# Scalar NSI: A unique tool for constraining absolute neutrino masses via
$\nu$–oscillations

Abinash Medhi  Arnab Sarker  Moon Moon Devi  Department of Physics, Tezpur
University, Napaam, Sonitpur, Assam-784028, India

###### Abstract

In the standard interaction scenario, a direct measurement of absolute
neutrino masses via neutrino oscillations is not feasible, as the oscillations
depend only on the mass-squared differences. However, the presence of scalar
non-standard interactions can introduce sub-dominant terms in the oscillation
Hamiltonian that can directly affect the neutrino mass matrix and thereby
making scalar NSI a unique tool for neutrino mass measurements. In this work,
for the first time, we constrain the absolute masses of neutrinos by probing
scalar NSI. We show that a bound on the lightest neutrino mass can be induced
in the presence of scalar NSI at DUNE. We find that the lightest neutrino mass
can be best constrained with $\eta_{\tau\tau}$ and $\eta_{\mu\mu}$ at
$2\sigma$ C.L. for normal and inverted hierarchy respectively. This study
suggests that scalar NSI can serve as an interesting avenue to constrain the
absolute neutrino masses in long-baseline neutrino experiments via neutrino
oscillations.

Introduction.– The discovery of neutrino ($\nu$) oscillations essentially
confirm that neutrinos are massive Fukuda et al. (1998); Ahmad et al. (2002)
which provides convincing evidence in favour of physics beyond the standard
model (BSM). Nonetheless, the absolute mass of neutrinos and the mass ordering
is still not precisely known. Neutrino oscillations imply that neutrino
flavors mix with each other indicating non-zero masses of neutrinos. However,
it cannot provide a direct measurement of the absolute $\nu$-masses as it
depends only on the mass-squared differences. The recent limits on the mass
squared splittings can be found in ref. Esteban et al. (2020a). Neutrino
masses are represented by $m_{i}$ with orderings $m_{3}>m_{2}>m_{1}$ termed as
normal hierarchy (NH) and $m_{2}>m_{1}>m_{3}$ as inverted hierarchy (IH).

Experiments around the world are in constant pursuit to provide a stringent
bound on the $\nu$-masses Gariazzo et al. (2018); Gerbino et al. (2016);
Drexlin et al. (2013). Different sources of data like cosmological Abazajian
et al. (2011); Vagnozzi et al. (2017); Di Valentino et al. (2021),
$\nu$–oscillations Esteban et al. (2020a) and beta-decay Otten and Weinheimer
(2008); Holzschuh (1992) will contribute towards imposing constraints on the
$\nu$-mass Cao et al. (2021). A bound on the sum of $\nu$-masses $\sum m_{i}<$
0.12 eV (at 95% C.L.) is established from cosmology Aghanim et al. (2020);
Dvorkin et al. (2019); Franco Abellán et al. (2022); Wong (2011); Lesgourgues
and Pastor (2006). However, due to various degeneracies and cosmological
assumptions, a stringent limit on the absolute mass of neutrinos is not
possible via the cosmological approach. This can be overcome using two
different approaches viz. direct and indirect. In direct approach, the
$\nu$-mass is estimated by measuring the endpoint of the electron energy
spectrum in $\beta-$decay process. The recent bound on the $\nu$-mass provided
by the KATRIN experiment in this approach is $m_{\nu}$ $\leq$ 0.8 eV$c^{–2}$
at 90 % C.L Aker et al. (2022). Whereas the neutrinoless double-beta decay
($2\nu\beta\beta$) is used as an indirect approach to measure the
$\nu$-masses. The mass difference between the parent and daughter nuclei in
$2\nu\beta\beta$ process is used to constrain the $\nu$-mass. The
$2\nu\beta\beta$ decay is only possible because neutrinos are Majorana
particles and it can provide a bound on the $\nu$-mass via $m_{\beta\beta}$ =
$\sum U_{ei}^{2}m_{i}$ and its current bound is $m_{\beta\beta}$ $<$ 75 - 180
meV Agostini et al. (2020); Adams et al. (2020); Gando et al. (2016). The mass
of neutrinos is five orders of magnitude smaller than any other Standard Model
fermions. The determination of $\nu$-mass and its underlying mechanism will
shed light on various fundamental unknowns of particle physics.

