# Impact of Nuclear effects in Energy Reconstruction Methods on Sensitivity of
Neutrino Oscillation Parameters at NO$\nu$A experiment

Paramita Deka1 111E-mail<EMAIL_ADDRESS>Jaydip Singh2 222E-mail:
<EMAIL_ADDRESS>Kalpana Bora1<EMAIL_ADDRESS>Department Of
Physics, Gauhati University, Assam, India1 Department Of Physics, Lucknow
University, Uttar Pradesh, India2

###### Abstract

Long baseline (LBL) neutrino experiments aim to measure the neutrino
oscillation parameters to high precision. These experiments use nuclear
targets for neutrino scattering and hence are inflicted with complexities of
nuclear effects. Nuclear effects and their percolation into sensitivity
measurement of neutrino oscillations parameters are not yet fully understood
and therefore need to be dealt with carefully. In a recent work Deka:2021qnw ,
we reported some results on this for NO$\nu$A experiment using the kinematic
method of neutrino energy reconstruction, where it was observed that the
nuclear effects are important in sensitivity analysis, and inclusion of
realistic detector set up specifications increases uncertainty in this
analysis as compared to ideal detector case. Precise reconstruction of
neutrino energy is an important component in the extraction of oscillation
parameters, and it is required that the neutrino energy is reconstructed with
very high precision. With this motivation, in this work, we use two methods of
neutrino energy reconstruction - kinematic and calorimetric, including the
nuclear effects, and study their impact on sensitivity analysis. We consider
nuclear interactions such as RPA and 2p2h and compare two energy
reconstruction methods with reference to events generation, measurement of
neutrino oscillation parameters $\Delta m_{32}^{2}$ and $\theta_{23}$ for
($\nu_{\mu}\rightarrow\nu_{\mu}$) disappearance channel, mass hierarchy
sensitivity, and CP-violation sensitivity for ($\nu_{\mu}\rightarrow\nu_{e}$)
appearance channel of the NO$\nu$A experiment. It is observed that with an
ideal detector setup, the kinematic method shows significant dependence on
nuclear effects, as compared to the calorimetric method. We also investigate
the impact of realistic detector setup for NO$\nu$A in these two methods (with
nuclear effects) and find that the calorimetric method shows more bias
(uncertainty increases) in sensitivity contours, as compared to the kinematic
method. This is found to be true for both the mass hierarchies and for both
neutrino and antineutrino incoming beams. The results of this study can offer
useful insights into future neutrino oscillation analysis at LBL experiments
(including leptonic CP phase measurement), especially when we are in the
precision measurement era for the same, particularly in the context of some
nuclear effects and energy reconstruction methods.

##  I Introduction

In LBL neutrino oscillation experiments Fukuda:1998mi ; Fukuda:2002pe ;
Ahmad:2002jz ; Eguchi:2002dm ; Michael:2006rx ; Abe:2011fz ; An:2012eh ;
Ahn:2012nd ; Abe:2017vif ; DUNE:2020txw ; Ayres:2004js , the precise
measurement of neutrino energy is one of the essential ingredients to extract
the neutrino oscillation parameters. As the neutrino oscillation parameters
such as mixing angles, mass splittings, and the CP-violating (CPV) Dirac phase
are determined from the collected energy-dependent event distribution, their
precision measurement requires an accurate reconstruction of neutrino energy
from the measured kinematics of the final state particles. The next generation
oscillation experiments are aiming to measure the mass splittings with a
precision exceeding 1%, and the CPV phase as well, for which they need to
measure the neutrino energy with a precision exceeding 1% Ankowski:2017uvv ;
DUNE:2015lol . Depending on the length of the baseline, the relevant neutrino
energies have a range from a few hundred MeV to a few GeV, at which the
dominant interaction processes change from Quasi-elastic (QE) to resonant and
non-resonant meson production Formaggio:2013kya . As neutrinos are produced in
a secondary decay of primarily produced hadrons, their beams are quite broad
in energy. Therefore the incoming neutrino energy can not be measured a priori
in ongoing experiments. The reconstructed energy of the neutrino is determined
on an event-by-event basis. These experiments depend on two methods for energy
reconstruction - kinematic and calorimetric method Ankowski:2015jya . The
calorimetric method plays an important role at higher energies, which is
estimated by adding the energies of all the observed particles in the detector
while the kinematic method has been used to determine the incoming neutrino
energy assuming quasi-free kinematics of outgoing lepton in the QE process at
lower energies (a few hundred MeV to a few GeV).

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 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) QE process is represented as $\nu n\rightarrow\ l^{-}p$,
where $l$ is a charged lepton. 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: $\nu p\rightarrow\mu^{-}p\pi^{+}$ or $\nu
p\rightarrow\mu^{-}\Delta^{++}$. 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. A second complication arises due to the presence of multi-nucleon events
in which the incoming neutrino interacts with, e.g, two nucleons (so-called
2p2h) 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 effect of these uncertainties in
reconstructed energy on extracting the neutrino oscillation parameters for
MiniBooNE and T2K experiments can be found in Refs. Lalakulich:2012hs ;
Meloni:2012fq ; Coloma:2013rqa .

Figure 1: Cross-section as a function of true neutrino energy for both
neutrino (solid line) and antineutrino (dashed line). The color scheme for
different interactions: QE(+RPA)+2p2h (black line), QE only (red line),
QE(+RPA) (blue line), and 2p2h only (magenta line).

We would like to mention that in Ankowski:2017uvv , the relevance of nuclear
effects (FSI) in the calorimetric reconstruction of neutrino energy was
studied and found that when detector effects are included, bias on sensitivity
analysis is more, than the case when detector effects are not included/ideal
detector set-up. In Ankowski:2015jya , they did a comparative study of the
calorimetric and kinematic methods of neutrino energy reconstruction in
disappearance experiments, and the chosen detector capabilities were not to
represent any existing detector but were indicative of the general level of
performance. They found that the calorimetric method is less sensitive to the
nuclear model, but is strongly affected by detector effects. While, the
kinematic method depends strongly on the nuclear model, and is less sensitive
to detector effects.

