aainstitutetext: Department of Physics, Tezpur University, Napaam, Sonitpur,
Assam-784028, Indiabbinstitutetext: Department of Physics, Indian Institute of
Technology, Guwahati, Assam-781039, India

# The Sensitivity of DUNE in Presence of Off-Diagonal Scalar NSI Parameters

Arnab Sarker a,2 Dharitree Bezboruah b,3 Abinash Medhi a,4 Moon Moon Devi
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Scalar non-standard interactions (NSI) presents an exciting pathway for
probing potential new physics that extends beyond the Standard Model (BSM).
The scalar coupling of neutrinos with matter can appear as a sub-dominant
effect that can impact the neutrino oscillation probabilities. The uniqueness
of these interactions is that it can directly affect the neutrino mass matrix.
This makes oscillations sensitive to the absolute neutrino mass. The effects
of scalar NSI scales linearly with matter density which motivates its
exploration in long-baseline sector. The presence of scalar NSI can influence
the key measurements in the field of neutrino physics, including the precise
determination of the leptonic CP phase ($\delta_{CP}$), neutrino mass ordering
and the octant of $\theta_{23}$. The precise determination of $\delta_{CP}$ is
one of the major goals of DUNE, which is an upcoming long-baseline experiment.
A better understanding of the impact of scalar NSI on CP measurement
sensitivities is crucial for accurate interpretation of $\delta_{CP}$ phase.
In this work, we have explored the impact of the complex off-diagonal scalar
NSI elements $\eta_{\alpha\beta}$ and their associated phases
$\phi_{\alpha\beta}$ on the CP-measurement sensitivities at DUNE. We have
explored the impact of the neutrino mass scale on these sensitivities. We look
for constraining these off-diagonal elements for different neutrino mass
scales. We also explore their correlation with $\delta_{CP}$, investigating
potential degeneracies that can arise due to additional phases. We also
perform a correlation study among different scalar NSI elements. We show that
the inclusion of the complex scalar NSI elements can significantly modify the
CP phase measurements.

###### Keywords:

CP-Violation, Non-Standard Interactions, Neutrino Physics, Beyond Standard
Model, Long Baseline Experiments

##  1 Introduction

Neutrino physics has undergone significant advancements over the past few
decades, with substantial progress made in understanding the properties of
neutrinos. The excellent ability of the Standard Model (SM) to explain the
phenomena around us makes it evident that SM is an effective theory and that
any new physics beyond SM (BSM) is suppressed at low energies. The
experimental confirmation of neutrino oscillations implies that neutrinos
($\nu^{\prime}s$) have mass and that neutrino flavors can change in the course
of their propagation. This was a milestone that reshaped our understanding and
provided evidence for physics beyond the Standard Model. The neutrinos are
described in the standard 3$\nu$ paradigm, with three active neutrino flavors
($\nu_{e},\nu_{\mu},\nu_{\tau}$) defined as the superposition of three
definite mass eigenstates ($\nu_{1}$, $\nu_{2}$, $\nu_{3}$). The production,
propagation and detection of these neutrinos are solely mediated by weak
interactions.

With high-precision accelerator experiments like T2K T2K:2014xyt  and NO$\nu$A
NOvA:2004blv ; NOvA:2016kwd , neutrino oscillation physics is at the dawn of
precision measurement era, allowing a robust test of the 3$\nu$ paradigm. Most
of the neutrino oscillation parameters are very precisely measured, in
particular, $|\Delta m_{31}^{2}|$, $\Delta m_{21}^{2}$, $\theta_{12}$ and
$\theta_{13}$. However, there exist three unknowns i.e. the leptonic CP-
violating phase ($\delta_{CP}$), the sign of $\Delta m_{31}^{2}$ and the
octant of $\theta_{23}$. Driven by significant advancements in beam
technology, detector capabilities and analysis methods, the upcoming long-
baseline (LBL) experiments like DUNE DUNE:2015lol , HK Hyper-
KamiokandeProto-:2015xww ; Hyper-Kamiokande:2016srs  and P2SO
Akindinov:2019flp  will have the capability to solve these three key unknowns
remaining in the field of neutrino physics. Currently, T2K data has shown a
preference for maximal CP violation with $\delta_{CP}\sim 3\pi/2$ T2K:2023smv
, whereas NO$\nu$A data has no preference for either CP violation or
conservation NOvA:2021nfi . The tension in T2K and NO$\nu$A data can be the
indication of new physics beyond non-zero neutrino mass. In order to quantify
leptonic CP-violation (CPV), a precise measurement of all the other
oscillation parameters is necessary. Many studies on the impact of precision
measurements and their interplay have been done as shown in references
Coloma:2012wq ; Minakata:2013eoa ; Coloma:2014kca ; Agarwalla:2022xdo ;
Denton:2023zwa .

While the three-flavor oscillation framework has been remarkably successful in
explaining most of neutrino data, there is growing interest in exploring
potential deviations from this paradigm that could arise from new physics
scenarios. With the increasing sensitivity of the detectors with state-of-the-
art technologies, it has now become crucial to probe and understand such
subdominant new physics scenarios. In the neutrino sector, the new physics
effects include the possibility of more than three neutrino states, neutrino
decay Berryman:2014qha ; PICORETI201670 ; SNO:2018pvg ; GOMES2015345 ;
Coloma:2017zpg ; Abrahao:2015rba , non-standard interactions (NSIs)
Liao:2016orc ; Friedland:2012tq ; Coelho:2012bp ; Rahman:2015vqa ;
Coloma:2015kiu ; deGouvea:2015ndi ; Liao:2016hsa ; Forero:2016cmb ;
Huitu:2016bmb ; Bakhti:2016prn ; Kumar:2021lrn ; Agarwalla:2015cta ;
Agarwalla:2014bsa ; Agarwalla:2012wf ; Blennow:2016etl ; Blennow:2015nxa ;
Deepthi:2016erc ; Masud:2021ves ; Soumya:2019kto ; Masud:2018pig ;
Masud:2017kdi ; Masud:2015xva ; Ge:2016dlx ; Fukasawa:2016lew ;
Chatterjee:2021wac ; Medhi:2023ebi ; Chaves:2021kxe ; Brahma:2023wlf ;
Davoudiasl:2023uiq ; Chatterjee:2020kkm ; Choubey:2014iia ; Singha:2021jkn ;
Denton:2018xmq ; Denton:2020uda ; Farzan:2015hkd ; Ge:2018uhz ; Medhi:2021wxj
, Lorentz Invariance Violation (LIV) Kostelecky:2003cr ; SNO:2018mge ;
Mewes:2019dhj ; Huang:2019etr ; ARIAS2007401 ; LSND:2005oop ; MINOS:2008fnv ;
MINOS:2010kat ; IceCube:2010fyu ; MiniBooNE:2011pix ; DoubleChooz:2012eiq ;
Sarker:2023mlz ; Sarkar:2022ujy ; Majhi:2022fed , quantum decoherence
PhysRevD.56.6648 ; Benatti:2000ph ; PhysRevD.100.055023 ; PhysRevD.95.113005 ;
Lisi:2000zt  etc. Wolfenstein first proposed the concept of NSI with generally
parameterized vector and axial vector currents in his seminal paper
Wolfenstein:1977ue  on matter effect in neutrinos. The vector-mediated NSI can
generically be of CC type which affects the production, propagation and
detection of neutrinos or NC type which affects only production and detection.
In last two decades, both theoretical and phenomenological exploration of
vector NSI have been done extensively in the literature. The presence of NSI
introduces various degeneracies in oscillation and scattering experiments,
which may highly affect the precision measurement of neutrino oscillation
parameters Miranda:2004nb ; Gonzalez-Garcia:2001snt ; Kikuchi:2008vq ;
Friedland:2012tq ; Rahman:2015vqa ; Masud:2015xva ; Coloma:2015kiu ;
Palazzo:2015gja ; deGouvea:2015ndi ; Masud:2016bvp ; Liao:2016hsa ;
deGouvea:2016pom ; Liao:2016orc ; Ge:2016dlx ; Agarwalla:2016fkh ;
Blennow:2016etl ; Fukasawa:2016lew ; Deepthi:2016erc ; Forero:2016cmb ;
Deepthi:2017gxg ; Coloma:2017egw ; Coloma:2017ncl ; Hyde:2018tqt ;
Esteban:2019lfo ; Kopp:2010qt . This makes the complementary measurement of
NSI parameters crucial for the robust interpretation of data from future
sophisticated experiments.

