr ; Berlin:2016woy ; Krnjaic:2017zlz ;
Brdar:2017kbt ; Liao:2018byh ; Capozzi:2018bps ; Huang:2018cwo ;
Farzan:2019yvo ; Cline:2019seo ; Dev:2020kgz ; Huang:2021kam ; Losada:2021bxx
; Chun:2021ief ; Dev:2022bae , neutrino decoherence or wave-packet separation
Liu:1997km ; Chang:1998ea ; Benatti:2000ph ; Lisi:2000zt ; Super-
Kamiokande:2004orf ; BalieiroGomes:2016ykp ; BalieiroGomes:2018gtd ;
Stuttard:2021uyw ; Hellmann:2021jyz , unitary violation Blennow:2016jkn ;
Parke:2015goa ; Denton:2021mso , non-standard neutrino interactions (NSI)
Wolfenstein:1977ue ; deGouvea:2015ndi ; Proceedings:2019qno ; Denton:2020uda ;
Bakhti:2020fde ; DUNE:2020fgq ; Chatterjee:2021wac ; Giarnetti:2021wur , and
sterile neutrinos Donini:2007yf ; Dighe:2007uf ; Berryman:2015nua ;
DUNE:2020fgq ; Acero:2022wqg . We focus on the last two: NSI and sterile
neutrinos. NSI provides a general framework for quantifying modifications to
the neutrino oscillation probability due to a new interaction with matter
particles. We further split NSIs into vector NSIs, which act like the regular
matter effect Wolfenstein:1977ue  but possibly in a different basis or with a
different dependence on the matter particles, and scalar NSI Ge:2018uhz ,
which acts like a new mass term for neutrinos sourced by matter particles. We
focus on NSI with off-diagonal couplings as these are the parameters for which
appearance measurements are particularly crucial to constrain. Diagonal NSI
are best constrained with solar and atmospheric neutrino oscillations and
neutrino scattering. Sterile neutrinos are well motivated extensions to the
standard three-flavor neutrino oscillation picture due to theoretical
arguments based on the fact that neutrinos have mass, as well as due to a host
of confusing anomalies LSND:2001aii ; T2K:2014xvp ; Giunti:2010zu ;
Mention:2011rk ; Barinov:2021asz ; MiniBooNE:2020pnu ; Denton:2021czb  which
could be explained by new light sterile neutrinos in the $m_{4}\sim 1$ eV
range. Sterile neutrino searches are also generally related to those involving
unitary violation but are a bit more comprehensive in their ability to
directly probe the new mass scale. In the following subsections we review the
formalism of each of these three cases as they apply for long-baseline
neutrino oscillations.

###  2.1 Vector Non-Standard Neutrino Interaction

Since the introduction of the matter effects in neutrino oscillations, the
possibility that neutrinos can undergo NSIs with matter has been widely
studied. Focusing only on neutral current vector NSI, which dominates over the
axial-vector current assuming comparable coupling strengths, we can describe
vector NSI using an effective theory approach. The Lagrangian now includes the
following terms:

${\cal L}^{eff}_{\rm vector\
NSI}=-2\sqrt{2}G_{F}\sum_{f,\alpha,\beta}\varepsilon^{f}_{\alpha\beta}(\bar{\nu}_{\alpha}\gamma_{\rho}\nu_{\beta})(\bar{f}\gamma^{\rho}f)\,,$
(1)

where $G_{F}$ is the Fermi constant, $\varepsilon_{\alpha\beta}^{f}$ is the
parameter which describes the strength of the NSI, $f$ is a first generation
SM charged-fermion ($e$, $u$, or $d$) and $\alpha$ and $\beta$ denote the
neutrino flavors $e$, $\mu$ or $\tau$. The $\varepsilon$ parameter can be
related to the parameters in a simplified model or even a UV complete
scenarios. Since these details do not affect oscillations within a single
experiment, we focus only on the $\varepsilon$ effective parameter. Notice
that in this subsection we will only consider interactions mediated by vector
particles.

The presence of such interactions modifies the neutrino oscillation
Hamiltonian to

$\displaystyle
H=\frac{1}{2E}\left[UM^{2}U^{\dagger}+a\left(\begin{array}[]{ccc}1+\varepsilon_{ee}&\varepsilon_{e\mu}&\varepsilon_{e\tau}\\\
\varepsilon_{e\mu}^{*}&\varepsilon_{\mu\mu}&\varepsilon_{\mu\tau}\\\
\varepsilon_{e\tau}^{*}&\varepsilon_{\mu\tau}^{*}&\varepsilon_{\tau\tau}\end{array}\right)\right]\,,$
(5)

where $U$ is the PMNS matrix Pontecorvo:1957cp ; Maki:1962mu ,
$M^{2}=\operatorname{diag}(0,\Delta m_{21}^{2},\Delta m_{31}^{2})$,
$a=2\sqrt{2}G_{F}N_{e}E$, and $N_{e}$ is the electron number density. Due to
the hermiticity of the Hamiltonian matrix, the diagonal NSI couplings
$\varepsilon_{\alpha\alpha}$ must be real, while the non-diagonal ones are in
general complex and can be written as
$\varepsilon_{\alpha\beta}=|\varepsilon_{\alpha\beta}|e^{i\phi_{\alpha\beta}}$.
Since we can subtract a matrix proportional to the identity without changing
the oscillation probabilities, only two of the diagonal NSI parameters are
independent. The Hamiltonian level NSI parameters relevant for neutrino
oscillations, those without superscripts $\varepsilon_{\alpha\beta}$, are
related to the Lagrangian level NSI terms via

$\varepsilon_{\alpha\beta}=\sum_{f\in\\{e,u,d\\}}\frac{N_{f}}{N_{e}}\varepsilon_{\alpha\beta}^{f}\,,$
(6)

where $N_{f}$ is the number density of fermion $f$. When looking at NSI in the
Sun and the Earth simultaneously one must consider the Lagrangian level NSI to
accurately translate between them. Here we are only considering experiments in
the Earth’s crust so we can safely work with the Hamiltonian level parameters.
Many studies of vector NSI exist in the literature, see e.g. deGouvea:2015ndi
; Coloma:2015kiu ; Denton:2020uda ; Chatterjee:2020kkm ; Esteban:2020cvm .

Several analyses of oscillation data444Scattering data is also sensitive to
NSI Coloma:2017egw ; Liao:2017uzy ; Farzan:2018gtr ; Denton:2018xmq ;
Denton:2020hop ; Denton:2022nol , although these data sets have a non-trivial
dependence on the mediator mass, while oscillation data is essentially
Joshipura:2003jh ; Grifols:2003gy ; Gonzalez-Garcia:2006vic ;
Bandyopadhyay:2006uh ; Samanta:2010zh ; Davoudiasl:2011sz ; Wise:2018rnb ;
Smirnov:2019cae ; Coloma:2020gfv  independent of it. have been considered
under various assumptions. A recent global analysis of oscillation data in the
context of NSIs has estimated the constraints on the NSI parameters in the
context of both LMA (Large Mixing Angle solution of the solar neutrino
problem) and LMA-Dark results are shown, with the difference mainly affecting
$\varepsilon_{ee}$. The LMA-Dark solution Miranda:2004nb ; Escrihuela:2009up ;
Gonzalez-Garcia:2013usa ; Bakhti:2014pva ; Coloma:2016gei ; Farzan:2017xzy ;
Coloma:2017egw ; Denton:2021vtf ; Denton:2022nol  is the solution with
$\varepsilon_{ee}\simeq-2$ and the opposite sign555We take the definition of
the three mass eigenstates as $|U_{e1}|>|U_{e2}|>|U_{e3}|$. Thus
$\theta_{12}<45^{\circ}$ by definition and the sign of $\Delta m^{2}_{21}$ has
been measured experimentally with solar neutrinos. Some define the mass
eigenstates by $m_{1}<m_{2}$, $|U_{e1}|>|U_{e3}|$, and $|U_{e2}|>|U_{e3}|$. In
this case $\Delta m^{2}_{21}>0$ by definition and the octant of $\theta_{12}$
is to be determined experimentally. See Denton:2020exu ; Denton:2021vtf . on
$\Delta m^{2}_{31}$, $\Delta m^{2}_{21}$, and $\delta$. For a recent
discussion of LMA-Dark in the context of the latest reactor constraints see
Denton:2022nol . We note that while the allowed values in the global analysis
Esteban:2018ppq  they find might seem to disfavor some of the values preferred
in recent analyses long-baseline data Denton:2020uda ; Chatterjee:2020kkm
used in this paper (see table 6), it is easy to see that the constraints on
real NSI and NSI with a large complex component can be quite different.

It might appear that charged lepton flavor violating probes would always be
stronger than those from oscillations, but numerous UV complete models with
large $\varepsilon_{\alpha\beta}\gtrsim 0.1$ exist in the literature where
oscillations provide the strongest probes Forero:2016ghr ; Denton:2018dqq ;
Dey:2018yht ; Babu:2017olk ; Farzan:2016wym ; Farzan:2015hkd ; Farzan:2015doa
; Babu:2019mfe ; Proceedings:2019qno . All of these models can be recast into
the language of NSI which is exactly what makes NSI such an attractive BSM
scenario to investigate.

In order to gain a good understanding of the impact of vector NSI on
oscillation experimental data, we derive approximate expressions for the
vector NSI contribution to neutrino oscillations in matter in appendix A by
performing a perturbative expansion in various parameters known to be small.