In this work, we consider realistic detector details of NO$\nu$A experiment,
and use detector resolution of NO$\nu$A and two values of detector efficiency
- 31.2% (33.9%) for $\nu_{\mu}$($\bar{\nu}_{\mu}$) (realistic value of
NO$\nu$A ) and 100%, and compare them. Our nuclear interaction models also
differ from those of Ankowski:2015jya . We consider the RPA suppression and
the multi-nucleon enhancement at the same time to obtain a complete model
QE(+RPA)+2p2h to describe the QE interactions for both neutrino and
antineutrino for both the mass hierarchies, which are believed to be more
appropriate and complete models for realistic neutrino-nucleus scattering
Deka:2021qnw . In Ankowski:2015jya  they used the Relativistic Fermi Gas (RFG)
model and spectral function (SF) approach, while we have used the local Fermi
gas model. We have used the Valencia model (Nieves et. al. Nieves:2012yz ),
while in Ankowski:2015jya  Dytman model Katori:2013eoa  was used. We have
shown migration matrices for both reconstruction methods, for the four
interaction processes QE(+RPA)+2p2h, QE, QE(+RPA), and 2p2h to have a clear
visual impression of the smearing effects. Feldman-Cousins approach
Feldman:1997qc  is used to compute the systematic uncertainties in sensitivity
of neutrino oscillation parameter measurement, calculate $\chi^{2}$ and
determine the confidence regions in ($\theta^{2}_{23}-\Delta m^{2}_{32}$)
plane. Hence, though qualitatively our results do not refute those of
Ankowski:2015jya , the results presented here add additional and new
information to those presented in Ankowski:2015jya , with specific reference
to NO$\nu$A, as we have utilized the latest nuclear models and realistic
detector specifications of NO$\nu$A. This work is an extension of our previous
work Deka:2021qnw  in which we investigated the uncertainties in the neutrino
oscillation parameters measurement due to multi-nucleon effects at NO$\nu$A
experiment, where we used only the kinematic method for the reconstruction of
neutrino energy. Since reconstructed neutrino energy may change with the
reconstruction method, therefore, in this work, we have undertaken to use and
compare kinematic and calorimetric methods to reconstruct the neutrino energy,
including nuclear effects such as RPA and 2p2h interactions. We compare near
detector (ND) and far detector (FD) events in the two cases, and then study
the sensitivity of these methods to nuclear effects in the $\nu_{\mu}$
disappearance channel. We compute and show the impact of 2p2h processes, RPA
corrections separately in cross-section, events, and sensitivity in
oscillation parameter measurement along with the QE process, using both the
reconstruction methods.

Figure 2: Comparison of ND events as a function of the calorimetric (solid
line) and kinematic (dashed line) reconstruction energy for neutrino (left
panel) and antineutrino (right panel) for different interactions:
QE(+RPA)+2p2h (black line), QE only (red line), QE(+RPA) (blue line) and 2p2h
only (magenta line).

The paper is organized as follows. In section II, we briefly review the
NO$\nu$A experiment and in section III, the principle of the two energy
reconstruction methods is outlined. The CP-violation sensitivity and the
determination of mass hierarchy are discussed in section IV. This is followed
by a description of simulation details of the neutrino event generator GENIE
in section V. In section VI, we perform a detailed comparison of both the
reconstruction methods in various interaction processes including the multi-
nucleon effects and present sensitivity contours in three regions of parameter
spaces - ($\theta^{2}_{23}-\Delta m^{2}_{32}$) plane, CP-violation, and mass
hierarchy determination. The paper is summarised in section VII.

##  II The NO$\nu$A Experiment

Here, some details about the NO$\nu$A (NuMI Off-Axis $\nu_{e}$ Appearance)
NOvA:2016vij  experiment are presented for the sake of completeness of this
work. It is an LBL neutrino oscillation experiment which is designed to
measure $\nu_{\mu}$ disappearance
P($\nu_{\mu}(\bar{\nu}_{\mu})\rightarrow\nu_{\mu}(\bar{\nu}_{\mu}$)) and
$\nu_{e}$ appearance
P($\nu_{\mu}(\bar{\nu}_{\mu})\rightarrow\nu_{e}(\bar{\nu}_{e})$) probability
with two functionally equivalent and identical detectors - 290 ton Near
Detector and 14 kton Far Detector. Both detectors are situated off the central
beam axis and FD is placed 14.6 mrad off-axis from the central beam so that
the narrow neutrino energy flux peaks at around 2 GeV, near the first
oscillation maximum driven by $\Delta m^{2}_{32}=2.5\times 10^{-3}eV^{2}$.
This off-axis configuration enhances the $\nu_{e}$ oscillation probability in
the appearance channel. The NuMI Adamson:2015dkw  (Neutrinos at the Main
Injector) beam produces neutrinos at the Fermi National Accelerator Laboratory
which is measured by the ND placed at a distance of 1 km from the source and
105 m underground. NO$\nu$A measures oscillations by comparing the un-
oscillated energy spectra of ND and oscillated spectra of FD located near Ash
River, Minnesota at a distance of 810 km from the production target. NuMI beam
is produced by using 120 GeV protons from Main Injector impinging on a fixed
graphite target producing pions and kaons. Using magnetic horns, pions and
kaons of desired charge and momentum are focused into a narrow beam where
these particles further spontaneously decay to the muons, anti-muons, and
their neutrino partners. Muons and anti-muons are filtered out from the decay
pipe with 240 meters thick rock wall. Both the

Figure 3: Comparison of two-dimensional migration matrices using the
calorimetric and kinematic reconstruction method for QE(+RPA)+2p2h and QE for
GENIE using carbon as the target nucleus for ND. Figure 4: Comparison of two-
dimensional migration matrices using the calorimetric and kinematic
reconstruction method for QE(+RPA) and 2p2h for GENIE using carbon as the
target nucleus for ND.

detectors are made from planes of extruded polyvinyl chloride (PVC) cells. The
flavor composition of NuMI beam is 97.5% $\nu_{\mu}$, 1.8% $\bar{\nu}_{\mu}$
and 0.7% $\nu_{e}+\bar{\nu}_{e}$. The primary physics goal of NO$\nu$A is to
make precise measurements of $\theta_{23}$ mixing angle and its octant,
neutrino mass hierarchy, and CP-violating phase $\delta_{CP}$ in the lepton
sector. The $\nu_{\mu}(\bar{\nu}_{\mu})$ disappearance channel is used to
measure $\Delta m^{2}_{32}$ and $\sin^{2}\theta_{23}$ while
$\nu_{e}(\bar{\nu}_{e})$ appearance channel allows to measure mass hierarchy,
$\theta_{13}$, $\theta_{23}$ and $\delta_{CP}$.

Figure 5: Comparison of extrapolated FD events as a function of the
calorimetric (solid line) and kinematic (dashed line) reconstruction energy
for neutrino for NH (left panel) and IH (right panel) for different
interactions: QE(+RPA)+2p2h (black line), QE (red line), QE(+RPA) (blue line)
and 2p2h (magenta line). Figure 6: Comparison of extrapolated FD events as a
function of the calorimetric (solid line) and kinematic (dashed line)
reconstruction energy for antineutrino for NH (left panel) and IH (right
panel) for different interactions: QE(+RPA)+2p2h (black line), QE (red line),
QE(+RPA) (blue line) and 2p2h (magenta line).

##  III Energy Reconstruction Methods

This work aims to compare the two methods for reconstructing the neutrino
energy-(i) the kinematic method, and (ii) the calorimetric method, with the
inclusion of nuclear effects, and how they percolate into the sensitivity of
neutrino oscillation parameters. The kinematic method is directly related to
the kinematics (energy $E_{\mu}$ and angle $\theta_{\mu}$) of the outgoing
lepton. It is based on the assumption of QE scattering when the nucleon is at
rest, and constant binding energy is taken into account. The reconstructed
neutrino energy with this approach is defined as Coloma:2013tba ,

$E^{kin}_{\nu}=\frac{2(M_{n}-E_{B})E_{\textit{l}}-(E^{2}_{B}-2M_{n}E_{B}+\Delta
M^{2})}{2(M_{n}-E_{B}-E_{\textit{lep}}+\textit{p}_{\textit{lep}}\cos\theta_{\textit{lep}})}$
(1)

where $\Delta M^{2}=M^{2}_{n}-M^{2}_{p}+\textit{m}^{2}_{\textit{lep}}$ and
$E_{B}$ is the average binding energy of the nucleon inside the nucleus.
$M_{n}$ represents the free neutron rest mass, $E_{\textit{lep}}$ and
$\theta_{\textit{lep}}$ are the energy and angle of the outgoing lepton
respectively. This equation is justified only for true QE interaction with the
neutron at rest. Any mixture of other interactions gives incorrect
reconstruction energy. For example, if a pion produced in the initial state is
absorbed or undetected, the energy reconstruction for QE is lower than the
true value by $\sim$ 300-350 MeV Ankowski:2015jya . We use the outgoing lepton
kinematics information also for non-QE events in the kinematic analysis.

