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

# Impact of scalar NSI on the neutrino mass hierarchy sensitivity at DUNE,
T2HK and T2HKK

Arnab Sarker a,2 Abinash Medhi a,3 Dharitree Bezboruah a,4 Moon Moon Devi b,5
and Debajyoti Dutta<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The study of neutrino non-standard interactions (NSI) is a well-motivated
phenomenological scenario to explore new physics beyond the Standard Model.
The possible scalar coupling of neutrinos ($\nu$) with matter is one of such
new physics scenarios which appears as a sub-dominant effect that can impact
the $\nu$-oscillations in matter. The presence of scalar NSI introduces an
additional contribution directly to the $\nu$-mass matrix in the interaction
Hamiltonian and subsequently to the $\nu$-oscillations. This indicates that
scalar NSI may have a significant impact on measurements related to
$\nu$-oscillations e.g. leptonic CP phase $(\delta_{CP})$, $\theta_{23}$
octant and neutrino mass hierarchy (MH). The linear scaling of the effects of
scalar NSI with matter density also motivates its exploration in long-baseline
(LBL) experiments. In this paper, we study the impact of a scalar-mediated NSI
on the MH sensitivity of DUNE, T2HK and T2HKK, which are upcoming LBL
experiments. We observe that the MH sensitivities of these experiments are
affected by scalar NSI. For certain chosen values of scalar NSI elements, we
note an enhancement in the standard MH sensitivities. We then combine the data
from DUNE with T2HK/T2HKK to explore possible synergy among these experiments
in a wider parameter space. We also observe a significant enhancement in the
MH sensitivities for the combined analysis.

###### Keywords:

Non-Standard Interactions, Neutrino Mass Hierarchy, Neutrino Physics, Beyond
Standard Model, Long Baseline Experiments

##  1 Introduction

The Standard Model (SM) of particle physics stands as one of the most
successful theories to explain the fundamental nature of our universe. Still,
there exist various phenomena that cannot be explained by theories within the
SM, such as the phenomenon of $\nu$-oscillations, the presence of dark matter
and dark energy, baryon asymmetry etc. This compels us to explore physics
beyond the Standard Model (BSM). Neutrino oscillations decisively affirm the
existence of non-zero masses for neutrinos, marking the initial experimental
indication of BSM physics SNO:2002tuh ; Super-Kamiokande:1998kpq . The current
& upcoming $\nu$-experiments are primarily focused on determining the three
major unknowns in the neutrino sector viz. neutrino mass hierarchy (MH),
octant of $\theta_{23}$ & leptonic CP phase. The various sub-dominant non-
standard effects, like non-standard interactions (NSI) 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 , 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 ,
neutrino decay Berryman:2014qha ; PICORETI201670 ; SNO:2018pvg ; GOMES2015345
; Coloma:2017zpg ; Abrahao:2015rba , neutrino decoherence PhysRevD.56.6648 ;
Benatti:2000ph ; PhysRevD.100.055023 ; PhysRevD.95.113005 ; Lisi:2000zt  etc.,
are also being explored through these experiments. The precise determination
of the unknowns is of immense significance for the complete understanding of
the standard 3$\nu$ paradigm. The presence of NSI may affect the physics
sensitivities of these experiments. In this study, we have mainly focused on
exploring the impact of a scalar-mediated NSI on the MH sensitivities in the
LBL sector, in a model-independent way. In our study, we include the upcoming
LBL experiments namely DUNE, T2HK and T2HKK. We have also performed a combined
analysis of the MH sensitivities for DUNE with T2HK/T2HKK.

The ordering of the absolute $\nu$-masses $(m_{1},m_{2},m_{3})$ are not yet
fully known. From the solar sector, we know that $m_{2}>m_{1}$ SNO:2001kpb ;
SNO:2002tuh ; SNO:2002hgz ; Harrison:2002er , however, the relative ordering
of $m_{3}$ is yet to be determined. We have two possible ordering of
$\nu$-masses i.e. normal hierarchy (NH) $(m_{3}>m_{2}>m_{1})$ and inverted
hierarchy (IH) $(m_{2}>m_{1}>m_{3})$ Esteban:2020cvm . Various current and
upcoming $\nu$-experiments from the atmospheric ICAL:2015stm ; IceCube-
PINGU:2014okk ; Super-Kamiokande:2002weg ; KM3Net:2016zxf ; IceCube-
Gen2:2020qha , reactor Ardellier:2004ui ; RENO:2010vlj ; DayaBay:2007fgu ;
JUNO:2021vlw  and LBL accelerator NOvA:2004blv ; T2K:2001wmr ; Abi:2020loh ;
Hyper-KamiokandeProto-:2015xww ; Hyper-Kamiokande:2016srs ; ESSnuSB:2013dql ;
Akindinov:2019flp  sectors are equipped with advanced technologies and
systematics which are expected to be highly sensitive to MH determination
using different oscillation channels. Present global fit from LBL (T2K +
NO$\nu$A), reactor (Daya Bay, RENO, Double Chooz) and atmospheric (Super-K)
sectors favour NH over IH at 2.7$\sigma$ Confidence Level (CL) Esteban:2020cvm
. In the current precision era, the future experiments are designed to have
much higher sensitivities towards MH determination. All these experiments can
individually achieve 3$\sigma$ CL in MH determination JUNO:2015zny ;
Devi:2014yaa ; DUNE:2020jqi ; Hyper-Kamiokande:2016srs ; Hyper-
KamiokandeProto-:2015xww ; ESSnuSB:2021azq . In addition, combining
experiments from different sectors can further enhance the sensitivities by
leveraging the synergy and a wider parameter space. In Fukasawa:2016lew , the
MH sensitivity up to 15$\sigma$ is obtained for synergy among T2HK, HK and
DUNE. Combining current experiments T2K and NO$\nu$A with DUNE and ESS$\nu$SB,
a 5$\sigma$ sensitivity for hierarchy determination is obtained in
Chakraborty:2019jlv . A combination of T2K-II, NO$\nu$A and JUNO can also
provide a similar sensitivity, as studied in reference Cao:2020ans . In
Ghosh:2012px , the authors have explored the MH sensitivity at INO-ICAL and
have also performed a combined analysis with T2K, NO$\nu$A and reactor
experiments. Combining JUNO with IceCube Upgrade, and PINGU can also enhance
the hierarchy sensitivity IceCube-Gen2:2019fet . In reference Choubey:2022gzv
, the MH sensitivity by combining JUNO and T2HK is studied. Though, JUNO alone
has the potential of discovering MH at $3\sigma$ CL with $3\%$ energy
resolution, a synergy between JUNO and T2HK increases it up to $9\sigma$. A
synergy among T2HK, JUNO and atmospheric $\nu$-experiment INO-ICAL also has
the potential of determining the MH with up to $7\sigma$ CL with $5\%$ energy
resolution of JUNO Raikwal:2023txr .

