aainstitutetext: Department of Physics, Tezpur University, Napaam, Sonitpur,
Assam-784028, Indiabbinstitutetext: Department of Physics, Assam Don Bosco
University, Kamarkuchi, Sonapur, Assam-782402, India

# Exploring the effects of Scalar Non Standard Interactions on the CP
violation sensitivity at DUNE

Abinash Medhi b Debajyoti Dutta a and Moon Moon Devi<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Neutrino oscillations have provided an excellent opportunity to study new
physics beyond the Standard Model or popularly known as BSM. The unknown
couplings involving neutrinos, termed as Non Standard Interactions (NSIs), may
appear as ‘new physics’ in different neutrino experiments. The neutrino NSIs
offer significant effects in neutrino oscillations and CP sensitivity, which
may be probed in various neutrino experiments. The idea of neutrinos coupling
with a scalar has evolved recently and looks promising. The effects of scalar
NSI may appear as a perturbation to the neutrino mass matrix in the neutrino
Hamiltonian. Also, as it modifies the neutrino mass matrix, it may provide a
direct possibility of probing neutrino mass models. As the scalar NSI affects
the neutrino mass matrix in the Hamiltonian, its effect is energy independent.
Moreover, the matter effects due to scalar NSI scales linearly with the matter
density.

In this work, we have performed a model independent study of the effects of
scalar NSI at long baseline neutrino experiments, taking DUNE as a case study.
We have done such a thorough study for the first time for DUNE. Various
neutrino parameters may get affected due to the inclusion of scalar NSI as it
modifies the effective mass matrix of neutrinos. We have explored the impact
of scalar NSI in neutrino oscillations and its impact on the measurements of
various mixing parameters. Also, we have probed the effects of scalar NSI on
different oscillation channels relevant to the experiment. We have also
explored the impact of various possible elements in the scalar NSI term at
DUNE.

###### Keywords:

Neutrino Physics, Beyond Standard Model, Non Standard Interactions, and CP
violation.

††preprint:

##  1 Introduction

The well established formalism of neutrino oscillations Super-
Kamiokande:1998kpq  have provided one of the first clear evidences of Physics
beyond the Standard Model (BSM). To explain the phenomena of neutrino
oscillations, which confirms neutrinos are not massless, needs an extension of
the Standard Model (SM) despite its unprecedented success. As we go beyond SM
to explain the neutrino mixing and associated phenomena, many BSM models
suggest additional interactions which are generally termed as the Non Standard
Interactions (NSIs). The NSIs may impact the production, propagation, and
detection of neutrinos in different neutrino experiments and thus necessitate
a thorough understanding of the possible impacts. The study of scalar NSI has
been a growing field to explore the possibility of new interactions in various
neutrino experiments. In this paper, we explore the impact of a general scalar
NSI on the neutrino mixing and for the first time, we have thoroughly studied
its effects on the CP-violation sensitivity at the proposed long baseline
neutrino experiment, DUNE Abi:2020loh .

In the standard interaction scenario, neutrinos may interact with matter via
charged current (CC) and/or neutral current (NC) interactions mediating a
$W^{\pm}$ and/or $Z$ bosons respectively. The idea of neutrino interaction
with matter was initially proposed in Wolfenstein:1977ue , where a new matter
potential term in the neutrino Hamiltonian due to neutrino matter interactions
was introduced. It was proposed that neutrino interactions with matter appear
as a matter potential term in neutrino Hamiltonian. Also, later in a few works
MSW1 ; MSW2 ; Mikheyev:1985zog  showed that the neutrino mixing can resonate
to a maximal value for some particular values of neutrino energy times the
matter density. This phenomenon has been termed as the Mikheyev-Smirnov-
Wolfenstein (MSW) effect MSW3 , which successfully explains the results of
various experiments Bahcall:1964gx ; Davis:1964hf ; deSalas:2017kay ;
SNO:2002tuh . The idea of neutrino matter effect was later explored widely and
it has been well established by different experiments. In the same paper
Wolfenstein:1977ue  where the idea of neutrino matter interactions were
proposed, the idea of neutrino NSIs were also indicated using vector boson as
mediators of the new non standard interactions.

The study of NSI is a well motivated phenomenological approach to probe new
physics beyond SM (BSM). The idea of NSI was introduced with a generally
parametrized vector and axial-vector currents Wolfenstein:1977ue , and later
explored by a large number of studies [for reviews Miranda:2015dra ;
Farzan:2017xzy ; Biggio:2009nt ; Babu:2019mfe ; Ohlsson:2012kf ]. The general
formalism for vector NSI which has been widely used in the literature are as
follows,

$V_{NSI}=V_{\mathrm{CC}}\begin{bmatrix}\varepsilon_{ee}&\varepsilon_{e\mu}&\varepsilon_{e\tau}\\\
\varepsilon_{e\mu}^{*}&\varepsilon_{\mu\mu}&\varepsilon_{\mu\tau}\\\
\varepsilon_{e\tau}^{*}&\varepsilon_{\mu\tau}^{*}&\varepsilon_{\tau\tau}\end{bmatrix}$
(1)

Where $V_{CC}$ = $\pm\sqrt{2}G_{F}n_{e}$ comes due to neutrino matter effects
and $\varepsilon_{\alpha\beta}$ represents the strengths of the vector NSI.

In Capozzi:2019iqn , the authors have discussed how one flavour changing NSI
element ($\epsilon_{e\tau}$) can significantly affect the mass ordering
sensitivities of NO$\nu$A NOvA:2004blv ; NOvA:2016kwd  and T2K T2K:2014xyt .
This can completely washed out the current $\sim$ 2.4$\sigma$ indication of
NO$\nu$A and T2K in favour of Normal Hierarchy (NH) of neutrino mass ordering.
In Agarwalla:2016fkh , the authors showed that the discovery potential of the
octant of $\theta_{23}$ of DUNE gets considerably affected in presence of off-
diagonal ($\epsilon_{e\mu}$ and $\epsilon_{e\tau}$) vector NSI elements. Also,
in presence of these NSI elements the transition probability $P_{\mu e}$
acquires an extra new interference term, which creates a degeneracy in the
measurement of $\delta_{CP}$. In reference Deepthi:2017gxg , the authors have
explored how diagonal and off-diagonal elements can severely affect the
sensitivity of determination of $\delta_{CP}$ at DUNE. Apart from DUNE, CP
sensitivity of other long baseline (LBL) neutrino experiments like NO$\nu$A,
T2K, and proposed T2HK Hyper-KamiokandeWorkingGroup:2014czz  may also be
severely get affected in presence of NSI PhysRevD.94.013014 . The mass
ordering sensitivities of different LBL experiments in presence of NSI have
been widely explored in this study Masud:2016nuj . For different studies on
the impact of NSI elements on the sensitivities of LBL experiments may be
referred to 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:2016bvp
; Masud:2015xva ; Ge:2016dlx ; Fukasawa:2016lew ; Chatterjee:2021wac . A
global status on the sensitivity of CP violation and neutrino mass ordering in
presence of NSI in LBL neutrino experiments has been summarised in
Esteban:2019lfo ; Coloma:2019mbs . The study of NSI opens up the potentiality
of using neutrino oscillations experiments to explore new physics scenarios
beyond the SM. Such a new interaction leads to a rich phenomenology in both
scattering experiments and neutrino oscillation experiments. Various
experimental data are being analyzed to see the bounds on such effects
Choubey:2015xha ; Khatun:2019tad ; Chatterjee:2014gxa ; Super-
Kamiokande:2011dam ; Davidson:2003ha ; Choubey:2014iia ; Denton:2018xmq ;
Farzan:2015hkd ; Farzan:2015doa ; Esmaili:2013fva ; Khan:2021wzy ; Liu:2020emq
; Chatterjee:2020kkm ; Denton:2020uda ; Babu:2020nna ; Flores:2020lji ;
Farzan:2019xor ; Pandey:2019apj .