Contrary to the kinematic method, in the calorimetric method, the incoming
neutrino energy is reconstructed by adding the energies of all the observed
particles in the final state, and no information is needed regarding the
direction of the outgoing particles. It is computed as
$E_{cal}$=$E_{\mu}+E_{had}$. Muon energy $E_{\mu}$ is reconstructed from the
length of the muon track in the detector and the hadron energy $E_{had}$ is
calculated from calorimetry by adding all the visible energy except the muonic
energy. Thus, this method requires the information of all the detectable FSIs
on an event-by-event basis and is therefore considered to be more precise than
the kinematic method which is based on only the kinematics of the outgoing
lepton. The calorimetric method is also applicable to all kinds of CC
interactions. This method works effectively if it can reconstruct the hadrons
precisely and there is no missing energy. Typically neutrons escape the
detection and the undetected mesons lower the energy reconstruction by the
value of the pion mass, at least $\sim$ 135 MeV. Thus, the calorimetric method
mainly depends on the capability and design of the detector to accurately
reconstruct the final state particles. And this justifies more impact of
detector effects in the calorimetric method. Using the calorimetric approach,
the energy of a CC neutrino, scattering off a nuclear target producing ’m’
mesons and knocking out ’n’ nucleons can be reconstructed as:

Figure 7: Comparison of ND events for different binding energies as a function
of the calorimetric (solid line) and kinematic (dashed line) reconstruction
energy for neutrino for QE(+RPA)+2p2h process.

$E^{cal}_{\nu}=E_{lep}+\sum_{i}(E_{p}-M)+\epsilon_{n}+\sum_{m}E_{m}$ (2)

where $E_{lep}$ is the energy of the outgoing lepton, $E_{p}$ and $M$ denote
the energy of the $i$-th knocked-out nucleon and mass of the nucleon (target
nucleus). $\epsilon_{n}$ represents the single-nucleon separation energies of
the outgoing nucleons and we keep it to a fixed value of 25 MeV for the carbon
target for both neutrino and antineutrino for both the reconstruction methods.
$E_{m}$ stands for the energy of the $m$-th produced meson. Keeping this in
view, we studied the variation in the ND event distribution with binding
energy, for its three randomly chosen values (taking an uncertainty of 10%,
i.e. 22.5 MeV, 25 MeV, and 27.5 MeV), for both the reconstruction methods.

##  IV Mass hierarchy and CP sensitivity

In LBL neutrino experiments, the appearance channel
$\nu_{\mu}\rightarrow\nu_{e}$ is very sensitive in exploring the CP-violation
effect which is the most challenging problem in neutrino physics today. The
oscillation probability of $\nu_{\mu}\rightarrow\nu_{e}$ through matter in the
standard three-flavor scenario and constant density approximation can be
expressed as DUNE:2020jqi ,

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

$\times\frac{\sin(aL)}{aL}\Delta_{21}\cos(\Delta_{31}+\delta_{CP})+\cos^{2}\theta_{23}\sin^{2}2\theta_{12}\frac{\sin^{2}(aL)}{(aL)^{2}}\Delta^{2}_{21}$
(3)

where
$a=\frac{G_{F}N_{e}}{\sqrt{2}}\approx\pm\frac{1}{3500km}\frac{\rho}{3.0g/cm^{3}}$,
$G_{F}$ is the Fermi constant, $N_{e}$ is the number density of electrons in
Earth’s crust, $\Delta m^{2}_{ij}$ is the difference between mass squares of
neutrinos of $i$-th and $j$-th family, $\Delta_{ij}=1.267\Delta
m^{2}_{ij}L/E_{\nu}$, L is the baseline in km (810 km for NO$\nu$A), and
$E_{\nu}$ is the neutrino energy in GeV, $\rho$=2.848 $gcm^{-3}$. As the true
value of $\delta_{CP},$ is not known, the analysis is done by scanning all
possible true values of $\delta_{CP}$ over the entire range
$-\pi<\delta_{CP}<+\pi$ and comparing them with the CP conserving values i.e.,
0 or $\pm\pi$. To detect the CP violation, the value of the CP phase must be
different from 0 or $\pm\pi$. Therefore, the CP-violation sensitivity can be
obtained at a given true value of $\delta_{CP}$, by minimizing $\chi^{2}$ at
its fixed test values of 0 and $\pi$. 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)

The significance of CP-violation is obtained using
$\sigma=\sqrt{\Delta}\chi^{2}$ and it is shown in Fig. 16 (left panel).

Another important topic of interest in the neutrino oscillation experiments is
to understand the true nature of neutrino mass ordering, i.e., whether it is
normal or inverted. The mass hierarchy sensitivity is calculated by assuming a
normal (inverted) hierarchy to be a true hierarchy and comparing it to a test
inverted (normal) 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)

The true values for the oscillation parameters are used from Table I and the
test values of $\Delta m^{2}_{32}$, $\sin^{2}\theta_{23}$ and
$\sin^{2}\theta_{13}$ are marginalized over their $3\sigma$ ranges in both the
cases. The result is shown in Fig. 16 (right panel).

Figure 8: Comparison of 1$\sigma$, 2$\sigma$ and 3$\sigma$ contours in $\Delta
m^{2}_{32}$ vs $\sin^{2}\theta_{23}$ plane using the calorimetric (solid line)
and kinematic methods (dashed line) of energy reconstruction for QE(+RPA)+2p2h
for NH (left panel) and IH (right panel) for neutrino (blue line) and
antineutrino (red line) without detector effect. The global best-fit point is
shown by a black marker.

##  V Physics and Simulation Details

In $\nu_{\mu}$ ($\bar{\nu}_{\mu})$ disappearance channel analysis, the
$\nu_{\mu}$($\bar{\nu}_{\mu})$-CC interaction channel is used as the signal.
In NO$\nu$A detectors, the $\nu_{\mu}$-CC interactions are identified from the
long track of muons and any associated hadronic activity at the vertex. The
un-oscillated spectra from the NuMI beam are first measured at the ND. Then
these ND spectra are extrapolated to predict the FD spectra. The detailed
extrapolation technique used in this work can be found in our previous work
Deka:2021qnw . For sensitivity analysis of neutrino oscillation parameters
$\theta_{23}$ and $\Delta m^{2}_{32}$, we have used the Feldman-Cousins method
Feldman:1997qc  to calculate the allowed confidence level in the parameter
space Deka:2021qnw . We have generated 1 million CC $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ events sample for carbon target in the energy range 0-5 GeV
using NO$\nu$A-ND neutrino and antineutrino flux. Simulation is done using
neutrino event generator GENIE (Generates Events for Neutrino Interaction
Experiments ) v3.0.6 Andreopoulos:2009rq . Currently, GENIE is used by many
neutrino baseline experiments, such as Miner$\nu$a MINERvA:2019ope , MINOS
Adamson:2007gu , MicroBooNE Chen:2007ae , T2K Abe:2018wpn  and NO$\nu$A
experiments.