Moreover, the MH sensitivities may be substantially affected by various BSM
physics scenarios which makes it crucial to explore and understand such
subdominant effects. In reference Capozzi:2019iqn ; Esteban:2019lfo , the data
from the ongoing LBL experiments T2K and NO$\nu$A are fitted in the presence
of vector NSI. The inclusion of vector NSI has completely washed out the
preference for NH as expected in the standard scenario. In reference
Masud:2016nuj , the MH sensitivity and discovery potential are studied for the
accelerator experiments T2K and NO$\nu$A along with DUNE. Though vector NSI
has a significant impact on MH sensitivity, DUNE is still able to determine
the MH for most of the values of $\delta_{CP}$ at about 5$\sigma$ CL. The
generalized MH degeneracy in the presence of vector NSI Liao:2016hsa
deteriorates the hierarchy sensitivities of all three LBL experiments T2HK,
T2HKK and DUNE as shown in the study Liao:2016orc . In Deepthi:2016erc , the
authors have studied the effect of vector NSI on the MH sensitivities,
considering the diagonal NSI parameter $\epsilon_{ee}$ to be equal to -1 so
that it can nullify the standard matter effect. In this scenario, if
additionally $\delta_{CP}$ is $\pm\pi/2$, there will be an intrinsic
degeneracy between MH and NSI that cannot be lifted even if we can precisely
measure the values of $\epsilon_{ee}$ and $\delta_{CP}$. Apart from vector
NSI, other BSM phenomena can also affect the MH determination in LBL
experiments. In reference Dutta:2016czj , the authors have discussed how the
non-unitarity in the $\nu$-mixing matrix can affect MH determination in the
accelerator experiments DUNE, NO$\nu$A and T2K. The non-unitary mixing is
found to decrease the MH sensitivity of these experiments. Though, DUNE will
be sensitive to hierarchy discrimination even in the presence of non-unitarity
for some values of $\delta_{CP}$; its sensitivity mostly decreases. In the
case of NO$\nu$A and T2K, the scenario worsens as a result of their shorter
baseline and less matter interaction contribution. The MH sensitivity of DUNE,
T2HK and T2HKK in a $(3+1)$ scenario, with one sterile neutrino is studied in
Choubey:2017cba . The MH sensitivity deteriorates for small mixing with the
sterile neutrino, but with increasing sterile mixing the MH sensitivity
improves for all the experiments. The impact of various BSM physics scenarios
on the MH sensitivities has been well explored for various LBL
$\nu$-experiments. In this paper, we particularly probe the impact of diagonal
scalar NSI elements on the MH sensitivities in the context of upcoming LBL
experiments. We also perform a combined analysis for a chosen combination of
LBL experiments as it can lead to increased statistics and broader parameter
space. It may also help in better constraining of the oscillation parameters.

The study of scalar NSI is a well-motivated phenomenological scenario to probe
new physics searches in the leptonic sector. The possible scalar coupling of
neutrinos with environmental fermions can modify the $\nu$-oscillation
probabilities and also affect the physics sensitivities of various
$\nu$-oscillation experiments Ge:2018uhz ; Yang:2018yvk ; Khan:2019jvr ;
Gupta:2023wct . This kind of unique coupling can manifest as medium-dependent
corrections directly to the $\nu$-mass matrix. Scalar NSI can be a unique
probe to explore new physics phenomena in the neutrino sector. The effects of
scalar NSI show a linear scaling relationship with environmental matter
density, which makes LBL experiments a promising candidate for probing its
effects. The idea of scalar NSI was explored in reference Ge:2018uhz  and was
also used to fit the Borexino data. Some studies like Babu:2019iml ;
Venzor:2020ova  has put bounds on the scalar NSI parameters under
astrophysical and cosmological limits, although they are not so stringent. The
authors in reference Medhi:2021wxj ; Medhi:2022qmu  have explored the impact
of scalar NSI on the CP-measurement sensitivities at DUNE and have also
performed a combined analysis with T2HK and T2HKK. In reference Denton:2022pxt
, the authors have investigated various new physics scenarios in LBL
experiments (NOvA, T2K and DUNE) and have found that in many cases DUNE will
have better model discrimination. The authors in reference Singha:2023set
have studied the effects of scalar NSI on the determination of CP-violation,
$\theta_{23}$ octant and MH in the context of P2SO and DUNE. In reference
Medhi:2023ebi , we have constrained the lightest $\nu$-mass in the presence of
scalar NSI as it has a direct dependence on the absolute $\nu$-mass.

In this paper, we have explored the impact of scalar-mediated NSI on the
neutrino MH sensitivity at the LBL experiments i.e. DUNE, T2HK and T2HKK. We
have probed the impact of diagonal scalar NSI elements
$(\eta_{ee},\eta_{\mu\mu},\eta_{\tau\tau})$ on the MH sensitivities for the
chosen LBL experiments by taking only one element at a time. We find that the
presence of scalar NSI can significantly improve/deteriorate the MH
sensitivity of LBL experiments. A positive $\eta_{ee}$, mostly enhances the MH
sensitivity as compared to the standard case for all three experiments. In
this study, for the first time, we have performed an MH analysis in the
presence of scalar NSI with the synergy of experiments viz. DUNE+T2HK and
DUNE+T2HKK. For the combined analysis of experiments, the overall MH
sensitivities have significantly improved. It can also help in resolving
underlying degeneracy in MH determination in the presence of scalar NSI. This
is expected as the synergy of the experiments improves the statistics as well
as widens the parameter region of the analysis. Currently, most of the
$\nu$-experiments aim at determining the neutrino MH unambiguously. The
determination of neutrino MH is crucial as it would help us to understand the
nature of neutrinos and could also provide insights into the origin of
$\nu$-mass.

The paper is organized into different sections as follows. We first discuss
the formalism of scalar NSI in section 2. The methodology and experimental
details of the analysis have been described in section 3. We present our
findings, along with associated discussions, in section 4. We then summarise
the study in section 5.

##  2 Formalism of Scalar NSI

In SM, neutrinos interact with the environmental fermions by weak interactions
while passing through matter. These interactions can either be charged-current
(CC) type, mediated by a W± or neutral-current (NC) type, mediated by a Z0
boson Linder:2005fc . The CC interactions can affect $\nu$-oscillations by
introducing a flavor-dependent matter potential, whereas NC interactions are
flavor-independent and do not affect the $\nu$-oscillations. Therefore, the
effective Lagrangian for standard matter effect in $\nu$-oscillations can be
written as Wolfenstein:1977ue ; Nieves:2003in ; Nishi:2004st ; Maki:1962mu ;
Bilenky:1987ty ,

$\mathcal{L}_{CC}^{eff}=-\frac{4G_{F}}{\sqrt{2}}[\bar{\nu_{e}}(p_{3})\gamma_{\mu}P_{L}\nu_{e}(p_{2})][\bar{e}(p_{1})\gamma^{\mu}P_{L}e(p_{4})].$
(1)