The formulation of possible non standard couplings of neutrino to a scalar
field has been explored recently Ge:2018uhz ; Yang:2018yvk . This type of
scalar interaction appears as a correction to the neutrino mass term
Babu:2019iml  and it can have different phenomenological consequences than
that of vector mediated NSI. Various studies are done to investigate the
effects of these scalar NSI elements taking different astrophysical and
cosmological constraints also with terrestrial and space based experimental
constraints Babu:2019iml ; Venzor:2020ova . The presence of scalar NSI has
been used to explain the existing Borexino data Borexino:2017rsf ; Ge:2018uhz
. Also, as the scalar NSI appears as an addition to the neutrino mass matrix,
its impact on neutrino mass models is highly interesting and promising. These
scalar couplings may also have an impact on the measurement of different
neutrino oscillation parameters in various neutrino oscillation experiments.
As the effects of scalar NSI is directly proportional to the environmental
matter density and hence it makes long baseline neutrino experiments one of
the suitable candidates to probe its effects.

In this work, we have performed a model independent study of the effects of
scalar NSI parameters on the long baseline neutrino experiments, taking DUNE
as a case study. It is one of the first thorough studies of such scalar NSI
effects at DUNE. We have found that scalar NSI may have a significant impact
on the measurements of the $\delta_{CP}$ phase at DUNE. We start with a
general formulation for scalar NSI as a matrix and probe its effects element-
wise. The details of the formalism of scalar NSI are extensively discussed in
the subsequent section. For this study, we have considered the diagonal NSI
parameters. We observe that the effects of scalar NSI are mostly significant
around the oscillation maxima. Also, we see the occurrence of various
degeneracies in determining $\delta_{CP}$ in presence of the scalar NSI
elements. Following that, we have further checked how NSI can impact the CP
violation measurement at DUNE. We see that the effects of diagonal NSI
parameters are notably significant at DUNE. We show that for some chosen
values of scalar NSI parameters the experiment’s sensitivities get enhanced.
In addition to that, we have also studied the CP precision measurement
potential of DUNE in presence of scalar NSI. Here also we have found that the
capability of the experiment to constrain $\delta_{CP}$ is significantly
affected by the inclusion of these NSI parameters. For some cases, it is
observed that the precision capability of the experiment is poor as compared
to the standard case. However, there were also enhancements in the precision
measurement of $\delta_{CP}$ in presence of another set of NSI parameters.
Hence, constraining these NSI parameters is extremely crucial for
$\delta_{CP}$ sensitivities at DUNE.

We organize the paper as follows: we discuss the formalism of scalar NSI in
section 2. Then in the methodology section 3, we present a brief discussion on
the oscillation probabilities in presence of scalar NSI in section 3.1.
Afterward, in section 3.2 we discuss our simulation methodology used in the
analysis. In addition in section 4 we present our results of the study and
finally we summarise our findings in section 5. In Appendix A.1, we give a
sample derivation of transformation equations, which will be used to obtain
the analytical expressions of probabilities in presence of scalar NSI. Also,
for completeness we have shown our analysis results for non zero
$\eta_{\tau\tau}$ in Appendix A.2.

##  2 Scalar NSI

The experiments Super-Kamiokande:2004orf ; KamLAND:2004mhv ; MINOS:2008kxu ;
MINOS:2011neo  covering a wide range of baselines and energies established the
theory of neutrino oscillations, i.e., neutrinos can change their flavours due
to their non zero masses Maki:1962mu . The data from such experiments prove
that the neutrino flavours ($\nu_{e},\nu_{\mu},\nu_{\tau}$) which are the
superposition of their mass states $\nu_{1},\nu_{2},\nu_{3}$ with masses
$m_{1},m_{2},m_{3}$ respectively. The mixing of these flavour states and mass
eigenstates are governed by a 3 $\times$ 3 matrix called Pontecorvo-Maki-
Nakagawa-Sakata (PMNS) matrix, $\mathcal{U}$ Pontecorvo:1957cp ;
Pontecorvo:1957qd ; Pontecorvo:1967fh ; Maki:1962mu .

$\displaystyle{\mathcal{U}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\ 0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\\\
\end{array}\right)\left(\begin{array}[]{ccc}c_{13}&0&s_{13}e^{-i\delta_{CP}}\\\
0&1&0\\\ -s_{13}e^{i\delta_{CP}}&0&c_{13}\\\
\end{array}\right)\left(\begin{array}[]{ccc}c_{12}&s_{12}&0\\\
-s_{12}&c_{12}&0\\\ 0&0&1\\\ \end{array}\right)\ ,$ (11)

Where $s_{ij}=\sin{\theta_{ij}},c_{ij}=\cos\theta_{ij}$ and $\delta_{CP}$ is
the Dirac-type CP phase. This is called the PDG parametrization
ParticleDataGroup:2020ssz  of the PMNS matrix. In addition, if neutrinos are
considered as Majorana particles then there may also come two additional
phases called Majorana phases. However, these phases do not affect the
neutrino oscillations as they can come as a common phase in the neutrino
Hamiltonian. We are going to use this parametrization of the PMNS matrix
throughout the study.

In the Standard Model interactions scenario, the neutrinos interact with
matter via weak interactions only, via a $W^{\pm}$ or Z boson mediator
Linder:2005fc . The expression of effective Lagrangian for these interactions
is given by Wolfenstein:1977ue ; Nieves:2003in ; Nishi:2004st ,

$\mathcal{L}^{\rm eff}_{\rm
cc}=-\frac{4G_{F}}{\sqrt{2}}\left[\overline{\nu_{e}}(p_{3})\gamma_{\mu}P_{L}\nu_{e}(p_{2})\right]\left[\overline{e}(p_{1})\gamma^{\mu}P_{L}e(p_{4})\right]$
(12)

Where, $P_{L}$ and $P_{R}$ are left and right chiral projection operators
respectively, with $P_{L}=(1-\gamma_{5})/2$ and $P_{R}=(1+\gamma_{5})/2$),
$p_{i}$’s are momentum of incoming and outgoing states and $G_{F}$ is the
Fermi constant.

Usually, the neutrino matter effects come from the forward scattering of
neutrinos, considering zero momentum transfer between initial and final
states. These effects appear as matter potentials in the neutrino Hamiltonian
viz. $V_{\rm CC}=\pm\sqrt{2}G_{F}n_{e}$ and $V_{\rm
NC}=\pm\frac{G_{F}n_{n}}{\sqrt{2}}$. Here, $V_{\rm CC}$ and $V_{\rm NC}$ are
the matter potentials due to CC and NC matter interactions of neutrinos with
the matter. Here, the positive sign arises due to interactions of neutrinos
with matter while the negative sign arises due to interactions of
antineutrinos. Note that the matter potential due to NC interactions ($V_{\rm
NC}$) does not affect the neutrino oscillations since it just appears as a
common phase in the neutrino Hamiltonian. The effective Hamiltonian
($\mathcal{H}_{matter}$) for neutrino oscillations in the matter thus can be
written as Bilenky:1987ty ,

$\mathcal{H}_{matter}\approx E_{\nu}+\frac{MM^{\dagger}}{2E_{\nu}}\pm V_{\rm
SI}\,$ (13)

Where $E_{\nu}$ is the neutrino energy, M is the mass matrix of neutrinos and
$V_{\rm SI}$ is the matter potential due to neutrino matter effects. We
emphasize here again that the positive/negative sign of the ‘$V_{SI}$’ term
arises due to neutrino/antineutrino modes.

Although a vector mediated NSI has been well explored, a non standard term may
also arise due to other factors. Since neutrinos can couple with a scalar
(Higgs boson) with non zero vacuum expectation values to generate its mass,
coupling of neutrinos with scalar is an interesting possibility. The effective
Lagrangian for such a typical scalar NSI can be framed as,

$\displaystyle{\cal L}_{\rm eff}^{\rm S}\ =\
\frac{y_{f}y_{\alpha\beta}}{m_{\phi}^{2}}(\bar{\nu}_{\alpha}(p_{3})\nu_{\beta}(p_{2}))(\bar{f}(p_{1})f(p_{4}))\,$
(14)

Where, $\alpha$,$\beta$ = e, $\mu$, $\tau$ refer to the neutrino flavours,
f = e, u, d indicate the matter fermions, (e: electron, u: up-quark, d: down-
quark),
and $\bar{f}$ is for corresponding anti fermions,
$y_{\alpha\beta}$ is the Yukawa couplings of the neutrinos with the scalar
mediator, $\phi$,
$y_{f}$ is the Yukawa coupling of the mediator $\phi$ with the environmental
fermions f
and $m_{\phi}$ is the mass of the scalar mediator $\phi$.