We have considered four interaction processes - QE, the resonance from
$\Delta$ resonant decay and contribution from higher resonances, nucleon-
nucleon correlations 2p2h, and DIS (deep inelastic scattering) in both
neutrino and antineutrino modes. We examined the simulated events with no
pions in the final state. In GENIE, by default QE scattering is modeled using
Llewellyn-Smith model LlewellynSmith:1971uhs  and the Local Fermi Gas (LFG)
model Nieves:2004wx  is used as the nuclear model in our work. LFG model of
nuclear ground state predicts a different initial nucleon momentum
distribution than the RFG model NOvA:2020rbg . When combined with the impact
of Pauli suppression, this difference changes the available kinematic space in
the QE interaction process. Resonance processes are predicted according to the
formalism attributed to Berger and Sehgal model Berger:2007rq . Further, to
include the effect of long-range nuclear charge screening according to RPA
correlations which also modify the kinematics of QE interaction, we simulate
the QE interactions with the Nieves et. al. model Nieves:2004wx . These
effects suppress the QE interaction process significantly at low invariant
four-momentum transferred to the nucleus ($Q^{2}$), and slightly enhance them
at higher $Q^{2}$, relative to the RFG prediction. DIS interaction which is
classified as a non-resonant process in GENIE is implemented following the
method of Bodek and Yang Bodek:2002ps . Using the Valencia model by Nieves et
al. Nieves:2012yz , two-nucleon knockout (2p2h) events are simulated. The
relevant oscillation probability for $\nu_{\mu}\rightarrow\nu_{\mu}$
disappearance channel NO$\nu$A is calculated as shown in Ref. Deka:2021qnw .
Oscillation parameters best fit values are taken from deSalas:2020pgw  and
shown in Table 1.

Figure 9: Comparison of 1$\sigma$, 2$\sigma$ and 3$\sigma$ contours in $\Delta
m^{2}_{32}$ vs $\sin^{2}\theta_{23}$ plane using the calorimetric (solid line)
and kinematic methods (dashed line) of energy reconstruction for QE for NH
(left panel) and IH (right panel) for neutrino (blue line) and antineutrino
(red line) without detector effect.

Detector response and efficiency are also included in the analysis, where the
measured energies are smeared with respect to the true values by finite
detector resolution. Incorrect detector capabilities produce a non-vanishing
probability such that an event with a true energy $E_{true}$ ends up being
reconstructed with different energy $E_{rec}$. These non-vanishing
probabilities are encoded in a set of migration matrices. We have considered
realistic specification for NO$\nu$A - 31.2% (33.9%) efficiency of selection
of $\nu_{\mu}$ ($\bar{\nu}_{\mu}$) events. For demonstration and comparison,
we also analyzed for 100% detector efficiency. 3.5% muon energy resolution and
25% hadron energy resolution are used respectively NOvA:2017ohq , which gives
an overall 7% energy resolution for $\nu_{\mu}$-CC events for both detectors.
We have simulated the true event rates with GENIE and global best fit values,
and the fitted rates are obtained for both the reconstruction methods by
varying the atmospheric oscillation parameters $\theta_{23}$ and $\Delta
m^{2}_{32}$ in their 3$\sigma$ range. After determining the minimum $\chi^{2}$
value, $\chi^{2}_{best-fit}$, the confidence regions are found from the
condition

$\Delta\chi^{2}(\theta_{23},\Delta
m^{2}_{32})\equiv\chi^{2}(\theta_{23},\Delta m^{2}_{32})-\chi^{2}_{best-
fit}<x$ (9)

where $x$= 2.30, 6.18, and 11.83 for the 1 $\sigma$, 2$\sigma$ and 3$\sigma$
confidence level, respectively.

parameter | best fit | 3$\sigma$ range
---|---|---
$\Delta m_{21}^{2}[10^{-5}eV^{2}]$ | $7.50$ | $6.94-8.14$
$|\Delta m_{31}^{2}|[10^{-3}eV^{2}]$(NH) | $2.56$ | $2.46-2.65$
$|\Delta m_{31}^{2}|[10^{-3}eV^{2}]$(IH) | $2.46$ | $2.37-2.55$
$\sin^{2}\theta_{23}/10^{-1}$(NH) | $5.66$ | $4.46-6.09$
$\sin^{2}\theta_{23}/10^{-1}$(IH) | $5.66$ | $4.41-6.09$
$\sin^{2}\theta_{13}/10^{-2}$(NH) | $2.225$ | $2.015-2.417$
$\sin^{2}\theta_{13}/10^{-2}$(IH) | $2.250$ | $2.039-2.441$
$\sin^{2}\theta_{12}/10^{-1}$(IH) | $3.18$ | $2.71-3.70$
Table 1: 3$\sigma$ values of neutrino oscillation parameters taken from
deSalas:2020pgw , which are used in this work.

Figure 10: Comparison of 1$\sigma$, 2$\sigma$ and 3$\sigma$ contours in
$\Delta m^{2}_{32}$ vs $\sin^{2}\theta_{23}$ plane using the calorimetric
(solid line) and kinematic methods (dashed line) of energy reconstruction for
QE(+RPA) for NH (left panel) and IH (right panel) for neutrino (blue line) and
antineutrino (red line) without detector effect

.

##  VI Results and Discussion

In Fig. 1, we have shown the total cross-sections per neutron on ${}^{12}C$
with no pions in the final state as a function of true neutrino energy for
four different interaction processes- QE(+RPA)+2p2h (black line), QE only (red
line), QE with RPA suppression QE(+RPA) (blue line), and 2p2h only (magenta
line) for neutrino (solid line) and antineutrino (dashed line). As we are
considering a complete model - QE with long-range RPA correlation and multi-
nucleon enhancement 2p2h, therefore QE(+RPA)+2p2h interaction shows a higher
contribution to the cross-section, which is also reflected in ND and FD event
distribution. A suppression in cross-section due to RPA correction is observed
in the energy range below 1 GeV for neutrino, after which both QE (without
RPA) and QE with RPA curve tend to overlap with a slight suppression in cross-
section for QE(+RPA). RPA correlations show maximum effect at forwarding
angles where the contribution of squared four-momentum transfer $Q^{2}$ is
small. They decrease with increasing angle and with increasing energy
transfer. RPA correlations tend to lower the cross-section in the QE-peak
region. Such effects have also been observed earlier too Nieves:2012yz ;
Singh:2019qac .

Figure 11: Comparison of 1$\sigma$, 2$\sigma$ and 3$\sigma$ contours in
$\Delta m^{2}_{32}$ vs $\sin^{2}\theta_{23}$ plane using the calorimetric
(solid line) and kinematic methods (dashed line) of energy reconstruction for
2p2h only for NH (left panel) and IH (right panel) for neutrino (blue line)
and antineutrino (red line) without detector effect.

