# Sensitivity of octant of $\theta_{23}$, CP violation and mass hierarchy in
NO$\nu$A with multinucleon and detector effects

Paramita Deka 111E-mail<EMAIL_ADDRESS>Kalpana Bora
<EMAIL_ADDRESS>Department Of Physics, Gauhati University, Assam,
India

###### Abstract

In this work, we investigate how multinucleon enhancement and RPA (Random
Phase Approximation) suppression can affect the measurement of three unknown
neutrino oscillation parameters - the CP-violating phase $\delta_{CP}$, the
octant of the atmospheric mixing angle $\theta_{23}$, and the determination of
the mass hierarchy, in the appearance channel of the NO$\nu$A experiment. We
include the presence of the detector effect as well in the analysis, which is
crucial for capturing realistic experimental scenarios. It is found that the
analysis using our comprehensive model (QE(+RPA)+2p2h) exhibits significantly
enhanced sensitivity compared to the pure QE interaction process, in all the
cases. Also, the higher octant of $\theta_{23}$, the lower half plane of
$\delta_{CP}$, and the normal mass hierarchy (HO-LHP-NH) exhibit improved
sensitivity, enabling a more precise determination of the corresponding
parameters. Furthermore, it is also noted that improving the performance of
the detector also improves the results. Thus, including multinucleon effects
and improving detector efficiency have the potential to enhance the
capabilities of the NO$\nu$A (and other long baseline) experiment in
conducting precise parameter studies.

Keywords: Multinucleon Effects, Octant Degeneracy, CP-Violating Phase, Mass
Hierarchy.

##  I Introduction

The currently unknown parameters in neutrino oscillations are - the mass
hierarchy, the octant of the mixing angle $\theta_{23}$, and the CP violating
phase $\delta_{CP}$, and it is widely acknowledged that the presence of
degeneracies related to the mass hierarchy-$\delta_{CP}$-octant of
$\theta_{23}$ can influence the determination of these parameters. Detecting
CP violation in the neutrino sector is a challenging objective for neutrino
experiments. Strong evidence of CP violation could be demonstrated by
observing an asymmetry in the oscillation rates of muon neutrinos and
antineutrinos that undergo a transition into electron neutrinos as in the
absence of CP violation, the rates for both are expected to be equal. To
conduct this test, both muonic neutrino and antineutrino beams are required.
In the context of neutrino interactions, CP violation can occur in the weak
force that governs the interaction of neutrinos with matter, and in neutrino
oscillation experiments, the phenomenon of CP violation can be observed
through differences in the oscillation probabilities between neutrinos and
antineutrinos. The discovery of CP violation in neutrino oscillations could
help explain the observed matter-antimatter asymmetry in the universe. Modern
neutrino experiments do not use free nucleons as the primary target, instead,
they use heavy nuclear targets (like carbon, oxygen, argon, etc.), in which
complexities regarding nuclear effects are unavoidable. Nuclear effects play
an important role and contribute significantly to the incorrect estimation of
neutrino energy as they lead to a significant amount of missing energy during
Final State Interactions (FSIs). Among all the uncertainties, nuclear effects
are considered one of the largest sources of systematic uncertainties in the
oscillation analysis of long-baseline (LBL) experiments. Nuclear effects can
be broadly divided into two categories - initial-state and final-state
effects. Initial-state effects affect the nucleon before the neutrino
interactions while hadrons produced by final-state effects influence the
outgoing final-state particles before their exit from the nucleus. A true
charged current (CC) quasielastic (QE) process is represented as
$\nu_{\mu}n\rightarrow\mu^{-}p$. These true interaction processes are
accompanied by some other processes where the outgoing proton re-interacts
inside the nucleus thus producing $\Delta$ resonance. This $\Delta$ then
decays to produce a pion which is then absorbed in the nucleus through FSIs.
This can be represented as: $\nu_{\mu}p\rightarrow\mu^{-}p\pi^{+}$ or
$\nu_{\mu}p\rightarrow\mu^{-}\Delta^{++}$. Thus the absence of the pion in the
final state (called ”stuck pion”) leads to missing energy and appears as a QE-
like event. Therefore because of FSI, a non-QE even may be wrongly identified
as QE Deka:2022idq . A second complication arises due to the presence of
multinucleon events in which the incoming neutrino interacts with, e.g, two
nucleons (so-called 2p2h) Deka:2021qnw ; Deka:2022lvy ; Martini:2011wp ;
Martini:2012uc ; Martini:2012fa ; Benhar:2013bwa ; Lalakulich:2012ac . The
neutrino energy reconstructed in such events differs significantly from their
true energy value. Though pion production contributes to the background in any
QE process, later it has been shown that 2p2h excitations and some other
processes also shift the reconstructed energy towards lower energy bins. In
MiniBooNE and K2K experiments, it was found that QE contains about 30%
contribution from 2p2h events Martini:2009uj ; Martini:2011wp ; Martini:2010ex
. The presence of multinucleon effects modifies the oscillation probabilities
and measurements of oscillation parameters and hence can affect the
measurements of CP-violating parameters too. Therefore, it is important to
understand and properly account for the multinucleon effects in the analysis
of neutrino-nucleus interactions. This requires the development of accurate
theoretical models of nuclear structure and dynamics, as well as high-
precision experimental measurements of neutrino-nucleus cross-sections.