###  2.2 Scalar Non-Standard Neutrino Interaction

In addition to a vector mediator, one can consider different Lorentz structure
for the underlying theory behind a new neutrino interaction. Scalar NSI has
been investigated in the context of some neutrino oscillation experiments as
well as early universe constraints Ge:2018uhz ; Babu:2019iml ; Smirnov:2019cae
; Venzor:2020ova ; Medhi:2021wxj ; Medhi:2022qmu ; Sarkar:2022ujy ;
Dutta:2022fdt . All previous studies, to our knowledge, focused on the
diagonal scalar NSI parameters; instead, we focus here on the off-diagonal
parameters. Early universe constraints and fifth-force probes may be stronger
than terrestrial probes in many cases, although not necessarily all, depending
primarily on the mediator mass Smirnov:2019cae . Given the highly disparate
environments between the early universe and terrestrial oscillations for which
an UV complete model may behave differently, in addition to some hints for a
new interaction in early universe data Kreisch:2019yzn ; Barenboim:2019tux ,
we consider this scenario in DUNE data nonetheless. That said, we do caution
the reader to be aware of important non-oscillation constraints on scalar NSI.

The effective Lagrangian for scalar NSI is:

${\cal L}^{eff}_{\rm scalar\
NSI}=\frac{y_{f}y_{\alpha\beta}}{m_{\phi}^{2}}(\bar{\nu}_{\alpha}\nu_{\beta})(\bar{f}f)\,,$
(7)

where the $y$’s are the Yukawa couplings to matter fermions and neutrinos and
$m_{\phi}$ is the mass of the scalar mediator. We note that this can no longer
be considered as a matter potential, but it can be seen as a Yukawa
interaction term for Dirac neutrinos that induces a correction to the mass
term that depends on the density of fermions sourcing the new interactions.
After the spin summation of the environmental fermion, the Dirac equation for
neutrinos becomes:

$\bar{\nu}_{\beta}\left[i\partial_{\mu}\gamma^{\mu}+\left(M_{\beta\alpha}+\frac{\sum_{f}N_{f}y_{f}y_{\alpha\beta}}{m_{\phi}^{2}}\right)\right]\nu_{\alpha}=0\,,$
(8)

where $M_{\beta\alpha}$ is the Dirac mass matrix of the neutrinos and $N_{f}$
is the number density of fermion $f$. The effect of such scalar NSI is thus a
modification of the neutrino mass matrix by a factor of $\delta M$. As a
consequence, the Hamiltonian governing neutrino oscillations is modified from
the diagonal $M^{2}$ term to $(M+\delta M)(M+\delta M)^{\dagger}$. We then
parameterize the correction term $\delta M$ as:

$\delta M=\sqrt{|\Delta
m_{31}^{2}|}\begin{pmatrix}\eta_{ee}&\eta_{e\mu}&\eta_{e\tau}\\\
\eta_{e\mu}^{*}&\eta_{\mu\mu}&\eta_{\mu\tau}\\\
\eta_{e\tau}^{*}&\eta_{\mu\tau}^{*}&\eta_{\tau\tau}\end{pmatrix}\,,$ (9)

where we have chosen to scale the size of $\delta M$ relative to
$\sqrt{|\Delta m^{2}_{31}|}$ to make the parameters of the model,
$\eta_{\alpha\beta}$, dimensionless666Note that this $\eta_{\alpha\beta}$
parameter is unrelated to the parameter sometimes used in the context of
unitarity violation, see e.g. Fernandez-Martinez:2007iaa .. We have also
chosen to make $\delta M$ Hermitian although it need not be (see e.g.
Sarkar:2022ujy ) depending on if the scalar mediator is real or complex; we
encourage further research into the non-Hermitian case. In addition, while we
have used the Dirac equation for Dirac neutrinos, the effect is the same for
Majorana neutrinos in the ultrarelativistic case, see e.g. Babu:2020ivd .

Unlike in the vector case parameterized by the dimensionless
$\varepsilon_{\alpha\beta}$ parameters, these dimensionless scalar NSI
parameters, $\eta_{\alpha\beta}$, are proportional to the matter density since
they depend on $N_{f}$. As in the vector case, the diagonal parameters are
real while the off-diagonal elements are complex,
$\eta_{\alpha\beta}=|\eta_{\alpha\beta}|e^{i\phi_{\alpha\beta}}$. To be
explicit, we can relate these $\eta_{\alpha\beta}$ parameters to the
parameters of the underlying theory as

$\eta_{\alpha\beta}=\frac{1}{m_{\phi}^{2}\sqrt{|\Delta
m^{2}_{31}|}}\sum_{f}N_{f}y_{f}y_{\alpha\beta}\,.$ (10)

Since the dimensionless parameters for scalar NSI depend on the density,
unlike for vector NSI, the density at which they are calculated must be
considered. For clarity, we will consider all values of $\eta_{\alpha\beta}$
presented to be at 3 g/cc as a benchmark, and then the corresponding value
used in NOvA, T2K, or DUNE will be appropriately rescaled to the density of
that experiment. The effect of this rescaling is small for long-baseline
experiments, but would be quite significant if atmospheric or solar neutrinos
were also considered.

One of the main feature of such a model is that in the neutrino oscillation
Hamiltonian we cannot subtract an identity matrix proportional to the lightest
neutrino mass. This procedure, in the standard oscillation, allows us to write
all the probabilities only in terms of mass splittings $\Delta m_{ij}^{2}$. In
the scalar NSI case, on the other hand, the absolute neutrino mass
(parameterized as the lightest neutrino mass which $m_{1}$ in the normal
ordering (NO) and $m_{3}$ in the inverted ordering (IO)) survives, and appears
in the probabilities. Thus, in presence of such interactions, the oscillation
probabilities are, in principle, sensitive to the neutrino mass scale. We find
that the impact, however, is quite small for realistic parameters.

Using the same expansion procedure described in the previous section and
expanding up to the second order also in the parameter $\eta_{ij}$ it is
possible to obtain approximate expressions also for the scalar NSI case see
Ref. Medhi:2021wxj . In order to avoid cumbersome expressions, we did not show
here the effect of the solar mass splitting and of the lightest neutrino mass.
See appendix A.

###  2.3 Sterile Neutrino

Sterile neutrinos are a simple, phenomenologically rich, and theoretically and
experimentally motivated extension to the standard three-flavor neutrino
scenario. Since neutrinos have mass, there are additional particles and
sterile neutrinos are present in many of the explanations. In addition, there
are numerous hints of various significances and robustness that indicates that
new light ($m_{4}\lesssim 10$ eV) neutrinos may exist LSND:2001aii ;
T2K:2014xvp ; Giunti:2010zu ; Mention:2011rk ; Barinov:2021asz ;
MiniBooNE:2020pnu ; Denton:2021czb  although strong constraints also exist
MINOS:2017cae ; IceCube:2020phf ; Hagstotz:2020ukm , see refs. Acero:2022wqg ;
Dentler:2018sju ; Diaz:2019fwt ; Boser:2019rta ; Machado:2019oxb ;
Berryman:2020agd ; Dasgupta:2021ies  for recent reviews. Sterile neutrinos
also play a role in long-baseline accelerator neutrino experiments, although
typically at somewhat lighter masses than the above hints $\sim 1$ eV
Donini:2007yf ; Dighe:2007uf ; Meloni:2010zr ; Bhattacharya:2011ee ;
Hollander:2014iha ; Klop:2014ima ; Gandhi:2015xza ; Palazzo:2015gja ;
Berryman:2015nua ; Agarwalla:2016mrc ; Agarwalla:2016xxa ; Dutta:2016glq ;
Kelly:2017kch ; Coloma:2017ptb ; Ghosh:2017atj ; Choubey:2017cba ;
Choubey:2017ppj ; Agarwalla:2018nlx ; Gupta:2018qsv ; KumarAgarwalla:2019blx ;
Majhi:2019hdj ; Reyimuaji:2019wbn ; Ghosh:2019zvl ; Chatterjee:2020yak ;
Penedo:2022etl . We focus on the scenario with a single light sterile neutrino
which modifies the neutrino oscillation Hamiltonian to

$H=\frac{1}{2E}\left[U_{4}\begin{pmatrix}0\\\ &\Delta m^{2}_{21}\\\ &&\Delta
m^{2}_{31}\\\ &&&\Delta
m^{2}_{41}\end{pmatrix}U_{4}^{\dagger}+a\begin{pmatrix}1\\\ &0\\\ &&0\\\
&&&\frac{1}{2}\frac{N_{n}}{N_{e}}\end{pmatrix}\right]\,,$ (11)

where we have chosen to parameterize the mixing matrix as

$U_{4}\equiv
R_{34}(\theta_{34})R_{24}(\theta_{24})R_{14}(\theta_{14})U_{23}(\theta_{23},\delta_{23})U_{13}(\theta_{13},\delta_{13})U_{12}(\theta_{12},\delta_{12})\,,$
(12)

the relevant $2\times 2$ submatrix of $U_{ij}$ is

$\begin{pmatrix}c_{ij}&s_{ij}e^{-i\delta_{ij}}\\\
-s_{ij}e^{i\delta_{ij}}&c_{ij}\end{pmatrix}\,,$ (13)

and $R_{ij}(\theta_{ij})=U_{ij}(\theta_{ij},0)$. In the three-flavor limit
where $\theta_{i4}=0$ we get that the regular complex phase is given by
$\delta=\delta_{13}-\delta_{12}-\delta_{23}$ Rodejohann:2011vc ;
Denton:2021vtf . Note that in Eq. 11 we have subtracted off the neutral
current potential leading to an apparent potential for sterile neutrinos.