In Equation 14, as the effective Lagrangian composed of Yukawa terms, as a
result of which, it can not be converted into vector currents. As a result of
which the scalar NSI will not appear as a contribution to the matter potential
term in the neutrino Hamiltonian. Instead, it may appear as a medium dependent
perturbation to the neutrino mass matrix of neutrino oscillationsGe:2018uhz .

The effective Hamiltonian ($\mathcal{H}_{NSI}$) for such scalar interactions
may be a modified form of Equation 13 as given below,

$\mathcal{H}_{NSI}\approx E_{\nu}+\frac{\left(M+\delta M\right)\left(M+\delta
M\right)^{\dagger}}{2E_{\nu}}\pm V_{\rm SI}\,$ (15)

Here, $\delta M\equiv\sum_{f}n_{f}y_{f}y_{\alpha\beta}/m^{2}_{\phi}$ ,
where, $n_{f}$ is the number density of the environmental fermions.

The perturbative term $\delta M$ comes due to inclusion of scalar NSI in
neutrino propagation. For defining an effective form of $\delta M$, which will
bring ease in element-wise study coupled with the neutrino mass matrix, we
have used the following parametrization,

$\delta M\equiv\sqrt{\Delta
m^{2}_{31}}\left\lgroup\begin{matrix}\eta_{ee}&\eta_{e\mu}&\eta_{e\tau}\\\
\eta_{\mu e}&\eta_{\mu\mu}&\eta_{\mu\tau}\\\ \eta_{\tau
e}&\eta_{\tau\mu}&\eta_{\tau\tau}\end{matrix}\right\rgroup\,$ (16)

Where, $\eta_{\alpha\beta}$ are dimensionless parameters and it quantifies the
size of the scalar NSI.

The hermicity of the neutrino Hamiltonian demands that the diagonal elements
are real and the off-diagonal elements are complex, which may be parameterized
as the following,

$\eta_{\alpha\beta}=|\eta_{\alpha\beta}|e^{i\phi_{\alpha\beta}};\qquad\alpha\neq\beta$
(17)

This formalism would enable the exploration of the scalar NSI elements through
different probability channels. The elements $\eta_{\alpha\beta}$ quantify the
strength of the interactions, they can be probed in various neutrino
experiments. Currently, there are not any definite bounds on these elements,
the bounds may be imposed from the results of different neutrino experiments.

##  3 Methodology

In this section, we have studied the modifications imposed by scalar NSI
elements on the analytical expressions of oscillation probabilities. In
section 3.1, we have derived the expressions of probabilities in presence of
NSI elements, $\eta_{ee}$, and $\eta_{\mu\mu}$. We have also discussed how
scalar NSI may affect the oscillation probabilities by looking at the modified
analytical expressions. Then in section 3.2 we have discussed the details of
the experiment and the simulation used to study the effects of scalar NSI.

###  3.1 Analytical expressions of oscillation probabilities

The phenomena of neutrino oscillations have played a crucial role in our
understanding of various properties of neutrinos from different neutrino
experiments. In general, in the Long Baseline (LBL) neutrino experiments the
most relevant neutrino oscillation channels are appearance probability
($\nu_{\mu}$ $\rightarrow$ $\nu_{e}$) channel and disappearance probability
($\nu_{\mu}$ $\rightarrow$ $\nu_{\mu}$) channel. Since the presence of scalar
NSI modifies the neutrino mass matrix, thus the Hamiltonian, so its effects on
neutrino oscillation probability may be probed.

$E$ | Neutrino energy
---|---
$L$ | Baseline
$V$ | $\sqrt{2}G_{F}n_{e}$
$G_{F}$ | Fermi’s constant
$n_{e}$ | Electron # density
$s_{ij}$ | $\sin\theta_{ij}$
$c_{ij}$ | $\cos\theta_{ij}$
$\Delta m^{2}_{ij}$ | $m_{i}^{2}-m_{j}^{2}$
$\Delta$ | $\Delta m^{2}_{31}L/(4E)$
$\alpha$ | $\Delta m^{2}_{21}/\Delta m^{2}_{31}$
$A$ |  $\frac{2EV}{\Delta m^{2}_{31}}$ = $\frac{VL}{2\Delta}$
Table 1: The various terms used in the paper. Natural units are assumed
throughout unless otherwise mentioned.

Usually, when neutrino passes through matter it experiences a matter
potential, commonly termed as Wolfenstein potential, as Wolfenstein
Wolfenstein:1977ue  was the first to point it out. The analytical expression
of neutrino oscillation probability (appearance probability $P_{\mu e}$), in
the matter, can be framed as Parke:2019vbs ; Akhmedov:2004ny ; Cervera:2000kp
,

$\displaystyle P_{\mu e}^{SI}$
$\displaystyle=\alpha^{2}\,\sin^{2}2\theta_{12}\,c_{23}^{2}\frac{\sin^{2}A\Delta}{A^{2}}+4\,s_{13}^{2}\,s_{23}^{2}\frac{\sin^{2}(A-1)\Delta}{(A-1)^{2}}$
$\displaystyle+2\,\alpha\,s_{13}\,\sin 2\theta_{12}\,\sin
2\theta_{23}\cos(\Delta+\delta_{\rm CP})\,\frac{\sin
A\Delta}{A}\,\frac{\sin(A-1)\Delta}{A-1}\,,$ (18)

Where, $\alpha\equiv\frac{\Delta m_{21}^{2}}{\Delta m_{31}^{2}}$,
$\Delta\equiv\frac{\Delta m^{2}_{31}L}{4E}$, $A\equiv\frac{2EV}{\Delta
m^{2}_{31}}=\frac{VL}{2\Delta}$

The details of the various terms used in the expression are listed in Table 1.

As the presence of scalar NSI reforms the neutrino Hamiltonian (Equation 15),
as a result of which the expression of appearance probability also acquire
some modifications. To derive the analytical expression of appearance
probability in presence of scalar NSI, we have considered only the diagonal
elements of $\Delta M$ matrix (Equation 16). Particularly, we have considered
$\eta_{ee}$ and $\eta_{\mu\mu}$ one at a time, to derive the probability
expressions. Considering all NSI parameters at a time can complicate the
expression too much to extract useful information. In this work, we
specifically looked into the diagonal parameters and another work is in
preparation taking into account the off diagonal elements. For the case of
$\eta_{ee}$ non zero, by performing the following transformations the
expression of appearance probability may be obtained. [See Appendix A.1]

$\displaystyle\begin{split}\Delta m_{21}^{2}\rightarrow\Delta
m_{21}^{2}-p_{1}\\\ \Delta m_{31}^{2}\rightarrow\Delta
m_{31}^{2}-p_{1}\end{split}$ (19)

Where, $p_{1}$ = $(\sqrt{\Delta m_{31}^{2}}\eta_{ee})^{2}+2m_{1}\sqrt{\Delta
m_{31}^{2}}\eta_{ee}$

Hence the following transformations of $\alpha$, $\Delta$ and A have to be
done to obtain the expression of appearance probability in presence of NSI
element $\eta_{ee}$,

$\displaystyle\begin{split}\alpha\rightarrow\alpha^{\prime}&=\frac{\Delta
m_{21}^{2}-p_{1}}{\Delta m_{31}^{2}-p_{1}}\\\
\Delta\rightarrow\Delta^{\prime}&=\frac{(\Delta m^{2}_{31}-p_{1})L}{4E}\\\
A\rightarrow A^{\prime}&=\frac{2EV}{\Delta
m^{2}_{31}-p_{1}}=\frac{VL}{2\Delta^{\prime}}\end{split}$ (20)

Hence the modified expression of appearance probability in presence of
$\eta_{ee}$ can be framed as,

$\displaystyle P_{\mu e}^{\eta_{ee}}$ $\displaystyle=\alpha^{\prime
2}\,\sin^{2}2\theta_{12}\,c_{23}^{2}\frac{\sin^{2}A^{\prime}\Delta^{\prime}}{A^{\prime
2}}+4\,s_{13}^{2}\,s_{23}^{2}\frac{\sin^{2}(A^{\prime}-1)\Delta^{\prime}}{(A^{\prime}-1)^{2}}$
$\displaystyle+2\,\alpha^{\prime}\,s_{13}\,\sin 2\theta_{12}\,\sin
2\theta_{23}\cos(\Delta^{\prime}+\delta_{\rm CP})\,\frac{\sin
A^{\prime}\Delta^{\prime}}{A^{\prime}}\,\frac{\sin(A^{\prime}-1)\Delta^{\prime}}{A^{\prime}-1}\,,$
(21)