The standard three-flavor neutrino oscillation probability, as expressed in
Eq. 1, depends on six parameters: $\theta_{12}$, $\theta_{23}$, $\theta_{13}$,
$\Delta m^{2}_{31}$, $\Delta m^{2}_{21}$, and $\delta_{CP}$. Experimental data
from various sources, including solar, atmospheric, reactor, and accelerator
experiments, have provided information about each of these oscillation
parameters, except for $\delta_{CP}$ Gonzalez-Garcia:2014bfa ; Capozzi:2013csa
; Forero:2014bxa . In LBL neutrino experiments, the
$\nu_{\mu}\rightarrow\nu_{e}$ appearance channel is highly sensitive to
exploring the CP-violation effect, which remains one of the most challenging
problems in neutrino physics today. The oscillation probability for
$\nu_{\mu}\rightarrow\nu_{e}$ in the standard three-flavor scenario and
constant density approximation can be described by the following expression
DUNE:2020jqi :

$P(\nu_{\mu}\rightarrow\nu_{e})=\sin^{2}\theta_{23}\sin^{2}2\theta_{13}\frac{\sin^{2}(\Delta_{31}-aL)}{(\Delta_{31}-aL)^{2}}\Delta^{2}_{31}+\sin
2\theta_{23}\sin 2\theta_{13}\sin
2\theta_{12}\frac{\sin(\Delta_{31}-aL)}{(\Delta_{31}-aL)}\Delta_{31}$

$\times\frac{\sin(aL)}{aL}\Delta_{21}\cos(\Delta_{31}+\delta_{CP})+\cos^{2}\theta_{23}\sin^{2}2\theta_{12}\frac{\sin^{2}(aL)}{(aL)^{2}}\Delta^{2}_{21}$
(1)

where

$a=\frac{G_{F}N_{e}}{\sqrt{2}}\approx\pm\frac{1}{3500km}\frac{\rho}{3.0g/cm^{3}}$
(2)

Here $G_{F}$ represents the Fermi constant and $N_{e}$ is the number density
of electrons in Earth’s crust. The value of $\rho$ is set to be 2.848
$g/cm^{3}$ Devi:2020lem . Additionally, $\Delta m^{2}_{ij}(\equiv
m^{2}_{i}-m^{2}_{j})$ represents the difference in mass squares between
neutrinos of the $i$-th and $j$-th families, $\Delta_{ij}=(1.267\Delta
m^{2}_{ij}L)/E_{\nu}$, where $L$ is the baseline in kilometers and $E_{\nu}$
is the neutrino energy in GeV.

For $\nu_{\mu}\rightarrow\nu_{e}$ and
$\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$ oscillations, the terms
$\delta_{CP}$ and $a$ have positive and negative signs, respectively. This
leads to a neutrino-antineutrino asymmetry due to both CP violation and the
matter effect $a$. The rates of neutrino oscillations, specifically
$\nu_{\mu}\rightarrow\nu_{e}$ and $\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$,
are influenced by the mass hierarchy as neutrinos travel through the Earth in
contrast to when they propagate through a vacuum. The presence of coherent
forward scattering with electrons in the Earth’s crust amplifies the
$\nu_{\mu}\rightarrow\nu_{e}$ transition while reducing the occurrence of
$\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$ in the normal hierarchy (NH).
Conversely, this amplification and suppression are reversed in the inverted
hierarchy (IH). This phenomenon, known as the matter effect Wolfenstein:1977ue
, modifies the oscillation probabilities for the NO$\nu$A experiment by
approximately 20% NOvA:2021nfi . The coupling between the mass state $\nu_{3}$
and the neutrino states $\nu_{\mu}$ and $\nu_{\tau}$ is primarily determined
by the angle $\theta_{23}$. When $\theta_{23}=\pi/4$, called maximal
mixing,Harrison:2002et , $\nu_{\mu}$ and $\nu_{\tau}$ are equally linked to
$\nu_{3}$. In the case of non-maximal mixing, $\theta_{23}$ can exist in
either the higher octant (HO) with $\theta_{23}>\pi/4$, or the lower octant
(LO) with $\theta_{23}<\pi/4$.

A non-zero value of $\delta_{CP}$, other than $0^{\circ}$ and $\pm
180^{\circ}$, would indicate CP violation in the lepton sector, and
$\delta_{CP}=\pm 90^{\circ}$ corresponds to maximum CP violation. It is often
convenient to divide the parameter space into the lower half-plane (LHP) with
$-180^{\circ}<\delta_{CP}<0^{\circ}$ and the upper half-plane (UHP) with
$0^{\circ}<\delta_{CP}<180^{\circ}$. The appearance channel
$P_{{\mu}\rightarrow e}$ often known as the golden channel can measure all
three unknown parameters. In LBL experiments, the measurement of neutrino
oscillation parameters is a challenging task due to the presence of
degeneracies. These degeneracies arise because different combinations of
parameter values can lead to the same oscillation probability
Agarwalla:2021bzs ; Barger:2001yr ; Minakata:2004pg . As a result, determining
the true values of the parameters becomes complicated and lacks unambiguous
resolution. In Ref. Capozzi:2021fjo , the authors observe a preference at a
significance level of 1.6$\sigma$ for $\theta_{23}$ in the LO compared to the
secondary best-fit in the HO. Assuming NH, they derive a best-fit value of
$\sin^{2}\theta_{23}$ = 0.455 in the LO, and at a significance of
approximately 1.8$\sigma$, they disfavor maximal $\theta_{23}$ mixing. In Ref.
deSalas:2020pgw , de Salas et al. find the best-fit value in the HO to be
around $\sin^{2}\theta_{23}\sim$ 0.57, assuming NH. On the other hand, Capozzi
et al. Capozzi:2021fjo  and Esteban et al. Esteban:2020cvm  obtain the best-
fit value around $\sin^{2}\theta_{23}\sim$ 0.45 in the LO. These degeneracies
make it difficult to uniquely determine the specific values of the parameters.
More advanced experimental techniques and additional data are needed to
resolve these degeneracies and obtain more accurate measurements of
$\delta_{CP}$, the neutrino mass hierarchy, and the octant of $\theta_{23}$
Bora:2014zwa .