In Fig. 2, the ND event distribution as a function of reconstructed energy,
for two energy reconstruction methods are shown for different interactions -
QE(+RPA)+2p2h (black line), QE only (red line), QE(+RPA) (blue line), and 2p2h
only (magenta line). From the comparison of the two methods, we notice that
event distribution with the kinematic method shifts towards the lower energy
side compared to the calorimetric method. This shift is more for QE(+RPA)+2p2h
and 2p2h while for QE and QE(+RPA), both the methods almost overlap in the
case of neutrino. A similar observation is also made in the case of the
antineutrino. The RPA suppression is less for neutrinos while it is distinctly
visible in the antineutrino case. Also, the amplitude of the distribution is
suppressed in the case of the calorimetric method (as compared to the
kinematic method). These features will lead to shifts in sensitivity contours
and can be attributed to different contributions in the energy reconstruction
formulae in the two methods.

The importance of the differences between the migration matrices for
calorimetric and kinematic reconstruction methods in the $\nu_{\mu}$
disappearance analysis for QE(+RPA)+2p2h, QE, QE(+RPA), and 2p2h interaction
processes using carbon as the target nucleus for ND can be understood from
Figs. 3 and 4. These matrices include FSI effects. Migration matrices are
drawn between true and reconstructed neutrino energies where each element
represents the probability that an event with specific true neutrino energy is
reconstructed with different neutrino energy, which corresponds to an energy
smearing as shown in Figs. 3 and 4. When the energy reconstruction method
works well, the probability distribution is nearly symmetric about the line
$E^{true}$=$E^{rec}$ and the probability of getting the energy reconstructed
at a different value is relatively small which gives a diagonal line without
any smearing. The broadening in the distribution comes from the Fermi motion
of the bound nucleons Lalakulich:2012hs . For all the events broadenings are
observed to the line $E^{true}$=$E^{rec}$. From Figs. 3 and 4, we observe that
the calorimetric method of energy reconstruction works rather well for all
four processes in comparison with the kinematic method. The QE(+RPA)+2p2h, QE,
QE(+RPA), and 2p2h events smear to a lesser extent when reconstructed with the
calorimetric method as compared to the kinematic method. The kinematic method
shows maximum smearing for QE(+RPA)+2p2h, QE, and QE(+RPA) which implies that
the calorimetric method reconstructs incoming neutrino energy more precisely
than the kinematic method. Smearing may be the effect of an incorrect
description of nuclear effects. Hence, to reduce the systematic uncertainties
in neutrino oscillation parameters, such errors in energy reconstruction in
these interactions should be taken care of carefully.

Figure 12: Comparison of 1$\sigma$, 2$\sigma$ and 3$\sigma$ contours in
$\Delta m^{2}_{32}$ vs $\sin^{2}\theta_{23}$ plane using the calorimetric
(solid line) and kinematic methods (dashed line) of energy reconstruction for
QE(+RPA)+2p2h for NH (left panel) and IH (right panel) with detector effect
for detector efficiency 31.2% ($\nu_{\mu}$) and 33.9% ($\bar{\nu}_{\mu}$) for
both neutrino (blue line) and antineutrino (red line). The black point
indicates the global best fit value. The magenta and black curves are for no
detector (ideal detector setup) case.

In Figs. 5 and 6, we have presented the FD events generated by the
extrapolation technique using migration matrices for both neutrino and
antineutrino for both mass hierarchies. We observe that FD events distribution
spectra broadens in the kinematic method while their amplitude increase in the
calorimetric method. The broadening of spectra may be related to that of
migration matrices in the kinematic reconstruction case (right panels in Figs.
3 and 4). Both the first and second oscillation maxima of the spectra with the
kinematic method shift towards a little lower energy side than the
calorimetric method. These spectra are the convolution of folding neutrino
flux, cross-section (Fig. 1), ND event distribution (Fig. 2), migration
matrices (Figs. 3 and 4), oscillation probability, nuclear effects, and
detector response. Multi-nucleon effects and reconstruction methods change the
amplitude as well as the position of the oscillation peaks in the FD events
spectrum. For $\nu_{\mu}$ CC events, the position, and amplitude of the
oscillation maximum in the FD energy spectra are strongly dependent on the
neutrino oscillation parameters $\Delta m^{2}$ and $\theta_{23}$, therefore
the differences in FD events distribution in the two construction methods (for
all the interaction processes) are reflected in sensitivity analysis of these
parameters.

How the choice of BE of target Carbon can affect event distribution in the two
energy reconstruction methods is shown in Fig. 7, at ND of NO$\nu$A. However,
almost no difference in a given reconstruction method is seen with a change in
BE.

Figure 13: Comparison of 1$\sigma$, 2$\sigma$ and 3$\sigma$ contours in
$\Delta m^{2}_{32}$ vs $\sin^{2}\theta_{23}$ plane using the calorimetric
(solid line) and kinematic methods (dashed line) of energy reconstruction for
QE for NH (left panel) and IH (right panel) for neutrino (blue line) and
antineutrino (red line) with detector effect for detector efficiency 31.2%
($\nu_{\mu}$) and 33.9% ($\bar{\nu}_{\mu}$). The black point indicates the
global best fit value.

In Figs. 8 and 9, we have shown the comparison of the calorimetric (solid
line) and kinematic method (dashed line) of energy reconstruction in
1$\sigma$, 2$\sigma$ and 3$\sigma$ confidence regions in the ($\Delta
m^{2}_{32}$, $\theta_{23}$) plane using the migration matrices for
QE(+RPA)+2p2h (Fig. 8) and QE (Fig. 9) for NH (left panel) and IH (right
panel) for both neutrino and antineutrino without detector effects. The
difference between calorimetric and kinematic methods is distinctly visible
for QE(+RPA)+2p2h interaction, for both the mass hierarchies and for both
neutrino and antineutrino. However, in the case of QE interaction, the
confidence regions for the kinematic method mostly coincide for neutrino and
antineutrino in 1$\sigma$ for both NH and IH. The difference between the two
reconstruction methods and the difference between neutrino and antineutrino is
less for QE than QE(+RPA)+2p2h interaction. The contour area decreases
significantly for the calorimetric method as compared to the kinematic method
which would imply an improvement in the precision measurement of oscillation
parameters using calorimetric reconstruction. Moreover, nuclear effects are
not significant for the calorimetric method when detector effects are not
included.

In Fig. 10, we have shown the comparison of the calorimetric (solid line) and
kinematic (dashed line) method in 1$\sigma$, 2$\sigma$ and 3$\sigma$ contours
in ($\Delta m^{2}_{32},\sin^{2}\theta_{23}$) plane for QE(+RPA) for NH (left
panel) and IH (right panel), for both neutrino and antineutrino (without
detector effects). We observe that in QE interaction with RPA enhancement, the
difference between calorimetric and kinematic methods is less for antineutrino
than neutrino which is a reflection of the observations from Fig. 2 and 6. For
the calorimetric method, the difference between neutrino and antineutrino
confidence regions is seen to be significant, but for the kinematic method,
they almost coincide with each other. A similar analysis for 2p2h is shown in
Fig. 11, in which it is observed that the calorimetric method shows less
uncertainty than the kinematic method.

Figure 14: Comparison of 1$\sigma$, 2$\sigma$ and 3$\sigma$ contours in
$\Delta m^{2}_{32}$ vs $\sin^{2}\theta_{23}$ plane using the calorimetric
(solid line) and kinematic methods (dashed line) of energy reconstruction for
QE(+RPA)+2p2h for NH (left panel) and IH (right panel) for neutrino (blue
line) and antineutrino (red line) with detector effect for detector efficiency
31.2% ($\nu_{\mu}$) and 33.9% ($\bar{\nu}_{\mu}$). The black point indicates
the global best fit value.