The challenges in measuring the absolute $\nu$-masses can arise from various
reasons. The weak interaction of neutrinos with matter makes it challenging to
measure the $\nu$-mass with high precision. The background noises are
generally very high in such mass measurement experiments. Also, the complexity
of the neutrino experiments leads to higher systematic errors. Despite these
challenges, a significant development can be seen in the $\nu$-mass bounds. In
recent years, there have been several important results, such as the first
direct measurement of the $\nu$-mass squared difference and the most stringent
limits on the $2\nu\beta\beta$ decay half-life Singh and Mirza (2023).

The neutrino oscillation experiments can probe the mass-squared differences of
the neutrinos but are not sensitive to the absolute masses. However, the
presence of scalar non-standard interactions (NSIs) can bring a direct
dependence of absolute $\nu$-masses via neutrino oscillations Ge and Parke
(2019); Yang and Kneller (2018); Khan et al. (2020); Babu et al. (2020); Gupta
et al. (2023). Neutrinos coupling via a scalar is interesting as it affects
the $\nu$-mass term which can be further probed to constrain the absolute
masses of neutrinos. In our earlier studies Medhi et al. (2022, 2023), we
found that the scalar NSI (SNSI) can significantly impact the physics
sensitivities of long baseline experiments. Also, it can introduce various
degeneracies in the measurement of oscillation parameters. The presence of
SNSI can provide a unique pathway to explore the absolute $\nu$-masses. A
stringent bound on the $\nu$-mass will not only provide a better understanding
of the properties of neutrinos but also help in explaining the physics BSM.

In this work, for the first time, we obtain a constrain on the lightest
$\nu$-mass by probing neutrino oscillation experiments. We have explored the
effects of SNSI in a model-independent way to constrain the $\nu$-mass for
both hierarchies. As the SNSI contribution scales linearly with matter
density, it makes long-baseline experiments a suitable candidate to explore
its effects. The impact of SNSI is probed element-wise, one at a time, at the
upcoming long baseline experiment DUNE Acciarri et al. (2015, 2016); Abi et
al. (2020, 2021). We point out that the SNSI elements can constrain the
$\nu$-mass with a marginally better constraining capability for NH as compared
to that of IH. We also note a slight variation in the constraining capability
at DUNE with respect to different choices of SNSI elements. The findings of
our analysis can help in estimating the absolute $\nu$-masses in
$\nu$-oscillation experiments and will have the potential to play a key role
in our quest to understand the nature of neutrinos.

Scalar NSI Formalism.– The interaction of neutrinos with matter occurs via the
mediators $W^{\pm}$, $Z^{0}$ and it impacts the $\nu$-oscillations Pontecorvo
(1957a, b, 1967); Zyla et al. (2020). This matter effects appear as an
additional potential term in the Hamiltonian for $\nu$-oscillations
Wolfenstein (1978); Nieves and Pal (2004); Nishi (2005); Maki et al. (1962).
The scalar interactions of neutrinos is an interesting possibility as
neutrinos can couple with a scalar (Higgs boson) with non-zero vacuum
expectation to generate its mass. The Lagrangian for such non-standard
coupling of $\nu$’s with the environmental fermions via a scalar can be
written as Ge and Parke (2019); Babu et al. (2020); Medhi et al. (2022, 2023),

$\mathcal{L}_{SNSI}=y_{f}Y_{\alpha\beta}\left[\overline{\nu_{\alpha}}(p_{3})\nu_{\beta}(p_{2})\right]\left[\overline{f}(p_{1})f(p_{4})\right]$
(1)

where, $y_{f}$ and $Y_{\alpha\beta}$ represent the Yukawa coupling of the
scalar mediator with environmental fermions $f$ and neutrinos respectively.
The modification in $\mathcal{L}$ leads to a correction in the Dirac equation
as seen below,

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

where, $m_{\phi}$ is the mass of the scalar mediator. As shown in ref. Ge and
Parke (2019); Babu et al. (2020), the scalar NSI can appear as a correction to
the $\nu$-mass matrix. The effective Hamiltonian in presence of SNSI can be
framed as,