In this work, we investigate the influence of the multinucleon enhancement and
RPA suppression on the determination of the CP-violating phase $\delta_{CP}$,
the octant of the atmospheric mixing angle $\theta_{23}$, and the
determination of the mass hierarchy in the appearance channel of the NO$\nu$A
experiment. We also include the presence of detector effects of NO$\nu$A
experiment, which is an important consideration for realistic experimental
scenarios. We consider the multinucleon enhancement and CP violating phase
($\delta_{CP}=\pm 90^{\circ}$) at the same time to know their impact on
sensitivity analysis. As seen in our works Deka:2021qnw ; Deka:2022lvy  the
RPA suppression and multinucleon enhancement play a significant role in
shaping the oscillation probabilities and can have a substantial impact on the
determination of the neutrino oscillation parameters, we have also considered
the pure QE interaction process in octant and mass hierarchy (MH) sensitivity
to observe the deviation from pure QE process to QE+multinucleon
enhancement+RPA suppression interaction process. Throughout our study, we have
considered the interplay between the multinucleon effect, the detector effect,
and the relevant oscillation parameters to gain insights into their combined
impact on the precision of our measurements.

The paper is structured as follows: in section II, we briefly discuss the
NO$\nu$A experiment, and section III provides an overview of the physics
principles and simulation details employed in our study. In section IV, we
present the results of our analysis and engage in a comprehensive discussion.
Specifically, we focus on the $\nu_{\mu}+\bar{\nu}_{\mu}$ channel,
investigating the effects on various neutrino oscillation parameters, as well
as the impact on QE(+RPA)+2p2h and pure QE interaction processes, along with
realistic NO$\nu$A detector effects. Finally, in section V, we summarize our
findings and provide a concluding remark on our work.

##  II The NO$\nu$A Experiment

The NO$\nu$A (NuMI Off-Axis $\nu_{e}$ Appearance) experiment NOvA:2016vij
focuses on neutrino oscillations, specifically measuring the probability of
$\nu_{\mu}$ disappearance
P($\nu_{\mu}(\bar{\nu}_{\mu})\rightarrow\nu_{\mu}(\bar{\nu}_{\mu}$)) and
$\nu_{e}$ appearance
P($\nu_{\mu}(\bar{\nu}_{\mu})\rightarrow\nu_{e}(\bar{\nu}_{e})$). To achieve
this, two functionally identical detectors are utilized: the 290-ton Near
Detector (ND) and the 14-kton Far Detector (FD). Both detectors are positioned
off the central beam axis, with the Far Detector placed 14.6 milliradians off-
axis. This off-axis configuration enables the narrow neutrino energy flux to
peak around 2 GeV, near the first oscillation maximum driven by $\Delta
m^{2}_{32}=2.5\times 10^{-3}eV^{2}$, thereby enhancing the oscillation
probability in the $\nu_{e}$ appearance channel. The NuMI (Neutrinos at the
Main Injector) beam Adamson:2015dkw , generated at the Fermi National
Accelerator Laboratory, serves as the source of neutrinos for the experiment.
The ND is positioned at a distance of 1 km from the source and is located 105
m underground. NO$\nu$A measures oscillations by comparing the un-oscillated
energy spectra captured by the ND and the oscillated spectra observed by the
FD, situated near Ash River, Minnesota, approximately 810 km away from the
production target. The NuMI beam is generated by directing 120 GeV protons
from the Main Injector at a fixed graphite target, which produces pions and
kaons. These particles are then focused into a narrow beam using magnetic
horns, and further decay into muons, anti-muons, and their associated
neutrinos. Muons and anti-muons are removed from the decay pipe by a
240-meter-thick rock wall. Both the ND and FD are composed of planes made from
extruded polyvinyl chloride (PVC) cells. The flavor composition of the NuMI
beam consists of 97.5% $\nu_{\mu}$, 1.8% $\bar{\nu}_{\mu}$, and 0.7%
$\nu_{e}+\bar{\nu}_{e}$. The primary objectives of the NO$\nu$A experiment are
to precisely measure the mixing angle $\theta_{23}$ and determine its octant,
determine the neutrino mass hierarchy, and investigate the CP-violating phase
$\delta_{CP}$ in the lepton sector. The $\nu_{\mu}(\bar{\nu}_{\mu})$
disappearance channel is utilized to measure $\Delta m^{2}_{23}$ and
$\sin^{2}\theta_{23}$, while the $\nu_{e}(\bar{\nu}_{e})$ appearance channel
allows for measurements of the mass hierarchy, $\theta_{13}$, $\theta_{23}$,
and $\delta_{CP}$.

##  III Physics and Simulation Details

In the analysis of the $\nu_{\mu}$ ($\bar{\nu}_{\mu})$ appearance channel, the
$\nu_{\mu}$($\bar{\nu}_{\mu})$-CC interaction channel is utilized as the
signal in NO$\nu$A detectors. In these detectors, $\nu_{\mu}$-CC interactions
are identified by detecting long muon tracks and any associated hadronic
activity at the vertex. The un-oscillated spectra from the NuMI beam are
initially measured at the ND. These ND spectra are then extrapolated to
predict the spectra at the FD. For a detailed description of the extrapolation
technique used in this study, refer to our previous work Deka:2021qnw ;
Deka:2022lvy .