Similarly, for the case of $\eta_{\mu\mu}$ following transformations have to
be made to obtain the analytical expression of appearance probability,

$\displaystyle\begin{split}\Delta m_{21}^{2}\rightarrow\Delta
m_{21}^{2}+p_{2}\\\ \Delta m_{31}^{2}\rightarrow\Delta
m_{31}^{2}+p_{2}\end{split}$ (22)

Where , $p_{2}$ = $(\sqrt{\Delta
m_{31}^{2}}\eta_{\mu\mu})^{2}+2m_{2}\sqrt{\Delta m_{31}^{2}}\eta_{\mu\mu}$

Hence the following transformations of $\alpha$, $\Delta$ and A have to be
done to obtain the expression of appearance probability in presence of
$\eta_{\mu\mu}$ elements,

$\displaystyle\begin{split}\alpha\rightarrow\alpha^{\prime\prime}&=\frac{\Delta
m_{21}^{2}+p_{2}}{\Delta m_{31}^{2}+p_{2}}\\\
\Delta\rightarrow\Delta^{\prime\prime}&=\frac{(\Delta
m^{2}_{31}+p_{2})L}{4E}\\\ A\rightarrow A^{\prime\prime}&=\frac{2EV}{\Delta
m^{2}_{31}+p_{2}}=\frac{VL}{2\Delta^{\prime\prime}}\end{split}$ (23)

So, the final modified expression of probability in presence of scalar NSI
element, $\eta_{\mu\mu}$ can be written as,

$\displaystyle P_{\mu e}^{\eta_{\mu\mu}}$ $\displaystyle=\alpha^{\prime\prime
2}\,\sin^{2}2\theta_{12}\,c_{23}^{2}\frac{\sin^{2}A^{\prime\prime}\Delta^{\prime\prime}}{A^{\prime\prime
2}}+4\,s_{13}^{2}\,s_{23}^{2}\frac{\sin^{2}(A^{\prime\prime}-1)\Delta}{(A^{\prime\prime}-1)^{2}}$
$\displaystyle+2\,\alpha^{\prime\prime}\,s_{13}\,\sin 2\theta_{12}\,\sin
2\theta_{23}\cos(\Delta^{\prime\prime}+\delta_{\rm CP})\,\frac{\sin
A^{\prime\prime}\Delta^{\prime\prime}}{A^{\prime\prime}}\,\frac{\sin(A^{\prime\prime}-1)\Delta^{\prime\prime}}{A^{\prime\prime}-1}\,,$
(24)

Furthermore, the analytical expression for disappearance probability
($P_{\mu\mu}$) in presence of neutrino matter effects may be formulated as
Parke:2019vbs ; Akhmedov:2004ny ; Cervera:2000kp ,

$\displaystyle P_{\mu\mu}^{SI}$
$\displaystyle=1-\sin^{2}2\theta_{23}\,\sin^{2}\Delta+\alpha\,c_{12}^{2}\,\sin^{2}2\theta_{23}\,\Delta\,\sin
2\Delta$
$\displaystyle-\alpha^{2}\,\sin^{2}2\theta_{12}\,c_{23}^{2}\,\frac{\sin^{2}A\Delta}{A^{2}}-\alpha^{2}\,c_{12}^{4}\,\sin^{2}2\theta_{23}\,\Delta^{2}\,\cos
2\Delta$
$\displaystyle+\frac{1}{2A}\,\alpha^{2}\,\sin^{2}2\theta_{12}\,\sin^{2}2\theta_{23}\left(\sin\Delta\,\frac{\sin
A\Delta}{A}\,\cos(A-1)\Delta-\frac{\Delta}{2}\,\sin 2\Delta\right)$
$\displaystyle-4\,s_{13}^{2}\,s_{23}^{2}\frac{\sin^{2}(A-1)\Delta}{(A-1)^{2}}$
$\displaystyle-\frac{2}{A-1}\,s_{13}^{2}\,\sin^{2}2\theta_{23}\left(\sin\Delta\,\cos
A\Delta\,\frac{\sin(A-1)\Delta}{A-1}-\frac{A}{2}\Delta\,\sin 2\Delta\right)$
$\displaystyle-2\,\alpha\,s_{13}\,\sin 2\theta_{12}\,\sin
2\theta_{23}\,\cos\delta_{\rm CP}\,\cos\Delta\,\frac{\sin
A\Delta}{A}\,\frac{\sin(A-1)\Delta}{A-1}$
$\displaystyle+\frac{2}{A-1}\,\alpha\,s_{13}\,\sin 2\theta_{12}\,\sin
2\theta_{23}\,\cos 2\theta_{23}\,\cos\delta_{\rm
CP}\,\sin\Delta\left(A\sin\Delta-\frac{\sin
A\Delta}{A}\,\cos(A-1)\Delta\right)\,,$ (25)

To obtain the analytical expression of disappearance probability $P_{\mu\mu}$
in presence of scalar NSI we have used the same transformations 19 and 20, as
we have used for the case with appearance probability. Thus the analytical
expression of $P_{\mu\mu}$ in presence of diagonal scalar NSI element
$\eta_{ee}$ attains the following form,

$\displaystyle P_{\mu\mu}^{\eta_{ee}}$
$\displaystyle=1-\sin^{2}2\theta_{23}\,\sin^{2}\Delta{{}^{\prime}}+\alpha{{}^{\prime}}\,c_{12}^{2}\,\sin^{2}2\theta_{23}\,\Delta{{}^{\prime}}\,\sin
2\Delta{{}^{\prime}}$
$\displaystyle-\alpha{{}^{\prime}}^{2}\,\sin^{2}2\theta_{12}\,c_{23}^{2}\,\frac{\sin^{2}A{{}^{\prime}}\Delta{{}^{\prime}}}{A{{}^{\prime}}^{2}}-\alpha{{}^{\prime}}^{2}\,c_{12}^{4}\,\sin^{2}2\theta_{23}\,\Delta{{}^{\prime}}^{2}\,\cos
2\Delta{{}^{\prime}}$
$\displaystyle+\frac{1}{2A{{}^{\prime}}}\,\alpha{{}^{\prime}}^{2}\,\sin^{2}2\theta_{12}\,\sin^{2}2\theta_{23}\left(\sin\Delta{{}^{\prime}}\,\frac{\sin
A{{}^{\prime}}\Delta{{}^{\prime}}}{A{{}^{\prime}}}\,\cos(A{{}^{\prime}}-1)\Delta{{}^{\prime}}-\frac{\Delta{{}^{\prime}}}{2}\,\sin
2\Delta{{}^{\prime}}\right)$
$\displaystyle-4\,s_{13}^{2}\,s_{23}^{2}\frac{\sin^{2}(A{{}^{\prime}}-1)\Delta{{}^{\prime}}}{(A{{}^{\prime}}-1)^{2}}$
$\displaystyle-\frac{2}{A{{}^{\prime}}-1}\,s_{13}^{2}\,\sin^{2}2\theta_{23}\left(\sin\Delta{{}^{\prime}}\,\cos
A{{}^{\prime}}\Delta{{}^{\prime}}\,\frac{\sin(A{{}^{\prime}}-1)\Delta{{}^{\prime}}}{A{{}^{\prime}}-1}-\frac{A{{}^{\prime}}}{2}\Delta{{}^{\prime}}\,\sin
2\Delta{{}^{\prime}}\right)$
$\displaystyle-2\,\alpha{{}^{\prime}}\,s_{13}\,\sin 2\theta_{12}\,\sin
2\theta_{23}\,\cos\delta_{\rm CP}\,\cos\Delta{{}^{\prime}}\,\frac{\sin
A{{}^{\prime}}\Delta{{}^{\prime}}}{A{{}^{\prime}}}\,\frac{\sin(A{{}^{\prime}}-1)\Delta{{}^{\prime}}}{A{{}^{\prime}}-1}$
$\displaystyle+\frac{2}{A{{}^{\prime}}-1}\,\alpha{{}^{\prime}}\,s_{13}\,\sin
2\theta_{12}\,\sin 2\theta_{23}\,\cos 2\theta_{23}\,\cos\delta_{\rm
CP}\,\sin\Delta{{}^{\prime}}\left(A{{}^{\prime}}\sin\Delta{{}^{\prime}}-\frac{\sin
A{{}^{\prime}}\Delta{{}^{\prime}}}{A{{}^{\prime}}}\,\cos(A{{}^{\prime}}-1)\Delta{{}^{\prime}}\right)\,,$
(26)

