aainstitutetext: Mitchell Institute for Fundamental Physics and Astronomy,
Department of Physics and Astronomy, Texas A$\&$M University, College Station,
Texas 77843, USAbbinstitutetext: School of Physics, Korea Institute for
Advanced Study, Seoul 02455, Koreaccinstitutetext: CAS Key Laboratory of
Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of
Sciences, Beijing, 100190, People’s Republic Chinaddinstitutetext: School of
Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049,
People’s Republic Chinaeeinstitutetext: Department of Physics and Astronomy,
Northwestern University, Evanston, IL 60208, USA

# Non-standard neutrino interactions mediated by a light scalar at DUNE

Bhaskar Dutta b Sumit Ghosh a Kevin J. Kelly c, d Tianjun Li a,e Adrian
Thompson a and Ankur Verma<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate the effect on neutrino oscillations generated by beyond-the-
standard-model interactions between neutrinos and matter. Specifically, we
focus on scalar-mediated non-standard interactions (NSI) whose impact
fundamentally differs from that of vector-mediated NSI. Scalar NSI contribute
as corrections to the neutrino mass matrix rather than the matter potential
and thereby predict distinct phenomenology from the vector-mediated ones.
Similar to vector-type NSI, the presence of scalar-mediated neutrino NSI can
influence measurements of oscillation parameters in long-baseline neutrino
oscillation experiments, with a notable impact on CP measurement in the case
of DUNE. Our study focuses on the effect of scalar NSI on neutrino
oscillations, using DUNE as an example. We introduce a model-independent
parameterization procedure that enables the examination of the impact of all
non-zero scalar NSI parameters simultaneously. Subsequently, we convert DUNE’s
sensitivity to the NSI parameters into projected sensitivity concerning the
parameters of a light scalar model. We compare these results with existing
non-oscillation probes. Our findings reveal that the region of the light
scalar parameter space sensitive to DUNE is predominantly excluded by non-
oscillation probes, except for scenarios with very light mediator mass.

###### Keywords:

Neutrino oscillation, NSI, Light scalars, Scalar NSI, DUNE

††preprint: MI-HET-823, KIAS-P23069

##  1 Introduction

The discovery of neutrino oscillations established that physics beyond the
Standard Model (SM) exists in the phenomenon of neutrino oscillations
Mikheev:1986gs ; Smirnov:2004zv ; Wolfenstein:1977ue ; SNO:2002tuh ;
KamLAND:2002uet ; T2K:2011ypd ; DoubleChooz:2011ymz ; DayaBay:2012fng ;
RENO:2012mkc . Since the earliest studies of neutrino oscillations, neutrino
interactions with SM matter and their impact on oscillations have been a prime
focus. Specifically, neutrino-electron interactions in the Sun lead to the MSW
effect Smirnov:2004zv ; Mikheev:1986gs , and neutrino-matter interactions in
the Earth modify oscillation probabilities significantly for neutrinos between
their production and detection in experiments such as T2K T2K:2011qtm , NOvA
NOvA:2007rmc , IceCube IceCube:2006tjp , and Super-Kamiokande Super-
Kamiokande:2004orf . Such effects are all predictions of the simplest
extensions of the SM to account for neutrino masses and do not require any
further beyond-the-SM (BSM) neutrino interactions.

The additional BSM neutrino interactions have been studied in great detail
under the moniker of neutrino non-standard interactions (NSIs) that posit
additional neutrino-SM (typically electrons and quarks) interactions that can
be detectable in laboratory environments. This includes modifications to
neutrino oscillations incurred by non-standard matter effects Gavela:2008ra ;
Antusch:2008tz ; Ohlsson:2012kf ; Proceedings:2019qno  as well as more direct
searches for modified neutrino-target scattering rates and spectra, e.g., in
coherent elastic neutrino-nucleus (CEvNS) scattering AristizabalSierra:2018eqm
; Giunti:2019xpr .

Typically, such NSIs are assumed to be mediated by vector/axial-vector
interactions à la those from the SM $W$\- and $Z$-bosons for neutrinos. In
this work, we focus instead on scalar non-standard interactions (SNSIs), where
a new scalar particle is introduced to mediate interactions between the SM
particles. Light and weakly-coupled scalar-mediated NSIs between neutrinos and
SM fermions can naturally emerge in numerous UV-complete models at low energy
scales (see, for example Ref. Dutta:2022fdt ). Various laboratory-based
experiments involving solar and reactor neutrino scattering, as well as
cosmological and astrophysical probes, enable searches in the parameter space
associated with light scalar fields. The existence of these new interactions
can also be explored through neutrino oscillation experiments.

The presence of SNSI manifests as a correction to the neutrino mass matrix in
the evolution Hamiltonian. Consequently, the BSM modifications to neutrino
oscillation probabilities follow a different structure (in terms of scaling
with matter densities, baseline lengths, and neutrino energies) than that
predicted by vector-mediated NSI which modify the matter potential in the
time-evolution Hamiltonian. Given that the impact of scalar NSI scales
linearly with the matter density of the medium, long-baseline experiments such
as Deep Underground Neutrino Experiment (DUNE) DUNE:2020ypp , JUNO
JUNO:2015zny , T2K T2K:2011qtm , and NOvA NOvA:2007rmc  provide an ideal
environment for probing these effects. In particular, it can mimic the CP
measurement efforts in oscillation experiments and prevent the accurate
measurement of the leptonic Dirac CP phase angle. Recently, such possibilities
have been extensively explored by formulating SNSI in a manner that primarily
allows for the exploration of one parameter at a time Ge:2018uhz ; Ge:2019tdi
; Denton:2022pxt ; Medhi:2021wxj ; Medhi:2022qmu ; Medhi:2023ebi ;
Singha:2023set ; Sarker:2023qzp ; Gupta:2023wct ; ESSnuSB:2023lbg .

In this work, we introduce a new method for studying SNSI in neutrino
oscillations, focusing on a parameterization that allows for more than one
nonzero SNSI parameters to be studied at a time, while focusing on the direct
impact on the neutrino oscillation Hamiltonian. We discuss how constraints and
sensitivities using this new phenomenological approach may be compared against
those depending on the underlying model parameters, i.e., Yukawa couplings and
the mass of the new particle. We use the upcoming DUNE experiment as a test
case in understanding this dynamic, as well as the mild T2K/NOvA tension
T2K:2021xwb ; T2K:2023smv ; NOvA:2021nfi ; NOvA:2023iam ; Kelly:2020fkv ;
Esteban:2020cvm ; deSalas:2020pgw  that has been highlighted as potential
motivation for SNSI in recent studies Denton:2022pxt .