$\mathcal{H}\thickapprox E_{\nu}+\frac{M_{eff}M_{eff}^{\dagger}}{2E_{\nu}}\pm
V_{SI}$ (3)

where, $M_{eff}$=M+$M_{SNSI}$ with $M_{SNSI}$
$\equiv$$\sum_{f}n_{f}y_{f}Y/m_{\phi}^{2}$. The $\nu$-mass matrix can be
diagonalized by a modified mixing matrix as $\mathcal{U}^{{}^{\prime}}$ =
$P\mathcal{U}Q^{\dagger}$, where the Majorana rephasing matrix Q can be
absorbed by $QD_{\nu}Q^{\dagger}=D_{\nu}=diag(m_{1},m_{2},m_{3})$. The
unphysical diagonal rephasing matrix, P can be rotated away into the SNSI
contribution as follows,

$M_{eff}=\mathcal{U}D_{\nu}\mathcal{U}^{\dagger}+P^{\dagger}M_{SNSI}P=M+\delta
M$ (4)

We parametrize the SNSI contribution $\delta M$ in a model-independent way as
follows Ge and Parke (2019); Medhi et al. (2022, 2023),

$\delta M\equiv\sqrt{|\Delta
m_{31}^{2}|}\left(\begin{array}[]{ccc}\eta_{ee}&\eta_{e\mu}&\eta_{e\tau}\\\
\eta_{e\mu}^{*}&\eta_{\mu\mu}&\eta_{\mu\tau}\\\
\eta_{e\tau}^{*}&\eta_{\mu\tau}^{*}&\eta_{\tau\tau}\end{array}\right)$ (5)

where, the dimensionless parameter $\eta_{\alpha\beta}$ quantifies the
strength of SNSI and $\sqrt{|\Delta m_{31}^{2}|}$ is used as a scaling. In the
standard case, a common $m_{i}^{2}$ term can be subtracted to obtain a
dependence of neutrino oscillations on the mass-squared splittings i.e.
$\Delta m_{21}^{2}$ and $\Delta m_{31}^{2}$. However, in the presence of SNSI,
the cross terms i.e. $M\delta M^{\dagger}$ and $M^{\dagger}\delta M$ bring a
direct dependence on the absolute masses of neutrinos as no common term can be
subtracted from the mass matrix. Hence, oscillation probabilities will also
directly depend on the absolute mass of neutrinos in presence of SNSI. In this
work, we utilize this direct dependence on the $\nu$-mass to put a bound on
the absolute $\nu$-mass. We test with a non-zero diagonal $\eta_{\alpha\beta}$
one at a time. The $M_{eff}$ for such a case when $\eta_{ee}\neq 0$ is shown
as an example below, where the explicit dependence on the absolute
$\nu$-masses is illustrated.

$M_{eff}=\mathcal{U}{\rm
diag}\left(m_{1},m_{2},m_{3}\right)\mathcal{U}^{\dagger}+\sqrt{|\Delta
m^{2}_{31}|}~{}\rm diag\left(\eta_{ee},0,0\right)$ (6)

Methodology.– In this work, we have investigated the possibility of
constraining the absolute $\nu$-masses by probing the effects of SNSI. From
the existing cosmological bound $\sum m_{i}<0.12$ eV (95$\%$ C.L.) Aghanim et
al. (2020); Dvorkin et al. (2019); Franco Abellán et al. (2022); Wong (2011);
Lesgourgues and Pastor (2006), we obtain the upper limits of the lightest
$\nu$-masses for both hierarchies. The upper limits of the ligtest neutrino
masses $m_{1}$ and $m_{3}$ for NH and IH are 0.03 eV and 0.015 eV respectively
according to the cosmological bound as in table 2 of supplementary. The
benchmark values of neutrino oscillation parameters used throughout the
analysis are Esteban et al. (2020b)\- $\theta_{12}=34.51^{\circ}$,
$\theta_{13}=8.44^{\circ}$, $\theta_{23}=47^{\circ}$, $\delta_{CP}=-\pi/2$,
$\Delta m_{21}^{2}=7.56\times 10^{-5}eV^{2}$, $\Delta m_{32}^{2}=-2.497\times
10^{-3}eV^{2}$ and $\Delta m_{31}^{2}=2.55\times 10^{-3}eV^{2}$.