Similar to the two NSI cases, we again perform a perturbative calculation of
the sterile neutrino contribution to the oscillation probability in appendix
A.

##  3 Benchmark Scenarios

In order to determine the level at which DUNE can differentiate different new
physics scenarios, we use several benchmark points in the parameter spaces of
each scenario. These points are motivated by existing long-baseline data from
NOvA and T2K. This serves two useful purposes. First, if the tension between
NOvA and T2K is due to new physics, then these are the parameters that DUNE
should be looking at. Second, even if this is not the first hint of new
physics, it represents a good estimate of the expected upper limit on new
physics parameters from the current generation of long-baseline accelerator
neutrino experiments, generally consistent with the constraints from IceCube
for $\varepsilon_{e\mu}$ and $\varepsilon_{e\tau}$
IceCubeCollaboration:2021euf .

While different benchmark points might yield somewhat different results in
sections 5 and 6 below, we have checked that the results do not drastically
change as these benchmarks are varied.

###  3.1 Vector NSI Motivated by NOvA and T2K

The first set of benchmark points we consider are from Ref. Denton:2020uda
which investigated the possibility that NOvA and T2K data could be described
by off-diagonal complex vector NSI. The fit was performed to both NOvA and T2K
disappearance and appearance data in both neutrino and anti-neutrino modes and
wrong sign lepton contributions were included. In order to fully constrain all
six regular oscillation parameters plus the new physics parameters, data from
Daya Bay were used for $\theta_{13}$ and $\Delta m^{2}_{31}$ and from KamLAND
for $\theta_{12}$ and $\Delta m^{2}_{21}$. Such data sets were chosen since
the effect of NSI would not modify these external constraints. This analysis
finds results generally consistent with other similar analyses in the
literature Kelly:2020fkv ; Chatterjee:2020kkm ; Esteban:2020cvm . The best fit
values can be found in the appendix in table 6.

###  3.2 Scalar NSI Motivated by NOvA and T2K

We reanalyze the NOvA and T2K data using the same assumptions as in the
previous section, but now in the context of scalar NSI. The best fit points
are shown in table 7 and the preferred regions are shown in Fig. 7 in appendix
B.2. The physics impact of this analysis is also discussed in appendix B.2.

To visually see the different effects of vector and scalar NSI at different
long-baseline experiments, we show the difference in the appearance
probabilities between to two benchmark NSI cases in Fig. 1, in particular
$|P_{\mu e}(\varepsilon_{e\mu})-P_{\mu e}(\varepsilon_{e\tau})|$ in NO and
$|P_{\mu e}(\eta_{e\mu})-P_{\mu e}(\eta_{e\tau})|$ in IO. These particular
benchmark cases have been chosen being among the most significant ones in the
T2K-NOvA fit (see appendix B). We checked that results obtained using other
scenarios are similar. It is possible to notice that DUNE’s broadband beam
allows to catch more features of the probabilities than the other long-
baseline experiments, and that the strength of the effects tend to be somewhat
higher at larger baselines and energies, while is very mild at the first
oscillation maximum.

Figure 1: The difference in neutrino appearance probabilities for two
benchmark cases (see table 6) as a function of baseline and neutrino energy.
On the left the difference is between the probabilities with vector NSI and
$\varepsilon_{e\mu}$ and $\varepsilon_{e\tau}$ in normal ordering and the
right is between probabilities with scalar NSI and $\eta_{e\mu}$ and
$\eta_{e\tau}$ in inverted ordering. The regions probed by the different long-
baseline experiments are indicated. The density is taken to be that for DUNE,
2.848 g/cm3, throughout.

###  3.3 Sterile Neutrino Motivated by NOvA and T2K

Light sterile neutrinos $m_{4}\lesssim 10$ eV are interesting due to numerous
anomalies. While the slight tension in the NOvA and T2K data isn’t
substantially improved by the addition of a light sterile neutrino, one can
nonetheless perform a fit to the experiments in the context of a 3+1 scenario.
In Ref. Chatterjee:2020yak  they found that the benchmark point of
$\theta_{14}=\theta_{24}=8^{\circ}$ and $\theta_{34}=0$ with $\Delta
m^{2}_{41}=1$ eV2 provides modest improvement to the NOvA and T2K data over
the three-flavor oscillation hypothesis. We note that long-baseline
experiments are not particularly sensitive to the value of $\Delta m^{2}_{41}$
as long as $\Delta m^{2}_{41}\gg|\Delta m^{2}_{31}|$. Meanwhile, these values
of the two non-zero sterile mixing angles, $\theta_{14}$ and $\theta_{24}$ are
near the existing limits from solar data Goldhagen:2021kxe  and long-baseline
disappearance data MINOS:2017cae  respectively. As they are not ruled out from
other experiments, we then consider two benchmark points, one for each mass
ordering. While the NOvA and T2K data are somewhat constraining for
$\delta_{13}$, the usual CP phase, they are not that constraining for the
other two new phases, notably $\delta_{23}$. Some information does exist,
however, for $\delta_{12}$. The benchmark points that provide the best fit to
NOvA and T2K data in a sterile neutrino picture are shown in table 1.

MO |  $\Delta m^{2}_{41}$ [eV2] |  $\theta_{14}$ [∘] |  $\theta_{24}$ [∘] | $\delta_{13}/\pi$ | $\delta_{12}/\pi$
---|---|---|---|---|---
NO | 1 | 8 | 8 | 1.9 | 0.7
IO | 1 | 8 | 8 | 0 | 0.5
Table 1: Benchmark sterile neutrino parameters from NOvA and T2K data from
Ref. Chatterjee:2020yak . Sterile parameters not shown are zero and standard
oscillation parameters not shown are taken to the standard values shown in
table 2.

##  4 DUNE Analysis Details

The DUNE (Deep Underground Neutrino Experiment) DUNE:2020ypp  is a next
generation long baseline accelerator experiment that is under construction in
the US. The near site, located at Fermilab, will host the Long Baseline
Neutrino Facility (LBNF) and the Near Detectors complex. The Far Detector, a
40kt fiducial volume Liquid Argon Time Projection Chamber (LAr-TPC), on the
other hand will be located at the Sanford Underground Research Facility (SURF)
in South Dakota, 1285 km away from Fermilab. According to the most recent
Technical Design Report (TDR), DUNE will use a 120 GeV proton beam of 1.2 MW
power, which will deliver $1.1\,\times\,10^{21}$ protons on target (POT) per
year. DUNE will use an intense, on-axis, broad-band muon neutrino beam, peaked
at 2.5 GeV (in order to sit around the first atmospheric oscillation maximum),
with a small intrinsic $\nu_{e}$ contamination. Such a beam can also operate
in antineutrino mode, providing an anti-muon neutrino flux.

For the simulation of the DUNE experiment, we use the GLoBES software
Huber:2004ka . Considering the latest version of the DUNE GLoBES files
DUNE:2021cuw , we take into account an exposure of 6.5 years in neutrino mode
and 6.5 years in antineutrino mode (total exposure of 312 kt-MW-years for each
mode). Accelerator upgrades to 2.4 MW would increase the statistics further or
get to the quoted exposure faster, while realistic far detector staging would
slow down the statistics somewhat. The two oscillation channels we consider
are the $\nu_{\mu}$ disappearance and the $\nu_{e}$ appearance. For the former
the DUNE collaboration suggests a systematic uncertainty of 5%; backgrounds to
this channel are misidentified $\nu_{e}$ and Neutral Current (NC) events. For
the latter, the systematic uncertainty is 2% and the backgrounds consists on
misidentified $\nu_{\tau}$ and NC events. Energy resolution and efficiency
functions, as well as smearing matrices are given by the collaboration. We
consider the Earth matter density to be 2.848 g/cm3.

In order to study the DUNE performances in the measurements of the new physics
parameters, we derive the statistical significance of our results using a
$\Delta\chi^{2}$ function calculation with priors based on the pull method
Fogli:2002pt . As true values for the standard oscillation parameters, except
for the CP-violating phase $\delta$ whose value depends on the given benchmark
scenario, we consider the ones from the global fit without SuperKamiokande
atmospheric data Esteban:2020cvm , summarized in table 2. In our simulations
we also marginalize over the cited parameters using Gaussian priors given by
their $1\sigma$ uncertainties shown in the above mentioned table; the mass
ordering is considered to be known. It is worth to notice that other global
fits deSalas:2020pgw ; Capozzi:2021fjo  obtained slightly different results
for the oscillation parameters. However, we checked in our simulations that
the results are not significantly affected by the choice of the sets of
parameters and their uncertainties. Moreover, the inclusion of the atmospheric
data in the fit, can change the preference of the $\theta_{23}$ octant from
the upper one to the lower one in NO. We also checked that the performances of
DUNE in constraining our benchmark scenarios are not drastically affected by
the atmospheric angle octant. This is because DUNE will have measurements of
$\theta_{23}$, $\Delta m^{2}_{31}$, and $\delta$ that will all be considerably
better than currently available, and is significantly less sensitive to the
other three oscillation parameters.