The remainder of this work is organized as follows. First, in section 2, we
introduce the SNSI scenario under study here, review how such SNSI impact
neutrino oscillations, and introduce the phenomenological parameterization for
studies of this type. In section 3, we demonstrate the DUNE sensitivity in
this approach and discuss how these constraints may be compared against other
results in the literature. Section 4 then performs a broad comparison of
existing experimental constraints on the SNSI parameter space, while our
results are then shown in section 5. Finally in Section 6 we offer some
remarks on the future study of SNSI. Additional details concerning the
numerical results are provided in the appendix.

##  2 Scalar NSI Formalism for Oscillations

In the standard three-neutrino mixing framework, neutrino oscillations are
described by a Hamiltonian for (unitary) time-evolution through a medium. This
Hamiltonian, along with the misalignment between neutrino mass and flavor
eigenstates (described by the leptonic mixing matrix $U$), determines
transition amplitudes and oscillation probabilities for neutrinos traveling
over some distance. The standard picture also allows for neutrino interactions
with matter induced by the SM $W-$ and $Z-$ boson exchange.

Such SM interactions, as well as vector-like non-standard interactions (NSI)
Proceedings:2019qno  add an effective potential to the propagation
Hamiltonian, e.g.,

$H_{ij}=H_{ij}^{0}+U^{\dagger}V_{\alpha\beta}U,$ (1)

where $H_{ij}^{0}$ is the Hamiltonian for neutrino evolution in vacuum in the
neutrino-mass basis and $V_{\alpha\beta}$ describes the vector-like
interactions between neutrinos and SM matter during propagation. In the SM,
$V_{\alpha\beta}=\mathrm{diag}(V_{\rm CC},0,0)$, $V_{\rm
CC}=\sqrt{2}G_{F}n_{e}$ ($G_{F}$ being the Fermi constant and $n_{e}$ being
the number density of electrons along the path).111Despite SM interactions
with protons and neutrons existing, they make no impact on the propagation
Hamiltonian because such interactions provide an overall diagonal contribution
to $V_{\alpha\beta}$ to which neutrino oscillations are insensitive.
Additional BSM vector-like NSIs amount to modifying $V_{\alpha\beta}$,
including the possibility of inducing off-diagonal, complex contributions that
lead to additional CP violation in neutrino oscillations. We refer the reader
to Refs. deGouvea:2015ndi ; Coloma:2015kiu  for further detail on vector-like
NSI in the context of long-baseline oscillation experiments such as DUNE.

In this work, we focus on the less-explored possibility that NSIs are
generated by scalar mediators. Scalar NSIs can be described by the following
Lagrangian that includes the Yukawa interaction terms for Dirac neutrinos and
SM fermions $f$

$\mathcal{L}_{\phi^{\prime}}=\bar{\nu}\left(i\gamma^{\mu}\partial_{\mu}-m_{\nu}\right)\nu-(y_{\nu})_{\alpha\beta}\bar{\nu}_{\alpha}\nu_{\beta}\phi^{\prime}-\left(y_{f}\right)_{\alpha\beta}\bar{f}_{\alpha}f_{\beta}\phi^{\prime}-\frac{1}{2}\left(\partial_{\mu}\phi^{\prime}\right)^{2}-\frac{m_{\phi^{\prime}}^{2}}{2}\phi^{\prime
2}$ (2)

The Yukawa interaction terms, responsible for the generation of the SNSI, can
arise naturally in a UV-complete model with an extended scalar sector. Here
$\phi^{\prime}$ is a light real scalar field that emerges from the CP-even
Higgs sector (see, for example, Refs. Dutta:2020scq ; Dutta:2022fdt ). As we
discussed above, vector-like NSIs modify $V_{\alpha\beta}$, where scalar
interactions will generate an effective mass for the neutrinos, e.g.,

$\mathbb{M}\to\mathbb{M}_{0}+\delta\mathbb{M}_{ij}$ (3)

We have implicitly assumed here that $\mathbb{M}_{0}$ (the standard, diagonal
mass matrix) and $\delta\mathbb{M}_{ij}$ are written in the neutrino mass
basis. However in the literature, the corrections to the mass terms are
usually parameterized in the flavor basis as follows

$\delta\mathbb{M}_{\alpha\beta}\equiv\sqrt{|\Delta
m_{31}^{2}|}\begin{pmatrix}\eta_{ee}&\eta_{e\mu}e^{i\phi_{e\mu}}&\eta_{e\tau}e^{i\phi_{e\tau}}\\\
\eta_{e\mu}e^{-i\phi_{e\mu}}&\eta_{\mu\mu}&\eta_{\mu\tau}e^{i\phi_{\mu\tau}}\\\
\eta_{e\tau}e^{-i\phi_{e\tau}}&\eta_{\mu\tau}e^{-i\phi_{\mu\tau}}&\eta_{\tau\tau}\end{pmatrix},$
(4)

such that
$\delta\mathbb{M}_{ij}\equiv(U^{\dagger})_{i\alpha}\delta\mathbb{M}_{\alpha\beta}U_{\beta
j}$. Here the elements of $\delta\mathbb{M}_{\alpha\beta}$ are expressed
relative to $\sqrt{|\Delta m_{31}^{2}|}$ in order to ensure that
$\eta_{\alpha\beta}$ are dimensionless. Hermiticity in the neutrino
Hamiltonian necessitates that the elements along the diagonal must be real,
while the off-diagonal elements are complex.

As we are only interested in coherent forward scattering of neutrinos in
matter with zero momentum transfer, the SNSI parameters $\eta_{\alpha\beta}$
can be related to the parameters of the underlying theory as follows

$\displaystyle\eta_{\alpha\beta}$
$\displaystyle=\dfrac{n_{f}y_{f}y_{\alpha\beta}}{\sqrt{|\Delta
m_{31}^{2}|}m_{\phi}^{2}}$ (5)

Effectively, scalar NSIs induce a correction to $\mathbb{M}$ that enters
$H_{ij}\propto\mathbb{M}^{2}_{ij}/2E_{\nu}$. Since the mapping between the
fundamental Lagrangian parameters $y_{\alpha\beta}$ or the dimensionless
$\eta_{\alpha\beta}$ to the impact on the propagation Hamiltonian (and
therefore neutrino oscillations) is relatively complicated, we choose to
introduce an effective parameterization for neutrino oscillations. We assume
that $\mathbb{M}^{2}_{\rm eff}$ takes the following form

$\mathbb{M}_{\rm eff}^{2}\equiv\left(\begin{array}[]{c c
c}m_{1}^{2}+\mu_{11}&\mu_{12}&\mu_{13}\\\
\mu_{12}^{*}&m_{2}^{2}+\mu_{22}&\mu_{23}\\\
\mu_{13}^{*}&\mu_{23}^{*}&m_{3}^{2}+\mu_{33}\end{array}\right).$ (6)

When $\mu_{ij}\to 0$, we recover the standard three-neutrino case. The
diagonal $\mu$ are real and the off-diagonal are complex. We are insensitive
to an overall constant times the identity appearing in this Hamiltonian, so we
subtract $m_{1}^{2}+\mu_{11}$ without loss of generality, recovering $\Delta
m_{21}^{2}+(\mu_{22}-\mu_{11})$ and $\Delta m_{31}^{2}+(\mu_{33}-\mu_{11})$
along the diagonal.