To analyze the sensitivity of the neutrino oscillation parameters
$\theta_{23}$ and $\Delta m^{2}_{23}$, we employ the Feldman-Cousins method
Feldman:1997qc  to calculate the confidence level allowed in the parameter
space Deka:2021qnw . We generate a sample of 1 million CC $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ events on a carbon target in the energy range of 0-5 GeV
using the NO$\nu$A-ND neutrino and antineutrino flux. The simulation is
performed using the GENIE (Generates Events for Neutrino Interaction
Experiments) v3.0.6 neutrino event generator Andreopoulos:2009rq . GENIE is
currently employed by several neutrino baseline experiments, including
Miner$\nu$a MINERvA:2019ope , MINOS Adamson:2007gu , MicroBooNE Chen:2007ae ,
T2K Abe:2018wpn , and the NO$\nu$A experiment. We consider four interaction
processes: QE scattering, resonances from $\Delta$ decay and higher
resonances, nucleon-nucleon correlations (2p2h), and deep inelastic scattering
(DIS) in both neutrino and antineutrino modes. We focus on simulated events
with no pions in the final state. The QE scattering in GENIE is modeled using
the Llewellyn-Smith model LlewellynSmith:1971uhs . Resonance processes are
implemented according to the formalism attributed to the Berger and Sehgal
model Berger:2007rq . Furthermore, to incorporate the effect of long-range
nuclear charge screening due to random-phase approximation correlations, which
also modifies the kinematics of QE interactions, we simulate the QE
interactions with the Nieves et al. model Nieves:2004wx . These effects
significantly suppress the QE interaction process at low invariant four-
momentum transferred to the nucleus ($Q^{2}$) and slightly enhance it at
higher $Q^{2}$, relative to the RFG prediction. The DIS interaction,
categorized as a non-resonant process in GENIE, is implemented following the
method of Bodek and Yang Bodek:2002ps . Two-nucleon knockout (2p2h) events are
simulated using the Valencia model developed by Nieves et al. Nieves:2012yz ,
and the Effective Spectral function (ESF) model is used as the nuclear model,
along with the Transverse Enhancement model (TEM) Bodek:2011ps . The ESF model
used in GENIE is based on the RFG model, which assumes that the nucleons
inside the nucleus behave like a gas of free, non-interacting particles with a
Fermi momentum that depends on the nuclear density. However, the RFG model has
limitations, as it neglects the effects of nuclear correlations, meson
exchange currents, and other complex nuclear dynamics that are important at
higher energies and in heavier nuclei. To overcome these limitations, the
effective spectral function model incorporates additional phenomenological
parameters that are tuned to match experimental data, such as the nuclear
binding energy, the nucleon momentum distribution, and the strength of the
nucleon-nucleon correlations. The TEM is used to describe the transverse
momentum distribution of the struck nucleon inside the nucleus. This model
takes into account the effects of final-state interactions between the struck
nucleon and other nucleons in the nucleus, which can lead to a broadening of
the transverse momentum distribution. The transverse enhancement model is
particularly important for high-energy neutrino interactions, where the struck
nucleon can have a large momentum and can interact with multiple nucleons in
the nucleus. By including the transverse enhancement model, GENIE can
accurately predict the transverse momentum distributions of the final state
particles in these high-energy interactions. The best-fit values for the
oscillation parameters are taken from deSalas:2020pgw .

The energy of a CC neutrino, which scatters off a nuclear target and produces
’m’ mesons while knocking out ’n’ nucleons, can be reconstructed using the
calorimetric approach as Deka:2022lvy :

$E_{\nu}^{\text{cal}}=E_{\text{lep}}+\sum_{i}(E_{p}-M)+\epsilon_{n}+\sum_{m}E_{m}$
(3)

In this equation, $E_{\text{lep}}$ represents the energy of the outgoing
lepton, $E_{p}$ and $M$ denote the energy of the i-th knocked-out nucleon and
the mass of the nucleon (target nucleus), respectively. The term
$\epsilon_{n}$ corresponds to the single-nucleon separation energy of the
outgoing nucleons, which we set to a fixed value of 25 MeV for the carbon
target in both neutrino and antineutrino reconstruction methods. Lastly,
$E_{m}$ represents the energy of the m-th produced meson.

We have taken into account, the detector response and efficiency, which
introduce smearing effects on the measured energies compared to their true
values due to finite detector resolution. This leads to a non-zero probability
for an event with a true energy $E_{\text{true}}$ to be reconstructed with a
different energy $E_{\text{rec}}$. These probabilities are represented by a
set of migration matrices. In our study, we consider realistic specifications
for the NO$\nu$A experiment, including a selection efficiency of 31.2% (33.9%)
for $\nu_{\mu}$ ($\bar{\nu}_{\mu}$) events. For demonstration and comparison,
we also analyze the case of 80% detector efficiency. We use a muon energy
resolution of 3.5% and a hadron energy resolution of 25% NOvA:2017ohq ,
resulting in an overall energy resolution of 7% for $\nu_{\mu}$-CC events in
both detectors.

##  IV Results and Discussion

Figure 1: $P(\nu_{\mu}\rightarrow\nu_{e})$(solid line) and
$P(\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e})$(dashed line) vs neutrino energy
for 810 km baseline of NO$\nu$A. Variation of $\delta_{CP}$ leads to variation
in oscillation probability.