In equation 1, $G_{F}$ is the Fermi coupling constant,
$P_{L}=(1-\gamma_{5})/2$ is the chiral projection operator for left-handed
neutrinos and $p_{i}$’s are momenta of incoming and outgoing fermions as well
as neutrino states. Therefore, the effective Hamiltonian for matter-effect in
$\nu$-oscillations expressed in flavor basis is,

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

where $\pm V_{SI}$ represents the matter potential, $E_{\nu}$ is neutrino
energy and M ($=\mathcal{U}D_{\nu}\mathcal{U}^{\dagger}$, $\mathcal{U}$ being
the PMNS mixing matrix Pontecorvo:1957cp ; Pontecorvo:1957qd ;
Pontecorvo:1967fh ; ParticleDataGroup:2020ssz ) is the $\nu$-mass matrix in
flavor basis, where $D_{\nu}=diag(m_{1},m_{2},m_{3})$ is the diagonal mass
matrix of neutrinos. The matrix $D_{\nu}$ can be expressed in terms of the
mass-squared differences ($\Delta m_{ij}^{2}=m_{i}^{2}-m_{j}^{2}$) to write
the effective Hamiltonian as

$\mathcal{H}_{eff}=E_{\nu}+\frac{1}{2E_{\nu}}\mathcal{U}.diag(0,\Delta
m_{21}^{2},\Delta m_{31}^{2}).\mathcal{U}^{\dagger}+diag(V_{CC},0,0).$ (3)

In BSM scenarios, the vector-mediated NSI has been extensively explored in the
past few years. However, the possibility of NSIs mediated by a scalar particle
is another intriguing scenario, which has very compelling phenomenological
consequences. The effective Lagrangian for $\nu$-fermion coupling mediated by
a scalar particle, $\phi$ having mass $m_{\phi}$ is,

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

where,

  * •

$\alpha,\beta=(e,\mu,\tau)$ refers to the three active $\nu$-flavors,

  * •

$f$($\bar{f}$) represents matter fermions (anti-fermions),

  * •

$y_{f}$ is the Yukawa coupling of scalar $\phi$ with matter fermions $f$, and,

  * •

$y_{\alpha\beta}$ is the Yukawa coupling of $\phi$ with the active
$\nu$-flavors.

The presence of Yukawa terms in equation 4 prevents us from converting it to
vector currents. Consequently, it does not manifest itself as an additional
matter potential. In contrast, the contribution of scalar NSI is a
perturbation to the mass term in the $\nu$-Hamiltonian. Thus, the presence of
scalar NSI modifies the Dirac equation as follows.

$\bar{\nu}_{\beta}\left[i\partial_{\mu}\gamma^{\mu}+\left(M_{\beta\alpha}+\frac{\sum_{f}n_{f}y_{f}y_{\alpha\beta}}{m_{\phi}^{2}}\right)\right]\nu_{\alpha}=0,$
(5)

where $n_{f}$ represents the number density of matter fermions. The effective
form of matter Hamiltonian with scalar NSI correction can be framed as,

$\mathcal{H}_{SNSI}\equiv E_{\nu}+\frac{M_{eff}M_{eff}^{\dagger}}{2E_{\nu}}\pm
V_{SI}.$ (6)

In equation 6, the effective $\nu$-mass matrix ($M_{eff}$) has contributions
from the genuine $\nu$-mass matrix (M) as well as the correction due to scalar
NSI i.e. $M_{SNSI}\equiv\sum_{f}n_{f}y_{f}y_{\alpha\beta}/m_{\phi}^{2}$. The
effective mass matrix can be expressed as $M_{eff}=M+M_{SNSI}$. The $\nu$-mass
matrix can be diagonalized by the mixing matrix $\mathcal{U}^{\prime}$.
$\mathcal{U}^{\prime}$ has the form $P\mathcal{U}Q^{\dagger}$, with P and Q
representing an unphysical rephasing matrix and the Majorana rephasing matrix
respectively. We can rotate away the Majorana rephasing matrix by
$QD_{\nu}Q^{\dagger}=D_{\nu}$, but the diagonal rephasing matrix P can not be
rotated away. Therefore, rotating the unphysical rephasing matrix P into
scalar NSI correction and denoting it as $\delta M$, we have $M_{eff}$ as,

$M_{eff}\equiv\mathcal{U}D_{\nu}\mathcal{U}^{\dagger}+P^{\dagger}M_{SNSI}P\equiv
M+\delta M,$ (7)

where,

$\delta M\equiv\sqrt{|\Delta
m_{31}^{2}|}\begin{pmatrix}\eta_{ee}&\eta_{e\mu}&\eta_{e\tau}\\\ \eta_{\mu
e}&\eta_{\mu\mu}&\eta_{\mu\tau}\\\ \eta_{\tau
e}&\eta_{\tau\mu}&\eta_{\tau\tau}\end{pmatrix}.$ (8)

We use a similar parameterization of $\delta M$ as shown in Ge:2018uhz ;
Medhi:2021wxj ; Medhi:2022qmu . Here, $\eta_{\alpha\beta}$ are dimensionless
parameters that quantify the strength of scalar NSI contribution. The
hermiticity of the Hamiltonian requires the diagonal parameters to be real and
the off-diagonal parameters to be complex. In this work, the effects of
diagonal scalar NSI parameters are explored by taking one element at once. The
form of $M_{eff}$ for the three cases is shown in the table 1.

Case | Scalar NSI parameter | ${M_{eff}}$
---|---|---
I | $\eta_{ee}$ | $\mathcal{U}\bm{\cdot}diag(m_{1},m_{2},m_{3})\bm{\cdot}\mathcal{U}^{\dagger}+\sqrt{|\Delta m_{31}^{2}|}diag(\eta_{ee},0,0)$
II | $\eta_{\mu\mu}$ | $\mathcal{U}\bm{\cdot}diag(m_{1},m_{2},m_{3})\bm{\cdot}\mathcal{U}^{\dagger}+\sqrt{|\Delta m_{31}^{2}|}diag(0,\eta_{\mu\mu},0)$
III | $\eta_{\tau\tau}$ | $\mathcal{U}\bm{\cdot}diag(m_{1},m_{2},m_{3})\bm{\cdot}\mathcal{U}^{\dagger}+\sqrt{|\Delta m_{31}^{2}|}diag(0,0,\eta_{\tau\tau})$
Table 1: Cases of $M_{eff}$ for the diagonal scalar NSI parameters.

An intriguing implication of scalar NSI is its dependence on the absolute
$\nu$-mass Medhi:2023ebi . The value of $m_{1}$ is considered to be $10^{-5}$
eV throughout this work and $m_{2}$, $m_{3}$ are calculated accordingly from
the current best-fit values of $\Delta m_{21}^{2}$ and $\Delta m_{31}^{2}$
respectively.

##  3 Methodology

We describe the methodology of our work here. We first give the details of the
simulation of the experiments in section 3.1. We then mention the details and
specifications of the three chosen LBL experiments in section 3.2.