Figure 1: Appearance probabilities for different values of lightest neutrino
mass with $\eta_{ee}=\pm 0.2$ (left-panel), $\eta_{\mu\mu}=\pm 0.2$ (middle-
panel) and $\eta_{\tau\tau}=\pm 0.2$ (right-panel) for normal hierarchy (top-
panel) and inverted hierarchy (bottom-panel). The solid (dashed) line
represents a positive (negative) value of $\eta_{\alpha\beta}$. Also, we
consider $\delta_{CP}$ = -$\pi$/2 and $\theta_{23}=47^{\circ}$ to generate the
plots.

The long-baseline neutrino experiments will help to explore new-physics
effects and bring stringer constraints on the $\nu$-mixing parameters. As SNSI
scales linearly with matter density, the long-baseline experiments are
suitable to probe its effects. This will in turn help in constraining the
absolute mass of neutrinos by putting bounds on the SNSI parameters. DUNE
Acciarri et al. (2015, 2016); Abi et al. (2020, 2021) is a next-generation
long baseline neutrino experiment with a 40 kton LArTPC-type detector for a
baseline of 1300 km. We have considered a total runtime of 7 years
($3.5\nu+3.5\bar{\nu}$). The main focus of the experiment is the precise
measurement of neutrino oscillation parameters, probing of CP-violation and
identifying the true mass hierarchy of neutrinos. This experiment can be a
good probe for exploring the effects of SNSI as it scales linearly with the
environmental matter density. We have used the GLoBES simulation package Huber
et al. (2005, 2007) to calculate the numerical probabilities and to develop
the statistical framework for exploring the physics sensitivities.

Impact on oscillation probabilities.– In Fig. 1, we explore the impact of
different allowed values of the lightest $\nu$-mass on $P_{\mu e}$ for
$\eta_{ee}$ (left-panel), $\eta_{\mu\mu}$ (middle-panel) and $\eta_{\tau\tau}$
(right-panel) at DUNE. For NH (top-panel), we observe a marginal deviation as
the scale of absolute $\nu$-mass shifts to a higher value. The effect on
$P_{\mu e}$ for different values of $m_{1}$ are different for both positive
and negative $\eta_{ee}$. In presence of positive (negative) $\eta_{\mu\mu}$,
we see a suppression (enhancement) of probabilities for higher values of
$\nu$-masses. We also note a significant suppression (enhancement) of $P_{\mu
e}$ for positive (negative) $\eta_{\tau\tau}$ as shown in the top-right panel.
For IH (bottom-panel), in presence of a positive $\eta_{ee}$, we observe a
suppression with increasing $\nu$-mass, $m_{3}$ while for a negative
$\eta_{ee}$ an enhancement is seen. For positive (negative) $\eta_{\mu\mu}$, a
marginal shift of the second peak towards lower (higher) energy is noticed for
increasing $\nu$-mass. We observe an enhancement (suppression) of $P_{\mu e}$
with the $\nu$-mass for a positive (negative) $\eta_{\tau\tau}$. Additionally,
the impact of NSI elements as a function of $\delta_{CP}$-$\nu$ mass and
energy-$\nu$ mass is shown in supplementary in Fig. 4 and Fig. 5 respectively.
This provides a concrete motivation to further explore the SNSI effects in
constraining the $\nu$-mass.