Using the details given above, and the technique explained in our papers
Deka:2022idq ; Deka:2021qnw ; Deka:2022lvy , we do the sensitivity analysis
for CPV phase measurement, octant of $\theta_{23}$ and mass hierarchy, with
multinucleon effects, and the results are shown in Figs. (1-8). In the left
panel of Fig. 1, we have presented the variation of $P_{{\mu}\rightarrow e}$
as a function of $E_{\nu}$ for both NH and IH, for $\delta_{CP}$ at
$-90^{\circ}$, $0^{\circ}$, and $90^{\circ}$. Generally, the values of
$P_{{\mu}\rightarrow e}$ are higher for NH and lower for IH and vice versa for
$P_{{\bar{\mu}}\rightarrow\bar{e}}$, which directly arises from the matter
effect parameter ‘a’. In the case of neutrino (left panel), for both NH and
IH, the curve corresponding to $\delta_{CP}=+90^{\circ}$ appears at the lowest
position within the band, while the curve for $\delta_{CP}=-90^{\circ}$ is
located at the highest position. This behavior can be easily understood by
referring to Eq. 1. When approaching the oscillation maximum, with
$\Delta_{31}\simeq 90^{\circ}$, the term $\cos(\Delta_{31}+\delta_{CP})$
becomes +1 for $\delta_{CP}=-90^{\circ}$ and -1 for $\delta_{CP}=+90^{\circ}$.
In the right panel of Fig. 1, we have displayed the corresponding antineutrino
probabilities. Here, $P_{{\bar{\mu}}\rightarrow\bar{e}}$ is higher for IH and
lower for NH due to the sign reversal of ‘a’. Since $\delta_{CP}$ is reversed
for antineutrinos, the upper curves are defined by $\delta_{CP}=+90^{\circ}$,
while the lower curves correspond to $\delta_{CP}=-90^{\circ}$.

Based on Fig. 1, we can establish the concept of a favorable half-plane for
each hierarchy. Assuming NH represents the true hierarchy, when $\delta_{CP}$
is within the LHP (ranging from $-180^{\circ}$ to $0^{\circ}$), all the curves
for $P_{{\mu}e}$ (NH, $\delta_{CP}$) must lie above the set of curves for
$P_{{\mu}\rightarrow e}$ (IH, $\delta_{CP}$) Ghosh:2015ena ; Prakash:2012az .
In the case of antineutrinos, $P_{{\bar{\mu}}\rightarrow\bar{e}}$ (NH,
$\delta_{CP}$) will be much lower than $P_{{\bar{\mu}}\rightarrow\bar{e}}$
(IH, $\delta_{CP}$). In this scenario, the NH can be determined solely by
NO$\nu$A. Therefore, we refer to the LHP as the favorable half-plane for NH.
Similar reasoning applies if IH is the true hierarchy and $\delta_{CP}$ lies
within the UHP. Hence, UHP is the favorable half-plane for IH. Consequently,
we have two possible combinations to choose from: (NH, LHP) or (IH, UHP). The
NO$\nu$A experiment can successfully determine the neutrino mass hierarchy in
two favorable scenarios Ghosh:2015ena :

  * •

When the hierarchy is NH and $\delta_{CP}$ lies in the LHP
($-180^{\circ}\leqslant\delta_{CP}\leqslant 0^{\circ}$).

  * •

When the hierarchy is IH and $\delta_{CP}$ lies in the UHP
($0^{\circ}\leqslant\delta_{CP}\leqslant 180^{\circ}$).

###  IV.1 Octant Sensitivity

Figure 2: The octant sensitivity is compared for different true values of
$\delta_{CP}$ and for QE(+RPA)+2p2h (solid line) and QE (dashed line)
interaction processes without considering the detector effect. The analysis is
performed for both NH (left panel) and IH (right panel) in the
$\nu_{\mu}$+$\bar{\nu}_{\mu}$ mode. The capability of the NO$\nu$A experiment
to establish a non-maximal $\theta_{23}$ at $2\sigma$ ($\Delta\chi^{2}=4$) and
$3\sigma$ ($\Delta\chi^{2}=9$) confidence levels are indicated by the dashed
red and green lines, respectively.

The octant sensitivity of an experiment is determined by comparing the
sensitivity of the experiment to measure the correct octant in the true
spectrum with the incorrect octant in the test spectrum. In Fig. 2 we show
$\Delta\chi^{2}$ as a function of true $\sin^{2}\theta_{23}$ without
considering the detector effect for QE(+RPA)+2p2h and QE interaction
processes, where for each true value of $\sin^{2}\theta_{23}$, we consider
test values of $\sin^{2}\theta_{23}$ in the opposite octant including
$\sin^{2}\theta_{23}$(test)=0.5 in the fit. To calculate the octant
sensitivity we simulate the data for a representative value of true
$\theta_{23}$ belonging to LO (HO) and test it by varying $\theta_{23}$ in the
opposite octant i.e., HO (LO) along with marginalization over $|\Delta
m^{2}_{31}|$, hierarchy and $\delta_{CP}$. The figure includes three curves:
the blue line represents the sensitivity analysis for $\delta_{CP}=0^{\circ}$,
the black line corresponds to $\delta_{CP}=90^{\circ}$ and the magenta line
represents the analysis for $\delta_{CP}=-90^{\circ}$. From the results shown
in this Fig. 2, it is observed that:

  * •

The inclusion of the complete model, which incorporates QE(+RPA)+2p2h
(multinucleon enhancement+RPA suppression), demonstrates significantly
improved octant sensitivity compared to the pure-QE interaction process. This
highlights the crucial role of multinucleon effects in the analysis of octant
sensitivity.

  * •

In the case of NH, there is a clear and noticeable distinction among the three
curves corresponding to $\delta_{CP}=0^{\circ},90^{\circ}$, and $-90^{\circ}$.
In terms of $\delta_{CP}$ value, among these curves, the one corresponding to
$\delta_{CP}=90^{\circ}$ exhibits higher sensitivity compared to the others,
which may be due to the addition of both neutrino and antineutrino modes.

  * •

On the other hand, in the case of IH, all three curves overlap which implies
that octant sensitivity is independent of the value of the specific value of
$\delta_{CP}$ for IH.

  * •

(NH, $\delta_{CP}=90^{\circ}$) shows octant sensitivity $>3\sigma$ range for
both the processes while (NH, $\delta_{CP}=-90^{\circ}$) and (NH,
$\delta_{CP}=0^{\circ}$) shows sensitivity $>2\sigma$.