###  3.1 Simulation Details

In this work, we mainly focused on exploring the impact of diagonal scalar NSI
parameters on the MH sensitivities at LBL experiments. We have used the
simulation package i.e. General Long Baseline Experiment Simulator (GLoBES)
Huber:2004ka ; Kopp:2006wp ; Huber:2007ji  for our numerical simulations of
the experiments. The values of oscillation parameters used in our analysis are
listed in table 2.

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: Benchmark values of $\nu$-oscillation parameters used in GLoBES
simulation NuFIT5.0 .

In order to study the sensitivity of experiments towards the determination of
MH, we define a statistical $\chi^{2}$ as

$\chi^{2}\equiv\min_{\eta}\sum_{i}\sum_{j}\frac{\left[N_{true}^{i,j}-N_{test}^{i,j}\right]^{2}}{N_{true}^{i,j}},$
(9)

where, $N_{true}^{i,j}$ ($N_{test}^{i,j}$) are the number of true (test)
events in the $\\{i,j\\}$-th bin. In our analysis, we have considered the
higher octant of $\theta_{23}$ as the true octant. In order to study the
impact of scalar NSI on the MH sensitivities, we have considered three
upcoming LBL experiments viz. DUNE, T2HK and T2HKK. The detailed experimental
setup and configuration of the detectors are discussed in the section 3.2.

###  3.2 Overview of the three LBL experiments

The experimental systematics and background information are taken from the
Technical Design Reports (TDR) of the corresponding experiments. The details
of the experiments are summarised in table 3. In the subsequent sections, we
briefly discuss the technical details of the three future LBL
$\nu$-experiments viz. DUNE, T2HK, and T2HKK.

####  3.2.1 DUNE

The Deep Underground Neutrino Experiment (DUNE) DUNE:2016hlj ; DUNE:2015lol ;
DUNE:2016rla ; DUNE:2020ypp ; DUNE:2021tad  is a flagship accelerator-based
$\nu$-experiment to be hosted by Fermilab. The Long-Baseline Neutrino Facility
(LBNF) of Fermilab which can deliver $1.1\times 10^{21}$ proton on target
(POT) per year with its 1.2 MW proton beam will produce the neutrinos for the
experiment. It will have two detectors, the near detector at Fermilab will be
situated 60m underground and will have a baseline of 574 meters. The far
detector with a 1300 km baseline will be situated 1.5 km below the earth’s
surface at the Sanford Underground Research Facility, South Dakota. The far
detector module will consist of four Liquid Argon Time Projection Chambers
(LArTPC) each having a fiducial mass of 10 kt. The excellent energy
reconstruction, 3D image tracking and particle identification capabilities of
LArTPC technology will provide exceptional precision to DUNE in probing
neutrino properties.

Experiment details | Channels | Normalization error
---|---|---
Signal | Background
DUNE |  |  |
Baseline = 1300 km |  $\nu_{e}(\bar{\nu}_{e})$ appearance | 2 % (2%) | 5 % (5 %)
L/E = 1543 km/GeV |  |  |
Fiducial mass = 40 kt (LArTPC) |  $\nu_{\mu}(\bar{\nu}_{\mu})$ disappearance | 5 % (5 %) | 5 % (5 %)
Runtime = 3.5 yr $\nu+3.5$ yr $\bar{\nu}$ |  |  |
T2HK |  |  |
Baseline = 295 km |  $\nu_{e}(\bar{\nu}_{e})$ appearance | 3.2 % (3.9 %) | 10 % (10 %)
L/E = 527 km/GeV |  |  |
Fiducial mass = $187\times 2$ kt (WC) |  $\nu_{\mu}(\bar{\nu}_{\mu})$ disappearance | 3.6 % (3.6 %) | 10 % (10 %)
Runtime = 2.5 yr $\nu+7.5$ yr $\bar{\nu}$ |  |  |
T2HKK |  |  |
Baseline = 295, 1100km |  $\nu_{e}(\bar{\nu}_{e})$ appearance | 3.2 % (3.9 %) | 10 % (10 %)
L/E = 527, 1964 km/GeV |  |  |
Fiducial mass = 187, 187 kt(WC) |  $\nu_{\mu}(\bar{\nu}_{\mu})$ disappearance | 3.6 % (3.6 %) | 10 % (10 %)
Runtime = 2.5 yr $\nu+7.5$ yr $\bar{\nu}$ |  |  |
Table 3: Detector details and systematic uncertainties for DUNE, T2HK and
T2HKK.

####  3.2.2 T2HK

Tokai to Hyper-Kamiokande (T2HK) Hyper-KamiokandeProto-:2015xww  will be an
LBL counterpart of the Hyper-Kamiokande (Hyper-K) experiment. Hyper-K is the
upgradation of the successful Super-Kamiokande (Super-K) experiment with Water
Cherenkov (WC) detectors having 16.8 times larger mass than Super-K. It will
use neutrinos produced by the 1.2 MW beam at the J-PARC beam facility in
Japan. It can produce 27$\times 10^{21}$ POT per year. The neutrinos will be
detected by the HyperK detectors situated 2.5∘ off-axis from the beam at the
baseline of 295 km. Each of the two WC detectors of the experiment has a
fiducial mass of 187kt.

####  3.2.3 T2HKK

T2HKK (Tokai to Hyper-Kamiokande to Korea) Hyper-Kamiokande:2016srs  is an
extension of Hyper-K. The second detector for this experiment is placed in
Korea at a distance of 1100 km from the J-PARC facility. With this setup,
T2HKK will consist of two WC detectors, one 187kt detector at 295 km at the
Hyper-K site, and another identical detector in Korea. The 2.5∘ off-axis beam
at a baseline of T2HKK will enable us to study the oscillations at second
oscillations maxima at an energy $\sim 0.6eV$.

##  4 Results and Discussion

We discuss the impact of diagonal scalar NSI elements ($\eta_{ee}$,
$\eta_{\mu\mu}$, $\eta_{\tau\tau}$) on the appearance $\nu$-oscillation
probability ($P_{\mu e}$), for NH and IH, and how it affects the MH
sensitivities in LBL experiments (DUNE, T2HK, T2HKK). The presence of scalar
NSI leads to a correction in the $\nu$-mass matrix which can modify values of
$P_{\mu e}$. In section 4.1, we first explore the impact of diagonal scalar
NSI parameters on $P_{\mu e}$ for varied neutrino energies. We also study its
effects on the MH-asymmetry parameter to further investigate its impact on MH
determination at the probability level in section 4.2. The impact of scalar
NSI on the MH sensitivities is presented in section 4.3. The MH sensitivities
for the synergy of DUNE with T2HK and T2HKK in the presence of scalar NSI are
shown in section 4.4. In section 4.5, we explore the $\Delta m_{31}^{2}$
constraining capability of the experiments in presence of scalar NSI.