We note here that oscillations are sensitive to eight additional real
parameters when dealing with the most general scalar NSI scenario, i.e., not
restricting ourselves to any specific flavor-structure in the
$\phi$-$\nu$-$\nu$ interaction. However, nine real parameters appear in Eq. 4,
foreshadowing a redundancy when we interpret constraints on $\mu_{ij}$ with
respect to $\eta_{\alpha\beta}$ – we will address this in Sec. 5.222We also
emphasize here that in standard three-flavor neutrino oscillations we are
insensitive to the overall mass scale of the (ultrarelativistic) neutrinos
$m_{0}$. However, when mapping between $\mu_{ij}$ and $\eta_{\alpha\beta}$ we
will be, since $\mathbb{M}_{\rm
eff}^{2}=\left(\mathbb{M}_{0}+\delta\mathbb{M}\right)^{2}$ will involve cross-
terms between $\mathbb{M}_{0}$ and $\delta\mathbb{M}$.

In the remainder of this work, we will study the capability of the upcoming
DUNE experiment to test the scalar NSI paradigm by constraining the $\mu_{ij}$
parameterization. Subsequently, we will discuss how these constraints are
related to flavor-specific constraints on $y_{\alpha\beta}$ and $m_{\phi}$
(from Eq. 5) so that we can compare against (a) constraints pertaining
directly to neutrino scattering and (b) constraints regarding light scalars
coupling to SM charged fermions via the Yukawa couplings $y_{f}$.

###  2.1 Mapping Between Parameter Spaces

In this section, we briefly discuss the parameter mapping back and forth
between the $\eta_{\alpha\beta}$ basis – directly related to the flavor-basis
Yukawa couplings present in Eq. 5 – adopted in the literature for scalar NSI
and neutrino oscillations and our $\mu_{ij}$ basis in Eq. 6. First let us
assume that
$\delta\mathbb{M}_{ij}=U^{\dagger}\delta\mathbb{M}_{\alpha\beta}U$, i.e. we
can rotate the flavor-basis modification to the neutrino masses in Eq. 4 into
the mass basis by using the standard leptonic mixing matrix.

We define $\delta\mathbb{M}_{ij}$ such that

$\delta\mathbb{M}_{ij}=U^{\dagger}\delta\mathbb{M}_{\alpha\beta}U=\left(\begin{array}[]{lll}\epsilon_{11}&\epsilon_{12}&\epsilon_{13}\\\
\epsilon_{12}^{*}&\epsilon_{22}&\epsilon_{23}\\\
\epsilon_{13}^{*}&\epsilon_{23}^{*}&\epsilon_{33}\end{array}\right),$ (7)

so that $\epsilon_{ij}$ carry mass dimension (relative to the dimensionless
$\eta_{\alpha\beta}$). We can then express $\mathbb{M}_{\rm eff}^{2}$ by
taking $\left(\mathbb{M}_{0}+\delta\mathbb{M}_{ij}\right)^{2}$, and setting
elements equal. Comparing against Eq. 6, we obtain the following relationships

$\displaystyle\mu_{11}$ $\displaystyle=$ $\displaystyle
2m_{1}\epsilon_{11}+\left|\epsilon_{11}\right|^{2}+\left|\epsilon_{12}\right|^{2}+\left|\epsilon_{13}\right|^{2},$
$\displaystyle\mu_{22}$ $\displaystyle=$ $\displaystyle
2m_{2}\epsilon_{22}+\left|\epsilon_{22}\right|^{2}+\left|\epsilon_{12}\right|^{2}+\left|\epsilon_{23}\right|^{2},$
$\displaystyle\mu_{33}$ $\displaystyle=$ $\displaystyle
2m_{3}\epsilon_{33}+\left|\epsilon_{33}\right|^{2}+\left|\epsilon_{13}\right|^{2}+\left|\epsilon_{23}\right|^{2},$
$\displaystyle\mu_{12}$ $\displaystyle=$
$\displaystyle\left(m_{1}+m_{2}+\epsilon_{11}+\epsilon_{22}\right)\epsilon_{12}+\epsilon_{13}\epsilon_{23}^{*},$
$\displaystyle\mu_{13}$ $\displaystyle=$
$\displaystyle\left(m_{1}+m_{3}+\epsilon_{11}+\epsilon_{33}\right)\epsilon_{13}+\epsilon_{12}\epsilon_{23}^{*},$
$\displaystyle\mu_{23}$ $\displaystyle=$
$\displaystyle\left(m_{2}+m_{3}+\epsilon_{22}+\epsilon_{33}\right)\epsilon_{23}+\epsilon_{13}\epsilon_{12}^{*}.$
(8)

For a given set of $\eta_{\alpha\beta}$, the $\epsilon_{ij}$ can be determined
given a mixing matrix and a set of $m_{1},$ $m_{2}$, $m_{3}$. Here we
demonstrate the final mapping into $\mathbb{M}_{\rm eff}^{2}$ that enters the
propagation Hamiltonian for neutrino oscillations. The inverse of this is less
straightforward – assuming an oscillation experiment can constrain the set of
$\\{\mu_{ij}\\}$, interpreting those constraints instead as being on
$\eta_{\alpha\beta}$ requires further assumptions. This is because oscillation
experiments are only sensitive to the differences $\mu_{22}-\mu_{11}$ and
$\mu_{33}-\mu_{11}$ instead of the overall scaling, so further assumptions on
$\mu_{ij}$ (or equivalently $\epsilon_{ij}$) are required to solve the system
of equations.