One of the primary goals of current and future neutrino experiments is the
precise measurement of the $\delta_{CP}$ phase. This measurement is crucial
for understanding the matter-antimatter asymmetry in the universe, as CP-
violation in the lepton sector could potentially contribute to the observed
dominance of matter over antimatter. A non-zero value of the CP phase in the
Pontecorvo-Maki-Nakagawa-Sakata (PNMS) Pontecorvo:1957cp ; Pontecorvo:1957qd ;
Pontecorvo:1967fh ; ParticleDataGroup:2020ssz  matrix is indispensable for a
viable explanation of the asymmetry observed in the universe based on
leptogenesis Fukugita:1986hr ; Davidson:2008bu . The appearance probability
channel in accelerator-based experiments is the most straightforward way to
investigate leptonic CPV T2K:2001wmr ; Marciano:2001tz ; Dick:1999ed ;
Cervera:2000kp ; Nunokawa:2007qh . In measuring the leptonic CP phase, the
presence of NSI can have a huge impact as they may introduce new sources of
CP-violation apart from the Dirac CP phase. The presence of new CP-violating
phases may alter the relationship between the measured oscillation parameters
and the Dirac CP phase Gonzalez-Garcia:2001snt . In the standard scenario,
DUNE has the potential to observe CP-violation at 5$\sigma$ confidence level
(CL) after 10 years of data taking for more than 50$\%$ of the $\delta_{CP}$
values DUNE:2020jqi . The presence of vector NSI can spoil the CPV
sensitivities at DUNE as shown in Masud:2016bvp . The NSI-SI degeneracies and
the interplay of their moduli and phases have been explored in Masud:2015xva
for precise CP phase measurement. The reference Rout:2017udo  explores the
impact of new physics scenarios on distinguishing between intrinsic and
extrinsic CP-violation at LBL experiments. However, by exploiting the
flexibility of the tunable beam line in DUNE, better discrimination of NSI
from SI can be achieved, as presented in Masud:2017bcf . In reference
Ghosh:2013pfa , it is shown that synergy among different experiments can lift
the $\delta_{CP}$-hierarchy degeneracy and hence provide better sensitivity
towards CP-violation. The authors in reference Choubey:2017cba  have studied
the sensitivity of DUNE+T2HKK towards $\delta_{CP}$ measurement in the
presence of one sterile neutrino. In reference Prakash:2012az , it is shown
that combining data from the T2K and NO$\nu$A could break the degeneracy
between neutrino mass hierarchies and $\delta_{CP}$. In a recent study
Agarwalla:2022xdo , authors have explored the possibility of enhanced CP-
violation sensitivity by combining data from DUNE, T2HK and T2HKK. In
reference Denton:2023qmd , the authors have suggested the exploration of the
disappearance channel for determining CP-violation which will help in the
robustness of $\delta_{CP}$ measurement.

Beyond the well-established vector NSI, scalar field-mediated NSI Ge:2018uhz ,
presents a fascinating avenue for exploring new physics beyond the Standard
Model. Unlike vector NSI, scalar NSI directly modifies the neutrino masses
through their Yukawa-like terms in the Lagrangian. This results in a distinct
phenomenological signature in neutrino experiments, as shown by various recent
studies. In references Medhi:2021wxj ; Medhi:2022qmu , the authors have
explored the CPV sensitivities at DUNE considering only the diagonal scalar
NSI elements and performed a synergy study of DUNE, T2HK and T2HKK. A similar
study is also performed for DUNE and P2SO with diagonal scalar NSI elements
for CPV and octant sensitivities in Singha:2023set . They have also explored
the constraining capabilities of DUNE and P2SO towards these diagonal scalar
NSI parameters. The authors in reference Medhi:2023ebi  have explored the
possibility of constraining the absolute neutrino mass in the presence of
scalar NSI at DUNE. The impact on the neutrino mass ordering is studied in
reference Sarker:2023qzp  in the context of LBL experiments i.e. DUNE, HK and
HK+KNO. The authors in reference Denton:2022pxt  have explored the possibility
of distinguishing between different new physics scenarios, such as the
presence of sterile neutrinos, vector and scalar NSI at NO$\nu$A, T2K and
DUNE. In reference ESSnuSB:2023lbg , the effect of scalar NSI on $\delta_{CP}$
measurement is studied in the context of the ESSnuSB experiment. The reference
Chaves:2021kxe  has explored scalar NSI using $\nu$($\bar{\nu}$)-disappearance
in oscillation data at different experiments. The studies like Babu:2019iml ;
Venzor:2020ova  have set bounds on scalar NSI within cosmological and
astrophysical limits, though these constraints are relatively loose. The
authors of Gupta:2023wct  have constrained scalar NSI parameters considering
the JUNO experiment. In reference Dutta:2024hqq , the authors have explored
the sensitivities at DUNE and compared DUNE’s constraint with other non-
oscillatory probes.

In this work, we build upon these efforts by conducting a comprehensive study
for the first time on the effects of off-diagonal scalar NSI parameters
$\eta_{\alpha\beta}$ on the CP-measurement sensitivities at DUNE. We perform a
model-independent analysis, exploring the impact of the off-diagonal elements
$\eta_{\alpha\beta}$ and their associated phases $\phi_{\alpha\beta}$ on the
$\nu$-oscillation probabilities. We will particularly explore the impact of
new additional CP phases in the presence of scalar NSI on the CP-measurement
sensitivities at DUNE. The presence of additional phases can lead to
degeneracy in the $\delta_{CP}$ measurement. Hence, it is necessary to study
the impact of these phases on the CP-measurement. This is particularly
important as it can introduce more complexities and degeneracies in the CP
measurements. A better understanding of the impact is essential for accurately
isolating the intrinsic CP-violating phase. This study will shed light on the
impact of additional phases due to scalar NSI and the reliability of CP phase
measurements at upcoming long-baseline neutrino experiments i.e. DUNE. We will
also set limits on the off-diagonal elements at a given statistical
significance. We will explore the correlation between different scalar NSI
elements that may lead to possible degeneracies.