###  4.1 Effects on the appearance channel $P_{\mu e}$

The impact of diagonal scalar NSI elements ($\eta_{ee}$, $\eta_{\mu\mu}$,
$\eta_{\tau\tau}$) on $P_{\mu e}$ for varying neutrino energy (E) at baselines
corresponding to DUNE, T2HK and T2HKK are shown in figure 1, 2 and 3
respectively. We observe that,

Figure 1: $P_{\mu e}$ vs E at the DUNE-baseline of 1300km in the presence of
scalar NSI, for NH (red band) and IH (blue band). Both the bands correspond to
$\delta_{CP}$ values $\in[-\pi,\pi]$. The panels correspond to no scalar NSI
(top-left), $\eta_{ee}$ (top-right), $\eta_{\mu\mu}$ (bottom-left) and
$\eta_{\tau\tau}$ (bottom-right).

  * •

For no scalar NSI case, the DUNE baseline can clearly distinguish between the
NH and IH in the energy range $\sim$ [1.5, 4] GeV. The presence of $\eta_{ee}$
widens the energy range as well as the separation between the hierarchies. The
$\eta_{\mu\mu}$ element marginally reduces the separation between the
hierarchies and the energy window of the hierarchy separation. For
$\eta_{\tau\tau}$ element, the overlapping of bands makes the experiment
insensitive to the neutrino MH.

  * •

For T2HK baseline, we observe an overlap between the NH and IH bands for the
no scalar NSI case. This can create degeneracy in discriminating the neutrino
MH. The presence of $\eta_{ee}$ can lift the degeneracy in the energy range
$\sim$ [0.4, 0.7] GeV. The element $\eta_{\mu\mu}$ shows a nominal change
compared to the no scalar NSI case. For $\eta_{\tau\tau}$, the IH band lies
above the NH band without any distinct separation between the hierarchies.

  * •

For T2HKK baseline, we observe a clear discrimination between the hierarchies
at the probability level in the energy window $\sim$ [1.5, 3] GeV in the
absence of scalar NSI. The presence of $\eta_{ee}$ element widens the band
separation as well as the energy window. The element $\eta_{\mu\mu}$ reduces
both the amplitude as well as the discriminating power of the neutrino MH. For
$\eta_{\tau\tau}$, both the NH and IH band completely overlap providing no
sensitivity to MH.

Figure 2: $P_{\mu e}$ vs E at the T2HK-baseline of 295km in the presence of
scalar NSI, for NH (red band) and IH (blue band). Both the bands correspond to
$\delta_{CP}$ values $\in[-\pi,\pi]$. The panels correspond to no scalar NSI
(top-left), $\eta_{ee}$ (top-right), $\eta_{\mu\mu}$ (bottom-left) and
$\eta_{\tau\tau}$ (bottom-right).

Figure 3: $P_{\mu e}$ vs E at the T2HKK-baseline of 1100km in the presence of
scalar NSI, for NH (red band) and IH (blue band). Both the bands correspond to
$\delta_{CP}$ values $\in[-\pi,\pi]$. The panels correspond to no scalar NSI
(top-left), $\eta_{ee}$ (top-right), $\eta_{\mu\mu}$ (bottom-left) and
$\eta_{\tau\tau}$ (bottom-right).

###  4.2 Imprints on the MH asymmetry

In order to quantify the effects of scalar NSI on the MH determination at the
probability level, we define an asymmetry parameter as

$A_{MH}=\frac{P_{\mu e}^{NH}-{P}_{\mu e}^{IH}}{P_{\mu e}^{NH}+{P}_{\mu
e}^{IH}}\,,$ (10)

where, $P_{\mu e}^{NH}$ and ${P}_{\mu e}^{IH}$ are the appearance
probabilities for NH and IH respectively. This asymmetry parameter may
quantify an experiment’s capability towards discriminating between the two
$\nu$-mass orderings.

In figure 4, we show the deviation of $A_{MH}$ as a function of $\delta_{CP}$
for the baselines of DUNE (left–panel), T2HK (middle–panel) and T2HKK
(right–panel) respectively. We have chosen the neutrino energies as their
respective peak energies i.e. 2.5 GeV (DUNE), 0.5 GeV (T2HK) and 0.66 GeV
(T2HKK). We note the following.

  * •

At DUNE, the presence of positive (negative) $\eta_{ee}$ can enhance
(suppress) $A_{MH}$ as compared to no scalar NSI case. The effect of
$\eta_{\mu\mu}$ is nominal on $A_{MH}$. A positive value of the
$\eta_{\tau\tau}$ suppresses $A_{MH}$, while a negative $\eta_{\tau\tau}$
nominally enhances $A_{MH}$.

  * •

For no scalar NSI, T2HK has nominal sensitivity towards the neutrino MH as
compared to DUNE. The presence of positive (negative) $\eta_{ee}$ can enhance
(suppress) $A_{MH}$. The $\eta_{\mu\mu}$ element shows a similar effect on
$A_{MH}$ as compared to no scalar NSI case. However, a negative (positive)
$\eta_{\tau\tau}$ enhances (suppresses) $A_{MH}$.

  * •

For T2HKK, we see a suppression in $A_{MH}$ for positive values of
$\eta_{\tau\tau}$ in the range $\delta_{CP}$ $\sim$ [$40^{\circ}$,
$180^{\circ}$]. A negative $\eta_{ee}$, mostly suppresses $A_{MH}$ except in
the range $\delta_{CP}$ $\sim$ [$0^{\circ}$, $90^{\circ}$]. A negative
$\eta_{\mu\mu}$ mostly suppresses $A_{MH}$, while a positive $\eta_{\mu\mu}$
enhances $A_{MH}$ in the range $\delta_{CP}$ $\sim$ [$-70^{\circ}$,
$60^{\circ}$]. The positive $\eta_{\tau\tau}$ reduces the experiment’s
sensitivity towards MH for the complete $\delta_{CP}$ parameter space.

We observe that the presence of diagonal scalar NSI elements leads to a
significant modification in the values of $P_{\mu e}$. As a result, it can
substantially enhance/hamper the MH sensitivities of these LBL experiments.

Figure 4: Impact of scalar NSI on $A_{MH}$ at DUNE (left–panel), T2HK
(middle–panel) and T2HKK (right–panel) baselines. The red solid line
corresponds to no scalar NSI case. The solid (dashed) lines correspond to
positive (negative) diagonal scalar NSI parameters as indicated in the
legends.