In Figs. 12 and 13, comparison of the calorimetric (solid line) and kinematic
method (dashed line) of energy reconstruction in 1$\sigma$, 2$\sigma$ and
3$\sigma$ contours in $\Delta m^{2}_{32}$ vs $\sin^{2}\theta_{23}$ plane for
QE(+RPA)+2p2h and QE for NH (left panel) and IH (right panel) for both
neutrino and antineutrino with detector effects with efficiency 31.2% and
33.9% are shown. We have also shown the no detector effect in 1$\sigma$
contribution to estimate the deviation from no detector to detector effect. In
Fig. 12, we observe that for both neutrino and antineutrino for efficiency
31.2% and 33.9%, the kinematic method shows better precision than the
calorimetric method. A similar analysis for 100% efficiency is shown in Fig.
14 and 15 respectively. These contours show how the confidence regions change
when detector effects (along with nuclear effects) are included in the
analysis. The quantum and nature of the above biases are almost the same in
the two mass hierarchies for neutrino, and uncertainty is generally more in
antineutrino. By comparison of respective figures, it is clear that in the no
detector case (ideal detector) and the case with 100% detector efficiency,
areas inside the contour are less in the calorimetric method - and hence less
uncertainty in parameter measurement with this method than with the kinematic
method. On the other hand, in cases with detector efficiency 31.2% and 33.9%,
the kinematic method shows less uncertainty.Thus, it is observed that the
calorimetric method of energy reconstruction is more sensitive to detector
effects when nuclear effects are included than the kinematic method. This can
be justified as the calorimetric method depends more on the performance of the
detector as energies of all particles in the final state need to be measured
accurately. Detector effects reduce the uncertainty in the sensitivity
analysis in both calorimetric and kinematic methods. This implies that
improvement in detector efficiency is important for improving precision. We
also observe that in the case of the kinematic method, the deviation from the
no detector effect to the detector effect is less than in the case of the
calorimetric method. This is because the kinematic method depends only on the
kinematics of the outgoing lepton and in modern neutrino detectors muons are
the most precisely reconstructed particles.

Figure 15: Comparison of 1$\sigma$, 2$\sigma$ and 3$\sigma$ contours in
$\Delta m^{2}_{32}$ vs $\sin^{2}\theta_{23}$ plane using the calorimetric
(solid line) and kinematic methods (dashed line) of energy reconstruction for
QE for NH (left panel) and IH (right panel) with detector effect for detector
efficiency 100% for both neutrino (blue line) and antineutrino (red line). The
black point indicates the global best fit value.

The dependence of sensitivity to the measurement of CPV phase $\delta_{CP}$,
and of mass hierarchy significance, on energy reconstruction methods are shown
in Fig. 16, for NH only for illustration, and here, detector effects have not
been included. In the left panel, we show the CP sensitivity variation for the
interaction processes QE(+RPA)+2p2h and QE, for neutrino+antineutrino mode.
The 2$\sigma(1\sigma)$ line in the left panel corresponds to $\Delta\chi^{2}$=
4(1) for one degree of freedom (dof). This indicates a 95.45%(68.3%)
probability of determining the CP-violation, and NH is considered as the true
hierarchy. The variation of CP sensitivity with energy reconstruction methods
is seen to be present at $\leqslant 1\sigma$ level, and the difference between
QE(+RPA)+2p2h and QE processes, for both the calorimetric and kinematic
methods is seen to be of the order $\sim 0.2\sigma$ at
$\delta_{CP}\sim-0.5\pi$ and $\delta_{CP}\sim 0.5\pi$. Though the sensitivity
is more for the complete model, i.e. QE(+RPA)+2p2h than only the QE process,
for a given interaction, however, the dependence on the energy reconstruction
method is not very strong. In Fig 16 (right panel) we have shown the mass
hierarchy significance plot with NH as the true hierarchy. The 2$\sigma$ and
1$\sigma$ lines in the right panel indicate approximately 95.45% and 68.3%
probability of determining the MH correctly. The $\delta_{CP}$ values for
which the curve is above 2$\sigma$ (3$\sigma$) are the values for which the
hierarchy can be determined with 95.45%(68.3%) confidence level (CL). We
observe the difference in CL in the two energy reconstruction methods is the
maximum of the order $\sim 0.3\sigma$ at $\delta_{CP}\sim-0.5\pi$. It may be
noted that qualitatively similar results on CP sensitivity MH significance
have been reported earlier also Pershey:2018gtf ; Singh:2019qac ;
Deepthi:2014iya . Thus, energy reconstruction methods can play important role
in MH measurements as well.