Statistical $\chi^{2}$-framework.– We present our results for constraining the
absolute $\nu$-masses for both hierarchies in presence of SNSI taking DUNE as
a case study. We place bounds on the $\nu$-masses by taking one diagonal NSI
element $(\eta_{\alpha\beta})$ at a time. We probe $\eta_{\alpha\beta}$ on the
appearance channel ($P_{\mu e}$) for some chosen values of the lightest
$\nu$-mass. We particularly focus only on the SNSI parameters $\eta_{ee}$,
$\eta_{\mu\mu}$ and $\eta_{\tau\tau}$ and examine DUNE’s capability towards
constraining the $\nu$-masses. We further study the correlation of the
lightest $\nu$-mass with the SNSI parameters. We use the same values of
neutrino oscillation parameters mentioned above. In order to constrain the
absolute mass of neutrinos, we define a statistical $\chi^{2}$ which is a
measure of sensitivity as,

$\small\chi_{pull}^{2}=\underset{\zeta_{j}}{min}\left(\min_{\eta}\sum_{i}\sum_{j}\frac{\left[N_{true}^{i,j}-N_{test}^{i,j}\right]^{2}}{N_{true}^{i,j}}+\sum_{i=1}^{k}\frac{\zeta_{i}^{2}}{\sigma_{\zeta_{i}}^{2}}\right),$
(7)

where, $N_{true}^{i,j}$ and $N_{test}^{i,j}$ represents the number of true and
test events in the $\\{i,j\\}$-th bin respectively. Using the pull method
described in Huber et al. (2005); Fogli et al. (2002), we incorporate the
systematic errors as additional parameters known as nuisance parameters
($\zeta_{k}$) with the systematical errors ($\sigma_{\zeta_{k}}^{2}$).

Figure 2: Constrain on lightest neutrino mass for normal (top-panel) and
inverted (bottom-panel) hierarchy for the diagonal NSI parameters $\eta_{ee}$
(left panel), $\eta_{\mu\mu}$ (middle-panel) and $\eta_{\tau\tau}$ (right-
panel). The solid (dashed) lines represent positive (negative) parameter
values. The magenta and green dashed horizontal lines represent the $3\sigma$
and $5\sigma$ C.L.

Constraining the neutrino masses.– In Fig. 2, we demonstrate the capability of
DUNE towards constraining the absolute $\nu$-masses for both hierarchies. In
eq. 3, we see that the SNSI parameters contribute directly to the standard
$\nu$-mass matrix which can be probed to place a bound on the $\nu$-masses.We
define the sensitivity as $\Delta\chi^{2}$ which can help in constraining the
lightest $\nu$-masses,

$\displaystyle\Delta\chi^{2}=min\bigr{[}\chi^{2}$
$\displaystyle\left(\eta_{\alpha\beta}^{test}\neq
0,m_{lightest}^{test}\right)-$
$\displaystyle\chi^{2}\left(\eta_{\alpha\beta}^{true}\neq
0,m_{lightest}^{true}\neq 0\right)\bigr{]}.$ (8)

We vary the test value of the lightest $\nu$-mass in the allowed range, while
fixing the true $\nu$-mass at $m_{1}^{true}=0.02$ eV and $m_{3}^{true}=0.01$
eV for NH and IH respectively. We have marginalized over the oscillation
parameters $\theta_{23}$ and $\delta_{CP}$ in the fit data. The sensitivity
$\sqrt{\Delta\chi^{2}}$ is plotted for an allowed range of the lightest
$\nu$-mass, where the solid (dashed) lines represent positive (negative)
$\eta_{\alpha\beta}$. We focus on the diagonal parameters $\eta_{ee}$ (left-
panel), $\eta_{\mu\mu}$ (middle-panel) and $\eta_{\tau\tau}$ (right-panel) for
NH (top-panel) and IH (bottom-panel) respectively. We have varied $m_{1}$
($m_{3}$) for NH (IH) in the allowed region. For NH, the presence of positive
(negative) $\eta_{ee}$ indicates an increase (decrease) in the constraining
capability. For $\eta_{\mu\mu}$, we see a significant enhancement in the
sensitivity with an increase in negative values. A similar trend is also
observed for higher negative values of $\eta_{\tau\tau}$. The effect of
positive $\eta_{\mu\mu}$ and $\eta_{\tau\tau}$ is relatively nominal.