Similarly, in presence of scalar NSI element $\eta_{\mu\mu}$, we have used the
transformation 22 and 23 in the standard probability expression 25 to obtain
the analytical expression of probability. Thus, eventually, the final
expression of probability after these transformations gets the following form,

$\displaystyle P_{\mu\mu}^{\eta_{\mu\mu}}$
$\displaystyle=1-\sin^{2}2\theta_{23}\,\sin^{2}\Delta^{\prime\prime}+\alpha^{\prime\prime}\,c_{12}^{2}\,\sin^{2}2\theta_{23}\,\Delta^{\prime\prime}\,\sin
2\Delta^{\prime\prime}$ $\displaystyle-\alpha^{\prime\prime
2}\,\sin^{2}2\theta_{12}\,c_{23}^{2}\,\frac{\sin^{2}A^{\prime\prime}\Delta^{\prime\prime}}{A^{\prime\prime
2}}-\alpha^{\prime\prime
2}\,c_{12}^{4}\,\sin^{2}2\theta_{23}\,\Delta^{\prime\prime 2}\,\cos
2\Delta^{\prime\prime}$
$\displaystyle+\frac{1}{2A^{\prime\prime}}\,\alpha^{\prime\prime
2}\,\sin^{2}2\theta_{12}\,\sin^{2}2\theta_{23}\left(\sin\Delta^{\prime\prime}\,\frac{\sin
A^{\prime\prime}\Delta^{\prime\prime}}{A^{\prime\prime}}\,\cos(A^{\prime\prime}-1)\Delta^{\prime\prime}-\frac{\Delta^{\prime\prime}}{2}\,\sin
2\Delta^{\prime\prime}\right)$
$\displaystyle-4\,s_{13}^{2}\,s_{23}^{2}\frac{\sin^{2}(A^{\prime\prime}-1)\Delta^{\prime\prime}}{(A^{\prime\prime}-1)^{2}}$
$\displaystyle-\frac{2}{A^{\prime\prime}-1}\,s_{13}^{2}\,\sin^{2}2\theta_{23}\left(\sin\Delta^{\prime\prime}\,\cos
A^{\prime\prime}\Delta^{\prime\prime}\,\frac{\sin(A^{\prime\prime}-1)\Delta^{\prime\prime}}{A^{\prime\prime}-1}-\frac{A^{\prime\prime}}{2}\Delta^{\prime\prime}\,\sin
2\Delta^{\prime\prime}\right)$
$\displaystyle-2\,\alpha^{\prime\prime}\,s_{13}\,\sin 2\theta_{12}\,\sin
2\theta_{23}\,\cos\delta_{\rm CP}\,\cos\Delta^{\prime\prime}\,\frac{\sin
A^{\prime\prime}\Delta^{\prime\prime}}{A^{\prime\prime}}\,\frac{\sin(A^{\prime\prime}-1)\Delta^{\prime\prime}}{A^{\prime\prime}-1}$
$\displaystyle+\frac{2}{A^{\prime\prime}-1}\,\alpha^{\prime\prime}\,s_{13}\,\sin
2\theta_{12}\,\sin 2\theta_{23}\,\cos 2\theta_{23}\,\cos\delta_{\rm
CP}\,\sin\Delta^{\prime\prime}\left(A^{\prime\prime}\sin\Delta^{\prime\prime}-\frac{\sin
A^{\prime\prime}\Delta^{\prime\prime}}{A^{\prime\prime}}\,\cos(A^{\prime\prime}-1)\Delta^{\prime\prime}\right)\,,$
(27)

To validate our analytical expressions of probabilities, we have checked the
probabilities with a neutrino oscillation probability generator. For this
purpose, we have used the NuOscProbExact package Bustamante:2019ggq  and
modified it to incorporate the scalar NSI parameters. Then we have generated
the probabilities for both SI and some chosen values of NSI elements. In
figure 1 the comparison of the probabilities obtained from the analytical
expressions and the probability generator is shown. The study has been
performed at fixed $\delta_{CP}$ = -$\pi$/2 and $\theta_{23}$ = 47∘ for a
baseline of 1300 km (DUNE). The other mixing parameter values used to generate
the probabilities are listed in Table 2. The solid and dashed lines represent
the probabilities generated from the probability generator and analytical
expressions respectively. We have performed the analysis for both SI (red
plots), as well as various NSI values $\eta_{ee}$ (blue plots) and
$\eta_{\mu\mu}$ (black plots). We have seen that the probabilities from both
the analytical and from the generator are matching well with each other.

Figure 1: A comparison of probabilities ($P_{\mu e}$) generated from the probability generator and the analytical expressions. The study is performed at fixed $\delta_{CP}$ = -$\pi$/2 and $\theta_{23}$ = 47∘ for the DUNE baseline (1300 km). Case 1 is the output from the probability generator (NuOscProbExact) and Case 2 is the output from the analytical expressions. Parameters | True Values
---|---
$\theta_{12}$ | 34.51∘
$\theta_{13}$ | 8.44∘
$\theta_{23}$ | 47∘
$\delta_{CP}$ | -$\pi$/2
$\Delta m_{21}^{2}$ | 7.56 $\times$ 10-5 $eV^{2}$
$\Delta m_{31}^{2}$ | 2.43 $\times$ 10-3 $eV^{2}$
Table 2: The benchmark values of oscillation parameters used.

Here, in this study, we have explored the effects of these diagonal scalar NSI
elements i.e. $\eta_{ee}$, $\eta_{\mu\mu}$ and $\eta_{\tau\tau}$ on the
oscillation probabilities. The oscillation parameter values used throughout
the analysis are listed in Table 2. The values used in the analysis are within
the range of the current global best fit data range NuFIT5.0 . Note that, we
have considered the normal hierarchy of neutrino mass as true hierarchy
throughout the analysis.

From the analytical expressions of probabilities, it can be observed that the
diagonal scalar NSI elements affect the amplitude of the oscillation
probabilities. Consequently, the effects of these scalar NSI parameters can be
seen around the oscillation maxima of the probabilities. Figure 2 shows the
appearance probability $P_{\mu e}$ with varied neutrino energy at a fixed
baseline of 1300km (i.e. DUNE baseline). We have varied the energy of the
neutrino beam from 0.05 to 15 GeV, which is the significant energy range for
DUNE. The mixing parameters values taken in our analysis are listed in Table2.
The plot shows the effects of diagonal scalar NSI element, $\eta_{ee}$ on the
appearance probability ($P_{\mu e}$) at DUNE for different values of
$\eta_{ee}$ $\in$ [-0.2, 0.2]. The solid red line is for the case of Standard
Interaction (SI) with $\eta_{ee}$ = 0. The black and blue solid curves are for
non zero positive scalar NSI values. Also, the dotted green and dotted magenta
colour curves are for negative non zero scalar NSI values. It is observed that
the effects of diagonal scalar NSI are reasonably significant on the
oscillation probabilities of DUNE especially around the oscillation peaks.
Similarly, the plot in the Figure 3 shows the effects of appearance
probability $P_{\mu e}$ of DUNE in presence of $\eta_{\mu\mu}$ element keeping
all other NSI parameters as zeros. The red solid line corresponds to
$\eta_{\mu\mu}$ = 0 i.e. matter SI case. We have varied $\eta_{\mu\mu}$ from
-0.2 to 0.2 to check its effects on the probabilities. Unlike the element
$\eta_{ee}$, $\eta_{\mu\mu}$ enhances the probabilities around

Figure 2: The effect of the scalar NSI element, $\eta_{ee}$ on neutrino
appearance probabilities ($P_{\mu e}$) for fixed $\delta_{CP}$ = -$\pi$/2 and
$\theta_{23}$ = 47∘. The red color line is for the SI case and black and blue
solid (dashed) color lines are for positive (negative) non zero values of
$\eta_{ee}$.
Figure 3: The effect of the scalar NSI element, $\eta_{\mu\mu}$ on neutrino
appearance probabilities ($P_{\mu e}$) for fixed $\delta_{CP}$ = -$\pi$/2 and
$\theta_{23}$ = 47∘. The red color line is for SI case and black and blue
solid (dashed) color lines are for positive (negative) non zero values of
$\eta_{\mu\mu}$.

the oscillation maxima. It is interesting to see that the probability plots
for positive and negative non zero $\eta_{ee}$ and $\eta_{\mu\mu}$ are
symmetric around zero. For example, the black solid and dashed green plots
correspond to $\eta_{\mu\mu}$ = 0.1 and -0.1 respectively and both the
probability plots are overlapping one another. Thus, only the absolute value
of diagonal $\eta_{\alpha\beta}$ elements is affecting the probabilities
irrespective of its sign.