MO |  $\theta_{12}$ (∘) |  $\theta_{23}$ (∘) |  $\theta_{13}$ (∘) |  $\Delta m_{21}^{2}$ [$10^{-5}$ eV2] |  $\Delta m_{3l}^{2}$ [$10^{-3}$ eV2]
---|---|---|---|---|---
NO | $33.44^{+0.77}_{-0.74}$ | $49.2^{+1.0}_{-1.3}$ | $8.57^{+0.13}_{-0.12}$ | $7.42^{+0.21}_{-0.20}$ | $2.515^{+0.028}_{-0.028}$
IO | $33.45^{+0.77}_{-0.74}$ | $49.5^{+1.0}_{-1.2}$ | $8.60^{+0.12}_{-0.12}$ | $7.42^{+0.21}_{-0.20}$ | $-2.498^{+0.028}_{-0.029}$
Table 2:  Preferred values from a recent global fit to oscillation data
Esteban:2020cvm . $\Delta m^{2}_{3l}$ is $\Delta m^{2}_{31}$ when positive and
$\Delta m^{2}_{32}$ when negative.

While DUNE will have a state-of-the-art near detector facility DUNE:2021tad
which will provide important information about sterile neutrinos depending on
the $\Delta m^{2}_{41}$ values, we conservatively consider only the far
detector in this study.

##  5 Single Scenario Results

In this section we present our results of DUNE’s sensitivity to just a single
new physics scenario. This is done to show how well DUNE can identify the
benchmarks we are considering when compared against the standard oscillation
picture. It also allows us to present the first results of scalar NSI at NOvA
and T2K. We find that, in general, DUNE has very good sensitivity to discover
new physics at the level indicated by current long-baseline oscillation
experiments, as expected.

###  5.1 Vector NSI

We discuss here the performances of the DUNE Far Detector in constraining the
NSI parameters in the benchmark scenarios discussed in section 3. Fig. 2 shows
the DUNE allowed regions at 68, 90, and 99% CL in the
($\varepsilon_{\alpha\beta}-\phi_{\alpha\beta}$) plane. The top (bottom)
panels show the results using the NO (IO) hypothesis. In order to obtain the
contours, we consider as true values for the two mass splittings and the
mixing angles the ones summarized in table 2 and the values for
$\varepsilon_{\alpha\beta}$, $\phi_{\alpha\beta}$ and $\delta$ from the
NOvA-T2K fit. For the fit we marginalize over the oscillation parameters with
pull terms. All the NSI parameters that do not appear in each plot are fixed
to 0 both in the theory and in the fit.

For the vector NSI case, we observe that the most interesting results are
obtained when the mass ordering is normal, since the $\Delta\chi^{2}$-s in
respect to the standard model are 4.44 and 3.65 when the fits are performed in
NO considering non-vanishing $\varepsilon_{e\mu}$ and $\varepsilon_{e\tau}$,
respectively (see table 6 taken from Denton:2020uda ); however, for the sake
of completeness, we show the results considering also the IO scenarios.

Figure 2: 68% (solid lines), 95% (dashed lines) and 99% (dotted lines)
contours in DUNE in the vector NSI $(|\varepsilon_{ij}|-\phi_{ij})$-planes
when data are generated using vector NSI best fits considering one parameter
at a time. The top (bottom) panels with red (blue) contours have been obtained
using NO (IO) best fits.

It is clear that DUNE is expected to be able to measure all the above
mentioned vector NSI parameter with a good precision. Indeed, when
$\varepsilon_{e\mu}$ is considered in NO, in the 2-dimensional plane (2
degrees of freedom), the 68% CL allowed region includes the intervals
[0.16,0.21] for $\varepsilon_{e\mu}$ and [1.3,1.8]$\pi$ for $\phi_{e\mu}$.
This means that in the first benchmark scenario, DUNE would be able to
determine both NSI parameters with a precision of roughly 10% at the given
benchmarks. In the IO case, since the best fit value for $\varepsilon_{e\mu}$
is smaller (0.04), the allowed region is bigger, but still excludes at 99% CL
the standard model.

In the middle panels, namely when the benchmark scenario presents non-zero
values for $\varepsilon_{e\tau}$ and $\phi_{e\tau}$, it is possible to observe
that the DUNE performances are similar to the previous case. In particular,
the allowed regions includes the intervals [0.22,0.34] for
$\varepsilon_{e\tau}$ and [1.4,1.8]$\pi$ for $\phi_{e\tau}$ which correspond
to a precision of roughly 20% for the magnitude of the parameter and 10% for
its phase for the NO case. When we consider the IO scenario, the precision on
the parameters remains basically the same, even though the best fit value for
the NSI coupling magnitude is reduced by almost a factor of 2.

The results are very different when $\varepsilon_{\mu\tau}$ is considered. In
this case, the best fit value for $\phi_{\mu\tau}$ from NOvA and T2K is very
close to $\pi/2$ in the NO case. Since the NSI correction to the $\nu_{\mu}$
disappearance probability depends at the leading order on the combination
$\varepsilon_{\mu\tau}\cos\phi_{\mu\tau}$, see Eq. 16, DUNE is expected to be
very sensitive to small variation of the phase around $3/2\pi$, but is not
adequate to constrain the magnitude $|\varepsilon_{\mu\tau}|$ with the same
precision reached for the other parameters. When we consider the IO
hypothesis, the situation is the opposite, since the phase best fit is close
to zero: the magnitude is tightly constrained while the phase can vary in a
relatively large interval. We note also that atmospheric data from SK and
IceCube is quite strong for this parameter Super-Kamiokande:2011dam ;
IceCube:2022ubv .

Then, instead of considering just a single NSI parameter at a time, we
marginalize over all NSI parameters (magnitudes and phases) with select input
priors: $|\varepsilon_{\mu\tau}|<0.02$ (when undisplayed)
IceCubeCollaboration:2021euf  and $\varepsilon_{ee}<0.3$ Esteban:2019lfo . The
results are shown in Fig. 3. Compared to the single parameter case the
contours are obviously enlarged, however DUNE is still be able to exclude
large portions of the parameters spaces in the studied benchmark scenarios and
easily discover new physics. The most interesting features that can be
observed are the following:

  * •

In both the NO and IO cases, when we consider $\varepsilon_{\mu\tau}$, the
$\cos\phi_{\mu\tau}$ degeneracy appears.

  * •

In the IO case, now the SM is allowed at 99% (95%) CL in the
$\varepsilon_{e\mu}$ ($\varepsilon_{e\tau}$) scenarios due partially to
slightly smaller benchmark parameters.

We note that the degeneracy is $\varepsilon_{\mu\tau}$ appears only in the
case where all NSI parameters are scanned over because $\varepsilon_{\mu\tau}$
does appear in the appearance probability at a subleading level which is
enough to break the degeneracy between
$\sin\phi_{\mu\tau}\leftrightarrow-\sin\phi_{\mu\tau}$. When all NSI
parameters are considered, however, there is more than enough freedom to
cancel out the effect in appearance mode without significantly affect the
disappearance measurement. See also Masud:2018pig  for more discussion of some
of these degeneracies.

Other studies have also investigated complex off-diagonal vector NSI at DUNE,
often using this benchmark approach deGouvea:2015ndi ; Bakhti:2020fde ;
DUNE:2020fgq ; Chatterjee:2021wac . As they have used different benchmarks,
either the SM or particular NSI values, it is not possibly to directly compare
them, but qualitatively they find comparable sensitivity to vector NSI at the
$|\varepsilon_{\alpha\beta}|\sim 0.1$ level with a sizable dependence on the
complex NSI phases and a stronger sensitivity and more precision for
$\varepsilon_{\mu\tau}$ than either of $\varepsilon_{e\beta}$.

Figure 3: 68% (solid lines), 95% (dashed lines) and 99% (dotted lines)
contours in DUNE in the vector NSI $(|\varepsilon_{ij}|-\phi_{ij})$-planes
when data are generated using NSI best fits marginalizing over all the NSI
parameters. The three top (bottom) plots with red (blue) contours have been
obtained using NO (IO) best fits.

###  5.2 Scalar NSI

For the sensitivities performed using the off-diagonal scalar NSI benchmark
scenarios, the first study of its kind, we use the same approach described in
the previous subsection. Differently from the previous case, the results with
more statistical significance in the NOvA-T2K fit are the IO ones. In Fig. 4
we show our results in the ($\eta_{\alpha\beta}-\phi_{\alpha\beta}$) planes in
the case in which the lightest neutrino mass is zero.

When the mass ordering is inverted, DUNE will to constrain scalar NSI at the
following levels at 68% CL: $|\eta_{e\mu}|\in[0.012,0.035]$ and
$|\eta_{e\tau}|\in[0.008,0.012]$. On the other hand, DUNE cannot set
remarkable bounds on the phases, being able to exclude at 68% CL only one
third of the possible values of $\phi_{e\mu}$ (from $0.44\pi$ to $1.1\pi$) and
one sixth of the possible values of $\phi_{e\tau}$ (from $1.6\pi$ to
$1.9\pi$). For the non-zero $\eta_{\mu\tau}$ scenario, in which the scalar NSI
coupling best fit is bigger, we have a different situation. Indeed, DUNE is
expected in this case to bound with good precision the phase, but is very
unconstraining in the magnitude. In the NO case, the NOvA-T2K results are
characterized by very small best values and small $\Delta\chi^{2}$-s. When we
perform sensitivity scans with DUNE, we can observe that this experiment is
not able to distinguish the new physics scenarios from the Standard Model not
even at 68% level. Moreover, the magnitudes of the three off-diagonal scalar
NSI parameters cannot be bounded from above and the phases are unconstrained
by DUNE.

We checked that when a full marginalization is performed, DUNE is not able to
exclude remarkable portions of the parameters space at a good confidence
level. Indeed, the scalar NSI scenarios produce very similar phenomenology at
DUNE (see section 6); thus, when we allow all the parameters to vary in the
fit, the effects of the $\eta$-s can be reduced by the presence of other non-
zero parameters.