###  2.2 Impact of SNSI on Long-Baseline Oscillation Probabilities

In the following sections, we will be interested in long-baseline experiments
(such as DUNE) and their measurements of appearance ($\nu_{\mu}\to\nu_{e}$)
and disappearance ($\nu_{\mu}\to\nu_{\mu}$) oscillation probabilities in a
muon-neutrino beam. In this subsection, we inspect the impact of nonzero
scalar NSIs on these oscillation probabilities, focusing on
$P(\nu_{\mu}\to\nu_{e})$ where the effects are most apparent.

| $\sin^{2}\theta_{12}$ | $\sin^{2}\theta_{13}$ | $\sin^{2}\theta_{23}$ | $\delta_{\rm CP}$ | $\Delta m_{21}^{2}$ | $\Delta m_{31}^{2}$
---|---|---|---|---|---|---
Normal MO | $0.318$ | $0.022$ | $0.574$ | $1.08\pi$ |  $7.506\times 10^{-5}$ eV2 |  $2.55\times 10^{-3}$ eV2
Inverted MO | $0.304$ | $0.022$ | $0.578$ | $1.58\pi$ |  $7.506\times 10^{-5}$ eV2 |  $-2.45\times 10^{-3}$ eV2
Table 1: Oscillation parameters used as assumed-true values for our numerical
studies deSalas:2020pgw .

In determining oscillation probabilities, we assume that the (constant) matter
density of the earth along the path of propagation is $2.84$ g/cm3 and that
the baseline length $L=1300$ km. Since the lightest neutrino mass factors in
to the mapping between $\eta_{\alpha\beta}$ and $\mu_{ij}$ (Eq. 8), we take
the normal mass-ordering parameters in Table 1 and $m_{1}=0.1$ eV for
demonstrative purposes.

In Fig. 1, we focus instead on the impact of scalar NSI and CP violation on
the oscillation probability near its peak, $E=2.5$ GeV. In the three-flavor
scenario (even with matter effects), the oscillation probability can be shown
to vary primarily with respect to $\sin\delta_{\rm CP}$ due to its dependence
on the Jarlskog invariant. In contrast, we see now with
$\eta_{\alpha\beta}\neq 0$, this trend can change (even with real NSI
parameters). Such degeneracies between the scalar NSI parameters and
$\delta_{\rm CP}$ will make robust extraction of the CP-violating parameters
Medhi:2021wxj ; Medhi:2022qmu ; Denton:2022pxt ; Sarker:2023qzp ;
ESSnuSB:2023lbg , as well as precision sensitivity to nonzero
$\eta_{\alpha\beta}$, more challenging as we will see in the coming results.

Figure 1:  The effects of $\eta_{ee}$ (left) and $\eta_{e\mu}$ (right) on
oscillation probabilities $P_{\mu e}$. We fix the normal mass ordering and
parameters from Table 1, varying $\delta_{\rm CP}$ in each panel, and fix
$E_{\nu}=2.5$ GeV. The black line corresponds to the standard interaction
case($\eta_{\alpha\beta}=0$) while other lines correspond to the presence of
non-zero NSI parameters. Similar effects on oscillation probabilities exist
for other $\eta_{\alpha\beta}$.

##  3 SNSI sensitivities at DUNE

The Deep Underground Neutrino Experiment (DUNE) is a forthcoming long-baseline
neutrino experiment in the US, with ${\sim}$few GeV muon-flavor neutrinos
produced via a $120$ GeV proton beam at Fermilab, travelling $1300$ km to
South Dakota. The far-detector site in South Dakota will host large liquid-
argon time-projection chamber detectors, with a target fiducial mass of 40
kiloton.333It is likely that this mass will not be achieved simultaneously
with the onset of beam operation. We will quote our sensitivities in terms of
kiloton$\times$MW$\times$year exposures, so that any rescaling for different
detector mass/beam power/time may be performed. We will perform analyses
assuming a baseline beam power of $1.2$ MW and seven years of data collection,
with equal operation in neutrino and antineutrino modes. This corresponds to
an assumed exposure of $336$ kt$\times$MW$\times$yr. We refer the reader to
Refs. Berryman:2015nua ; deGouvea:2015ndi ; Berryman:2016szd ;
deGouvea:2016pom ; deGouvea:2017yvn  for further details regarding how the
event spectra in DUNE are simulated.

To evaluate DUNE’s capability in constraining the scalar NSI parameters, we
simulate the expected event rate (in four distinct oscillation channels) at
DUNE assuming the standard oscillation parameters in Table 1 and no new
physics present. We then construct a test statistic
$\chi^{2}=-2\ln\mathcal{L}$ (where $\ln\mathcal{L}$ is the bin-by-bin
Poissonian log-likelihood, summed over all experimental bins) by comparing
this expected event rate against a rate determined using test parameters
allowing the scalar NSI to be nonzero. We employ the Bayesian inference tool
MultiNest Feroz:2008xx ; Buchner:2014nha  to study the parameter space and
derive expected constraints on the new-physics parameters given expected DUNE
data. In calculating the test statistic, we include 5% normalization
systematic uncertainties in each oscillation channel.

In Eq. 6, we see that there is also an unavoidable degeneracy between the
“vacuum” $\Delta m_{j1}^{2}$ parameters and the diagonal $\mu_{jj}$ ones. Such
degeneracies are present in different parameterizations of scalar NSI (e.g.
the $\eta_{\alpha\beta}$ one) when considering multiple parameters
simultaneously. To this end, we fix $\Delta m_{j1}^{2}$ to their best-fit
values in our analyses and allow $(\mu_{jj}-\mu_{11})$ ($j=2,$ $3$) to vary.
The off-diagonal parameters and their CP-violating phases are also free to
vary simultaneously, providing us with a thorough, conservative analysis of
these future prospects at DUNE. The standard mixing angles
$\sin^{2}\theta_{ij}$ and the CP-violating phase $\delta_{\rm CP}$ are also
allowed to vary in our analyses.

Using MultiNest, we obtain the expected constraints on all varied parameters –
in Appendix A we display the resulting corner plots demonstrating this
sensitivity with respect to the $\mu_{ij}$ parameters assuming that the truth
data are consistent with the normal mass ordering (Fig. 5) or the inverted
mass ordering (Fig. 6). In Sec. 2.1, we showed the nontrivial parameter
mapping between $\mu_{ij}$ and $\eta_{\alpha\beta}$, including the dependence
on the lightest neutrino mass $m_{0}$. To translate our results into
constraints on $\eta_{\alpha\beta}$, we assume $m_{0}=0.1$ eV and
$\epsilon_{11}=0$, resulting in the corner plots shown in Fig. 7 (NO) and Fig.
8 (IO), also in Appendix A. In Sec. 5 we will discuss the impact of this
$m_{0}$, $\epsilon_{11}$ assumption on the parameter mapping and on the
results we obtain.