###  4.3 Impact on the MH sensitivities

One of the major goals of the current and upcoming LBL $\nu$-oscillation
experiments is the determination of neutrino MH. The impact of scalar NSI on
the MH sensitivities at three different LBL experiments i.e. DUNE, T2HK and
T2HKK is explored here. We have considered two different cases in our analysis
i.e.,

  * •

Considering NH as the true hierarchy for which $\Delta\chi^{2}$ can be defined
as,

$\displaystyle\Delta\chi^{2}_{\text{MH}}$ $\displaystyle=$
$\displaystyle\chi^{2}_{\text{NH}}-\chi^{2}_{\text{IH}}~{}~{}~{}~{}(\text{for
true NH}).$ (11)

  * •

Considering IH as the true hierarchy for which $\Delta\chi^{2}$ can be defined
as,

$\displaystyle\Delta\chi^{2}_{\text{MH}}$ $\displaystyle=$
$\displaystyle\chi^{2}_{\text{IH}}-\chi^{2}_{\text{NH}}~{}~{}~{}~{}(\text{for
true IH}).$ (12)

To obtain the MH sensitivities, the statistical significance which is defined
as $\sigma=\sqrt{\Delta\chi^{2}}$ is plotted for varying true $\delta_{CP}$ in
[-$\pi$, $\pi$]. The other $\nu$-oscillation parameters are fixed at values
listed in table 2. We have marginalized over the test hierarchy as well as
$\delta_{CP}$ in [-$\pi$, $\pi$] and $\theta_{23}$ in [$40^{\circ}$,
$50^{\circ}$]. We have also additionally marginalized over the scalar NSI
parameters and the systematic uncertainties.

Figure 5: The MH sensitivity of DUNE, in the presence of scalar NSI, for true
NH (top–panel) and true IH (bottom–panel). The effects of diagonal scalar NSI
parameters $\eta_{ee}$, $\eta_{\mu\mu}$ and $\eta_{\tau\tau}$ are shown in
left, middle and right panels respectively. The solid red line is for
no–scalar NSI case. The solid (dashed) lines are for non-zero positive
(negative) scalar NSI parameters.

In figure 5, we show the effects of scalar NSI on the MH sensitivity of DUNE
considering NH (IH) as the true hierarchy in the top–panel (bottom–panel). The
solid red line corresponds to the case with no scalar NSI effects. The solid
(dashed) lines correspond to the positive (negative) values of scalar NSI
parameters. The dashed magenta and green lines represent the 5$\sigma$ and
3$\sigma$ CL respectively. We observe that,

  * •

For true NH, positive (negative) values of $\eta_{ee}$ and $\eta_{\mu\mu}$
elements enhance (suppress) the MH sensitivities as compared to no scalar NSI
case. For $\eta_{\tau\tau}=0.2$, the sensitivities show an enhancement only in
the lower half plane of $\delta_{CP}\sim[-180^{\circ},0^{\circ}]$, while for
$\eta_{\tau\tau}=0.1$, the sensitivities lie below the no scalar NSI case for
the whole $\delta_{CP}$ parameter space. For the chosen negative values of
$\eta_{\tau\tau}$, the sensitivities lie mostly below the no scalar NSI case.

  * •

For true IH, positive (negative) values of $\eta_{ee}$ and $\eta_{\tau\tau}$,
enhance (suppress) the MH sensitivities as compared to when there is no scalar
NSI present. However, for a positive (negative) $\eta_{\mu\mu}$ value the
sensitivities get mostly suppressed (enhanced).

Figure 6: The MH sensitivity of T2HK, in the presence of scalar NSI, for true
NH (top–panel) and true IH (bottom–panel). The effects of diagonal scalar NSI
parameters $\eta_{ee}$, $\eta_{\mu\mu}$ and $\eta_{\tau\tau}$ are shown in
left, middle and right panels respectively. The solid red line is for
no–scalar NSI case. The solid (dashed) lines are for non-zero positive
(negative) scalar NSI parameters.

In figure 6, we present the MH sensitivities of T2HK in the presence of
diagonal scalar NSI parameters $\eta_{ee}$ (left–panel), $\eta_{\mu\mu}$
(middle–panel) and $\eta_{\tau\tau}$ (right–panel) assuming NH (IH) as true
hierarchy in the top–panel (bottom–panel). The no scalar NSI case is depicted
by the red line, whereas the positive (negative) scalar NSI parameters are
shown by the solid (dashed) lines. The 5$\sigma$ (3$\sigma$) CL are
represented by the horizontal magenta (green) lines. The observations made are
listed below.

  * •

For true NH, a positive value of the $\eta_{\alpha\beta}$ element mostly
enhances the sensitivity towards MH determination. Even with
$\eta_{\alpha\beta}=0.1$, above $5\sigma$ sensitivity is observed in the
negative half-plane of $\delta_{CP}\sim[-180^{\circ},0^{\circ}]$. For most
values of $\delta_{CP}$, the negative value of scalar NSI parameters also
enhances the sensitivities as compared to no scalar NSI case. However, for
some combinations of $\delta_{CP}$ and negative $\eta_{\alpha\beta}$, a
suppression in the sensitivity is observed.

  * •

For true IH, a significant enhancement in the hierarchy sensitivity is
observed only for $\eta_{\alpha\beta}=0.2$, particularly for $\eta_{\mu\mu}$
and $\eta_{\tau\tau}$. For $\eta_{\mu\mu}=0.1$, the sensitivity curves with
and without scalar NSI cases are seen to nearly overlap for the entire range
of $\delta_{CP}$ values.

Figure 7: The MH sensitivity of T2HKK, in the presence of scalar NSI, for true
NH (top–panel) and true IH (bottom–panel). The effects of diagonal scalar NSI
parameters $\eta_{ee}$, $\eta_{\mu\mu}$ and $\eta_{\tau\tau}$ are shown in
left, middle and right panels respectively. The solid red line is for
no–scalar NSI case. The solid (dashed) lines are for non-zero positive
(negative) scalar NSI parameters.

In figure 7, we have shown the effects of diagonal scalar NSI parameters on
the MH sensitivities at T2HKK. The top (bottom) panel corresponds to MH
sensitivity for true NH (IH) with non-zero $\eta_{ee}$ (left–panel),
$\eta_{\mu\mu}$ (middle–panel) and $\eta_{\tau\tau}$ (right–panel). The red
solid line is in the absence of scalar NSI, whereas, the black and blue lines
correspond to non-zero scalar NSI parameters. The solid (dashed) lines
correspond to positive (negative) values of scalar NSI parameters. The
$5\sigma$ (dashed magenta) and $3\sigma$ (dashed green) CL are drawn as a
reference. The following are the observations from figure 7.

  * •

For NH as the true hierarchy, the sensitivity is enhanced for both positive
and negative values of scalar NSI parameters. In general, with scalar NSI we
get above $5\sigma$ MH sensitivity except for the case when $\eta_{\mu\mu}$
has the value of $-0.1$ and $\delta_{CP}$ falls within the range of
$[-120^{\circ},-60^{\circ}]$.

  * •

For IH as the true hierarchy, a substantial enhancement in sensitivity is
observed only when $\eta_{\mu\mu}$ and $\eta_{\tau\tau}$ is equal to 0.2. The
effect of scalar NSI, for all other cases, is nominal. Although, certain
values $\eta_{ee}$ can improve the sensitivity beyond 5$\sigma$ for the
positive half plane of $\delta_{CP}$ i.e. $\delta_{CP}\sim[0,180^{\circ}]$.

###  4.4 The MH sensitivity for the Synergy of the LBL experiments

We here show the effects of scalar NSI on the MH sensitivities after combining
the data from two LBL experiments viz. DUNE+T2HK (in section 4.4.1) and
DUNE+T2HKK (in section 4.4.2) respectively.