Figure 4: 68% (solid lines), 95% (dashed lines) and 99% (dotted lines)
contours in DUNE in the scalar NSI $(|\eta_{ij}|-\phi_{ij})$-planes where
$\eta_{ij}$ is the value of $\eta_{ij}$ at $\rho=3$ g/cm3. The true data are
generated using scalar NSI best fits (with $m_{\rm lightest}=0$) considering
one NSI parameter at a time. The three top (bottom) plots with red (blue)
contours have been obtained using NO (IO) best fits.

We have mentioned that one of the most interesting features of the scalar NSI
model is that the oscillation probabilities in this case depend on the
absolute neutrino mass scale. A non-zero $m_{\rm lightest}$ can slightly
change the NOvA-T2K best fits. In particular, increasing the neutrino mass
scale, the best fit complex NSI phases are almost the same, while the
magnitudes of the $\eta_{ij}$ parameters decrease. The most significant
reduction of the best fits can be observed in the IO case, in which, as
already pointed out, the phenomenology would allow us to distinguish the
scalar NSI parameters from the SM oscillations. In Fig. 5 we show the 68%
contours using the best fits for $m_{\rm lightest}=0,0.05,0.1$ eV. It is clear
that the contours shapes are not drastically altered by the decrease of the
scalar NSI parameters magnitudes. However, when the lightest neutrino is not
massless, DUNE is expected to be able to set upper limits on $\eta_{\mu\tau}$:
$|\eta_{\mu\tau}|<0.08\,(0.06)$ at 68% CL when $m_{\rm lightest}=0.05\,(0.1)$
eV. Combining information from very different densities such as the crust of
the Earth and the center of the Sun could allow for a detection of the
absolute neutrino mass scale.

Figure 5: 68% contours in DUNE in the $(|\eta_{ij}|-\phi_{ij})$-planes when
data are generated using scalar NSI best fits in the IO case considering one
NSI parameter at a time. The red curve is obtained with $m_{\rm lightest}=0$,
the blue one with $m_{\rm lightest}=0.05$ eV and the green one with $m_{\rm
lightest}=0.1$ eV.

###  5.3 Sterile neutrinos

Figure 6: 68% (solid lines), 95% (dashed lines) and 99% (dotted lines)
contours in DUNE in the $(\theta_{14}-\theta_{24})$ and
$(\theta_{14}-\delta_{12})$ planes when data are generated using 3+1 model
best fits. The three top (bottom) plots with red (blue) contours have been
obtained using NO (IO) best fits.

In Fig. 6 we show the performances of the DUNE Far Detector in constraining
the 3+1 parameters if the best fits are the ones reported in table 1 from NOvA
and T2K data. In the analysis, we marginalize over the undisplayed parameters,
using the upper bound $\theta_{i4}<25^{\circ}$ on the non-standard mixing
angles777 This range of $\theta_{i4}$ is motivated by the existing constraints
on sterile neutrino mixing angles ranging from $\theta_{14}\lesssim 1^{\circ}$
to $\theta_{34}\lesssim 23^{\circ}$ Dentler:2018sju , although we note that
these constraints depend considerably on the experiments included and there
are numerous anomalies, some of which seem to point to mixing angles in excess
of those constraints.. In both IO and NO, in the $(\theta_{24}-\theta_{14})$
planes, it is evident that there is a modest amount of anti-correlation
between the two mixing angles, which is not surprising given that they both
play a similar role. Related, DUNE is expected to have a similar sensitivity
to both the mixing angles for similar true values. Since we are taking both
sterile mixing angles to be $8^{\circ}$, the 68% allowed ranges are [5-12]∘
for $\theta_{14}$ and [4-10]∘ for $\theta_{24}$. When we investigate the DUNE
capabilities in measuring the new complex phases, we observe in Fig. 6 that,
if $\delta_{12}\sim 0.5\pi$ as suggested by NOvA and T2K data, then there
exists two local $\Delta\chi^{2}$ minima, one around the best fit and one
around $\delta_{12}\sim 1.5\pi$ for both the IO and NO case. Thus DUNE would
not be able to determine the sign of $\sin\delta_{12}$, but would be able to
determine that a new source of CP violation exists. We checked that, on the
other hand, given the best fits in table 1, the DUNE experiment will not be
sensitive to $\delta_{13}$ and $\delta_{23}$.

Other studies have also investigated sterile neutrinos at DUNE Donini:2007yf ;
Dighe:2007uf ; Berryman:2015nua ; Krasnov:2019kdc ; DUNE:2020fgq . Most
studies assume the SM or different benchmarks from our study, so a direct
comparison is not possible, but Berryman:2015nua  does consider a similar
benchmark point in terms of mixing angles and finds very similar
sensitivities.

##  6 Differentiating the Models

###  6.1 Method

We now get to the primary result of this paper, where we explore the ability
for DUNE to correctly differentiate among different benchmark scenarios. Our
methodology is to assume that one of the benchmark new physics points
represents reality and then attempt to reconstruct it in a different new
physics scenario and determine the test statistic for the model comparison
test to show the significance at which the data would prefer the correct
answer over an incorrect answer. We compare each of the three classes of
models against each other for the most comprehensive such study to date.

The minimum $\Delta\chi^{2}$’s are shown in tables 3, 4, and 5. The true
scenarios are listed along the left column and then the statistical
significance in which the SM can be disfavored is shown in the second column
labeled SM. The remaining columns show at what level a different scenario can
be disfavored relative to the true scenario which is denoted with a slash as
the $\Delta\chi^{2}$’s in those cases are exact zero.

For the $\chi^{2}$ computations we marginalize over all the standard
oscillation parameters. We also marginalize over one NSI parameter, magnitude
up to 0.5 and phase, for the vector and scalar NSI columns. For the sterile
columns, labeled 3+1, we marginalize over $\Delta m^{2}_{41}$ in the range
$[10^{-5}-10]$ eV2, and the mixing angles in the range $[0-30]^{\circ}$.
Naturally the $\Delta\chi^{2}$ in the new physics scenarios will be less than
that of the SM as the new physics scenarios include all the degrees of freedom
of the SM plus several new ones.

###  6.2 Discussion

####  6.2.1 True Vector NSI

In the case in which data are generated using the best fit points for vector
NSI, for both the NO and IO, the data could not be well fit with the three
neutrino standard probabilities ($\chi^{2}>60$). Moreover, all the models
obtained generating data with vector NSI ($\varepsilon$) best fits from the
NOvA and T2K data can be distinguished at least $7\sigma$ from the scalar NSI
models. When we try to fit the vector NSI data with other vector NSI models or
3+1 models, the only interesting case (namely the only case in which DUNE
would struggle in differentiating the models) is when data are generated using
$\varepsilon_{e\mu}$ best fits in the IO case. In this framework, the minima
of the $\chi^{2}$ are 10, 13 and 3 when we fit the data with the
$\varepsilon_{e\tau}$, $\varepsilon_{\mu\tau}$ and 3+1 models respectively.
These results are generally compatible with deGouvea:2015ndi  which fit
benchmark NSI points in a sterile neutrino framework.

####  6.2.2 True Scalar NSI

Let us now focus on the case in which data are generated using the best fits
related to the scalar NSI model. It is clear that, for NO, the minimum
$\chi^{2}$ values are very small even in the standard model scenario (when all
the new physics parameters are fixed to zero). This is because the fit at the
NOvA and T2K data, if performed adding the scalar NSI in NO, gives very small
best fit values for the $\eta$ couplings. For this reason, we expect that DUNE
would not be able in this case to distinguish at any confidence level these
scenarios with the standard three neutrino framework. If we fit data with
models in which other $\eta$ parameters are used in the fit, we obtain even
smaller minima for the $\chi^{2}$, as expected. The same happens in the case
in which we fit the scalar NSI data with sterile neutrino models. On the other
hand, when we try to fit them with vector NSI models, apart from the
$\eta_{e\mu}$ case, the best fit point is always found when
$\varepsilon_{ij}=0$. The best fit for the scalar NSI parameters in the IO
case give even more interesting results. In this case, the three neutrino
framework is completely excluded ($\chi^{2}>30$ in all three cases). However,
we can find minima of the $\chi^{2}$ much smaller than the standard model ones
when we fit data with other scalar NSI models. In particular, if data are
generated using the best fits for $\eta_{e\tau}$, DUNE would have very limited
discrimination capabilities when we switch on only $\eta_{e\mu}$ or
$\eta_{\mu\tau}$ at more than $1.5\sigma$. On the other hand, when data are
generated with $\eta_{e\mu}$ or $\eta_{\mu\tau}$ best fits, the $\chi^{2}$
related to the fits with the other $\eta$ models are in the range $4.6-6.3$.
Thus, DUNE would also have some ability to distinguish different scalar NSI
scenarios in the IO if the true values for the $\eta$-s are the ones
compatible with the NOvA-T2K tension. If we try to fit scalar NSI data with
single parameter vector NSI models, DUNE would differentiate them at a good
confidence level. Only in the $\eta_{\mu\tau}$ case, the vector NSI models
could be rejected at less than $5\sigma$. Finally, the IO scalar NSI models
generated with $\eta_{e\tau}$ and $\eta_{\mu\tau}$ best fits cannot be
distinguished at more than $3.2\sigma$ from the sterile neutrino models. On
the other hand, we reach almost $5\sigma$ in the case of the $\eta_{e\mu}$
model.