This article is organised as follows. In section 2, we cover the theoretical
background of scalar non-standard interactions. We then provide details of the
technical aspects of DUNE and the simulation framework in section 3. The
impact of off-diagonal scalar NSI elements $\eta_{\alpha\beta}$ and
$\phi_{\alpha\beta}$ at the probability level is explored in section 4. In
section 5, we examine how scalar NSI affects the CP-measurement sensitivities
at DUNE. We particularly explore the impact of additional scalar NSI phases on
the sensitivities. The bounds on the off-diagonal elements are presented in
section 6, and the correlation between different scalar NSI elements is
discussed in section 7. We then summarize our findings in section 8.

##  2 Theoretical background

The interaction of neutrinos with fermions via a scalar-mediated particle can
have interesting phenomenological consequences. In the presence of scalar non-
standard interactions, the effective Lagrangian can be written as
Wolfenstein:1977ue ; Nieves:2003in ; Nishi:2004st ; Maki:1962mu ;
Bilenky:1987ty

$\mathcal{L}_{eff}^{s}=\frac{y_{f}y_{\alpha\beta}}{m_{\phi}^{2}}[\bar{\nu}_{\alpha}(p_{3})\nu_{\beta}(p_{2})][\bar{f}(p_{1})f(p_{4})],$
(1)

where $y_{\alpha\beta}$ and $y_{f}$ represents the coupling of scalar mediator
$\phi$ with neutrinos ($\nu_{\alpha,\beta}$ with $\alpha,\beta$ =
$e,\mu,\tau$) and fermions/anti-fermions ($f$/$\bar{f}$). The presence of
scalar NSI leads to a modification in the Dirac equation as shown in equation
2. The quantity $n_{f}$ represents the number density of matter fermions.

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

The Hamiltonian for standard neutrino-matter interactions in the flavor basis
can be written as,

$\mathcal{H}_{matter}=E_{\nu}+\frac{MM^{\dagger}}{2E_{\nu}}\pm V_{SI}$ (3)

Here, M represents the neutrino mass matrix in the flavor basis and
$\mathcal{U}$ represents the PMNS neutrino mixing matrix. The equation 3 can
also be expressed as,

$\mathcal{H}_{eff}=E_{\nu}+\frac{1}{2E_{\nu}}\mathcal{U}.diag(0,\Delta
m_{21}^{2},\Delta m_{31}^{2}).\mathcal{U}^{\dagger}+diag(V_{CC},0,0).$ (4)

As seen in equation 2, the contribution of scalar NSI arises as an addition to
the neutrino mass matrix. The uniqueness of scalar NSI is that it can mimic
the dependence on the true neutrino mass matrix. In the presence of scalar
NSI, the Hamiltonian governing the $\nu$-oscillations can be rewritten in the
form given below as,

$\mathcal{H}_{SNSI}\equiv E_{\nu}+\frac{M_{eff}M_{eff}^{\dagger}}{2E_{\nu}}\pm
V_{SI}.$ (5)

Here, $M_{eff}=M+\delta M$, with $\delta
M\equiv\frac{\sum_{f}n_{f}y_{f}y_{\alpha\beta}}{m_{\phi}^{2}}$. We have
followed the parameterization of $\delta M$ as shown in the references
Ge:2018uhz ; Medhi:2021wxj ; Medhi:2022qmu  and the form of the matrix is
shown in equation 6, where $\eta_{\alpha\beta}$ are dimensionless quantities.

$\delta M\equiv
S_{m}\left(\begin{array}[]{ccc}\eta_{ee}&\eta_{e\mu}&\eta_{e\tau}\\\
\eta_{e\mu}^{*}&\eta_{\mu\mu}&\eta_{\mu\tau}\\\
\eta_{e\tau}^{*}&\eta_{\mu\tau}^{*}&\eta_{\tau\tau}\end{array}\right)$ (6)

The parameters $\eta_{\alpha\beta}$ quantify the strength of scalar NSI and
the scaling factor $S_{m}$ is merely introduced to re-express $\delta M$ in
mass dimensions. In our work, the scaling factor is fixed at
$S_{m}=\sqrt{|\Delta m_{31}^{2}|}$ which corresponds to the magnitude of a
typical atmospheric mass-squared splitting. The Hamiltonian is hermitian in
nature, which allows the diagonal parameters to be real and the off-diagonal
parameters to be complex, as
$\eta_{\alpha\beta}=|\eta_{\alpha\beta}|e^{-i\phi_{\alpha\beta}}$. Here,
$|\eta_{\alpha\beta}|$ are the magnitudes of the off-diagonal scalar NSI
elements, and $\phi_{\alpha\beta}$ are the corresponding phases.

In the presence of scalar NSI, the neutrino oscillation probabilities will not
only depend on the mass-squared differences but also on the absolute neutrino
masses. As scalar NSI scales linearly with matter density, we have defined the
values of scalar NSI parameters at matter density $\rho=2.9g/cm^{3}$ in this
work. Considering the long-baseline sector, the impact of rescaling
$\eta_{\alpha\beta}$ for varying density is nominal. The presence of off-
diagonal phases $\phi_{\alpha\beta}$ can play an important role in the
measurement of the Dirac CP phase $\delta_{CP}$. The phases
$\phi_{\alpha\beta}$ may significantly modify the $\nu$-oscillation
probabilities. It may also affect DUNE’s sensitivity towards CP measurement.
This is particularly significant as it can introduce additional complexities
and degeneracies in CP-measurements. A deeper understanding of these impacts
is crucial for accurately isolating the intrinsic CP-violating phase. This
study will illuminate the effects of additional phases due to scalar NSI,
enhancing the reliability of CP phase measurements at upcoming long-baseline
neutrino experiments such as DUNE. In this study, we have investigated the
effects of off-diagonal elements and their associated phases on DUNE’s
sensitivity to CP-violation.

##  3 Experimental details and Numerical Analysis

We first describe the specifics of the experimental configurations of DUNE and
then elaborate on the details of the simulation procedure. Our study has
specifically addressed the Deep Underground Neutrino Experiment (DUNE)
DUNE:2016hlj ; DUNE:2015lol ; DUNE:2016rla ; DUNE:2020ypp ; DUNE:2021tad ,
which is a leading accelerator-based $\nu$-experiment set to be hosted at
Fermilab. Long-Baseline Neutrino Facility (LBNF) at Fermilab is capable of
providing $1.1\times 10^{21}$ protons on target (POT) annually through its 1.2
MW proton beam, which will generate the neutrinos for the experiment. The
project encompasses two detectors, with the near detector located 60 meters
underground at Fermilab, with a baseline of 574 meters. Meanwhile, the far
detector, with a baseline of 1300 km, will be positioned 1.5 km beneath the
Earth’s surface at the Sanford Underground Research Facility in South Dakota.
The far detector module will have four Liquid Argon Time Projection Chambers
(LArTPC), each with a fiducial mass of 10 kt. The cutting-edge technologies
will deliver outstanding capabilities for energy reconstruction, particle
identification and tracking, enabling unparalleled precision in the study of
neutrino properties. The details of detection channels and systematic
uncertainties for DUNE are described in table 1.

Experimental Detail | Detection Channels | Normalization error
---|---|---
Signal | Background
DUNE, Baseline = 1300 km |  |  |
L/E = 1543 km/GeV |  $\nu_{e}(\bar{\nu}_{e})$ appearance | 2 % (2%) | 5 % (5 %)
Fiducial mass = 40 kt (LArTPC) |  $\nu_{\mu}(\bar{\nu}_{\mu})$ disappearance | 5 % (5 %) | 5 % (5 %)
Runtime = 3.5 yr $\nu+3.5$ yr $\bar{\nu}$ |  |  |
Table 1: Detection channels and systematic uncertainties for DUNE.