####  4.4.1 Combining DUNE with T2HK

Figure 8: The impact of diagonal scalar NSI parameters on the MH sensitivities
for DUNE+T2HK for true NH (top–panel) and IH (bottom–panel) in the presence of
diagonal scalar NSI parameters: $\eta_{ee}$ (left–panel), $\eta_{\mu\mu}$
(middle–panel) and $\eta_{\tau\tau}$ (right–panel). The solid red line is for
no–scalar NSI case. The solid (dashed) lines are for non-zero positive
(negative) scalar NSI parameters.

In figure 8, the neutrino MH sensitivity of the combination of DUNE and T2HK
in the presence of scalar NSI is shown. The top panel represents the true NH,
whereas the bottom panel represents the true IH case. The non-zero positive
(negative) values of diagonal scalar NSI parameters are depicted by the blue
and black solid (dashed) curves. The red solid curve corresponds to the no
scalar NSI case, with $5\sigma$ and $3\sigma$ CL represented by dashed magenta
and green lines respectively. The observations from figure 8 are,

  * •

In case of DUNE+T2HK, the MH sensitivity is above $5\sigma$ CL for all values
of $\delta_{CP}$. In presence of scalar NSI, for the true NH case, positive
(negative) $\eta_{ee}$ and $\eta_{\mu\mu}$ mostly improve (deteriorate) the MH
sensitivity. However, for $\eta_{\tau\tau}$, the negative value of scalar NSI
parameters also enhances the sensitivity towards MH determination in
$\delta_{CP}$ range $[-30^{\circ},100^{\circ}]$.

  * •

For IH as the true hierarchy, the positive (negative) $\eta_{ee}$ and
$\eta_{\tau\tau}$ enhances (suppresses) MH sensitivity for DUNE+T2HK. The
effect is more prominent for $\eta_{\tau\tau}$ as compared to $\eta_{ee}$. For
$\eta_{\mu\mu}=0.2$, the sensitivity is enhanced, whereas $\eta_{\mu\mu}=0.1$
suppresses the MH sensitivity. In the case of negative $\eta_{\mu\mu}$, the
sensitivity curves largely overlap across various values of $\delta_{CP}$.

####  4.4.2 Combining DUNE with T2HKK

Figure 9: The impact of scalar NSI on MH sensitivities for DUNE+T2HKK in the
presence of diagonal scalar NSI parameters: $\eta_{ee}$ (left–panel),
$\eta_{\mu\mu}$ (middle–panel) and $\eta_{\tau\tau}$ (right–panel). The
top–panel corresponds to true NH and the bottom–panel corresponds to true IH.
The solid red line is for no–scalar NSI case. The solid (dashed) lines are for
non-zero positive (negative) scalar NSI parameters.

The MH sensitivity of DUNE$+$T2HKK for true NH (top-panel) and true IH
(bottom-panel) is shown in figure 9. The red solid line represents the case
without scalar NSI and other coloured lines represent non-zero values of
$\eta_{ee}$ (left–panel), $\eta_{\mu\mu}$ (middle–panel) and $\eta_{\tau\tau}$
(right–panel) as indicated in the legends. The $5\sigma$ and $3\sigma$ CL are
drawn in magenta and green colours. We observe that,

  * •

For true NH, a positive (negative) $\eta_{ee}$ and $\eta_{\mu\mu}$ enhances
(suppresses) the MH sensitivities. For negative $\eta_{\tau\tau}$, the MH
sensitivities are mostly suppressed except in the region of $\delta_{CP}$
$\in$ [$-40^{\circ},100^{\circ}$].

  * •

For true IH, the effect of $\eta_{\tau\tau}$ is prominent as compared to
$\eta_{ee}$ and $\eta_{\mu\mu}$. A negative value of $\eta_{\alpha\beta}$
mostly suppresses the sensitivity in all the cases.

  * •

In both the scenarios of true NH and true IH, we observe a similar trend to
those of the DUNE+T2HK configuration. This similarity is expected since the
detectors used in both setups are identical. However, in the case of
DUNE+T2HKK, the effects are more prominent due to its longer baseline, as
scalar NSI varies linearly with the environmental matter density.

###  4.5 Precision measurement of $\Delta m_{31}^{2}$

We look into the impact of diagonal scalar NSI elements $\eta_{ee}$,
$\eta_{\mu\mu}$ and $\eta_{\tau\tau}$ on the precision measurements of $\Delta
m_{31}^{2}$ at DUNE, T2HK and T2HKK. We have also performed combined analyses
for DUNE+T2HK and DUNE+T2HKK. We present the results in figure 10.

The magenta and green dashed lines represent the $5\sigma$ and $3\sigma$ CL
respectively. The red solid line represents the DUNE case. In the top (bottom)
panel, the black and blue solid lines represent the case T2HK (T2HKK) and
DUNE+T2HK (DUNE+T2HKK) respectively. We have considered the higher
$\theta_{23}$ octant as the true octant and the NH as the true hierarchy. The
true value of $\Delta m_{31}^{2}$ is fixed at 2.55$\times 10^{-3}eV^{2}$ and
the test value is varied in $[2.42,2.59]\times 10^{-3}eV^{2}$. The value of
$\eta_{\alpha\beta}$ is fixed at 0.15 for all the experimental configurations.
The findings obtained are stated below.

Figure 10: The impact of scalar NSI on $\Delta m_{31}^{2}$ constraining
capability in the presence of diagonal scalar NSI parameters $\eta_{ee}$
(left–panel), $\eta_{\mu\mu}$ (middle–panel) and $\eta_{\tau\tau}$
(right–panel) for true NH. The top–panel corresponds to DUNE (red), T2HK
(black) and the combined analysis of DUNE+T2HK (blue), whereas the
bottom–panel corresponds to DUNE (red), T2HKK (black) and DUNE+T2HKK (blue).

  * •

In the presence of $\eta_{ee}$, the $\Delta m_{31}^{2}$ constraining
capability of DUNE+T2HK and DUNE+T2HKK configuration is better compared to
DUNE, T2HK and T2HKK. The synergies lead to a better constraining of $\Delta
m_{31}^{2}$ in the presence of $\eta_{ee}$.

  * •

In the presence of $\eta_{\mu\mu}$, we observe a significant second dip
roughly around $\Delta m_{31}^{2}\sim 2.48\times 10^{-3}eV^{2}$ for all the
considered experimental configurations. In Appendix 6.1, we have explored the
variation of $\Delta P_{\mu e}$ in the $\Delta m_{31}^{2}$-energy parameter
space to further study the degeneracy in $\Delta m_{31}^{2}$-measurement in
the presence of scalar NSI.

  * •

The synergy configurations show better constraining capability in the presence
of $\eta_{\tau\tau}$ but a second dip around $\Delta m_{31}^{2}\sim 2.5\times
10^{-3}eV^{2}$ is also seen.