NSI Parameter | NH Case, $m_{1}(eV)$ | IH Case, $m_{3}(eV)$
---|---|---
$\eta_{ee}$ | [0.003,0.03] | -
$\eta_{\mu\mu}$ | [0.012,0.028] | [0.006,0.014]
$\eta_{\tau\tau}$ | [0.016,0.024] | [0.005,0.0145]
Table 1: $3\sigma$ bounds on the lightest neutrino mass in presence of
diagonal scalar NSI parameters. The true neutrino mass is $m_{1}=0.02$ eV
($m_{3}=0.01$eV) for NH (IH) with $\eta_{\alpha\beta}=0.2$.

For IH (bottom-panel), we observe that the constraining capability gets
improved for negative $\eta_{ee}$. The positive $\eta_{ee}$ shows minimal
changes in the sensitivities. For $\eta_{\mu\mu}$, we see an enhancement for
both the positive and negative values. In presence of $\eta_{\tau\tau}$, a
significant improvement can be seen with increase in the negative strength of
the parameter though only nominal changes are seen for positive values. In
table 1, we have summarized the constraints on the lightest $\nu$-mass at
$3\sigma$ CL for both hierarchies. We note that the mass of the lightest
neutrino is better constrained in the presence of $\eta_{\tau\tau}$ and
$\eta_{\mu\mu}$ for NH and IH respectively.

Figure 3: Allowed region of lightest neutrino mass for normal (top-panel) and
inverted (bottom-panel) hierarchy for the diagonal NSI parameters $\eta_{ee}$
(left-panel), $\eta_{\mu\mu}$ (middle-panel) and $\eta_{\tau\tau}$ (right-
panel). The blue, red and magenta lines represent the 1$\sigma$, 2$\sigma$ and
3$\sigma$ confidence level. The best fit values of $(\eta_{\alpha\beta},m)$
are $(0.2,0.02eV)$ and $(0.2,0.015eV)$ for NH and IH respectively.

Correlation in $(\eta_{\alpha\beta}-m)$ parameter space.– In Fig. 3, we show
the allowed region for $1\sigma$, $2\sigma$ and $3\sigma$ CL in
$\eta_{\alpha\beta}-m$ planes for DUNE. We set the true values of
$(\eta_{\alpha\beta}$, m) at (0.2, 0.02 eV) and (0.2, 0.015 eV) for NH and IH
respectively. The test values of $\eta_{\alpha\beta}$ are varied in the range
$[0.1,0.3]$ and the lightest $\nu$-mass is varied in the allowed range for NH
and IH. We focus on the diagonal elements - $\eta_{ee}$ (left-panel),
$\eta_{\mu\mu}$ (middle-panel) and $\eta_{\tau\tau}$ (right-panel) where the
top (bottom) panel represents the NH (IH) case. The blue, red and black solid
lines symbolize the $1\sigma$, $2\sigma$ and $3\sigma$ confidence regions
respectively. The true point is shown by a black star. For NH, the lightest
$\nu$-mass can be constrained as $m_{1}\in$ $[0.009,0.03]$ eV at $1\sigma$ CL
for $\eta_{ee}$. The element $\eta_{\mu\mu}$ can constrain the $\nu$-mass as
$m_{1}$ $\in$ $[0.016,0.024]$ eV at $1\sigma$ CL. For $\eta_{\tau\tau}$, we
see a similar constrain on $m_{1}$ $\in$ $[0.017,0.023]$ eV at $1\sigma$ CL.
The allowed region for all the diagonal elements in the IH (bottom-panel) is
larger as compared to the NH (top-panel). The constraint on the lightest mass
in presence of $\eta_{ee}$ worsens for IH. For $\eta_{\mu\mu}$ and
$\eta_{\tau\tau}$, the allowed region at $1\sigma$ CL for the lightest mass
$m_{3}$ is $\sim$ $[0.007,0.013]$ eV.

Concluding Remarks.– The neutrino oscillation parameters are being measured
with unprecedented precision in current experiments. However, sub-dominant
effects like scalar NSI can have a significant impact on the detector
sensitivities. Scalar NSI offers an intriguing way to probe the absolute
$\nu$-masses via neutrino oscillations. The central idea of this work is to
showcase the capability of constraining the lightest $\nu$-mass using neutrino
oscillation data in presence of scalar NSI. In this letter, we have
constrained the lightest $\nu$-mass while accounting for the cosmological
bound on the sum of $\nu$-masses. We find that, the scalar NSI parameter
$\eta_{\alpha\beta}$ can provide a significant bound on the absolute mass of
neutrinos. Particularly, the presence of $\eta_{\tau\tau}$ or $\eta_{\mu\mu}$
makes the constraining capability marginally better than that of $\eta_{ee}$
on the lightest mass of neutrinos for both the hierarchies. The constraining
of $\nu$-mass can be crucial to shed light on the $\nu$-mass generation
mechanisms. It may provide a major breakthrough in our understanding of the
universe and its underlying physics.