We simulate the DUNE experiment using the General Long Baseline Experiment
Simulator (GLoBES) Huber:2004ka ; Kopp:2006wp ; Huber:2007ji ; a sophisticated
C-based framework used for simulating long-baseline neutrino oscillation
experiments. The numerical calculation of low-level information, i.e., event
rates and oscillation probabilities is possible with GLoBES. In addition, the
framework also allows the extraction of statistical $\Delta\chi^{2}$ for any
simulated long-baseline experiment. The computation of statistical
$\Delta\chi^{2}$ helps in the sensitivity studies of long-baseline
experiments. In our work, the statistical significance for the sensitivity
studies considering the DUNE experiment is extracted from the GLoBES
framework. The statistical $\chi^{2}$ for the distinction between the standard
interaction and the effects due to scalar non-standard interactions can be
defined as,

$\chi^{2}\equiv\min_{\eta}\sum_{i}\sum_{j}\frac{\left[N_{true}^{i,j}-N_{test}^{i,j}\right]^{2}}{N_{true}^{i,j}},$
(7)

$\chi_{pull}^{2}=\underset{\zeta_{j}}{min}\left(\chi^{2}+\sum_{i=1}^{k}\frac{\zeta_{i}^{2}}{\sigma_{\zeta_{i}}^{2}}\right),$
(8)

The variables $N_{true}^{i,j}$ and $N_{test}^{i,j}$ indicate the count of true
events and test events, respectively. We have included the systematic errors
using the pull method described in Huber:2004ka ; Fogli:2002pt . To account
for these errors, we introduced nuisance parameters represented by
$\zeta_{k}$, where the systematic uncertainties associated with $\zeta_{k}$
are denoted by $\sigma_{\zeta_{k}}^{2}$. The overall statistical significance
involves minimization over all the considered systematic errors, as shown in
equation 8.

We have modified the GLoBES simulation framework accordingly to incorporate
the effects of scalar NSI. We have used the effective neutrino Hamiltonian in
the presence of scalar NSI as shown in equation 5. The true values of the
standard neutrino oscillation parameters are fixed at values as summarized in
table 2. We have also marginalized over the oscillation parameters
$\theta_{23}$, $\Delta m_{31}^{2}$ and $\delta_{CP}$ in the range shown in
table 2. The neutrino mass ordering is considered to be known and fixed to
true normal mass ordering (NO). The marginalization of the scalar NSI
parameters $|\eta_{\alpha\beta}|$ and $\phi_{\alpha\beta}$ has also been
performed wherever necessary. In the complete analysis, we have defined the
values of scalar NSI $\eta_{\alpha\beta}$ at matter density
$\rho=2.9g/cm^{3}$. For $\delta_{CP}$-measurement studies, we consider only
one non-zero off-diagonal scalar NSI parameter at a time for the statistical
analysis. However, in the correlation studies, we have considered two non-zero
scalar NSI parameters at a time. The scalar NSI introduces a dependence of
neutrino oscillations on the absolute neutrino masses. Therefore, in this
study, we set the lightest neutrino mass at $10^{-5}eV$.

We first study the impact of off-diagonal scalar NSI elements
($|\eta_{e\mu}|$, $|\eta_{e\tau}|$, $|\eta_{\mu\tau}|$) and their
corresponding phases ($\phi_{e\mu}$, $\phi_{e\tau}$, $\phi_{\mu\tau}$) on the
neutrino oscillation probabilities in section 4. We particularly focus on the
dominant oscillation channel, i.e., $P_{\mu e}$ in order to explore the
effects at the probability level. We consider only one non-zero
$\eta_{\alpha\beta}$ element at a time to understand the role of each
individual off-diagonal parameter unless otherwise mentioned. We also explore
DUNE’s sensitivity towards simultaneous measurement of $\delta_{CP}$ and
$\eta_{\alpha\beta}$ in section 5.1. We study the impact of
$\eta_{\alpha\beta}$ and $\phi_{\alpha\beta}$ on the CP-violation
sensitivities at DUNE in section 5.2. The DUNE’s capability to constrain the
off-diagonal scalar NSI parameters is explored in section 6. We finally
explore the correlation among different scalar NSI elements considering the
DUNE experiment in section 7.

Parameters |  $\theta_{12}$ $[^{\circ}]$ |  $\theta_{13}$ $[^{\circ}]$ |  $\theta_{23}$$[^{\circ}]$ |  $\Delta m_{21}^{2}$$[10^{-3}eV^{2}]$ |  $\Delta m_{31}^{2}$$[10^{-5}eV^{2}]$ | $\delta_{CP}$
---|---|---|---|---|---|---
Values | 33.45 | 8.62 | 47 | 7.42 | 2.55 | $-\pi/2$
Marginalization | Fixed | Fixed | 40-50 | Fixed | 2.25-2.65 | [$-\pi$, $\pi$]
Table 2: Neutrino oscillation parameters are presented with their
corresponding marginalization range Esteban:2020cvm .

##  4 Exploration at probability level with $P_{\mu e}$

In figure 1, we have explored the impact of off-diagonal scalar NSI parameters
($|\eta_{e\mu}|$, $|\eta_{e\tau}|$, $|\eta_{\mu\tau}|$) and their associated
phases ($\phi_{e\mu}$, $\phi_{e\tau}$, $\phi_{\mu\tau}$) on the appearance
channel ($P_{\mu e}$). We study for the DUNE experiment which is with a
baseline of 1300km. Here, we have considered only a single non-zero parameter
at a time. We have listed all the values of neutrino oscillation parameters
used for the simulation in table 2. We have plotted $P_{\mu e}$ as a function
of the neutrino energy which is varied in the range $0.5-10$ GeV. The
probabilities are numerically calculated using the modified GLoBES framework.
The red and blue solid lines represent the standard and the scalar NSI case
($|\eta_{\alpha\beta}|$=0.1) respectively. The impact on $P_{\mu e}$ due to
the presence of corresponding phases $\phi_{\alpha\beta}$ is shown by the
shaded grey band, where $\phi_{\alpha\beta}$ is varied in $[-\pi,\pi]$.

Figure 1: Impact on $P_{\mu e}$ due to the presence of off-diagonal elements
$\eta_{e\mu}$, $\eta_{e\tau}$ and $\eta_{\mu\tau}$ for varying neutrino energy
from 0.5-10 GeV for DUNE experiment i.e. a baseline of 1300km. The red and
blue solid lines represent the SI and the scalar NSI case respectively. The
shaded grey region showcases the variation due to the presence of scalar NSI
$\phi_{\alpha\beta}$ phase.

  * •

In presence of $\eta_{e\mu}$, we see an enhancement in the probability values
for varying neutrino energies. The non-zero values of the $\phi_{e\mu}$ phase
can significantly modify the probabilities, as shown by the grey-shaded
region. Depending on the value of the phase $\phi_{e\mu}$, we see either
enhancement or suppression.

  * •

Only nominal changes in the probabilities can be seen in the presence of
$\eta_{e\tau}$. Although, the phase $\phi_{e\tau}$ can considerably enhance or
suppress the values of $P_{\mu e}$.

  * •

In presence of $\eta_{\mu\tau}$, we observe a noticeable suppression in the
probability for $E<4$ GeV. In contrast to the presence of $\phi_{e\mu}$ and
$\phi_{e\tau}$, the phase $\phi_{\mu\tau}$ results in a narrower band,
indicating a smaller effect.