Figure 3: The comparison of octant sensitivity as a function of the true value
of $\theta_{23}$ is depicted in the figure only for the QE(+RPA)+2p2h
interaction process. The detector effect is taken into account, and the
results are shown for efficiency 31.2% ($\nu_{\mu}$) and 33.9%
($\bar{\nu}_{\mu}$) represented by the solid line, as well as for 80%
efficiency shown by the dashed line, for the $\nu_{\mu}$+$\bar{\nu}_{\mu}$
mode. The analysis is performed for both NH (left panel) and IH (right panel).
The capability of the NO$\nu$A experiment to establish a non-maximal
$\theta_{23}$ at $2\sigma$ ($\Delta\chi^{2}=4$) and $3\sigma$
($\Delta\chi^{2}=9$) confidence levels are indicated by the dashed red and
green lines, respectively.

In Fig. 3 we show the comparison of octant sensitivity $\chi^{2}$ as a
function of the true value of $\theta_{23}$ taking into account the detector
effect only for QE(+RPA)+2p2h process. The results are presented for the
$\nu_{\mu}$+$\bar{\nu}_{\mu}$ mode for efficiency of 31.2% ($\nu_{\mu}$) and
33.9% ($\bar{\nu}_{\mu}$) denoted by the solid line. Additionally, the results
for 80% efficiency are shown with a dashed line. The left panel corresponds to
NH, while the right panel corresponds to IH. From this figure, one can
determine the range of $\theta_{23}$ values for which the octant can be
determined for $\delta_{CP}=\pm 90^{\circ}$ at a specific confidence level.
Though detector effects enhance the discrimination among LHP and UHP of the
CPV phase, however, degeneracy with respect to octant of $\theta_{23}$ is
still present (better for NH as true MH), and other methods are needed to
break this degeneracy Bora:2014zwa . Nonetheless, with the inclusion of
detector effects, sensitivity increases slightly in HO for both MH, at more
than 2$\sigma$ level. And sensitivity to measure $\theta_{23}$ is improved as
detector efficiency is improved. Comparison between two panels of Fig. 3 shows
that for IH, these sensitivities are less.

Figure 4: Comparison of octant sensitivity as a function of the true value of
$\delta_{CP}$ for two different true values of $\theta_{23}$. In the top
panel, where $\theta_{23}=42^{\circ}$, and in the bottom panel, where
$\theta_{23}=48^{\circ}$, the analysis is performed for both NH (left panel)
and IH (right panel) considering the $\nu_{\mu}$+$\bar{\nu}_{\mu}$ mode. The
curves are shown for two different detector efficiencies: 31.2% (33.9%)
represented by the solid black line and 80% represented by the solid magenta
line.

In Fig. 4, we compare the octant sensitivity as a function of the true value
of $\delta_{CP}$ for two different true values of $\theta_{23}$. The top panel
corresponds to $\theta_{23}=42^{\circ}$ (LO), while the bottom panel
corresponds to $\theta_{23}=48^{\circ}$ (HO). The analysis is performed for
both the NH and IH hierarchies, displayed in the left and right panels, for
the $\nu_{\mu}$+$\bar{\nu}_{\mu}$ mode. The curves are shown for two different
detector efficiencies: 31.2% (33.9%) represented by the black line and 80%
represented by the magenta line. In addition to this variation, we also
account for uncertainties by marginalizing over $|\Delta m^{2}_{31}|$,
hierarchy, and $\delta_{CP}$ in our analysis. From this figure, it is observed
that:

  * •

In (LO-NH) the sensitivity improves to approximately 2$\sigma$ in the LHP and
reaches its peak at around $\delta_{CP}=-90^{\circ}$, while for (LO-IH) the
sensitivity improves to approximately 2$\sigma$ in the UHP and reaches its
maximum around $\delta_{CP}=90^{\circ}$.

  * •

(HO-NH) shows much better sensitivity ($\geqslant 3\sigma$) in the LHP while
for (HO-IH) shows two peaks at $\delta_{CP}=-180^{\circ}$ and
$\delta_{CP}=180^{\circ}$.
Above two points can be justified as LHP is the favorable plane for NH.

  * •

80% detector efficiency shows sensitivity close to no detector effect (100%
efficiency and no resolution function) at $\geqslant 2\sigma$ while NO$\nu$A
efficiency shows less sensitivity $\leqslant 1\sigma$ for (LO-NH) and (LO-IH).

  * •

(LHP-HO-NH) shows the highest sensitivity among the four panels.

###  IV.2 CP violation Sensitivity

Since the true value of $\delta_{CP}$ is unknown, the analysis involves
scanning all possible true values of $\delta_{CP}$ over the range
$-\pi<\delta_{CP}<+\pi$ and comparing them with CP-conserving values, such as
0 or $\pm\pi$. To detect CP violation, the value of the CP phase must differ
from 0 or $\pm\pi$. To investigate CPV sensitivity, we compute and study two
quantities,

$\Delta\chi^{2}_{0}=\chi^{2}(\delta_{CP}=0)-\chi^{2}_{true}$ (4)

$\Delta\chi^{2}_{\pi}=\chi^{2}(\delta_{CP}=\pi)-\chi^{2}_{true}$ (5)

and then we consider

$\Delta\chi^{2}=min(\Delta\chi^{2}_{0},\Delta\chi^{2}_{\pi})$ (6)

In Figs. 5 and 6, we have presented the CPV discovery potential of the
NO$\nu$A experiment. CP violation discovery potential is defined as the
capability to distinguish a value of $\delta_{CP}$ from the CP conserving
values of $0^{\circ}$ and $180^{\circ}$. The comparison of CPV sensitivity for
different true values of $\theta_{23}$ with (black, magenta line) and without
(blue line) detector effects are shown in the left, middle, and right panels
for both NH (Fig. 5) and IH (Fig. 6). The variation of $\chi^{2}$ vs
$\delta_{CP}$ (true) is presented in the left, middle, and right panel for
$\theta_{23}=42^{\circ}$, $45^{\circ}$ and $48^{\circ}$. From these two
figures, it is observed that

  * •

For both NH and IH, $\theta_{23}=42^{\circ}$ shows slightly improved
sensitivity than $\theta_{23}=45^{\circ}$ and $48^{\circ}$.