Figure 4: The effect of the scalar NSI element, $\eta_{ee}$ on neutrino
appearance probabilities ($P_{\mu e}$) for fixed $\delta_{CP}$ = -$\pi$/2,
$\theta_{23}$ = 47∘ and neutrino energy, E = 2.5 GeV. The red line is for
$\eta_{\alpha\beta}=0$ i.e SI case, and other solid (dashed) coloured lines
are for positive (negative) non zero values of $\eta_{ee}$.
Figure 5: The effect of the scalar NSI element, $\eta_{\mu\mu}$ on neutrino
appearance probabilities ($P_{\mu e}$) for fixed $\delta_{CP}$ = -$\pi$/2,
$\theta_{23}$ = 47∘ and neutrino energy, E = 2.5 GeV. The red line is for
$\eta_{\alpha\beta}=0$ i.e SI case, and other solid (dashed) coloured lines
are for positive (negative) non zero values of $\eta_{\mu\mu}$.

In Figures 4 and 5 the dependence of appearance probability ($P_{\mu e}$) with
varied $\delta_{CP}$ are shown. For this analysis, the neutrino beam energy is
kept at fixed E = 2.5 GeV and $\theta_{23}$ = 47∘ and NH is considered as true
hierarchy. Here, the elements $\eta_{ee}$ and $\eta_{\mu\mu}$ are varied from
-0.2 to 0.2 for both the Figures. The solid red line represents the SI case
i.e. $\eta_{ee}$ or $\eta_{\mu\mu}$ = 0. With the increasing value of the
element $\eta_{ee}$, the probability is suppressed whereas for the case of
$\eta_{\mu\mu}$ the probability is enhanced around the oscillation peaks. It
can also be seen that there are various degeneracies present for different
values of $\eta_{ee}$ and $\eta_{\mu\mu}$ elements which may affect the
experiment’s sensitivity towards measuring the $\delta_{CP}$ phase. Also, we
observed that the probability values do not change with the sign of NSI
parameters.

In Figure 6, we have shown the binned events as a function of reconstructed
neutrino energy. The mixing parameter values used to generate the event plots
are listed in Table 2. The solid blue line represents the SI case with all NSI
parameters set as zeros. The red and black line represents the binned events
in presence of scalar NSI elements $\eta_{ee}$ and $\eta_{\mu\mu}$
respectively. The behavior of the binned events is in good agreement with the
probabilities. In presence of non zero $\eta_{ee}$, the number of events
decreases compared to the SI case for each bin, while for $\eta_{\mu\mu}$ the
number of events increases for each bin.

Figure 6: Variation of binned event rates with neutrino energy for fixed
$\delta_{CP}$ = -$\pi$/2 and $\theta_{23}$ = 47∘. The blue plot corresponds to
zero NSI elements ($\eta_{\alpha\beta}$ = 0) and the red and black plot is for
NSI case with non zero $\eta_{ee}$ and $\eta_{\mu\mu}$ respectively.

###  3.2 The details of the simulation used

Deep Underground Neutrino Experiment (DUNE) DUNE1 ; DUNE2 ; DUNE3 ; DUNE4 ;
DUNE5  is a proposed super-beam neutrino experiment that will be located in
the USA. The beam for this experiment would be generated at Fermilab, USA and
the neutrinos are expected to be detected at a distance of 1300 km from the
source at Homestake Mine in South Dakota. The detector is made up of a 35 - 40
kt Liquid Argon Time projection chamber (TPC). The super-beam produced by
Fermilab is planned to have a power of 1.2 MW-120 GeV and will deliver 1021
proton-on-target (POT) per year. The experiment is expected to start data
taking around 2025. These flagship configurations of this experiment make it a
suitable candidate to tackle all the three unknowns of the neutrino sector i.e
the neutrino mass hierarchy, octant of $\theta_{23}$, and CP violation in the
leptonic sector. As scalar NSI scales linearly with matter density, the large
baseline of this experiment makes it a suitable candidate to probe its
effects.

We have used the General Long Baseline Experiment Simulator (GLoBES)
Huber:2004ka ; Kopp:2006wp ; Huber:2007ji  to simulate our experiment. GLoBES
is a neutrino experiment simulator, which is used to study the long baseline
neutrino oscillation experiments. In this work, we have taken a liquid argon
detector with a baseline of 1300 km, which is the baseline of DUNE. We have
considered 5 years of neutrino and 5 years of antineutrino mode to simulate
our experiments, which gives a total exposure of 35 $\times$ 1022 kt-POT-yr.
We have used the combined appearance and disappearance channel in our
analysis. The signal normalization for neutrino and antineutrino mode is 2%
and 5% respectively and the background normalization error is 10% and 10%
respectively. The energy resolution for the $\mu$ ($R_{\mu}$) and e ($R_{e}$)
is taken to be 20%/$\sqrt{E}$ and 15%/$\sqrt{E}$ respectively. Also, the
energy calibration error for neutrino and antineutrino mode is taken as 5% and
5% each respectively. We have used the Technical Design Report (TDR) of DUNE
DUNE3  for background and choice of systematics. The details of the
experimental configuration and the systematic uncertainties taken in our
simulations are listed in Table 3.

Detector details | Normalisation error | Energy calibration error
---|---|---
| Signal | Background | Signal | Background
Baseline = 1300 km |  |  |  |
Runtime (yr) = 5 $\nu$ \+ 5 $\bar{\nu}$ | $\nu_{e}:5\%$ | $\nu_{e}:10\%$ | $\nu_{e}:5\%$ | $\nu_{e}:5\%$
35 kton, LArTPC |  |  |  |
$\varepsilon_{app}=80\%$, $\varepsilon_{dis}=85\%$ | $\nu_{\mu}:5\%$ | $\nu_{\mu}:10\%$ | $\nu_{\mu}:5\%$ | $\nu_{\mu}:5\%$
$R_{e}=0.15/{\sqrt{E}}$, $R_{\mu}=0.20/{\sqrt{E}}$ |  |  |  |
Table 3:  Details of detector configurations, efficiencies, resolutions, and
systematic uncertainties for DUNE. Here, $\varepsilon_{app}$ and
$\varepsilon_{dis}$ are signal efficiencies for $\nu_{e}^{CC}$ and
$\nu_{\mu}^{CC}$ respectively. Also, $R_{e}$ and $R_{\mu}$ is energy
resolutions for signal $\nu_{e}^{CC}$ and $\nu_{\mu}^{CC}$ respectively.

##  4 Results and Discussion

To check whether an experiment can distinguish between CP conserving (
$\delta_{CP}$ = 0, $\pi$) and CP violating values ( $\delta_{CP}$ $\neq$ 0,
$\pi$), we have defined statistical $\chi^{2}$ likelihood parameters as in the
following form,

$\chi^{2}\equiv\min_{\eta}\sum_{i}\sum_{j}\frac{\left[N_{true}^{i,j}-N_{test}^{i,j}\right]^{2}}{N_{true}^{i,j}}$
(28)

Where, $N_{true}^{i,j}$ and $N_{test}^{i,j}$ are the number of true and test
events in the $\\{i,j\\}$-th bin respectively. In this analysis, our prime
goal is to point out the effects of diagonal scalar NSI elements on the CP
violation measurement at DUNE. Thus, we have studied CP violation as well as
CP precision measurement capabilities of the experiment in presence of scalar
NSI. The capability of an experiment to differentiate between CP conserving
and CP violating values of $\delta_{CP}$ is a measure of its CP sensitivity.
We have considered normal hierarchy (NH) as true hierarchy and higher octant
(HO) as true octant (true $\theta_{23}$ = 47∘) throughout the analysis. Also,
we have marginalized over the systematic uncertainties. In addition, the
mixing parameter values used in the analysis are taken from Table 2 unless
otherwise mentioned.