Figure 2: Plot of $\rm\Delta P_{\mu e}=\rm P_{\mu e}^{NSI}-P_{\mu e}^{SI}$ in
the ($\eta_{\alpha\beta}$-$\delta_{CP}$) parameter space with neutrino energy
fixed at the first oscillation maximum (i.e. 2.5 GeV) for DUNE with a baseline
of 1300km. The vertical axes correspond to $\delta_{CP}$ and the horizontal
axes correspond to $\eta_{\alpha\beta}$. The left, middle and right panels
represent the case of $\eta_{e\mu}$, $\eta_{e\tau}$ and $\eta_{\mu\tau}$
respectively.

In order to quantitatively explore the impact of scalar NSI on $P_{\mu e}$
over the $\delta_{CP}$ parameter space, we define a quantity $\Delta P_{\mu
e}$ as,

$\rm\Delta P_{\mu e}=\rm P_{\mu e}^{NSI}-P_{\mu e}^{SI}.$ (9)

Here, $P_{\mu e}^{NSI}$ is the numerically calculated appearance probability
in the presence of scalar NSI and $P_{\mu e}^{SI}$ is the appearance
probability in the absence of scalar NSI. In figure 2, we study the impact of
($\eta_{e\mu}$, $\eta_{e\tau}$, $\eta_{\mu\tau}$) on $P_{\mu e}$ in the
complete $\delta_{CP}$ parameter space. We examine the effects of scalar NSI
on $P_{\mu e}$ for the whole $\delta_{CP}$ parameter space to look for
potential regions of degeneracies at the probability level. We have plotted
$\Delta P_{\mu e}$ with $\delta_{CP}$ varied $\in[-\pi,\pi]$ and
$\eta_{\alpha\beta}$ in [-0.1,0.1]. Note that the other mixing parameters have
been set to the values described in the table 2. The left, middle and right
panels represent the $\eta_{e\mu}$, $\eta_{e\tau}$ and $\eta_{\mu\tau}$ cases,
respectively.

  * •

In presence of $\eta_{e\mu}$ (left panel), we observe different regions in the
$(\delta_{CP}$-$\eta_{e\mu})$ plane having similar impact of scalar NSI on
$\Delta P_{\mu e}$. The degeneracies observed at the probability level
indicate a possible degeneracy in the measurement of $\delta_{CP}$.

  * •

We observe a similar pattern of degenerate regions in the presence of
$\eta_{e\tau}$ (middle panel). The effects of $\eta_{e\mu}$ and $\eta_{e\tau}$
appear symmetric about $\delta_{CP}=0^{\circ}$.

  * •

For $\eta_{\mu\tau}$ (right panel), we observe an opposite impact on $\Delta
P_{\mu e}$ in comparison to $\eta_{e\mu}$ and $\eta_{e\tau}$. Additionally,
the magnitude of the impact of $\eta_{\mu\tau}$ is also relatively less.

##  5 CP-measurement sensitivities in presence of $\eta_{\alpha\beta}$

We observe that the presence of off-diagonal scalar NSI elements along with
their associated phases can significantly modify the oscillation
probabilities. Several regions of degeneracy at the probability level are also
observed. This may affect the $\delta_{CP}$ measurement sensitivities at the
upcoming long-baseline experiment DUNE. In section 5.1, we first explore the
$\delta_{CP}$-constraining capability at DUNE in the presence of scalar NSI
elements $\eta_{\alpha\beta}$. We also study the impact of scalar NSI on the
CPV sensitivities at DUNE in section 5.2.

###  5.1 $\delta_{CP}$-Constraining capability

We explore the $\delta_{CP}$ constraining capability in the presence of off-
diagonal scalar NSI elements ($\eta_{e\mu}$, $\eta_{e\tau}$, $\eta_{\mu\tau}$)
at DUNE. We investigate the impact of $|\eta_{\alpha\beta}|$ and corresponding
phases $\phi_{\alpha\beta}$ on the $\delta_{CP}$-measurement. The confidence
levels (CL) of 3$\sigma$, 2$\sigma$ and 1$\sigma$ are represented by the
black, red and blue solid lines, respectively. We have assumed normal mass
ordering as the true ordering. All the oscillation parameters are fixed at the
values shown in table 2.

In figure 3, we have a two-dimensional plane of
($\delta_{CP}$-$|\eta_{\alpha\beta}|$) with varying $\delta_{CP}$ along the
horizontal axes and $|\eta_{\alpha\beta}|$ along the vertical axes. We have
varied the test values of $\delta_{CP}$ in the complete parameter space
$[-\pi,\pi]$ and $|\eta_{\alpha\beta}|$ in the range [0,0.1]. We have also
marginalized over the oscillation parameters $\theta_{23}$, $\Delta
m_{31}^{2}$ and corresponding phases $\phi_{\alpha\beta}$. We have considered
two cases where, in the first case, the true value of
($\delta_{CP}^{true}$,$|\eta_{\alpha\beta}^{true}|$) is fixed at
($-90^{\circ}$,0.05) and the corresponding CL’s are represented by the solid
lines. In the second case, the true value is fixed at ($0^{\circ}$,0.05) and
the CL’s are represented by the dashed lines. The black star signifies the
true values. The observations are listed below as:

Figure 3: Sensitivity of DUNE to $|\eta_{e\mu}|$ (left-panel),
$|\eta_{e\tau}|$ (middle-panel), $|\eta_{\mu\tau}|$ (right-panel) and the
leptonic CP-phase $\delta_{CP}$. The blue, red and black colored lines
represent the 1$\sigma$, 2$\sigma$ and 3$\sigma$ CL respectively. However, the
solid CL lines represent the case when
($\delta_{CP}^{true}$,$|\eta_{\alpha\beta}^{true}|$) is fixed at
($-90^{\circ}$,0.05) and the dashed CL lines represent the case when
($\delta_{CP}^{true}$,$|\eta_{\alpha\beta}^{true}|$) is fixed at
($0^{\circ}$,0.05). The black star represents the true values.

  * •

In the left panel, with true values of
($\delta_{CP}^{true}$,$|\eta_{\alpha\beta}^{true}|$) fixed at
$(-90^{\circ},0.05)$, we observe that $\delta_{CP}$ is well constrained in the
range $\delta_{CP}\in[-110^{\circ},-60^{\circ}]$ in presence of $\eta_{e\mu}$
but spreads widely over the $|\eta_{e\mu}|$ values. The allowed region remains
constrained in the negative half-plane of $\delta_{CP}$. For
$\delta_{CP}^{true}=0^{\circ}$, we observe a relatively better constraint
about $|\eta_{e\mu}|$ but the constraining capability along $\delta_{CP}$
reduces.

  * •

In the middle panel, we observe a similar constraint on $\delta_{CP}$
restricted only in the negative plane within the range
$\delta_{CP}\in[-110^{\circ},-10^{\circ}]$ for $\eta_{e\tau}$ with
($\delta_{CP}^{true}$,$|\eta_{\alpha\beta}^{true}|$) fixed at
$(-90^{\circ},0.05)$. However, it spreads in the complete $\eta_{e\tau}$
range. For $\delta_{CP}^{true}=0^{\circ}$, we observe a weaker constraint on
the $\delta_{CP}$ with range $\delta_{CP}\in[-80^{\circ},80^{\circ}]$.