In comparing our results constraining $\mu_{ij}$ to those constraining
$\eta_{\alpha\beta}$, it’s worth noting the non-linearity of the conversion
between the two parameterizations. This leads to an additional layer of
degeneracy (because the $\eta_{\alpha\beta}$ modify $\mathbb{M}$ where the
true sensitivity lies in $\mathbb{M}^{2}\propto\mu_{ij}$) in determining
constraints on $\eta_{\alpha\beta}$ when allowing all parameters to vary
simultaneously. This leads to a degradation in DUNE sensitivity to scalar NSI.
As a simple example, the transformation

$\displaystyle\epsilon_{11}$ $\displaystyle\rightarrow$
$\displaystyle-\epsilon_{11}-2m_{1}$ $\displaystyle\epsilon_{22}$
$\displaystyle\rightarrow$ $\displaystyle-\epsilon_{22}-2m_{2}$
$\displaystyle\epsilon_{33}$ $\displaystyle\rightarrow$
$\displaystyle-\epsilon_{33}-2m_{3}$ $\displaystyle\epsilon_{ij}$
$\displaystyle\rightarrow$ $\displaystyle-\epsilon_{ij};\leavevmode\nobreak\
\leavevmode\nobreak\ i\neq j$ (9)

leaves $\mathbb{M}^{2}$ and therefore the propagation Hamiltonian invariant.
This leads to additional islands of degeneracy. In light of this, we attempt
to prune these degenerate branches in order to consider the most optimistic
constraints DUNE can hope to achieve, subject to this consideration.

##  4 Experimental bounds

In order to compare the DUNE projections against other experimental efforts,
we must consider other searches attempting to discover new light scalars
interacting with SM particles. For a light neutral scalar to impact
terrestrial neutrino oscillations, it must couple to neutrinos as well as some
earth matter, i.e. electrons or nucleons. This means that neutrino
oscillations are sensitive to the combination $\sqrt{y_{e}y_{\nu}}$ or
$\sqrt{y_{N}y_{\nu}}$ as a function of $m_{\phi}$, where other constraints may
probe the individual Yukawa couplings as a function of $m_{\phi}$
independently.

For the remainder of this section, we will summarize the dominant constraints
on these Yukawa interactions as a function of $m_{\phi}$. Then, in Sec. 5 we
will compare our DUNE sensitivity against these.

#### Fifth Force:

Scalar interactions between electrons or nucleons can lead to a modification
from the expected Newton’s law of gravitation on small scales (e.g.,
Damour:2010rm ; Brzeminski:2022sde  etc.). Strong constraints on both $y_{e}$
and $y_{N}$, particularly for $m_{\phi}\lesssim$ 0.1 eV, have been derived
from a lack of this deviation in various experiments. Specifically, torsion
balance experiments Schlamminger:2007ht  and space-based experiments for very
small mass Berge:2017ovy ; MICROSCOPE:2022doy  provide the strongest
constraints.

#### CMB Observations:

Neutrino-scalar interactions can have multiple effects in the early universe,
including delaying neutrino free-streaming and adding additional, thermalized,
light degrees of freedom around the epoch of the formation of the Cosmic
Microwave Background. Constraints of $\Delta N_{\rm eff}$ at this epoch lead
to very strong constraints on $y_{\nu}$ down to very small $m_{\phi}\lesssim$
eV Sandner:2023ptm ; NeffEscudero .

#### BBN:

Similar to the above, such new mediators can spoil the successful prediction
of light element abundances formed during big-bang nucleosynthesis. This can
be spoiled via a variety of Yukawa couplings, either among electrons
(modifying annihilation processes during this epoch) or neutrinos (modifying
their free-streaming as above). We refer the reader to Ref. Babu:2019iml  for
further discussion of these effects, and use the constraints therein.

#### Stellar and Supernova Cooling:

New, light mediators can be emitted in stellar environments, leading to
significantly higher cooling rates than expected. The strongest constraints of
this type come from red giant and horizontal branch stars Hardy:2016kme  for
very light mediators, requiring $y_{e}\lesssim 10^{-15}$ and $y_{N}\lesssim
10^{-12}$ Babu:2019iml . Supernovae provide constraints from similar
arguments, and the observation of neutrinos from SN1987A allow one to exclude
the regions $10^{-9}\lesssim y_{e}\lesssim 10^{-7}$ and $10^{-10}\lesssim
y_{N}\lesssim 10^{-7}$ if $m_{\phi}<m_{e}$ Babu:2019iml .

#### In-medium Neutrino Mass:

Just as the scalar NSI contribute to an effective mass term for neutrino
oscillations, they can generate an effective mass for neutrinos produced or
propagating in a sufficiently dense environment. Constraints on the effective
mass of neutrinos produced in dense environments (particularly the sun and in
a supernova explosion) have been translated into constraints on the Yukawa
couplings and mediator mass in Ref. Babu:2019iml , and yield some of the most
severe bounds for very small mediator masses. Specifically, Borexino data
Borexino:2017rsf  were used to put bounds in the case of sun, while data from
the SN1987A event Smirnov:2019cae ; Babu:2019iml  were used to set constraints
in the case of supernova. We will discuss the possibility of relaxing such
constraints in Sec. 5 when placing our results in context.

#### Neutrino Scattering Experiments:

Neutrino scattering off nuclei (and electrons nearby) allow experiments to
constrain the product $y_{\nu}y_{e/N}$. Various lab based experiments: elastic
neutrino-electron scattering such as TEXONO TEXONO:2009knm , BOREXINO
BOREXINO:2018ohr , GEMMA Beda:2013mta , XENONnT XENON:2022ltv , LZ LZ:2022lsv
and CEvNS experiment such as COHERENT Akimov:2017ade ; Akimov:2018vzs ;
Akimov:2018ghi ; Akimov:2019xdj ; Akimov:2020pdx ; COHERENT:2021pvd  put
bounds on this through measurements of neutrino-electron/neutrino-nucleon
elastic scattering rates. In neutrino-electron scattering experiments, XENONnT
provides the most stringent constraint, restricting
$\sqrt{y_{\nu}y_{e}}\lesssim 8\times 10^{-7}$ for $m_{\phi}\lesssim 10$ keV.
In the case of neutrino-nucleus scattering, COHERENT has placed bounds on
$\sqrt{y_{\nu}y_{N}}\lesssim 2\times 10^{-5}$ for $m_{\phi}\lesssim 10$ MeV.

Note that only the neutrino scattering experiments can put constraints on the
product coupling $\sqrt{y_{\nu}y_{e/N}}$, while all other constraints are on
the individual couplings. To obtain constraints on the product coupling
$\sqrt{y_{\nu}y_{e/N}}$, we combine the individual constraints on $y_{e/N}$
and $y_{\nu}$.