  * •

There is a significant difference among no detector, NO$\nu$A efficiency, and
80% efficiency for both NH and IH.

  * •

The (UHP-IH) peak shows slightly more sensitivity than the (LHP-IH) peak,
though this distinction is not seen there in NH.

Figure 5: The comparison of CPV sensitivity as a function of
$\delta_{CP}$(true) for different true values of $\theta_{23}$ with (black and
magenta line) and without (blue line) detector effects is shown for efficiency
31% ($\nu_{\mu}$) and 33.9%($\bar{\nu}_{\mu}$) for NH for
$\nu_{\mu}+\bar{\nu}_{\mu}$ mode for QE(+RPA)+2p2h interaction process. The
left, middle, and right panel show $\theta_{23}=42^{\circ}$, $45^{\circ}$ and
$48^{\circ}$. Figure 6: The comparison of CPV sensitivity as a function of
$\delta_{CP}$(true) for different true values of $\theta_{23}$ with (black and
magenta line) and without (blue line) detector effects is shown for efficiency
31% ($\nu_{\mu}$) and 33.9%($\bar{\nu}_{\mu}$) for IH for
$\nu_{\mu}+\bar{\nu}_{\mu}$ mode for QE(+RPA)+2p2h interaction process. The
left, middle, and right panel show $\theta_{23}=42^{\circ}$, $45^{\circ}$ and
$48^{\circ}$.

###  IV.3 Mass Hierarchy Sensitivity

parameter | best fit | Test 3 $\sigma$ range
---|---|---
$\Delta m_{21}^{2}[10^{-5}eV^{2}]$ | $7.50$ | fixed
$|\Delta m_{31}^{2}|[10^{-3}eV^{2}]$(NH) | $2.56$ | $2.46:2.65$
$|\Delta m_{31}^{2}|[10^{-3}eV^{2}]$(IH) | $2.46$ | $2.37:2.55$
$\sin^{2}\theta_{12}/10^{-1}$ | $3.18$ | fixed
$\theta_{23}/10^{-1}$(NH) |  $42^{\circ}$(LO), $48^{\circ}$(HO) | $41.63^{\circ}:51.32^{\circ}$
$\theta_{23}/10^{-1}$(IH) |  $42^{\circ}$(LO), $48^{\circ}$(HO) | $41.88^{\circ}:51.30^{\circ}$
$\sin^{2}\theta_{13}/10^{-2}$(NH) | $2.225$ | fixed
$\sin^{2}\theta_{13}/10^{-2}$(IH) | $2.250$ | fixed
$\delta_{CP}$ | $-180^{\circ}:180^{\circ}$ |  $0^{\circ}$(fixed)
Table 1: The true and test values of the oscillation parameters used in mass
hierarchy sensitivity.

The hierarchy sensitivity of an experiment is determined by taking the true
spectrum with the correct hierarchy and the test spectrum with the wrong
hierarchy. Following formulae are used to determine the $\Delta\chi^{2}$ for
mass hierarchy sensitivity:

$\Delta\chi^{2}_{MH}=\chi^{2}_{IH}-\chi^{2}_{NH}$ (7)

$\Delta\chi^{2}_{MH}=\chi^{2}_{NH}-\chi^{2}_{IH}$ (8)

In Fig. 7 and 8, the hierarchy sensitivity, in terms of $\chi^{2}$ vs. true
$\delta_{CP}$, is examined for NH (IH) as the true hierarchy and IH (NH) as
the test hierarchy in the $\nu$+$\bar{\nu}$ mode for the QE(+RPA)+2p2h and
pure QE interaction processes. The true and test values of the oscillation
parameters used in mass hierarchy sensitivity are shown in Table 1. From Figs.
7 and 8 it is observed that:

  * •

The inclusion of multinucleon effects has a pronounced impact on the
sensitivity to the neutrino mass hierarchy, as evident from the comparison of
the QE(+RPA)+2p2h (solid line) and QE (dashed line) interaction processes.

  * •

The difference between the two interaction processes is large in the case of
NH taken as the true hierarchy for both LO and HO. Hierarchy sensitivity is
seen at $\leqslant 1\sigma$ for QE while for QE(+RPA)+2p2h it is seen at
$\leqslant 2\sigma$.

  * •

The hierarchy sensitivity is highest when $\delta_{CP}=-90^{\circ}$ and lowest
when $\delta_{CP}=90^{\circ}$ when NH is the true hierarchy and IH is the test
hierarchy. Conversely, the opposite behavior is observed in the left panel
when IH is the true hierarchy and NH is the test hierarchy.

  * •

For true MH NH, $\delta_{CP}=-90^{\circ}$ and true MH IH,
$\delta_{CP}=90^{\circ}$ sensitivity $\leqslant 2\sigma$ but sensitivity is
$<1\sigma$ for true MH NH, $\delta_{CP}=90^{\circ}$ and true MH IH,
$\delta_{CP}=-90^{\circ}$. It is evident that in the favorable half-plane
(LHP-NH and UHP-IH) of $\delta_{CP}$ values, the ability to differentiate the
correct neutrino mass hierarchy is significantly enhanced. This indicates that
the sensitivity to determine the true hierarchy is greater when $\delta_{CP}$
falls within the favorable half-plane.