  * •

In the right panel, we see that $\delta_{CP}$ is much better constrained in
the presence of $\eta_{\mu\tau}$ compared to $\eta_{e\mu}$ and $\eta_{e\tau}$
for both scenarios. However, the constraint is much better for the scenario
$\delta_{CP}^{true}=0^{\circ}$ in comparison to
$\delta_{CP}^{true}=-90^{\circ}$.

In figure 4, we plot a two-dimensional plane spanning the
($\delta_{CP}$-$\phi_{\alpha\beta}$) parameter space, where the color scheme
is the same as used in figure 3. Similar to the previous figure, we consider
two different scenarios in this analysis. In the first scenario, we consider
the true values of ($\delta_{CP}^{true}$,$\phi_{\alpha\beta}^{true}$) are
fixed at ($-90^{\circ}$,$-90^{\circ}$) and the resulting CL’s are shown by the
solid lines. In the second scenario, the true value is fixed at
($0^{\circ}$,$-90^{\circ}$) and the CL’s are shown by dashed lines. The black
star signifies the true values. All the other oscillation parameters used for
simulation are as described in table 2. We have fixed
$|\eta_{\alpha\beta}^{true}|=0.05$ for all the cases and varied
$\phi_{\alpha\beta}$ in the complete parameter space.

Figure 4: Sensitivity of DUNE to $\phi_{e\mu}$ (left-panel), $\phi_{e\tau}$
(middle-panel), $\phi_{\mu\tau}$ (right-panel) and the leptonic CP-phase
$\delta_{CP}$. The blue, red and black lines represent the 1$\sigma$,
2$\sigma$ and 3$\sigma$ CL respectively. In all the panels, the solid lines
represent the case when ($\delta_{CP}^{true}$,$\phi_{\alpha\beta}^{true}$) is
fixed at ($-90^{\circ}$,$-90^{\circ}$) and the dashed lines represent the case
when ($\delta_{CP}^{true}$,$\phi_{\alpha\beta}^{true}$) is fixed at
($0^{\circ}$,$-90^{\circ}$). The black star represents the true values.

  * •

In left panel with ($\delta_{CP}^{true}$,$\phi_{\alpha\beta}^{true}$) fixed at
$(-90^{\circ},-90^{\circ})$, we see that $\delta_{CP}$ is well constrained in
the negative half-plane within the range
$\delta_{CP}\in[-160^{\circ},-40^{\circ}]$. For
$\delta_{CP}^{true}=0^{\circ}$, we observe a much better constraint on
$\delta_{CP}$ with $\delta_{CP}\in[-50^{\circ},50^{\circ}]$. However, we find
a degenerate region in the positive $\delta_{CP}$ plane around
$\delta_{CP}\sim 90^{\circ}$ at $3\sigma$ CL.

  * •

In presence of $\phi_{e\tau}$, the constraint on $\delta_{CP}$ is relatively
weaker in comparison to $\phi_{e\mu}$ as seen in the middle panel. In this
case, we observe multiple degenerate regions for both scenarios, i.e.,
$\delta_{CP}^{true}=-90^{\circ}$ and $0^{\circ}$.

  * •

The presence of $\phi_{\mu\tau}$ leads to degenerate regions which can be
prominently seen in the right panel. The degenerate regions in positive and
negative $\phi_{\mu\tau}$ plane is observed for $\phi_{\mu\tau}\sim\pm
90^{\circ}$. This can compromise DUNE’s sensitivity towards CP-violation in
the presence of scalar NSI. We make similar observations for both
$\delta_{CP}^{true}$ scenarios. However, the constraining is significantly
better for $\delta_{CP}^{true}=0^{\circ}$.

We see that the presence of off-diagonal scalar NSI can significantly affect
the $\delta_{CP}$ measurement. The non-zero phases $\phi_{\alpha\beta}$ can
give rise to degeneracies and further hamper the CP-measurement sensitivities
at DUNE.

###  5.2 Impact on CP-violation sensitivities

We explore the sensitivity of DUNE towards measuring the leptonic CP-violating
phase $\delta_{CP}$ in the presence of off-diagonal scalar NSI parameters
($\eta_{e\mu}$, $\eta_{e\tau}$, $\eta_{\mu\tau}$). CPV sensitivity quantifies
how well an experiment can distinguish between CP-conserving
($\delta_{CP}=[0,\pm\pi$]) and CP-violating values of $\delta_{CP}$. For the
analysis, we can define the sensitivity for CPV as

${\Delta\chi}^{2}_{\rm CPV}~{}(\delta_{\rm CP})={\rm
min}~{}\left[\chi^{2}~{}(\delta_{CP},\delta^{\text{test}}_{CP}=0),~{}\chi^{2}(\delta_{CP},\delta^{\text{test}}_{CP}=\pm\pi)\right].$
(10)

We obtain the minimum of $\Delta\chi^{2}_{CPV}$ for all the $\delta_{CP}$
values to measure CP-violation in the complete $\delta_{CP}$ parameter space.
We assume the normal mass ordering as the true ordering and a higher
$\theta_{23}$ octant as the true octant. We define
$\sigma=\sqrt{{\Delta\chi}^{2}}$ as the CPV significance, where 3$\sigma$
(5$\sigma$) CL are represented by the horizontal lines at
$\sqrt{{\Delta\chi}^{2}}$ = 9 (25) respectively. We have marginalized the
standard oscillation parameters in their 3$\sigma$ allowed range as shown in
table 2. We have also additionally marginalized over the scalar NSI parameters
$|\eta_{\alpha\beta}|\in[0,0.1]$. In all panels, the red solid line
corresponds to the SI case, whereas the blue solid line represents the
$|\eta_{\alpha\beta}|=0.05$ case. The green (black) dashed line representing
3$\sigma$ (5$\sigma$) CL is used as a reference. In figure 5, we have plotted
the significance, i.e., $\sigma=\sqrt{\Delta\chi^{2}}$ along the vertical axes
and varied the $\delta_{CP}^{True}$ along the horizontal axes. In each panel,
we have additionally varied the corresponding phase $\phi_{\alpha\beta}$ in
the range $[-\pi,\pi]$. The impact on the CPV sensitivities for varying
$\phi_{\alpha\beta}$ in the range $\in[-\pi,\pi]$ is represented by the shaded
grey band.

Figure 5: CPV sensitivity of DUNE in the presence of off-diagonal scalar NSI
parameters $|\eta_{e\mu}|$ (left-panel), $|\eta_{e\tau}|$ (middle-panel),
$|\eta_{\mu\tau}|$ (right-panel). The red and blue solid lines correspond to
SI and $|\eta_{\alpha\beta}|=0.05$. The grey band signifies the variation of
$\phi_{\alpha\beta}$ in the range $[-\pi,\pi]$. The dashed green and black
lines represent $3\sigma$ and $5\sigma$ CL, respectively.

  * •

In presence of $\eta_{e\mu}$ (left panel), we observe an enhancement in the
sensitivities. We also note that the presence of $\phi_{e\mu}$ can
significantly alter the CPV sensitivities depending on its value.

  * •

The presence of $\eta_{e\tau}$ (middle panel) deteriorates the sensitivities
in comparison to the SI case for the complete $\delta_{CP}$ space. And the
presence of the corresponding phase $\phi_{e\tau}$ can enhance/suppress the
sensitivities as also seen for $\phi_{e\mu}$.