##  5 Results

As we begin assessing the results of our DUNE scalar NSI analysis, we first
address the question of CP violation when considering an analysis with more
parameters than in the standard oscillation case. In Fig. 1 we demonstrated
the degeneracy between allowing nonzero SNSI parameters and the standard CP
violating phase $\delta_{\rm CP}$, alluding to the fact that analyses of this
sort will lead to diminished power of determining this parameter – Ref.
Medhi:2022qmu  explored some aspects of this question.

Figure 2:  DUNE’s potential towards CP precision measurements in the presence
of SNSI considering all parameters at the same time taking
$m_{\text{lightest}}=0.1$ eV, $\epsilon_{11}=0.0(\epsilon_{33}=0.0)$ for NO
(IO). The solid blue (orange) line corresponds to the NO (IO) case in the
presence of SNSI while the dashed blue (orange) line is the corresponding
curve in the absence of any SNSI.

Fig. 2 demonstrates the reduction in measurement power of $\delta_{\rm CP}$
assuming either set of true parameters given in Table 1. All other parameters
in our analysis (see Sec. 3 for details) are marginalized in constructing this
figure. Here, the blue (orange) curves represent our expected measurement
potential by DUNE assuming that nature has chosen the normal (inverted) mass
ordering. The solid lines represent our analysis considering all SNSI
parameters to be simultaneously nonzero, whereas the dashed ones are for a
“three-flavor-only” analysis, i.e. $\mu_{ij}=0$. For each truth scenario, we
see a significant reduction in the ability to measure $\delta_{\rm CP}$, as
well as the capability to exclude $\delta_{\rm CP}=0$, $\pi$, i.e. to
determine whether or not CP is conserved in the lepton sector. We highlight
here too that additional local minima appear ($\delta_{\rm CP}\sim 0$ for NO
and $\delta_{\rm CP}\sim\pi/2$ for IO) in these measurement plots due to the
degeneracy between “fake” solutions when nonzero SNSI are allowed. We note
however, one advantage of having several next-generation experiments that are
uniquely sensitive to SNSI. Combined analyses, for instance with JUNO
Gupta:2023wct  and Hyper-Kamiokande, should in principle guard against such
dramatic parameter degeneracies and allow for robust measurements/constraints
on all parameters. We leave a dedicated study of such combinations to future
work.

Figure 3: Left: Upper limits of various SNSI parameters $\eta_{\alpha\beta}$
as a function of $\epsilon_{11}$ at 95% C.L. for the normal mass ordering case
with the lightest mass taken as $m_{1}=0.1$ eV. Similar trends would follow
for the inverted mass ordering scenario in relation to the variable
$\epsilon_{33}$ for $m_{3}=0.1$ eV. Right: Same but fixing $\epsilon_{11}=0$
and varying $m_{1}$.

Next, it is our goal to take the results of our constraints on the $\mu_{ij}$
parameters (see Appendix A for corner plots and more discussion) into
constraints on the oft-studied $\eta_{\alpha\beta}$ ones. First, as a
benchmark for comparison, we perform analyses studying the sensitivity to
nonzero $\eta_{\alpha\beta}$ one-parameter-at-a-time. Such analyses require a
choice of $m_{1}$ (or some overall mass scale), and we set $m_{1}=0.1$ eV
($m_{3}=0.1$ eV) when considering the NO (IO). We provide the expected
sensitivity at 95$\%$ Confidence Level (C.L.) to the various
$\eta_{\alpha\beta}$ in this approach in Table 2.

NSI | NO ($m_{1}$ = 0.1 eV) | IO ($m_{3}=0.1$ eV)
---|---|---
$\eta_{ee}$ | $[-0.006,0.006]$ | $[-0.0077,0.0059]$
$\eta_{\mu\mu}$ | $[-0.004,0.004]$ | $[-0.0036,0.004]$
$\eta_{\tau\tau}$ | $[-0.004,0.004]$ | $[-0.0037,0.0036]$
$|\eta_{e\mu}|$ | [0,0.0017] | [0,0.0017]
$|\eta_{e\tau}|$ | [0,0.0019] | [0,0.0015]
$|\eta_{\mu\tau}|$ | [0,0.0042] | [0,0.0035]
Table 2: Summary of expected sensitivities of DUNE to SNSI parameters
$\eta_{\alpha\beta}$ at 95% C.L. when considered one at a time for
$m_{\text{lightest}}=0.1$ eV.

When allowing all parameters to vary simultaneously (as discussed in Sec. 2),
the larger number of free parameters in the $\eta_{\alpha\beta}$
parameterization relative to the $\mu_{ij}$ one means that choices must be
made in this mapping. In addition to fixing the lightest neutrino mass, this
amounts to choosing $\epsilon_{11}$ ($\epsilon_{33}$) in the NO (IO), one of
the mass-basis SNSI parameters. In Fig. 3(left), we demonstrate how the
constraints on $\mu_{ij}$ from our normal-mass-ordering analysis translate
into those on $\eta_{\alpha\beta}$ at $95\%$ CL, when $\epsilon_{11}$ is
varied and $m_{1}$ is fixed to be $0.1$ eV. The results when assuming the
inverted mass ordering are qualitatively similar. In both cases, when
$\epsilon_{11}$ (or $\epsilon_{33}$ is allowed to be large, degeneracies in
the mapping between parameter spaces (see Eq. 8) allow several parameters to
be simultaneously large leading to large $\eta_{\alpha\beta}$ (note that here
we are presenting upper-limit constraints on $|\eta_{\alpha\beta}|$, where the
degeneracies may rely on the diagonal $\eta_{\alpha\alpha}$ being negative
and/or the off-diagonal ones being complex).

We can repeat the same procedure, fixing $\epsilon_{11}$ (or $\epsilon_{33}$
in the IO) and varying the lightest neutrino mass $m_{1}$ ($m_{3}$), and
analyze how the constraints on $\eta_{\alpha\beta}$ vary. This is presented in
Fig. 3 (right) for the six different magnitudes under the NO assumption and
$\epsilon_{11}=0$. Again, the qualitative results are similar when considering
the IO and varying $m_{3}$ – we find that the most optimistic constraints we
can hope to obtain by DUNE are when $\epsilon_{11}$ ($\epsilon_{33}$) is zero
and $m_{1}$ ($m_{3}$) is taken to be as large as reasonably possible,
approximately $0.1$ eV. However, the constraints, unless we end up in a
highly-degenerate region, do not get more than an order-of-magnitude or so
worse. For these reasons, we will take the optimistic scenario that the
lightest neutrino mass is $0.1$ eV and $\epsilon_{11}=0$ ($\epsilon_{33}=0$)
in the NO (IO).