All these results have been obtained considering only one parameter at-a-time
for each model, namely switching on only one off-diagonal scalar or vector
parameter for each fit model. We checked that, if we vary all the off-diagonal
parameters together, varying their phases in the entire allowed range, the
results are not changing drastically. This means that, for instance, if we fit
data generated with one of the $\eta$’s best fits using the vector NSI model
in which we scan over the whole
$(\varepsilon_{e\mu},\varepsilon_{e\tau},\varepsilon_{\mu\tau})$ parameter
space, the minimum of the $\chi^{2}$ is very close to the smaller minima of
the $\chi^{2}$-s obtained using only one parameter at-a-time. This suggests
that there are no strong correlations between parameters that could mimic data
generated with a different model.

$\Delta\chi^{2}$ | SM | $\eta_{e\mu}$ | $\eta_{e\tau}$ | $\eta_{\mu\tau}$ | $\varepsilon_{e\mu}$ | $\varepsilon_{e\tau}$ | $\varepsilon_{\mu\tau}$ | 3+1
---|---|---|---|---|---|---|---|---
$\varepsilon_{e\mu}$ NO | 200 | 140 | 140 | 170 | / | 180 | 160 | 80
$\varepsilon_{e\tau}$ NO | 60 | 48 | 50 | 45 | 50 | / | 50 | 40
$\varepsilon_{\mu\tau}$ NO | 200 | 180 | 170 | 180 | 160 | 180 | / | 80
$\varepsilon_{e\mu}$ IO | 170 | 80 | 75 | 90 | / | 10 | 13 | 3
$\varepsilon_{e\tau}$ IO | 70 | 50 | 50 | 45 | 45 | / | 60 | 20
$\varepsilon_{\mu\tau}$ IO | 500 | 400 | 400 | 400 | 300 | 350 | / | 160
Table 3:  $\Delta\chi^{2}$ obtained fitting to each case all the models taken into account using as true values the best fits of the vector NSI case. $\Delta\chi^{2}$ | SM | $\eta_{e\mu}$ | $\eta_{e\tau}$ | $\eta_{\mu\tau}$ | $\varepsilon_{e\mu}$ | $\varepsilon_{e\tau}$ | $\varepsilon_{\mu\tau}$ | 3+1
---|---|---|---|---|---|---|---|---
$\eta_{e\mu}$ NO | 0.14 | / | 0.005 | 0.088 | 0.071 | 0.033 | 0.055 | 0.02
$\eta_{e\tau}$ NO | 0.08 | 0.003 | / | 0.041 | SM | SM | SM | 0.01
$\eta_{\mu\tau}$ NO | 0.60 | 0.48 | 0.48 | / | SM | SM | SM | 0.02
$\eta_{e\mu}$ IO | 100 | / | 4.7 | 6.3 | 80 | 70 | 90 | 21
$\eta_{e\tau}$ IO | 60 | 1.0 | / | 1.5 | 44 | 38 | 50 | 11
$\eta_{\mu\tau}$ IO | 30 | 4.6 | 4.8 | / | 23 | 20 | 29 | 12
Table 4:  Same as table 3 but with true values set to the best fits of scalar NSI. $\Delta\chi^{2}$ | SM | $\eta_{e\mu}$ | $\eta_{e\tau}$ | $\eta_{\mu\tau}$ | $\varepsilon_{e\mu}$ | $\varepsilon_{e\tau}$ | $\varepsilon_{\mu\tau}$
---|---|---|---|---|---|---|---
3+1 NO | 20 | 8.2 | 7.9 | 6.7 | 5.2 | 6.6 | 10
3+1 IO | 20 | 13 | SM | 18 | 7.4 | 6.2 | 9.5
Table 5:  Same as table 3 but with true values set to the best fits of 3+1
sterile neutrino model.

####  6.2.3 True Sterile Neutrino

Finally, when the data are obtained in the 3+1 framework (see table 5), with
the best fits shown in table 1, we observe that the SM solution is excluded
with $\Delta\chi^{2}=20$, namely at 4.5$\sigma$. On the other hand, when the
fit is performed with the NSI models, it is clear that in both NO and IO, the
$\Delta\chi^{2}$ is reduced to 5.2 in the vector case. The vector model which
can be distinguished more easily from the 3+1 is the one where we turn on
$\varepsilon_{\mu\tau}$ in NO ($\Delta\chi^{2}=10$). This is because
$\varepsilon_{\mu\tau}$ in DUNE, as already mentioned, is the one that mostly
modifies the disappearance probability. In the scalar case, on the other hand,
the $\Delta\chi^{2}$ are not drastically reduced in respect to the Standard
Model case if the mass ordering is inverted. Indeed, if we perform the fit
allowing $\eta_{e\mu}$ to vary we reach $\Delta\chi^{2}=13$, while in the
opposite case, when we change the value of $\eta_{e\tau}$, the best fit is
achieved when the scalar NSI parameter vanishes, namely in the Standard Model
case. This suggests us that in inverted ordering, if the sterile neutrino
mixing angles are as small as $8^{\circ}$ and $\theta_{34}$ is fixed to zero,
the phenomenology of 3+1 and scalar NSI models are moving in the opposite
direction. On the contrary, in the scalar NSI NO case, the $\Delta\chi^{2}$
are as low as the vector NSI ones. We checked also in this case that a full
marginalization over the NSI parameter spaces do not alter significantly the
$\Delta\chi^{2}$-s.

##  7 Conclusions

DUNE will be the state-of-the-art neutrino oscillation experiment making high
precision long-baseline measurements with a modest matter effect, and with a
fairly broad energy spectrum. As has been discussed in the literature, this
will enable crucial tests of many new physics scenarios that affect
oscillations. Some common new physics scenarios are light sterile neutrinos,
vector non-standard neutrino interactions (NSI), and scalar NSI. To
realistically understand DUNE’s sensitivity, we considered a number of
benchmark scenarios motivated by existing long-baseline accelerator neutrino
oscillation data from NOvA and T2K. We confirmed in section 5 that, in fact,
DUNE does have very good sensitivity to a majority of relevant new physics
cases at the edge of current limits. For example, we showed, for the first
time, that DUNE has sensitivity to off-diagonal scalar NSI.

While identifying any kind of additional new physics in the neutrino sector is
extremely exciting, in order to confirm it, its nature must be understood. In
order to do that, one must check if multiple different new physics scenarios
can describe the same data, or if the data can differentiate among these
scenarios. Our main results, that is the ability of DUNE to distinguish among
different new physics frameworks, are reported in tables 3, 4 and 5. We find
that, in general, DUNE can differentiate among vector NSI, scalar NSI, and
sterile neutrinos at modest to excellent significance for these interesting
benchmark points. It is not always possible, however, to easily identify which
of the new physics parameters are responsible for the new physics as there are
some degeneracies among e.g. $\varepsilon_{e\mu}$ and $\varepsilon_{e\tau}$
and among all the off-diagonal scalar NSI parameters. Nonetheless, additional
measurements from HK, IceCube, KM3NeT, and other experiments would even
further resolve the flavor structure of the new physics due to the presence of
more baselines and energies.

###### Acknowledgements.

We thank Julia Gehrlein for helpful comments. PBD acknowledges support from
the US Department of Energy under Grant Contract DE-SC0012704.

##  Appendix A Approximate Expressions for Neutrino Oscillation Probabilities

In this appendix we report the relevant transition probabilities for both
vector and scalar NSI, for the transition channels used in our numerical
computation.

The analytical behavior of the probabilities are put in a better readable form
expanding up to the second order in some small parameters, namely
$\alpha=\Delta m^{2}_{21}/\Delta m^{2}_{31}$ and the deviation of the mixing
angles from the tri-bimaximal mixing $r$, $s$, $t$, defined as King:2007pr ;
Pakvasa:2007zj :

$s_{13}=\frac{r}{\sqrt{2}}\,,\qquad s_{12}=\frac{1}{\sqrt{3}}(1+s)\,,\qquad
s_{23}=\frac{1}{\sqrt{2}}(1+t)\,.$ (14)

From the current global fits on the oscillation data, all previous parameters
$r$, $s$, and $t$ are of ${\cal{O}}(0.1)$ or smaller Capozzi:2021fjo . Since
at the DUNE baseline the matter effects are expected to be of
$\mathcal{O}(10\%)$, we can further expand in $V=aL/(4E\Delta_{31})=a/\Delta
m^{2}_{31}$.