  * •

Similar to $\eta_{e\mu}$, we observe that the presence of $\eta_{\mu\tau}$
(right panel) also suppresses the sensitivities in the complete $\delta_{CP}$
parameter space. Depending on the value of $\phi_{\mu\tau}$, we observe
enhancement/suppression of the sensitivities. We also note that the variation
of sensitivities due to $\phi_{\mu\tau}$ is relatively small compared to
$\phi_{e\mu}$ and $\phi_{e\tau}$.

Figure 6: Impact of different neutrino mass scale on CPV sensitivity at DUNE
in the presence of off-diagonal scalar NSI parameters $|\eta_{e\mu}|$ (left-
panel), $|\eta_{e\tau}|$ (middle-panel), $|\eta_{\mu\tau}|$ (right-panel). The
red solid line represents the SI case. The other coloured lines correspond to
a different value of the lightest $\nu$-mass and $|\eta_{\alpha\beta}|=0.05$.
The dashed green (black) line represents the $3\sigma$ ($5\sigma$) CL.

In figure 6, we have explored the impact of different neutrino mass scale on
the CPV sensitivities at DUNE. We have plotted for the scalar NSI parameters
$|\eta_{e\mu}|$, $|\eta_{e\tau}|$ and $|\eta_{\mu\tau}|$ on the left, middle,
and right panels, respectively. The values of the oscillation parameters are
the same as those used in figure 5. The red solid line represents the SI case.
The blue, black and cyan solid lines correspond to the different choices of
the lightest neutrino mass, i.e., $m_{1}=10^{-5},0.01,0.02$ eV, respectively.
For the above cases, the scalar NSI parameter is fixed at
$\eta_{\alpha\beta}=0.05$.

  * •

We observe that the increase in the value of the lightest neutrino mass
enhances the overall CPV sensitivities in the presence of parameters
$\eta_{e\mu}$ and $\eta_{e\tau}$.

  * •

As shown in figure 5, the presence of $\eta_{\mu\tau}$ (right-panel) can
suppress the sensitivities in the complete $\delta_{CP}$ space. However,
increasing the neutrino mass scale tends to restore the sensitivities to the
SI case.

##  6 Sensitivity to off-diagonal scalar NSI elements

We constrain the off-diagonal scalar NSI elements ($|\eta_{e\mu}|$,
$|\eta_{e\tau}|$, $|\eta_{\mu\tau}|$) by considering the simulation of the
DUNE experiment in the modified GLoBES framework. In the presence of scalar
NSI, neutrino oscillation experiments can become sensitive to the neutrino
mass scale. We have also probed the impact of different choices of the
lightest neutrino mass on constraining the parameters. The oscillation
parameters used for simulation are tabulated in table 2. We have followed a
similar marginalization for the standard oscillation parameters. In figure 7,
we have constrained the off-diagonal scalar NSI elements $\eta_{e\mu}$,
$\eta_{e\tau}$ and $\eta_{\mu\tau}$ which are represented by the black, red
and blue solid lines, respectively. We have plotted the $\Delta\chi^{2}$
profiles about the vertical axes after marginalizing over the mixing
parameters $\theta_{23}$, $\Delta m_{31}^{2}$. We have varied
$|\eta_{\alpha\beta}|$ along the horizontal axes. We have considered normal
ordering as the true ordering and the higher $\theta_{23}$ octant is
considered as the true octant. The 3$\sigma$ and 5$\sigma$ CL are represented
by the dashed lines in magenta and green, respectively. We have set the
lightest neutrino mass at $m_{1}=10^{-5}$ eV in the left panel. On the other
hand, we have it fixed at $m_{1}=0.02$ eV in the right panel.

Figure 7: DUNE’s sensitivity to the elements $|\eta_{e\mu}|$, $|\eta_{e\tau}|$
and $|\eta_{\mu\tau}|$ are shown by black, red and blue solid lines
respectively. The dashed lines in magenta and green indicate the 3$\sigma$ and
5$\sigma$ CL. The left (right) panel corresponds to the lightest $\nu$-mass,
$m_{1}=10^{-5}(0.02)$ eV.

  * •

We observe that DUNE is sensitive to the off-diagonal elements and can put
strong constraints on the parameters. For $m_{1}=10^{-5}$ eV, we see that
$\eta_{\mu\tau}$ is better constrained in comparison to $\eta_{e\mu}$ and
$\eta_{e\tau}$ at $5\sigma$ CL. The constraining of $\eta_{e\mu}$ is
comparable to that of $\eta_{e\tau}$ element.

  * •

When the lightest neutrino mass is fixed at $m_{1}=0.02$ eV, we see that the
constrain on $\eta_{\mu\tau}$ remains almost same as before. But the
constraint on $\eta_{e\mu}$ significantly improves at $5\sigma$ CL.

Parameters | Bounds for $m_{1}=10^{-5}$ eV | Bounds for $m_{1}=0.02$ eV
---|---|---
$|\eta_{e\mu}|$ | 0.13 | 0.02
$|\eta_{e\tau}|$ | 0.12 | 0.06
$|\eta_{\mu\tau}|$ | 0.045 | 0.035
Table 3: Bounds on the parameters at $5\sigma$ CL obtained from the simulation
of DUNE.

The bounds on the off-diagonal scalar NSI parameters at $5\sigma$ CL as
obtained from the simulation of DUNE experiment are shown in table 3. Overall,
the increase in the neutrino mass scale leads to a better constraining of the
off-diagonal elements. In addition, the constraining is significantly better,
particularly for the elements $\eta_{e\mu}$ and $\eta_{e\tau}$.

##  7 Correlation among the $\eta_{\alpha\beta}$ parameters

Figure 8: Correlation between diagonal
($\eta_{ee}$,$\eta_{\mu\mu}$,$\eta_{\tau\tau}$) and off-diagonal
($|\eta_{e\mu}|$,$|\eta_{e\tau}|$,$|\eta_{\mu\tau}|$) scalar NSI elements at
DUNE. The blue, cyan and green colors represent the $1\sigma$, $2\sigma$ and
$3\sigma$ allowed regions, respectively. The best-fit point is represented by
the white solid star.

In this section, we explore the correlation between different scalar NSI
elements, taking the DUNE experiment as a case study. To study the
correlation, we consider two non-zero scalar NSI parameters at a time out of
the diagonal ($\eta_{ee}$,$\eta_{\mu\mu}$,$\eta_{\tau\tau}$), off-diagonal
($|\eta_{e\mu}|$,$|\eta_{e\tau}|$,$|\eta_{\mu\tau}|$) and associated phases
($\phi_{e\mu}$,$\phi_{e\tau}$,$\phi_{\mu\tau}$). We have explored four
different scenarios: a) both elements are diagonal; b) both elements are off-
diagonal; c) one element is diagonal and the other is off-diagonal; d) one
element is off-diagonal $(\eta_{\alpha\beta})$ and the other is the associated
phase $(\phi_{\alpha\beta})$. In all cases, we have plotted the allowed
regions at $1\sigma$, $2\sigma$ and $3\sigma$ C.L. on the two-dimensional
planes. The values of the oscillation parameters used for the simulation are
as described in table 2. We have chosen the normal mass ordering as the true
mass ordering and higher $\theta_{23}$ octant as the true octant.