In comparison to Table 2, we provide expected constraints on the
$\eta_{\alpha\beta}$ when all parameters are allowed to vary simultaneously in
Table 3. We do so for three demonstrative choices for the lightest neutrino
mass, $0$ eV, $0.05$ eV, and $0.1$ eV (where the expected constraints tend to
improve with increasing $m_{\rm lightest}$, as demonstrated in Fig. 3 right).

$\begin{array}[]{|c|c||c|}\lx@intercol\hfil m_{\text{lightest
}}=0\leavevmode\nobreak\ \text{eV}\hfil\lx@intercol\\\ \hline\cr\text{ MO
}&\text{ NSI }&\eta_{\alpha\beta}\\\ \hline\cr&\eta_{ee}&[-0.133,0.063]\\\
&\eta_{\mu\mu}&[-0.107,0.059]\\\ \text{NO}&\eta_{\tau\tau}&[-0.12,0.075]\\\
&\eta_{e\mu}&[0,0.074]\\\ &\eta_{e\tau}&[0,0.072]\\\
&\eta_{\mu\tau}&[0,0.079]\\\ \hline\cr&\eta_{ee}&[-0.017,0.019]\\\
&\eta_{\mu\mu}&[-0.028,0.044]\\\ \text{ IO }&\eta_{\tau\tau}&[-0.031,0.041]\\\
&\eta_{e\mu}&[0,0.025]\\\ &\eta_{e\tau}&[0,0.024]\\\
&\eta_{\mu\tau}&[0,0.052]\\\ \hline\cr\end{array}$

$\begin{array}[]{|c|c||c|}\lx@intercol\hfil m_{\text{lightest
}}=0.05\leavevmode\nobreak\ \text{eV}\hfil\lx@intercol\\\ \hline\cr\text{ MO
}&\text{ NSI }&\eta_{\alpha\beta}\\\ \hline\cr&\eta_{ee}&[-0.015,0.005]\\\
&\eta_{\mu\mu}&[-0.028,0.015]\\\ \text{ NO }&\eta_{\tau\tau}&[-0.027,0.016]\\\
&\eta_{e\mu}&[0,0.013]\\\ &\eta_{e\tau}&[0,0.015]\\\
&\eta_{\mu\tau}&[0,0.02]\\\ \hline\cr&\eta_{ee}&[-0.012,0.013]\\\
&\eta_{\mu\mu}&[-0.019,0.012]\\\ \text{ IO }&\eta_{\tau\tau}&[-0.014,0.017]\\\
&\eta_{e\mu}&[0,0.012]\\\ &\eta_{e\tau}&[0,0.013]\\\
&\eta_{\mu\tau}&[0,0.021]\\\ \hline\cr\end{array}$

$\begin{array}[]{|c|c||c|}\lx@intercol\hfil m_{\text{lightest
}}=0.1\leavevmode\nobreak\ \text{eV}\hfil\lx@intercol\\\ \hline\cr\text{ MO
}&\text{ NSI }&\eta_{\alpha\beta}\\\ \hline\cr&\eta_{ee}&[-0.0074,0.0026]\\\
&\eta_{\mu\mu}&[-0.016,0.009]\\\ \text{ NO }&\eta_{\tau\tau}&[-0.016,0.008]\\\
&\eta_{e\mu}&[0,0.007]\\\ &\eta_{e\tau}&[0,0.008]\\\
&\eta_{\mu\tau}&[0,0.016]\\\ \hline\cr&\eta_{ee}&[-0.0077,0.0082]\\\
&\eta_{\mu\mu}&[-0.011,0.0067]\\\ \text{ IO
}&\eta_{\tau\tau}&[-0.0081,0.0096]\\\ &\eta_{e\mu}&[0,0.0073]\\\
&\eta_{e\tau}&[0,0.0077]\\\ &\eta_{\mu\tau}&[0,0.012]\\\ \hline\cr\end{array}$

Table 3: Summary of expected sensitivities of DUNE to SNSI parameters
$\eta_{\alpha\beta}$ at 95% C.L. when considered all at the same time for
$\epsilon_{11}=0.0$(NO) and $\epsilon_{33}=0.0$(IO)

Our final goal is to evaluate how DUNE’s sensitivity to SNSI parameters
compares with that of various non-oscillation-based probes mentioned in
section. 4. To facilitate a meaningful comparison, we begin by converting
DUNE’s projected sensitivities on SNSI parameters into projected sensitivities
on the model parameters, which include Yukawa couplings and mass. This
translation is achieved using the relationship established in Eq. 5.
Specifically, we consider values for the $\eta_{\alpha\beta}$ parameters that
correspond to the $95\%$ C.L. contours depicted in Fig. 7 and Fig. 8 for the
NO and IO scenarios, respectively. Note that these contours correspond to the
scenario where the lightest neutrino mass is fixed at $0.1$ eV. We then
superimpose these projected lines onto the bounds obtained from various
terrestrial experiments, as well as astrophysical and cosmological probes in
coupling vs mass plane $(\sqrt{y_{f}y_{\alpha\beta}},m_{\phi})$. Specifically,
in Figure 4(a), we present the results for electron couplings, while in Figure
4(b), we show the outcomes for nucleon couplings.

(a)

(b)

Figure 4:  The sensitivity of DUNE experiment at 95% C.L. (dashed blue) on the
light scalar mediator $\phi$ mass and couplings in the
$(m_{\phi},\sqrt{y_{\nu}y_{e/N}})$ plane. The left panel is for electron
couplings $\sqrt{y_{\nu}y_{e}}$ while the right panel is for nucleon couplings
$\sqrt{y_{\nu}y_{N}}$. We assume the NO, $m_{1}=0.1$ eV, and $\epsilon_{11}=0$
(constraints are nearly identical in the IO). Also shown are other relevant
constraints as shaded region, which are based on a single NSI parameter while
the DUNE sensitivity incorporates the full set of NSI parameters. In
particular the “Others” represents combination of constraints on $y_{e/N}$ and
$y_{\nu}$ coming from BBN, $\Delta N_{\rm eff}$, fifth force search, stellar
and supernova cooling. $\Delta m_{\rm Sun}$ ($\Delta m_{\rm SN}$) represents
constraints due to the in-medium mass of the neutrinos inside the Sun
(SN1987A).