Figure 7: Hierarchy sensitivity as a function of true $\delta_{CP}$
with/without detector effect for different combinations for $\nu$+$\bar{\nu}$
mode for QE(+RPA)+2p2h (solid line) and QE (dashed line) interaction processes
for $\theta_{23}=42^{\circ}$. Left panel: NH is considered as true hierarchy
and IH is taken as a test hierarchy. Right panel: IH is considered as true
hierarchy and NH is taken as a test hierarchy. Figure 8: Hierarchy sensitivity
as a function of true $\delta_{CP}$ with/without detector effect for different
combinations for $\nu$+$\bar{\nu}$ mode for QE(+RPA)+2p2h (solid line) and QE
(dashed line) interaction processes for $\theta_{23}=48^{\circ}$. Left panel:
NH is considered as true hierarchy and IH is taken as a test hierarchy. Right
panel: IH is considered as true hierarchy and NH is taken as a test hierarchy.

  * •

For (true MH NH, HO, LHP) and (true MH IH, LO, UHP) cases, the hierarchy
sensitivity can reach close to 2$\sigma$ significance.

  * •

The sensitivity to determine the MH shows a slight advantage in the higher
octant for true NH and lower octant for true IH.

Octant Sensitivity | CPV Sensitivity | MH Sensitivity
---|---|---
With no detector effects for NH $\delta_{CP}=90^{\circ}$ shows maximum octant sensitivity, while for IH there is no distinction among curves for $\delta_{CP}=0^{\circ},90^{\circ}$, and $-90^{\circ}$ |  LO shows slightly better CPV sensitivity for both MH than HO |  For MH NH, LHP shows better sensitivity than UHP, i.e., sensitivity to determine true MH is better when $\delta_{CP}$ falls within favorable half plane
Better detector efficiency implies better octant sensitivity |  UHP-IH shows slightly better CPV sensitivity than LHP-IH, distinction not seen in NH case |  For true NH, LHP-HO and true IH, UHP-LO case MH sensitivity can reach close to $\sim 2\sigma$ significance
Octant sensitivity for UHP-HO-NH is $\geqslant 3\sigma$ and for LHP-LO-NH $\geqslant 2\sigma$ |  Significant difference among no detector, NO$\nu$A efficiency and 80% efficiency for both NH and IH |  The sensitivity is slightly higher in HO-NH, and LO-IH cases

Table 2: Summary of the work done in this work.

##  V Summary and Conclusion

In this work we investigated the potential for determining the octant of the
atmospheric mixing angle $\theta_{23}$, the CP-violating phase $\delta_{CP}$,
and the sensitivity to the mass hierarchy in the appearance channel of
NO$\nu$A when multinucleon and detector effects are included. The concept of
octant degeneracy traditionally refers to the ambiguity between $\theta_{23}$
and $\pi/2-\theta_{23}$. We performed a $\chi^{2}$ analysis of the octant,
$\delta_{CP}$ and hierarchy sensitivity. It was observed that multinucleon and
detector effects significantly improve the Octant, CP phase, and MH
measurement sensitivities. After a careful analysis, the results of this work
can be summarised as shown in Table 2.

Preference of Octant Sensitivity | Preference of CPV Sensitivity | Favored MH Sensitivity
---|---|---
LBL gives two degenerate solutions for both LO and HO |  Values of CPV phase in the vicinity of $\delta_{CP}=\pi/2$ are excluded by 3$\sigma$ for IH, values around $\delta_{CP}=3\pi/2$ in NH are disfavored at 2$\sigma$ |  Independent analysis of both T2K and NO$\nu$A does not show specific preference of MH
LBL+ATM and LBL+ Reactor shifts towards HO |  From LBL+reactor data CP conserving value $\delta_{CP}=0$ is disfavored but $\delta_{CP}=\pi$ is still allowed |  As a consequence of tension in T2K and NO$\nu$A data all LBL data favor IO over NO

Table 3: Summary of the latest values for oscillation parameters and the
favored octant of $\theta_{23}$, $\delta_{CP}$, and the mass hierarchy
obtained from multiple global experiments NOvA:2021nfi ; deSalas:2020pgw .

Our findings revealed that the higher octant of $\theta_{23}$, the lower half
plane of $\delta_{CP}$, and the normal mass hierarchy (HO-LHP-NH) exhibit
improved sensitivity, enabling a more precise determination of the
corresponding parameters. The analysis using our comprehensive model
(QE(+RPA)+2p2h) demonstrated significantly enhanced sensitivity compared to
the pure QE interaction process. Furthermore, it is worth noting that the
curves representing 80% efficiency (overestimated) closely align with the
curves corresponding to no detector effects. These no-detector effect curves
represent an ideal scenario with a 100% efficient detector and no
consideration of a resolution function. On the other hand, the curves obtained
with 31.2% efficiency for $\nu_{\mu}$ and 33.9% efficiency for
$\bar{\nu}_{\mu}$ display more noticeable deviations from the no detector
effect curves. In conclusion, incorporating multinucleon effects and improving
detector efficiency, which constitutes the novelty of the research presented
in this paper, have the potential to enhance the capabilities of the NO$\nu$A
(and other long-baseline) experiments in conducting precise parameter studies.
This approach also provides deeper insights into the significant issue of
parameter degeneracies present in the measurements of long-baseline neutrino
experiments. Thus, we conclude that multinucleon and detector effects should
be carefully included in the analyses of LBL experimental data, for precise
determination of unknown neutrino oscillation parameters and for resolving
parameter degeneracies as well. If the results of this work can be
successfully applied in such future analyses, they can help us improve our
understanding of some unsolved crucial physics issues such as baryon asymmetry
of the universe (BAU), the contribution of neutrinos to dark matter, etc.

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