##  5 Summary and concluding remarks

The upcoming $\nu$-oscillation experiments are centered on achieving precise
measurements of $\nu$-mixing parameters and addressing the unknowns in the
field of neutrino physics like the determination of neutrino MH, octant of
$\theta_{23}$ and the leptonic CP phase. The determination of the true
neutrino MH is crucial as it holds the potential to significantly enhance our
understanding of the fundamental nature of neutrinos. The existence of NSIs of
neutrinos has the potential to influence the physics sensitivities of these
LBL $\nu$-experiments. The potential scalar coupling of neutrinos is one such
non-standard physics scenario which may affect the neutrino MH sensitivities.
Scalar NSIs of neutrinos bring a direct contribution to the $\nu$-mass matrix
which is sub-dominant in nature.

In this work, we have explored the effects of diagonal scalar NSI parameters
on the neutrino MH sensitivities at three upcoming LBL experiments i.e. DUNE,
T2HK and T2HKK. We also explored the potential improvement in the MH
sensitivities in the presence of diagonal scalar NSI parameters for the
synergy of DUNE+T2HK and DUNE+T2HKK. We mainly focused on the impact of the
presence of scalar coupling of neutrinos on the standard MH sensitivities. We
observe that the presence of scalar NSI can significantly alter the
$\nu$-oscillation probabilities which in turn affects the sensitivities of LBL
experiments toward MH determination. At the probability level, we observe that
the discrimination between the mass hierarchies is enhanced in the presence of
scalar NSI parameter $\eta_{ee}$ for all the experiments. For true NH, the
presence of positive diagonal scalar NSI parameters mostly enhances the
sensitivities. For certain combinations of negative $\eta_{\alpha\beta}$ and
$\delta_{CP}$, the sensitivity appears to deteriorate. We also observe a
significant overlapping of sensitivities for the standard case with the scalar
NSI case. However, the combined analysis of DUNE+T2HK and/or DUNE+T2HKK can
remove the overlapping for $\eta_{\tau\tau}$ parameter due to the increased
statistics over an enhanced parameter region. In presence of scalar NSI, we
observe that T2HK and T2HKK show better constraining capability of $\Delta
m_{31}^{2}$ in comparison to DUNE. The combined analyses for DUNE+T2HK and
DUNE+T2HKK can enhance the overall constraining capability for all non-zero
scalar NSI parameters.

The neutrino mass hierarchy stands as one of the fundamental mysteries in
particle physics, the determination of which would mark a significant
breakthrough in our understanding of the universe. Therefore, understanding
the impact of subdominant effects arising from scalar NSIs on the MH
sensitivities of LBL experiments is imperative. It is also crucial to impose
constraints on the scalar NSI parameters to ensure accurate interpretation of
experimental data.

## Acknowledgments

The authors acknowledge the DST SERB grant CRG/2021/002961. AS would also
acknowledge the fellowship received from CSIR-HRDG
(09/0796(12409)/2021-EMR-I). AM thanks the support of the Research and
Innovation Grant 2021 (DoRD/RIG/10-73/ 1592-A) funded by Tezpur University. DB
acknowledges DST for the financial support provided through the INSPIRE
fellowship.

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##  6 Appendix

###  6.1 Variation of $\Delta P_{\mu e}$ in $\Delta m_{31}^{2}$-E parameter
space

In order to quantify the impact of scalar NSI, we define a quantity $\Delta
P_{\mu e}$ as

$\Delta P_{\mu e}=P_{\mu e}^{NSI}-P_{\mu e}^{SI}$

where, $P_{\mu e}^{NSI}$ and $P_{\mu e}^{SI}$ represent the appearance
probabilities in the presence of scalar NSI and no scalar NSI cases
respectively. We have plotted $\Delta P_{\mu e}$ for varying neutrino energy
and $\Delta m_{31}^{2}$ for true NH. We have shown the variation in presence
of scalar NSI for all the considered experiments and the combined analysis as
well. We consider the diagonal scalar NSI parameters i.e. $\eta_{ee}$
(left–panel), $\eta_{\mu\mu}$ (middle–panel) and $\eta_{\tau\tau}$
(right–panel), one element a time. In figure 11, 12 and 13, we have shown the
variation of $\Delta P_{\mu e}$ at DUNE, T2HK and T2HKK respectively. The
plots for the combined analysis DUNE+T2HK and DUNE+T2HKK are shown in figures
14 and 15. In all the figures, we have fixed the value of $\eta_{\alpha\beta}$
at 0.15.

Figure 11: Variation of $\Delta P_{\mu e}$ in $\Delta m_{31}^{2}$-E parameter
space in the presence of scalar NSI elements $\eta_{ee}$ (left–panel),
$\eta_{\mu\mu}$ (middle–panel) and $\eta_{\tau\tau}$ (right–panel) for DUNE.

We observe that for higher values of neutrino energies, the value of $\Delta
P_{\mu e}$ changes in the complete $\Delta m_{31}^{2}$ range for DUNE, T2HK
and T2HKK. This shows the possible degeneracy arising in the $\Delta
m_{31}^{2}$ measurement in the presence of scalar NSI. In the case of DUNE,
the same trend is seen in the presence of all scalar NSI parameters.

In case of T2HK & T2HKK, the impact on $\Delta P_{\mu e}$ is more prominent in
the presence of $\eta_{ee}$ in comparison to $\eta_{\mu\mu}$ and
$\eta_{\tau\tau}$. For a fixed value of energy, $\Delta P_{\mu e}$ remains
almost the same for the whole $\Delta m_{31}^{2}$ range. In case of the
synergy of DUNE with T2HK & T2HKK, we observe a significant impact of scalar
NSI on $\Delta P_{\mu e}$ for all the diagonal scalar NSI elements.

Figure 12: Variation of $\Delta P_{\mu e}$ in $\Delta m_{31}^{2}$-E parameter
space in the presence of scalar NSI elements $\eta_{ee}$ (left–panel),
$\eta_{\mu\mu}$ (middle–panel) and $\eta_{\tau\tau}$ (right–panel) for T2HK.

Figure 13: Variation of $\Delta P_{\mu e}$ in $\Delta m_{31}^{2}$-E parameter
space in the presence of scalar NSI elements $\eta_{ee}$ (left–panel),
$\eta_{\mu\mu}$ (middle–panel) and $\eta_{\tau\tau}$ (right–panel) for T2HKK.

Figure 14: Variation of $\Delta P_{\mu e}$ in $\Delta m_{31}^{2}$-E parameter
space in the presence of scalar NSI elements $\eta_{ee}$ (left–panel),
$\eta_{\mu\mu}$ (middle–panel) and $\eta_{\tau\tau}$ (right–panel) for
DUNE+T2HK.

Figure 15: Variation of $\Delta P_{\mu e}$ in $\Delta m_{31}^{2}$-E parameter
space in the presence of scalar NSI elements $\eta_{ee}$ (left–panel),
$\eta_{\mu\mu}$ (middle–panel) and $\eta_{\tau\tau}$ (right–panel) for
DUNE+T2HKK.