Figure 16: Left panel: Plots for CP-violation sensitivity without detector effect for neutrino+antineutrino mode. Normal hierarchy (inverted hierarchy) is considered as a true hierarchy for QE(+RPA)+2p2h and QE interactions. Right panel: Mass hierarchy significance as a function of true $\delta_{CP}$ without detector effect for neutrino+antineutrino mode. Normal hierarchy (inverted hierarchy) is considered as true hierarchy and inverted (normal) is taken as test hierarchy for QE(+RPA)+2p2h and QE interactions. Details of the Process | $\sin^{2}\theta_{23,min}$ | $\Delta m^{2}_{32,min}[eV^{2}]$ | $\chi^{2}_{min}$ |  $\sin^{2}\theta_{23},\Delta m^{2}_{32}$(Global fit) | Fig. no.
---|---|---|---|---|---
$\nu_{\mu}$ QE(+RPA)+2p2h (Cal) (NH) | $0.524$ | $2.48\times 10^{-3}$ | $2.57892$ |  | $(8)$
$\nu_{\mu}$ QE(+RPA)+2p2h (Kine) (NH) | $0.46$ | $2.56\times 10^{-3}$ | $1.7667$ |  | $(8)$
$\nu_{\mu}$ QE (Cal) (NH) | $0.524$ | $2.48\times 10^{-3}$ | $2.06898$ |  | $(9)$
$\nu_{\mu}$ QE (Kine) (NH) | $0.54$ | $2.56\times 10^{-3}$ | $1.74289$ |  | $(9)$
$\nu_{\mu}$ QE(+RPA) (Cal) (NH) | $0.524$ | $2.48\times 10^{-3}$ | $2.02843$ |  $0.566,2.47\times 10^{-3}$ (NH) | $(10)$
$\nu_{\mu}$ QE(+RPA) (Kine) (NH) | $0.54$ | $2.56\times 10^{-3}$ | $1.62356$ |  | $(10)$
$\nu_{\mu}$ 2p2h (Cal) (NH) | $0.524$ | $2.48\times 10^{-3}$ | $0.55554$ |  | $(11)$
$\nu_{\mu}$ 2p2h (Kine) (NH) | $0.524$ | $2.56\times 10^{-3}$ | $0.393395$ |  | $(11)$
$\nu_{\mu}$ QE(+RPA)+2p2h (Cal) (NH) (Eff=31.2%) | $0.54$ | $2.56\times 10^{-3}$ | $0.649116$ |  | $(12)$
$\nu_{\mu}$ QE(+RPA)+2p2h (Kine) (NH) (Eff=31.2%) | $0.492$ | $2.48\times 10^{-3}$ | $1.58913$ |  | $(12)$
$\nu_{\mu}$ QE (Cal) (NH) (Eff=31.2%) | $0.476$ | $2.48\times 10^{-3}$ | $0.580956$ |  $0.566,2.47\times 10^{-3}$ (NH) | $(13)$
$\nu_{\mu}$ QE (Kine) (NH) (Eff=31.2%) | $0.492$ | $2.56\times 10^{-3}$ | $1.47534$ |  | $(13)$
$\nu_{\mu}$ QE(+RPA)+2p2h (Cal) (NH) (Eff=100%) | $0.54$ | $2.56\times 10^{-3}$ | $2.23225$ |  | $(14)$
$\nu_{\mu}$ QE(+RPA)+2p2h (Kine) (NH) (Eff=100%) | $0.508$ | $2.56\times 10^{-3}$ | $1.48601$ |  | $(14)$
$\nu_{\mu}$ QE (Cal) (NH) (Eff=100%) | $0.54$ | $2.56\times 10^{-3}$ | $2.05989$ |  $0.566,2.47\times 10^{-3}$ (NH) | $(15)$
$\nu_{\mu}$ QE (Kine) (NH) (Eff=100%) | $0.476$ | $2.56\times 10^{-3}$ | $1.50631$ |  | $(15)$
$\nu_{\mu}$ QE(+RPA)+2p2h (Cal) (IH) | $0.476$ | $2.4\times 10^{-3}$ | $2.68461$ |  | $(8)$
$\nu_{\mu}$ QE(+RPA)+2p2h (Kine) (IH) | $0.508$ | $2.48\times 10^{-3}$ | $1.81933$ |  | $(8)$
$\nu_{\mu}$ QE (Cal) (IH) | $0.476$ | $2.4\times 10^{-3}$ | $2.16304$ |  | $(9)$
$\nu_{\mu}$ QE (Kine) (IH) | $0.524$ | $2.56\times 10^{-3}$ | $1.25265$ |  | $(9)$
$\nu_{\mu}$ QE(+RPA) (Cal) (IH) | $0.476$ | $2.4\times 10^{-3}$ | $2.12139$ |  | $(10)$
$\nu_{\mu}$ QE(+RPA) (Kine) (IH) | $0.46$ | $2.48\times 10^{-3}$ | $1.76127$ |  $0.566,2.38\times 10^{-3}$(IH) | $(10)$
$\nu_{\mu}$ 2p2h (Cal) (IH) | $0.524$ | $2.4\times 10^{-3}$ | $0.571004$ |  | $(11)$
$\nu_{\mu}$ 2p2h (Kine) (IH) | $0.54$ | $2.48\times 10^{-3}$ | $0.435696$ |  | $(11)$
$\nu_{\mu}$ QE(+RPA)+2p2h (Cal) (IH) (Eff=31.2%) | $0.492$ | $2.4\times 10^{-3}$ | $0.695153$ |  | $(12)$
$\nu_{\mu}$ QE(+RPA)+2p2h (Kine) (IH) (Eff=31.2%) | $0.492$ | $2.48\times 10^{-3}$ | $1.4102$ |  | $(12)$
$\nu_{\mu}$ QE (Cal) (IH) (Eff=31.2%) | $0.476$ | $2.48\times 10^{-3}$ | $0.574578$ |  $0.566,2.38\times 10^{-3}$(IH) | $(13)$
$\nu_{\mu}$ QE (Kine) (IH) (Eff=31.2%) | $0.508$ | $2.48\times 10^{-3}$ | $1.05391$ |  | $(13)$
$\nu_{\mu}$ QE(+RPA)+2p2h (Cal) (IH) (Eff=100%) | $0.508$ | $2.4\times 10^{-3}$ | $2.32447$ |  | $(14)$
$\nu_{\mu}$ QE(+RPA)+2p2h (Kine) (IH) (Eff=100%) | $0.46$ | $2.48\times 10^{-3}$ | $1.48444$ |  | $(14)$
$\nu_{\mu}$ QE (Cal) (IH) (Eff=100%) | $0.508$ | $2.40\times 10^{-3}$ | $2.08782$ |  $0.566,2.38\times 10^{-3}$(IH) | $(15)$
$\nu_{\mu}$ QE (Kine) (IH) (Eff=100%) | $0.476$ | $2.48\times 10^{-3}$ | $1.55938$ |  | $(15)$
Table 2: Summary of best-fit values and corresponding $\chi^{2}$ values of the
oscillation parameters along with global best-fit values for neutrino for
different scenarios studied in this work.

In Table 2 and Table 3, we present the summary of best-fit values and
corresponding $\chi^{2}$ values of the oscillation parameters, along with
global best-fit values for neutrino and antineutrino for different scenarios
studied in this work. We followed the Feldman-Cousins Feldman:1997qc  unified
approach to determine confidence intervals within the range of likelihood
ratios in pseudo experiments. From the results in these Tables, it is depicted
how the best fit values change in different cases from no detector to detector
effect, from NH to IH, and from calorimetric to kinematic methods for
different interaction processes. The NO$\nu$A $\nu_{\mu}$ disappearance
results constrains $\sin^{2}\theta_{23}$ around degenerate best fit points of
0.404 and 0.624. From NO$\nu$A result, it is observed that the best fit to the
data gives $\Delta m^{2}_{32}=(+2.67\pm 0.11)\times 10^{-3}eV^{2}$ and
$\sin^{2}\theta_{23}$ at the two statistically degenerate values
$0.404^{+0.030}_{-0.022}$ and $0.624^{+0.022}_{-0.030}$ both at the 68% C.L.
in the NH. For IH, $\Delta m^{2}_{32}=(-2.72\pm 0.11)\times 10^{-3}eV^{2}$ and
$\sin^{2}\theta_{23}=0.398^{+0.030}_{-0.022}$ or $0.618^{+0.022}_{-0.030}$ at
68% C.L NOvA:2017ohq . In NO$\nu$A, the statistical significance of these
contours has been determined using the same Feldman-Cousins unified approach.

##  VII Summary and Conclusion

Nuclear effects in neutrino interactions are one of the leading sources of
systematic uncertainties in neutrino oscillation experiments. In recent T2K
appearance results Abe:2019vii , they are shown as among the largest
contributors to the overall systematic errors. Also, neutrino energy
reconstruction directly influences the precision of measurement of $\Delta
m^{2}$ and $\theta_{23}$ parameters in LBL experiments. With this as
motivation, in this work, we estimated the systematic uncertainties associated
with the theoretical models of nuclear effects with two energy reconstruction
methods - kinematic and calorimetric methods. We analyzed the impact of these
two neutrino energy reconstruction methods on the extraction of neutrino
oscillation parameters along with multi-nucleon enhancement in
$\nu_{\mu}\rightarrow\nu_{\mu}$ disappearance channel, and CP sensitivity and
mass hierarchy significance in $\nu_{\mu}\rightarrow\nu_{e}$ appearance
channel of LBL NO$\nu$A experiment, in the energy range (0-5) GeV. We computed
the contribution of RPA corrections and multi-nucleon effect 2p2h separately
along with QE and the QE(+RPA)+2p2h interaction processes to neutrino-nucleus
scattering cross-sections, ND event, and FD events, and migration matrices,
using both the reconstruction methods. How detector efficiencies, resolutions,
and nuclear model uncertainties affect the parameter oscillation analysis
simulating the event distributions in FD using the realistic detector scenario
for NO$\nu$A was also analyzed. The observed differences in the energy
distributions of events both at ND and FD for the two energy reconstruction
methods have important consequences for the oscillation analysis. The
inaccuracies that appear in the description of nuclear effects are translated
into a bias in the reconstruction of neutrino energy which further produces a
bias in the extraction of neutrino oscillation parameters. The main results,
i.e. the comparison of sensitivity analysis using the calorimetric and
kinematic methods with the inclusion of nuclear effects in interactions and
realistic detector effects of NO$\nu$A, were shown in Figs. 8-16.