This combined representation provides a comprehensive view of the parameter
space. We observe that the parameter space that DUNE will robustly explore
with its oscillation programme appears to already be constrained by current
data from various non-oscillation probes. We note that, this being the first
study that allows for all SNSI parameters varying simultaneously,
cancellations have occurred that limit DUNE’s sensitivity in our approach.
This can be seen by comparing the relative sensitivities presented in Table 2
(one-parameter-at-a-time) with Table 3 (all-varying-simultaneously). If this
high-dimension parameter space is explored thoroughly in the context of the
other competing constraints, a decent relaxation of those is expected as well
– especially when considering the in-medium contribution to effective masses
shown as $\Delta m_{\rm Sun}$ and $\Delta m_{\rm SN}$ in Fig. 4. With this in
mind, we find it likely that DUNE can contribute new information regarding
SNSI with very light mediator masses, e.g. $m_{\phi}\lesssim 10^{-12}$ keV.

Recent tensions in the T2K and NOvA data in the measurement of the CP-
violating phase $\delta_{CP}$ have prompted a reevaluation of our
understanding of neutrino oscillations. One possible solution to reconcile
these tensions is to consider the effects of complex non-diagonal NSIs as has
been in shown in Ref. Denton:2022pxt . However, our analysis suggests that the
parameter space required to explain these tensions using SNSIs is largely
ruled out by the existing constraints. The vector case, on the other hand,
could potentially accommodate the required parameter space that explains this
discrepancy. In our research, we are primarily interested in renormalizable
gauge theory models. Consequently, we cannot construct a vector boson model
with CP-violating complex coupling. So this study is not applicable for
vector-mediated NSI. However, it’s worth noting that theories with an
effective Lagrangian can accommodate such complex couplings which provide a
framework to study the CP-violating effects in the context of vector NSIs.

##  6 Conclusion

We have studied NSIs between neutrinos and SM fermions mediated by light
neutral scalar fields. Neutrino models incorporating light scalar fields can
accommodate such new interactions. The parameter space associated with these
light scalars is constrained by various non-oscillation probes, including
neutrino-electron (nucleon) scattering, fifth force searches, CMB studies,
BBN, supernova and stellar cooling observations. Moreover, the presence of
these new interactions would also influence the in-medium masses of neutrinos
within dense environments like supernovae and the Sun.

Another natural method for probing these new interactions is through neutrino
oscillation experiments. The presence of scalar NSIs emerges as a correction
to the neutrino mass matrix in the neutrino propagation Hamiltonian. The
oscillation phenomenology of scalar NSIs exhibit distinct characteristics
compared to the SM weak interactions of neutrinos and vector boson-mediated
NSIs. As the effect of scalar NSI scales linearly with the matter density of
the medium, long-baseline oscillation experiments are suitable candidates for
their exploration. In our analysis, we utilized DUNE as a case study. In
particular, scalar NSIs can mimic the CP effect and hinder the measurement
efforts of the leptonic Dirac CP phase at DUNE.

We introduced a new model-independent parameterization scheme to study the
effect of SNSI that enables us to consider more than one nonzero SNSI
parameters at a time. We performed a thorough analysis to find DUNE
sensitivity on the SNSI parameter space when all the parameters were nonzero.
We transformed these sensitivities on the SNSI parameters into projected
sensitivities in the parameter space of the underlying light scalar models and
compared them with the existing bounds. We found that the scalar parameter
space, which DUNE can probe, is entirely constrained by current data. A
comprehensive analysis considering all the SNSI parameters simultaneously
could potentially unveil unexplored parameter space in the low mass regime,
for instance, $m_{\phi}\lesssim 10^{-12}$ keV or so.

As scalar NSIs depend on the matter density of the medium, a synergy between
various long baseline experiments would be highly beneficial in enhancing our
understanding of the parameter space associated with SNSI and guarantee a
correct measurements of CP phase angle. Moreover, a reevaluation of the
existing constraints within a higher dimensional parameter space, considering
all the SNSI parameters simultaneously, would facilitate comparisons between
DUNE’s sensitivity and other non-oscillatory probes.

###### Acknowledgements.

B.D., A.T., K.J.K. and A.V. receive partial support from DOE grant DE-
SC0010813. S.G.’s work is funded by the National Research Foundation of Korea
(NRF) Grant No. NRF-2019R1A2C3005009 (SG). T.L. acknowledges support from the
National Key Research and Development Program of China Grant No.
2020YFC2201504, by the Projects No. 11875062, No. 11947302, No. 12047503, and
No. 12275333 supported by the National Natural Science Foundation of China, as
well as the Key Research Program of the Chinese Academy of Sciences, Grant NO.
XDPB15. We also recognize that portions of this research were conducted using
the advanced computing resources provided by Texas A&M High Performance
Research Computing.

##  Appendix A MultiNest Corner plots

Here we show the results of our MultiNest analysis in the form of corner
plots. First we show the sensitivities on $\mu_{ij}$ parameters in Fig. 5
(Fig. 6) for the normal (inverted) mass ordering. We use flat priors for both
phase and magnitude of $\mu_{ij}$ as well as for $\delta_{CP}$. We marginalize
over $\sin^{2}\theta_{23}$ and fix other parameters at their central values as
discussed in the main text. We then translate these to sensitivities on the
$\eta_{\alpha\beta}$ parameters using the mapping discussed in subsection.
2.1, setting $m_{1}=0.1$ eV ($m_{3}=0.1$ eV) and $\epsilon_{11}=0\
(\epsilon_{33}=0)$ for NO (IO) as benchmark. The obtained sensitivities in
terms of $\eta_{\alpha\beta}$ is depicted in Fig. 7 (Fig. 8) for NO(IO).
Within these figures, the diagonal subplots show marginal distributions as
function of just one parameter while the off-diagonal subplots show the 2D
marginalized plots with the dark (light) shaded regions showing the 68% (95%)
confidence intervals for the expected sensitivities to oscillation and NSI
parameters that DUNE experiment will be able to probe.

Figure 5:  Corner plot showcasing DUNE sensitivity to the $\mu_{ij}$
parameters where the truth data are simulated assuming no new physics and
according to the normal mass ordering. Figure 6:  Corner plot showcasing DUNE
sensitivity to the $\mu_{ij}$ parameters where the truth data are simulated
assuming no new physics and according to the inverted mass ordering. Figure 7:
Corner plot demonstrating the sensitivity to the $\eta_{\alpha\beta}$
parameters for normal mass ordering. While obtaining these sensitivities, we
have set the following parameters as fixed: $m_{1}=0.1$ eV and
$\epsilon_{11}=0$ eV. Figure 8:  Corner plot demonstrating the sensitivity to
the $\eta_{\alpha\beta}$ parameters for inverted mass ordering. While
obtaining these sensitivities, we have set the following parameters as fixed:
$m_{3}=0.1$ eV and $\epsilon_{33}=0$ eV.

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