Figure 7 shows the result of the $\chi^{2}$ analysis, where for each plot one
diagonal element of the scalar NSI matrix is considered, keeping others zero.
For example, the red solid line represents the case with true $\eta_{ee}$ =
0.01 and the test value of $\eta_{ee}$ is varied from -0.15 to 0.15. A similar
kind of methodology has been performed to produce the plots for non zero
$\eta_{\mu\mu}$ (dashed black line) and $\eta_{\tau\tau}$ (dashed blue line)
values. The horizontal dashed magenta and dashed black line corresponds to
5$\sigma$ and 3$\sigma$ CL respectively. This analysis has been carried out to
check the $\chi^{2}$ sensitivity of the experiment towards the diagonal scalar
NSI elements. It can be seen that indeed the sensitivities get affected by the
inclusion of these diagonal scalar NSI elements. Also, the plot for
$\eta_{\mu\mu}$ and $\eta_{\tau\tau}$ coincide with one another, while only
the effect of $\eta_{ee}$ is different.

Figure 7: $\chi^{2}$ analysis for DUNE in presence of diagonal scalar NSI
elements for true $\delta_{CP}$ = -$\pi$/2 and true $\theta_{23}$ = 47∘. Solid
red and dashed black and blue plots are for cases with non zero $\eta_{ee}$,
$\eta_{\mu\mu}$, and $\eta_{\tau\tau}$ respectively. Figure 8: The plot
represents CP violation sensitivity in presence of scalar NSI. The red line is
for the SI case whereas blue, green, and violet plots are for non zero NSI
parameters with true $\eta_{ee}$ = 0.01, 0.5 and 0.15 respectively. Figure 9:
The plot represents CP violation sensitivity in presence of scalar NSI. The
red line is for the SI case whereas blue, green, and violet plots are for non
zero NSI parameters with true $\eta_{\mu\mu}$ = 0.01, 0.5 and 0.15
respectively.

We show in Figures 8 and 9 the CP violation (CPV) sensitivity of DUNE in the
presence of scalar NSI elements $\eta_{ee}$ and $\eta_{\mu\mu}$ respectively.
To perform this analysis, we have excluded CP non conserving values in the
test $\delta_{CP}$ values. Also, we have marginalized the test values of
$\eta_{ee}$ keeping all the true NSI parameters values fixed. To generate the
sensitivity plots, we have considered only one non zero elements of the scalar
NSI matrix, keeping others as zeros. In both the Figures, the red solid line
represents the case with no NSI parameters ($\eta_{ij}=0$), which is the SI
case. The blue, green, and violet lines represent the cases with non zero NSI
values. The horizontal dashed blue and dashed black line corresponds to
5$\sigma$ and 3$\sigma$ CL respectively. From Figure 8, it is observed that
the CP sensitivity of the experiment is significantly suppressed in presence
of the diagonal $\eta_{ee}$ element. Especially with a value $\eta_{ee}$
$\geq$ 0.1, it lies below the reach of 5$\sigma$ sensitivity. The larger the
values of $\eta_{ee}$ more it will deteriorate the CP sensitivity of DUNE. In
contrary to that, in presence of diagonal scalar NSI elements $\eta_{\mu\mu}$
the CP violation sensitivity of DUNE gets enhanced. For the non zero values of
$\eta_{\mu\mu}$ that we have considered, the sensitivities always lie above
the SI sensitivity. Also, note that for $\eta_{\mu\mu}$ = 0.05 case the SI and
NSI sensitivities overlap in the region $\sim$ [-90∘, 0∘]. So, in this region,
it is difficult to pin point whether the CP sensitivity is coming from SI or
NSI cases with $\eta_{\mu\mu}$ = 0.05. Hence if nature has scalar NSI then CP
sensitivity measured at the DUNE far detector will get curbed in the presence
of NSI.

In Figures 10 and 11 the CP precision measurement capability of DUNE in
presence of diagonal scalar NSI elements $\eta_{ee}$ and $\eta_{\mu\mu}$
respectively are shown. To perform this study, we have fixed all the true
values of the parameters and marginalized the test $\delta_{CP}$ and
$\theta_{23}$ over the allowed range i.e [-$\pi$, $\pi$] and [$40^{\circ}$,
$50^{\circ}$] respectively. We have kept the value of true $\delta_{CP}$ fixed
at -$90^{\circ}$, which is the best-fit value from global analysis of current
world neutrino data NuFIT5.0 . Apart from that for the case with NSI we have
also marginalized both test $\eta_{ee}$ and $\eta_{\mu\mu}$ from -0.15 to 0.15
and we have plotted $\sigma$ (= $\sqrt{\Delta\chi^{2}}$) as a function of test
$\delta_{CP}$. The blue and black dotted line represents the $\Delta\chi^{2}$
corresponding to 5$\sigma$ and 3$\sigma$ CL respectively. The analysis
signifies, knowing the true values of $\delta_{CP}$, how precisely DUNE can
constrain the test $\delta_{CP}$ values. Here, from Figure 10 it can be seen
that in presence of $\eta_{ee}$, the capability of DUNE to constrain the
$\delta_{CP}$ phase gets reduced. For the SI case (red plot) DUNE should be
able to measure the phase $\delta_{CP}$ with a precision of $\sim$
-$90^{\circ^{+}45^{\circ}}_{-48^{\circ}}$ at 3$\sigma$ CL. However, in
presence of scalar NSI, the precision worsens both at 3$\sigma$ as well as
5$\sigma$ CL. In addition for the values of $\eta_{ee}$ = 0.15 and higher its
CP precision measurement ability lies below 3$\sigma$. This implies that for
$\eta_{ee}$ = 0.15, its ability to constrain the CP phase ($\delta_{CP}$) lies
below the 3$\sigma$ CL. Whereas in Figure 11, the presence of NSI element
$\eta_{\mu\mu}$ improves the sensitivities to constrain the $\delta_{CP}$
values. Here also for the SI case (red plot) DUNE’s ability to constrain the
$\delta_{CP}$ phase is of the order $\sim$
-$90^{\circ^{+}45^{\circ}}_{-48^{\circ}}$. In addition, for all the non zero
values of $\eta_{\mu\mu}$ the ability to constrain the $\delta_{CP}$ phase
gets enhanced. For example, for the case with $\eta_{\mu\mu}$ = 0.15 (violet
line) the precision of $\delta_{CP}$ gets further improved by $\sim$
$90^{\circ+25^{\circ}}_{-22^{\circ}}$ at 3$\sigma$ CL as compared to the SI
case. Here also, it can be seen that for $\eta_{\mu\mu}$ = 0.05 (blue plot)
the sensitivity lies below the SI case (red plot). This was expected as the
CPV sensitivity shows an overlapping region $\sim$ [-90∘, 0∘] for the
$\eta_{\mu\mu}$ case, as discussed at the end of the previous paragraph. As
the sensitivities overlap for the region $\sim$ [-90∘, 0∘], so the ability of
the experiment to constrain the $\delta_{CP}$ decreases for this region.

Figure 10: CP precision measurement in presence of diagonal scalar NSI element
($\eta_{ee}$) for true $\delta_{CP}$ = -$\pi$/2 and true $\theta_{23}$ = 47∘.
The red line is for the SI case whereas blue, green, and violet plots are for
non zero NSI parameters with true $\eta_{ee}$ = 0.01, 0.5 and 0.15
respectively. Figure 11: CP precision measurement in presence of diagonal
scalar NSI element ($\eta_{\mu\mu}$) for true $\delta_{CP}$ = -$\pi$/2 and
true $\theta_{23}$ = 47∘. The red line is for the SI case whereas blue, green,
and violet plots are for non zero NSI parameters with true $\eta_{\mu\mu}$ =
0.01, 0.5 and 0.15 respectively.