###  A.1 Vector NSI

At first order in $\mathcal{O}(\varepsilon_{\alpha\beta})$ and with the
identification
$\varepsilon_{\alpha\beta}\to\varepsilon_{\alpha\beta}\,e^{i\phi_{\alpha\beta}}$,
the NSI contribution to the electron appearance probability is:

$\displaystyle P(\nu_{\mu}\to\nu_{e})^{NSI}={}$
$\displaystyle\frac{4}{3}V\alpha\Delta_{31}[-\Delta_{31}(\varepsilon_{e\tau}\cos{\phi_{e\tau}}-\varepsilon_{e\mu}\cos{\phi_{e\mu}})$
$\displaystyle-\sin\Delta_{31}\varepsilon_{e\tau}(\cos\Delta_{31}\cos\phi_{e\tau}-\sin\Delta_{31}\sin\phi_{e\tau})$
(15)
$\displaystyle-\sin\Delta_{31}\varepsilon_{e\mu}(\cos\Delta_{31}\cos\phi_{e\mu}-\sin\Delta_{31}\sin\phi_{e\mu})]$
$\displaystyle+2Vr\sin\Delta_{31}[\Delta_{31}\cos\Delta_{31}(\varepsilon_{e\mu}\cos(\delta-\phi_{e\mu})-\varepsilon_{e\tau}\cos(\delta-\phi_{e\tau}))$
$\displaystyle+\Delta_{31}\sin\Delta_{31}(\varepsilon_{e\tau}\sin(\delta-\phi_{e\tau})-\varepsilon_{e\mu}\sin(\delta-\phi_{e\mu}))$
$\displaystyle+\sin\Delta_{31}(\varepsilon_{e\mu}\cos(\delta-\phi_{e\mu})+\varepsilon_{e\tau}\cos(\delta-\phi_{e\tau}))]\,.$

We observe that, at the perturbative order taken into account, the probability
depends on $\varepsilon_{e\mu}$ and $\varepsilon_{e\tau}$ only. In particular,
if $\phi_{e\tau}=\phi_{e\mu}$, the only relevant combinations are
$\varepsilon_{e\mu}+\varepsilon_{e\tau}$ and
$\varepsilon_{e\mu}-\varepsilon_{e\tau}$. The same happens if
$\phi_{e\mu}=\pi+\phi_{e\tau}$. For the muon disappearance channel, the NSI
contribution is:

$\displaystyle P(\nu_{\mu}\to\nu_{\mu})^{NSI}={}$ $\displaystyle
V\bigg{\\{}-8\Delta_{31}\varepsilon_{\mu\tau}\cos\phi_{\mu\tau}\cos\Delta_{31}\sin\Delta_{31}+\frac{4}{3}\alpha\Delta_{31}^{2}[\varepsilon_{e\tau}\cos\phi_{e\tau}\cos^{2}\Delta_{31}$
$\displaystyle-\varepsilon_{e\mu}\cos\phi_{e\mu}\cos^{2}\Delta_{31}+4\varepsilon_{\mu\tau}\cos\phi_{\mu\tau}(1-2\sin^{2}\Delta_{31})]$
$\displaystyle+4t\Delta_{31}\varepsilon_{\tau\tau}\cos\Delta_{31}\sin\Delta_{31}$
(16)
$\displaystyle+4r\Delta_{31}\varepsilon_{e\mu}\cos(\delta-\phi_{e\mu})\cos\Delta_{31}\sin\Delta_{31}-4a\varepsilon_{\tau\tau}\sin^{2}\Delta_{31}$
$\displaystyle+\frac{4}{3}\alpha\Delta_{31}[\varepsilon_{e\tau}\cos\phi_{e\tau}\cos\Delta_{31}\sin\Delta_{31}-\varepsilon_{e\mu}\cos\phi_{e\mu}\cos\Delta_{31}\sin\Delta_{31}]\bigg{\\}}\,.$

Compared to the previous case, all NSI parameters but $\varepsilon_{ee}$
contribute. However, the $\varepsilon_{\mu\tau}$ term is not suppressed by any
of the small parameters $r$, $s$ and $t$. Thus, this probability is expected
to be more sensitive to $\varepsilon_{\mu\tau}$ and its phase.

In light of such formulae, we can try to understand the results in table 3. It
is clear that, due to the strong dependence of the disappearance probability
on $\varepsilon_{\mu\tau}$, if we generate data with this parameter’s true
value different from zero, we expect very large $\chi^{2}$. Indeed, it is very
difficult to find values for the other parameters which fit well the
disappearance data. Moreover, even if the appearance probability has the same
dependence on $\varepsilon_{e\mu}$ and $\varepsilon_{e\tau}$, the
disappearance probability contains one more term in $\varepsilon_{e\mu}$. For
this reason, it is easy to fit data generated with $\varepsilon_{e\tau}\neq 0$
with a theory that contains $\varepsilon_{e\mu}$, but the opposite is more
complicated. This is clear from table 3, since $\chi^{2}$ in the
$\varepsilon_{e\mu}$ rows are bigger in respect to the $\varepsilon_{e\tau}$
ones.

###  A.2 Scalar NSI

In the scalar NSI case, we neglect for simplicity the solar mass splitting,
which would elongate in a considerable way the probabilities, making them
difficult to understand. For the electron appearance, and with the
identification
$\eta_{\alpha\beta}\to\eta_{\alpha\beta}\,e^{i\phi_{\alpha\beta}}$, we have:

$\displaystyle P(\nu_{\mu}\to\nu_{e})^{NSI}={}$
$\displaystyle[\eta_{e\mu}^{2}+\eta_{e\tau}^{2}+2r\eta_{e\mu}\cos(\delta+\phi_{12})+2r\eta_{e\tau}\cos(\delta+\phi_{13})]\sin^{2}\Delta_{13}+$
$\displaystyle+2\eta_{e\mu}\eta_{e\tau}\cos(\phi_{12}-\phi_{13})\sin^{2}\Delta_{13}+$
$\displaystyle-2V[\eta_{e\mu}^{2}+\eta_{e\tau}^{2}+2r\eta_{e\mu}\cos(\delta+\phi_{12})+2r\eta_{e\tau}\cos(\delta+\phi_{13})]\times$
(17)
$\displaystyle\times[\Delta_{13}\cos\Delta_{13}\sin\Delta_{13}-\sin^{2}\Delta_{13}]+$
$\displaystyle-2V\eta_{e\mu}\eta_{e\tau}\cos(\phi_{12}-\phi_{13})\sin^{2}\Delta_{13}[\Delta_{13}\cos\Delta_{13}\sin\Delta_{13}-\sin^{2}\Delta_{13}]\,.$

It can be noticed that the probability depends only on $\eta_{e\mu}$ and
$\eta_{e\tau}$. Both parameters appear in second order terms, proportional to
$\eta^{2}$ or to $r\eta$.

The $\nu_{\mu}$ disappearance probability, on the other hand, can be written
as:

$\displaystyle P(\nu_{\mu}\to\nu_{\mu})^{NSI}={}$
$\displaystyle-\Delta_{13}(\eta_{\mu\mu}+\eta_{\tau\tau}+2\eta_{\mu\tau}\cos\phi_{\mu\tau})\sin
2\Delta_{13}+$
$\displaystyle-\frac{1}{2}\Delta_{13}(\eta_{e\mu}^{2}+\eta_{e\tau}^{2}+4\eta_{\mu\tau}^{2}\sin\phi_{\mu\tau}^{2})\sin
2\Delta_{13}+$
$\displaystyle-4\Delta_{13}^{2}\eta_{\mu\tau}^{2}\cos\phi_{\mu\tau}^{2}\sin
2\Delta_{13}+$
$\displaystyle+\frac{1}{2}(\eta_{\mu\mu}^{2}+\eta_{\tau\tau}^{2})[1-(1-\Delta_{13}^{2})\cos
2\Delta_{13}-\Delta_{13}\sin 2\Delta_{13}]+$ (18)
$\displaystyle-2\Delta_{13}[r\eta_{e\mu}\cos(\delta-\phi_{e\mu})+r\eta_{e\tau}\cos(\delta+\phi_{e\tau})]\sin
2\Delta_{13}+$ $\displaystyle+2t(\eta_{\tau\tau}-\eta_{\mu\mu})[-1+\cos
2\Delta_{13}+\Delta_{13}\sin 2\Delta_{13}]+$
$\displaystyle-3\Delta_{13}\eta_{e\mu}\eta_{e\tau}\cos(\phi_{e\mu}-\phi_{e\tau})\sin
2\Delta_{13}+$
$\displaystyle-2\Delta_{13}\eta_{\mu\mu}\eta_{\mu\tau}\cos\phi_{\mu\tau}(2\Delta_{13}\cos
2\Delta_{13}+\sin 2\Delta_{13})+$
$\displaystyle+\eta_{\mu\mu}\eta_{\tau\tau}[-1+(1-2\Delta_{13}^{2})\cos
2\Delta_{13}+\Delta_{13}\sin 2\Delta_{13}]+$
$\displaystyle-2\Delta_{13}\eta_{\mu\tau}\eta_{\tau\tau}\cos\phi_{\mu\tau}(2\Delta_{13}\cos
2\Delta_{13}+\sin 2\Delta_{13})\,.$

In this case, all the scalar NSI parameters but $\eta_{ee}$ appear.
$\eta_{\mu\mu}$, $\eta_{\tau\tau}$ and $\eta_{\mu\tau}$ modify the probability
at the first order in perturbation theory; we expect them to affect the
$\Delta m_{13}^{2}$ measurement, since they are strongly correlated to $\sin
2\Delta_{13}$. Apart from the atmospheric mass splitting, the scalar NSI
parameters appear coupled also to $r$ and $t$.

Notice that, in our analytical approach, the probabilities are symmetric under
the $\eta_{e\tau}$-$\eta_{e\mu}$ exchange. However, from table 4, it is clear
that this symmetry cannot be exact. Indeed, we checked that, when we include
in the expansion also the solar mass splitting, the symmetry is broken, as
expected. From table 4 we can also interestingly see that, even though the
disappearance probability strongly depends on $\eta_{\mu\tau}$, since the best
fit for $\phi_{\mu\tau}$ is similar to $\pi/2$, the leading term in the
probability is suppressed. Thus, the fits to the data generated with
$\eta_{\mu\tau}$ are better than the ones where data are generated using the
other scalar NSI parameters.