Figure 9: Top panel: Correlation among the diagonal elements ($\eta_{ee}$,
$\eta_{\mu\mu}$, $\eta_{\tau\tau}$) only.
Middle panel: Correlation among the off-diagonal elements ($|\eta_{e\mu}|$,
$|\eta_{e\tau}|$, $|\eta_{\mu\tau}|$) only.
Bottom panel: Correlation among the off-diagonal elements ($|\eta_{e\mu}|$,
$|\eta_{e\tau}|$, $|\eta_{\mu\tau}|$) and phases ($\phi_{e\mu}$,
$\phi_{e\tau}$, $\phi_{\mu\tau}$). The blue, cyan and green colours represent
the $1\sigma$, $2\sigma$ and $3\sigma$ allowed regions, respectively. The
white solid star represents the best-fit point.

In figure 8, we have explored the correlation between the diagonal
($\eta_{ee}$,$\eta_{\mu\mu}$,$\eta_{\tau\tau}$) and off-diagonal
($|\eta_{e\mu}|$,$|\eta_{e\tau}|$,$|\eta_{\mu\tau}|$) elements. We have
marginalized over the oscillation parameters $\Delta m_{31}^{2}$,
$\theta_{23}$ and the associated scalar NSI phases $\phi_{\alpha\beta}$. We
then obtain the minimum $\Delta\chi^{2}$ by varying the scalar NSI parameters
in the fit data. We have plotted the diagonal elements in the vertical axes
and varied the values of the parameters in the range [-0.1,0.15]. We varied
the off-diagonal elements in the horizontal axes in the range [0,0.15]. For
each case, the true value of the parameters is fixed at (0.05,0.05). The
observations are listed below as

  * •

In the top-left panel, we observe a correlation between the elements
$\eta_{ee}$ and $|\eta_{e\mu}|$. We see a similar correlation between
$\eta_{ee}$ and $|\eta_{e\tau}|$.

  * •

We also observe a weaker but similar correlation of the element
$|\eta_{\mu\tau}|$ with $\eta_{\mu\mu}$ and $\eta_{\tau\tau}$. For all the
other parameters, we do not observe any notable correlation. The multiple
degenerate regions seen at $1\sigma$ regions are due to the marginalization of
the oscillation parameters.

In top panels of figure 9, we have explored the correlation among the diagonal
elements ($\eta_{ee}$,$\eta_{\mu\mu}$,$\eta_{\tau\tau}$) only. For the
diagonal element case, we marginalize only over the oscillation parameters
$\Delta m_{31}^{2}$ and $\theta_{23}$. In middle panels of figure 9, we have
examined the correlation among the off-diagonal elements
($|\eta_{e\mu}|$,$|\eta_{e\tau}|$,$|\eta_{\mu\tau}|$) only. For the off-
diagonal case, we have additionally marginalized over the associated phases
$\phi_{\alpha\beta}$. For each scenario, the true values of the scalar NSI
parameters are fixed at (0.05,0.05).

  * •

We observe a correlation between the diagonal elements $\eta_{\mu\mu}$ and
$\eta_{\tau\tau}$ as shown in the top-right panel. We do not see any strong
correlation among the off-diagonal elements, as shown in the middle panels of
figure 9.

In bottom panels of figure 9, we have explored the correlation between off-
diagonal ($|\eta_{e\mu}|$,$|\eta_{e\tau}|$,$|\eta_{\mu\tau}|$) elements and
their corresponding phases ($\phi_{e\mu}$,$\phi_{e\tau}$,$\phi_{\mu\tau}$).
The true values are fixed at
$(\phi_{\alpha\beta},|\eta_{\alpha\beta}|)\sim(-90^{\circ},0.05)$.

  * •

In the case of $\phi_{\mu\tau}$ and $\eta_{\mu\tau}$, we observe a degeneracy
for $\phi_{\mu\tau}\sim\pm 90^{\circ}$ as shown in the bottom-right panel. No
significant correlation is seen for the other parameters.

##  8 Summary and Conclusion

In this work, we have considered the upcoming long-baseline experiment DUNE
which has a baseline of 1300km with a fiducial volume of 40kton to be situated
at SURF. It will have a peak energy of around 2.5 GeV. It offers very high
sensitivity to the neutrino mass ordering, leptonic CP phase and the octant of
$\theta_{23}$. We have discussed the capability of this upcoming long-baseline
experiment to determine the leptonic CP phase in the presence of scalar non-
standard interactions. We first discuss how the presence of off-diagonal
scalar NSI elements and their associated phases can deviate the probabilities
from the standard interaction scenario. We find that the impact of
$|\eta_{e\mu}|$ and $|\eta_{\mu\tau}|$ on $P_{\mu e}$ channel is significant.
However, for a given value of $|\eta_{\alpha\beta}|$, the phases $\phi_{e\mu}$
and $\phi_{e\tau}$ have a greater impact on the probabilities in comparison to
$\phi_{e\mu}$. We illustrate the differences in the probabilities by defining
a quantity $\rm\Delta P_{\mu e}$ in the two-dimensional
$(\eta_{\alpha\beta}$-$\delta_{CP})$ plane at the first oscillation maximum.
In presence of $\eta_{e\mu}$ and $\eta_{e\tau}$, we find interesting
degenerate regions at the probability level. The impact of $\eta_{\mu\tau}$ on
$\Delta P_{\mu e}$ is relatively low.

The presence of additional phases $\phi_{\alpha\beta}$ can further complicate
the measurement capability of $\delta_{CP}$ at DUNE. We have explored the
$\delta_{CP}$-constraining capability in the presence of off-diagonal scalar
NSI elements $|\eta_{\alpha\beta}|$ and corresponding phases
$\phi_{\alpha\beta}$. We observe that the constraining is much better in the
presence of $|\eta_{\mu\tau}|$ compared to $|\eta_{e\mu}|$ and
$|\eta_{e\tau}|$. The presence of phases $\phi_{e\mu}$ and $\phi_{e\tau}$
shows similar constraining of the leptonic CP-phase. However, we observe a
degeneracy in the presence of scalar NSI phase $\phi_{\mu\tau}$ around $\pm
90^{\circ}$. We have also explored the impact of scalar NSI on the CPV
sensitivities at DUNE. We find that the phases $\phi_{e\mu}$ and
$\phi_{e\tau}$ can affect the sensitivities strongly in comparison to
$\phi_{\mu\tau}$. We have also explored how the absolute neutrino mass scale
can affect the CPV sensitivities. We observe an overall enhancement with the
increase in the lightest neutrino mass value. We also constrain the off-
diagonal scalar NSI elements at $5\sigma$ CL. We observe that the bounds on
the elements $|\eta_{e\mu}|$ and $|\eta_{e\tau}|$ improves significantly for
higher neutrino mass scale. We have also examined the correlation of the
scalar NSI parameters across different elements and phases, revealing
correlations among various parameters. In particular, we find that the
correlation between $|\eta_{\mu\tau}|$ and $\phi_{\mu\tau}$ leads to the
degenerate regions observed in the $\delta_{CP}$ measurement. This study has
shed light on the impact of off-diagonal elements and additional phases on the
CP-measurement sensitivities at DUNE.

## Acknowledgments

The authors acknowledge the DST SERB grant CRG/2021/002961. AS acknowledges
the CSIR SRF fellowship (09/0796(12409)/2021-EMR-I) received from CSIR-HRDG.
DB would thank DST INSPIRE Fellowship (DST/INSPIRE Fellowship/2022/IF220161)
for providing financial support. AM acknowledge the support of DST SERB grant
xPHYSPNx DST00627xxBB006-ZBSA-3237 titled “Indian Institutions-Fermi lab
Collaboration in Neutrino Physics” for the financial support.

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