Acknowledgments.– The authors acknowledge the Science and Engineering Research
Board (SERB), DST for the grant CRG/2021/002961. The authors thank Debajyoti
Dutta for his help and suggestions with the GLoBES framework. AS acknowledges
the fellowship received from CSIR-HRDG (09/0796(12409)/2021-EMR-I). AM
acknowledges the support of the Research and Innovation Grant 2021
(DoRD/RIG/10-73/1592-A) funded by Tezpur University.

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##  I Supplementary Materials

Allowed mass range for NH and IH.– Considering the best-fit values of mass-
squared splittings from table 3, we obtain the upper limit of the lightest
$\nu$-mass by taking the cosmological bound on the sum of $\nu$-masses. In
table 2, we show the limit on the $\nu$-masses $m_{1}$ and $m_{3}$ while
following the cosmological bound $\sum m_{i}<0.12eV$. We observe that $m_{1}$
and $m_{3}$ should be less than $0.03$ eV and $0.015$ eV for NH and IH
respectively. The values of mass-squared splitting $\Delta m_{2l}^{2}$ and
$\Delta m_{3l}^{2}$ for normal $(l=1)$ and inverted hierarchy $(l=2)$ with
$3\sigma$ range are shown in table 3.

Normal Hierarchy | Inverted Hierarchy
---|---
Parameters | True Values | Parameters | True Values
$m_{1}(eV)$ | 0.01 | 0.02828 | 0.03 | $m_{3}(eV)$ | 0.01 | 0.015 | 0.02
$m_{2}(eV)$ | 0.01319 | 0.02956 | 0.03121 | $m_{1}(eV)$ | 0.050227 | 0.051456 | 0.053130
$m_{3}(eV)$ | 0.0511 | 0.05757 | 0.05843 | $m_{2}(eV)$ | 0.050960 | 0.052172 | 0.053823
$\sum m_{i}(eV)$ | 0.0743 | 0.11541 | 0.11965 | $\sum m_{i}(eV)$ | 0.111187 | 0.118628 | 0.126953
Table 2: Table representing the restricted range of neutrino masses for normal
and inverted hierarchy. Normal Hierarchy
---
Mass Squared Splitting | Best Fit value | Lower $(3\sigma)$ | Higher $(3\sigma)$
$\Delta m_{21}^{2}(10^{-5}eV^{2})$ | 7.42 | 6.82 | 8.04
$\Delta m_{31}^{2}(10^{-3}eV^{2})$ | 2.514 | 2.431 | 2.599
Inverted Hierarchy
Mass Squared Splitting | Best Fit value | Lower $(3\sigma)$ | Higher $(3\sigma)$
$\Delta m_{21}^{2}(10^{-5}eV^{2})$ | 7.42 | 6.82 | 8.04
$\Delta m_{32}^{2}(10^{-3}eV^{2})$ | -2.497 | -2.525 | 2.469
Table 3: Mass-squared splitting for normal and inverted hierarchy with
$3\sigma$ ranges.