##  5 Summary and Concluding Remarks

In this precision era of neutrino physics, it is very crucial to identify the
subdominant effects like NSI in the neutrino experiments and their effects on
the physics potential of different experiments. In this work, we have
primarily taken DUNE as a LBL candidate to study the effects of scalar NSI.
Here we have considered only the diagonal elements of the scalar NSI matrix
and studied its impact on the experiment. We have done a $\chi^{2}$ analysis
to check the effects of diagonal scalar NSI elements. We have seen that the
sensitivity of DUNE towards finding the $\eta_{ee}$ is little less as compared
to the sensitivity of $\eta_{\mu\mu}$ and $\eta_{\tau\tau}$ elements. On the
other hand, the sensitivity of the $\eta_{\mu\mu}$ and $\eta_{\tau\tau}$
exactly matches with each other. Thereafter, we have also done a CP violation
sensitivity of DUNE in presence of these scalar NSI elements. We have seen
that the inclusion of scalar NSI can severely impact the CP sensitivities at
DUNE. For example, for a chosen value of $\eta_{ee}$ $\geq$ 0.10 the CP
sensitivity of DUNE lies below the 5$\sigma$ CL. Also, for the value of
$\eta_{ee}$ $\geq$ 0.15, the CP sensitivities of DUNE further deteriorate
below the 3$\sigma$ CL. Hence, the effects of scalar NSI can not be ignored in
the LBL experiments like DUNE, given the remarkable accuracy and precision
provided by the experiment. Similarly from our CP precision study, we have
observed that the ability of the experiment to constrain the CP phase gets
affected by the presence of scalar NSI. The presence of $\eta_{\mu\mu}$
improved the ability to measure the $\delta_{CP}$, whereas $\eta_{ee}$ worsen
the ability of the experiment to constrain $\delta_{CP}$ values. It is
extremely crucial to put some constraints on these scalar NSI parameters for
the correct interpretation of data from various LBL neutrino experiments. For
this combined global efforts of all the LBL experiments are necessary. In
addition, it is also very interesting to probe scalar NSI to various neutrino
mass models and it directly affects the mass term in the neutrino Hamiltonian.

## Acknowledgment

The work is supported by the Research and Innovation grant 2021
(DoRD/RIG/10-73/ 1592-A), Tezpur University received by AM. MMD would like to
acknowledge the support of the DST SERB grant EMR/2017/001436 for this work.
We also acknowledge the support of the DST FIST grant SR/FST/PSI-211/2016(C)
of the Department of Physics, Tezpur University.

##  Appendix A Appendices

###  A.1 A sample derivation of transformation equations for oscillation
probabilities

This method is only applicable to diagonal non zero elements of the $\Delta$M
matrix.

The parametrization of $\delta M$ used in the analysis is:

$\delta M\equiv\sqrt{\Delta
m^{2}_{31}}\left\lgroup\begin{matrix}\eta_{ee}&\eta_{e\mu}&\eta_{e\tau}\\\
\eta_{\mu e}&\eta_{\mu\mu}&\eta_{\mu\tau}\\\ \eta_{\tau
e}&\eta_{\tau\mu}&\eta_{\tau\tau}\end{matrix}\right\rgroup\,$ (29)

Let us take $\eta_{ee}$ non zero and other elements zero.

We have

$M=\begin{pmatrix}m_{1}&0&0\\\ 0&m_{2}&0\\\ 0&0&m_{3}\\\ \end{pmatrix}$ (30)

After some reduction

$MM^{\dagger}=\begin{pmatrix}0&0&0\\\ 0&\Delta m_{21}^{2}&0\\\ 0&0&\Delta
m_{31}^{2}\\\ \end{pmatrix}$ (31)

Then,

$M+\delta M=\begin{pmatrix}m_{1}+\sqrt{\Delta m_{31}^{2}}\eta_{ee}&0&0\\\
0&m_{2}&0\\\ 0&0&m_{3}\\\ \end{pmatrix}$ (32)

So,

$(M+\delta M)(M+\delta M)^{\dagger}=\begin{pmatrix}(m_{1}+\sqrt{\Delta
m_{31}^{2}}\eta_{ee})^{2}&0&0\\\ 0&m_{2}^{2}&0\\\ 0&0&m_{3}^{2}\\\
\end{pmatrix}$ (33)

After reduction

$(M+\delta M)(M+\delta M)^{\dagger}=\begin{pmatrix}(\sqrt{\Delta
m_{31}^{2}}\eta_{ee})^{2}+2m_{1}\sqrt{\Delta m_{31}^{2}}\eta_{ee}&0&0\\\
0&\Delta m_{21}^{2}&0\\\ 0&0&\Delta m_{31}^{2}\\\ \end{pmatrix}$ (34)

Then after further simplifications,

$(M+\delta M)(M+\delta M)^{\dagger}=\begin{pmatrix}0&0&0\\\ 0&\Delta
m_{21}^{2}-p&0\\\ 0&0&\Delta m_{31}^{2}-p\\\ \end{pmatrix}$ (35)

Where p = $(\sqrt{\Delta m_{31}^{2}}\eta_{ee})^{2}+2m_{1}\sqrt{\Delta
m_{31}^{2}}\eta_{ee}$

Hence the expression for the analytical expression probabilities in presence
of $\eta_{ee}$ element can be obtained by performing the following
transformation to the mass square differences values,

$\displaystyle\begin{split}\Delta m_{21}^{2}\rightarrow\Delta m_{21}^{2}-p\\\
\Delta m_{31}^{2}\rightarrow\Delta m_{31}^{2}-p\\\ \end{split}$ (36)

Similar calculations can be done to achieve the analytical expression of
probabilities in presence of non zero $\eta_{\mu\mu}$ element. For that case
the transformation equation will be,

$\displaystyle\begin{split}\Delta m_{21}^{2}\rightarrow\Delta
m_{21}^{2}+p_{2}\\\ \Delta m_{31}^{2}\rightarrow\Delta
m_{31}^{2}+p_{2}\end{split}$ (37)

Where , $p_{2}$ = $(\sqrt{\Delta
m_{31}^{2}}\eta_{\mu\mu})^{2}+2m_{2}\sqrt{\Delta m_{31}^{2}}\eta_{\mu\mu}$.

###  A.2 Analysis results for non zero $\eta_{\tau\tau}$

For the sake of completeness, we have also performed a similar analysis for
non zero $\eta_{\tau\tau}$ elements. The same procedure has been followed to
generate these plots as we have done for $\eta_{ee}$ and $\eta_{\mu\mu}$
cases. For example, Figure 12 shows the effect of $\eta_{\tau\tau}$ element on
the appearance probability at DUNE. It can be observed that the effect of
$\eta_{\tau\tau}$ is not notable. Similarly, in Figure 13 the CP violation
sensitivity of DUNE in presence of $\eta_{\tau\tau}$ is shown. Here also it is
seen that the effect of $\eta_{\tau\tau}$ on the CP violation sensitivity is
not significant. Also, Figure 14 shows the CP precision measurement capability
of DUNE in presence of $\eta_{\tau\tau}$. It is observed that the potentiality
of the experiment to constrain the CP values does not get highly affected by
the presence of $\eta_{\tau\tau}$ element. From all the figures it can be
noted that the non zero $\eta_{\tau\tau}$ does not affect significantly the
sensitivities of DUNE and its effects may be ignored.

Figure 12: The effect of the diagonal scalar NSI element, $\eta_{\tau\tau}$ on
neutrino appearance probabilities ($P_{\mu e}$) for fixed $\delta_{CP}$ =
-$\pi$/2 and $\theta_{23}$ = 47∘. The red color line is for the SI case and
black and blue solid (dashed) color lines are for positive (negative) non zero
values of $\eta_{\tau\tau}$.  Figure 13: The plot represents CP violation
sensitivity in presence of scalar NSI element $\eta_{\tau\tau}$. The red line
is for the SI case whereas blue, green, and violet plots are for non zero NSI
parameters with true $\eta_{\tau\tau}$ = 0.01, 0.5 and 0.15 respectively.
Figure 14: CP precision measurement in presence of diagonal scalar NSI element
($\eta_{\tau\tau}$) for true $\delta_{CP}$ = -$\pi$/2 and true $\theta_{23}$ =
47∘. The red line is for the SI case whereas blue, green, and violet plots are
for non zero NSI parameters with true $\eta_{\mu\mu}$ = 0.01, 0.5 and 0.15
respectively.

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