###  A.3 Sterile neutrinos

In this case, expanding in the small $s_{i4}=\sin\theta_{i4}$ quantities up to
the second order and averaging out the fast oscillations driven by $\Delta
m_{41}^{2}$, we obtain

$\displaystyle P_{\nu_{\mu}\to\nu_{e}}^{3+1}=0\,,$ (19)

and

$\displaystyle P_{\nu_{\mu}\to\nu_{\mu}}^{3+1}=$
$\displaystyle-2s_{24}^{2}\cos^{2}\Delta_{31}-2V^{\prime}\Delta_{31}s_{24}s_{34}\cos\delta_{23}\sin
2\Delta_{31}\,,$ (20)

where $V^{\prime}$ is the NC matter potential, which appears in the
probabilities due to the presence of a fourth sterile state. Notice that given
our PMNS parameterization, in the standard three-flavor oscillation
probabilities we substitute the usual complex phase $\delta$ with the
combination $\delta_{13}-\delta_{12}-\delta_{23}$. It is clear that, at our
expansion order, only $\theta_{24}$ modifies the disappearance probabilities,
multiplied however by the small (at the atmospheric peak)
$\cos^{2}\Delta_{31}$. This explains why in table 5 the Standard Model is
excluded only at 4.5$\sigma$ even if the true values for $\theta_{14}$ and
$\theta_{24}$ are relatively large (8∘). Moreover, it is clear that, since
$\delta_{12}\sim 1.9\pi$ or $0$ in the two best fits, the DUNE sensitivity to
the CP violating phase translates directly to the sensitivity to
$\delta_{12}$, whose best fits are maximal. Another interesting feature of the
disappearance probability is that it is always smaller for $\theta_{24}\neq 0$
. This explains why, for instance, a scalar NSI model in NO fits better the
3+1 data than the same model in IO. We can indeed observe that, the third line
in eq. A.2 reduces the disappearance probability in NO, while it enhances the
probability in IO.

##  Appendix B NSI Fits To NOvA, T2K, and Reactor Data

In this section we revisit the fits to NOvA and T2K data presented in
Denton:2020uda  for vector NSI and provide the first fits to NOvA and T2K data
in the presence of scalar NSI. The details of the fit are described there, but
we will briefly summarize the inputs here.

The fit includes neutrino and anti-neutrino appearance data from both NOvA and
T2K assuming a single energy bin. The fit also includes disappearance data
from both experiments where it is then effectively mapped onto a Hamiltonian
to derive the relevant constraints. To constrain the remaining parameters,
information from Daya Bay DayaBay:2018yms  was used for $\theta_{13}$ and
$\Delta m^{2}_{31}$ for both mass orderings and information from KamLAND
KamLAND:2013rgu  was used for the solar parameters $\theta_{12}$ and $\Delta
m^{2}_{21}$, along with the fact that $\Delta m^{2}_{21}>0$ from SNO,
Borexino, and SuperK. A fit was then performed to the NSI parameters while
minimizing the test statistic over $\Delta m^{2}_{31}$, $\theta_{23}$, and
$\delta$; the other three oscillation parameters were determined to have no
appreciable effect.

###  B.1 Vector NSI

In Denton:2020uda  a fit to NOvA and T2K data was performed in the context of
vector NSI. We reproduce the main result in table 6.

MO | NSI | $|\varepsilon_{\alpha\beta}|$ | $\phi_{\alpha\beta}/\pi$ | $\delta/\pi$ | $\Delta\chi^{2}$
---|---|---|---|---|---
NO | $\varepsilon_{e\mu}$ | 0.19 | 1.50 | 1.46 | 4.44
$\varepsilon_{e\tau}$ | 0.28 | 1.60 | 1.46 | 3.65
$\varepsilon_{\mu\tau}$ | 0.35 | 0.60 | 1.83 | 0.90
IO | $\varepsilon_{e\mu}$ | 0.04 | 1.50 | 1.52 | 0.23
$\varepsilon_{e\tau}$ | 0.15 | 1.46 | 1.59 | 0.69
$\varepsilon_{\mu\tau}$ | 0.17 | 0.14 | 1.51 | 1.03
Table 6: Best fit values to NOvA and T2K data and
$\Delta\chi^{2}=\chi^{2}_{\rm SM}-\chi^{2}_{\rm NSI}$ for a fixed MO
considering one complex vector NSI parameter at a time. (For the SM,
$\chi^{2}_{\rm NO}-\chi^{2}_{\rm IO}=2.3\,$.)

###  B.2 Scalar NSI

We redo the analysis of NOvA and T2K data as described above in the context of
scalar NSI. The results are reported in Fig. 7 and table 7.

Figure 7: The preferred and disfavored regions of scalar NSI given by NOvA and
T2K data with information from Daya Bay and KamLAND data as well. $\Delta
m^{2}_{31}$, $\theta_{23}$, and $\delta$ are minimized over and $\theta_{13}$,
$\theta_{12}$, and $\Delta m^{2}_{21}$ are fixed to the best fit values from
Daya Bay and KamLAND. The top, middle, and bottom rows correspond to
$\eta_{e\mu}(3)$, $\eta_{e\tau}(3)$, and $\eta_{\mu\tau}(3)$ respectively, and
the left and right columns correspond to the NO and IO respectively. The mass
of the lightest neutrino is taken to be zero here. The $(3)$ refers to the
fact that the scalar NSI parameters are plotted as rescaled to a density of 3
g/cm3. The blue stars are the best fit points, the light gray regions are
slightly disfavored and the dark gray regions are disfavored at 68% CL. The
successive orange colors represent integer units of $\Delta\chi^{2}$. $m_{\rm
lightest}=0$ eV
---
MO | NSI | $|\eta_{\alpha\beta}(3)|$ | $\phi_{\alpha\beta}/\pi$ | $\delta/\pi$ | $\Delta\chi^{2}$
NO | $\eta_{e\mu}(3)$ | 0.009 | 1.40 | 1.17 | 0.04
$\eta_{e\tau}(3)$ | 0.016 | 1.42 | 1.10 | 0.02
$\eta_{\mu\tau}(3)$ | 0.006 | 1.22 | 1.11 | 0.08
IO | $\eta_{e\mu}(3)$ | 0.016 | 1.82 | 1.86 | 2.33
$\eta_{e\tau}(3)$ | 0.013 | 0.66 | 1.89 | 2.20
$\eta_{\mu\tau}(3)$ | 0.057 | 1.60 | 1.85 | 2.33
$m_{\rm lightest}=0.05$ eV
MO | NSI | $|\eta_{\alpha\beta}(3)|$ | $\phi_{\alpha\beta}/\pi$ | $\delta/\pi$ | $\Delta\chi^{2}$
NO | $\eta_{e\mu}(3)$ | 0.002 | 1.66 | 1.18 | 0.10
$\eta_{e\tau}(3)$ | 0.003 | 0.62 | 1.13 | 0.08
$\eta_{\mu\tau}(3)$ | 0.009 | 0.56 | 1.17 | 0.06
IO | $\eta_{e\mu}(3)$ | 0.010 | 1.72 | 1.88 | 2.21
$\eta_{e\tau}(3)$ | 0.010 | 0.58 | 1.90 | 2.18
$\eta_{\mu\tau}(3)$ | 0.033 | 1.58 | 1.79 | 2.36
$m_{\rm lightest}=0.10$ eV
MO | NSI | $|\eta_{\alpha\beta}(3)|$ | $\phi_{\alpha\beta}/\pi$ | $\delta/\pi$ | $\Delta\chi^{2}$
NO | $\eta_{e\mu}(3)$ | 0.001 | 1.74 | 1.17 | 0.12
$\eta_{e\tau}(3)$ | 0.002 | 0.64 | 1.14 | 0.11
$\eta_{\mu\tau}(3)$ | 0.006 | 0.56 | 1.19 | 0.06
IO | $\eta_{e\mu}(3)$ | 0.006 | 1.72 | 1.86 | 2.20
$\eta_{e\tau}(3)$ | 0.006 | 0.60 | 1.88 | 2.19
$\eta_{\mu\tau}(3)$ | 0.024 | 1.56 | 1.83 | 2.36
Table 7: Best fit values to NOvA and T2K data and
$\Delta\chi^{2}=\chi^{2}_{\rm SM}-\chi^{2}_{\rm NSI}$ for a fixed MO
considering one complex scalar NSI parameter at a time, rescaled to what it
would be for a density of 3 g/cm3 for various values of $m_{\rm lightest}$.
(For the SM, $\chi^{2}_{\rm NO}-\chi^{2}_{\rm IO}=2.3\,$.)

We make some comments on these results. We note that the IO is always
preferred over the NO, but at even lower significance than for vector NSI. In
addition, we also recall that SuperK’s atmospheric data prefers the NO
Esteban:2020cvm ; Kelly:2020fkv ; Denton:2020uda  and will be affected by
scalar NSI in quite a different fashion to long-baseline data Ge:2018uhz ;
Medhi:2021wxj . Thus we find that scalar NSI is not a satisfactory improvement
to the slight NOvA and T2K tension, but we can still regard the best fit
points as valuable benchmarks moving forward.

Unlike in vector NSI, for scalar NSI the absolute neutrino mass scale plays a
role. In general the effect of $m_{\rm lightest}$, even to values quite a bit
larger than allowed by cosmology DiValentino:2021hoh  where the upper limit is
$\sim$few$\times 10^{-2}$ eV, on the results is quite small. That said, in
some cases we see that the best fit value changes by a fair amount as $m_{\rm
lightest}$ changes; this is due to the existence of multiple quasi-degenerate
local minima that slightly change as $m_{\rm lightest}$ changes.

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