Effect of SNSI with varying $\delta_{CP}$ and $\nu$-mass.– We study the
variation of $\nu$-oscillation probabilities as a function of $\delta_{CP}$
and the lightest $\nu$-mass for both hierarchies. To quantify the impact of
SNSI elements on the oscillation probabilities, we define a parameter $\Delta
P_{\mu e}=P_{\mu e}^{NSI}-P_{\mu e}^{SI}$ with $P_{\mu e}^{NSI}$ and $P_{\mu
e}^{SI}$ as the appearance probabilities with and without NSI respectively. In
Fig. 4, we have explored the effect of SNSI for varying $\delta_{CP}$ and
$\nu$-mass within the allowed range. The values of the oscillation parameters
used are listed in the methodology of the draft. Here, we have fixed the
neutrino energy at $E=2.5$ GeV and SNSI parameter at $\eta_{\alpha\beta}=0.2$.
We then vary the mass in the allowed range and $\delta_{CP}$ $\in$
$[-\pi,\pi]$. We have plotted $\Delta P_{\mu e}$ for varying $\delta_{CP}$ and
$\nu$-mass. We take the SNSI parameter $\eta_{ee}$, $\eta_{\mu\mu}$ and
$\eta_{\tau\tau}$ in the left, middle and right panel respectively. We show
the effects for true NH (IH) in the top (bottom) panel. We observe that,

Figure 4: Effect of SNSI in $\delta_{CP}$–$m$ plane for $\eta_{\alpha\beta}$ =
0.2 at DUNE. We plot $\Delta P_{\mu e}$ = $P_{\mu e}^{NSI}$ \- $P_{\mu
e}^{SI}$ with varying $\delta_{CP}$ and $\nu$-mass. The top–panel
(bottom–panel) represents NH (IH) for energy = 2.5, $\theta_{23}$ =
$47^{\circ}$.

  * •

For NH case, the element $\eta_{ee}$ enhances the probabilities for all values
of $\delta_{CP}$. The enhancement is significant in negative $\delta_{CP}$
plane. The elements $\eta_{\mu\mu}$ and $\eta_{\tau\tau}$ suppresses the
probabilities for the complete $\delta_{CP}$ space. While for
$\eta_{\tau\tau}$ the suppression is more as compared to that of
$\eta_{\mu\mu}$.

  * •

For IH case, the elements $\eta_{ee}$ and $\eta_{\mu\mu}$ suppress the
probabilities for complete $\delta_{CP}$ space. However, the suppression is
higher for $\eta_{ee}$. The element $\eta_{\tau\tau}$ enhances the
probailities and the enhancement is higher for $\delta_{CP}$ $\in$
[$-60^{\circ}$, $40^{\circ}$].

Figure 5: Plot of $\rm\Delta P_{\mu e}=P_{\mu e}^{NSI}-P_{\mu e}^{SI}$ for
varying neutrino energies and masses for NH and IH. The top (bottom) panel
showcases the effects of normal (inverted) hierarchy. The left, middle and
right panels represent the effect of SNSI parameters $\eta_{ee}$,
$\eta_{\mu\mu}$ and $\eta_{\tau\tau}$ respectively.

Effect of SNSI with varying $\nu$-energy and $\nu$-mass.– In Fig. 5, we have
explored the range of energy where the effect of SNSI is maximum for various
choices of lightest $\nu$-mass for both hierarchies within the allowed upper
range. To quantify the effects, we use the quantity $\Delta P_{\mu e}$ as
defined earlier. Here, we have fixed the baseline at $L=1300$ km and NSI
parameter at $\eta_{\alpha\beta}=0.2$. We then vary the value of the lightest
$\nu$-mass in the allowed range and the neutrino energy in 0.5-10 GeV. We have
plotted $\Delta P_{\mu e}$ for varying neutrino energies and masses. We
consider the SNSI parameters $\eta_{ee}$, $\eta_{\mu\mu}$ and
$\eta_{\tau\tau}$ in the left, middle and right panel respectively. The top
(bottom) panel represents the normal (inverted) hierarchy case.

  * •

In NH case (top–panel), the presence of scalar NSI elements mostly enhances
the probabilities. With element $\eta_{ee}$ (top–left), we observe a
significant enhancement, roughly around an energy of 1.5 GeV with $\nu$-mass
greater than 0.02 eV. For other elements, suppression of probabilities can be
seen around this energy value.

  * •

For IH case (bottom panel), the maximal enhancement in probabilities can be
seen for $\eta_{\mu\mu}$ (bottom-middle) and $\eta_{\tau\tau}$ (bottom-right).
Whereas, the presence of $\eta_{ee}$ (bottom-left) shows a significant
suppression of $P_{\mu e}$ at lower energies.