Details of the Process | $\sin^{2}\theta_{23,min}$ | $\Delta m^{2}_{32,min}[eV^{2}]$ | $\chi^{2}_{min}$ |  $\sin^{2}\theta_{23},\Delta m^{2}_{32}$(Global fit) | Fig. no.
---|---|---|---|---|---
$\bar{\nu}_{\mu}$ QE(+RPA)+2p2h (Cal) (NH) | $0.54$ | $2.56\times 10^{-3}$ | $1.68726$ |  | $(8)$
$\bar{\nu}_{\mu}$ QE(+RPA)+2p2h (Kine) (NH) | $0.524$ | $2.56\times 10^{-3}$ | $1.13777$ |  | $(8)$
$\bar{\nu}_{\mu}$ QE (Cal) (NH) | $0.54$ | $2.56\times 10^{-3}$ | $1.45461$ |  | $(9)$
$\bar{\nu}_{\mu}$ QE (Kine) (NH) | $0.54$ | $2.56\times 10^{-3}$ | $1.29659$ |  | $(9)$
$\bar{\nu}_{\mu}$ QE(+RPA) (Cal) (NH) | $0.54$ | $2.56\times 10^{-3}$ | $1.25655$ |  $0.566,2.47\times 10^{-3}$ (NH) | $(10)$
$\bar{\nu}_{\mu}$ QE(+RPA) (Kine) (NH) | $0.54$ | $2.56\times 10^{-3}$ | $1.11804$ |  | $(10)$
$\bar{\nu}_{\mu}$ 2p2h (Cal) (NH) | $0.54$ | $2.56\times 10^{-3}$ | $0.439069$ |  | $(11)$
$\bar{\nu}_{\mu}$ 2p2h (Kine) (NH) | $0.524$ | $2.56\times 10^{-3}$ | $0.287889$ |  | $(11)$
$\bar{\nu}_{\mu}$ QE(+RPA)+2p2h (Cal) (NH) (Eff=33.9%) | $0.492$ | $2.56\times 10^{-3}$ | $0.564969$ |  | $(12)$
$\bar{\nu}_{\mu}$ QE(+RPA)+2p2h (Kine) (NH) (Eff=33.9%) | $0.492$ | $2.48\times 10^{-3}$ | $0.940515$ |  | $(12)$
$\bar{\nu}_{\mu}$ QE (Cal) (NH) (Eff=33.9%) | $0.524$ | $2.56\times 10^{-3}$ | $0.433007$ |  $0.566,2.47\times 10^{-3}$ (NH) | $(13)$
$\bar{\nu}_{\mu}$ QE (Kine) (NH) (Eff=33.9%) | $0.492$ | $2.48\times 10^{-3}$ | $1.24153$ |  | $(13)$
$\bar{\nu}_{\mu}$ QE(+RPA)+2p2h (Cal) (NH) (Eff=100%) | $0.54$ | $2.56\times 10^{-3}$ | $1.41253$ |  | $(14)$
$\bar{\nu}_{\mu}$ QE(+RPA)+2p2h (Kine) (NH) (Eff=100%) | $0.476$ | $2.56\times 10^{-3}$ | $0.907663$ |  | $(14)$
$\bar{\nu}_{\mu}$ QE (Cal) (NH) (Eff=100%) | $0.46$ | $2.56\times 10^{-3}$ | $1.3784$ |  $0.566,2.47\times 10^{-3}$ (NH) | $(15)$
$\bar{\nu}_{\mu}$ QE (Kine) (NH) (Eff=100%) | $0.476$ | $2.56\times 10^{-3}$ | $1.00896$ |  | $(15)$
$\bar{\nu}_{\mu}$ QE(+RPA)+2p2h (Cal) (IH) | $0.46$ | $2.48\times 10^{-3}$ | $1.88138$ |  | $(8)$
$\bar{\nu}_{\mu}$ QE(+RPA)+2p2h (Kine) (IH) | $0.524$ | $2.56\times 10^{-3}$ | $1.16509$ |  | $(8)$
$\bar{\nu}_{\mu}$ QE (Cal) (IH) | $0.524$ | $2.4\times 10^{-3}$ | $1.56191$ |  | $(9)$
$\bar{\nu}_{\mu}$ QE (Kine) (IH) | $0.476$ | $2.4\times 10^{-3}$ | $1.47151$ |  | $(9)$
$\bar{\nu}_{\mu}$ QE(+RPA) (Cal) (IH) | $0.46$ | $2.48\times 10^{-3}$ | $1.40621$ |  $0.566,2.38\times 10^{-3}$(IH) | $(10)$
$\bar{\nu}_{\mu}$ QE(+RPA) (Kine) (IH) | $0.54$ | $2.48\times 10^{-3}$ | $1.31729$ |  | $(10)$
$\bar{\nu}_{\mu}$ 2p2h (Cal) (IH) | $0.46$ | $2.48\times 10^{-3}$ | $0.477686$ |  | $(11)$
$\bar{\nu}_{\mu}$ 2p2h (Kine) (IH) | $0.46$ | $2.48\times 10^{-3}$ | $0.322048$ |  | $(11)$
$\bar{\nu}_{\mu}$ QE(+RPA)+2p2h (Cal) (IH) (Eff=33.9%) | $0.46$ | $2.48\times 10^{-3}$ | $0.500182$ |  | $(12)$
$\bar{\nu}_{\mu}$ QE(+RPA)+2p2h (Kine) (IH) (Eff=33.9%) | $0.492$ | $2.48\times 10^{-3}$ | $0.977105$ |  $0.566,2.38\times 10^{-3}$(IH) | $(12)$
$\bar{\nu}_{\mu}$ QE (Cal) (IH) (Eff=33.9%) | $0.476$ | $2.56\times 10^{-3}$ | $0.481189$ |  | $(13)$
$\bar{\nu}_{\mu}$ QE (Kine) (IH) (Eff=33.9%) | $0.492$ | $2.32\times 10^{-3}$ | $1.15407$ |  | $(13)$
$\bar{\nu}_{\mu}$ QE(+RPA)+2p2h (Cal) (IH) (Eff=100%) | $0.444$ | $2.48\times 10^{-3}$ | $1.69252$ |  | $(14)$
$\bar{\nu}_{\mu}$ QE(+RPA)+2p2h (Kine) (IH) (Eff=100%) | $0.508$ | $2.48\times 10^{-3}$ | $0.913228$ |  | $(14)$
$\bar{\nu}_{\mu}$ QE (Cal) (IH) (Eff=100%) | $0.46$ | $2.48\times 10^{-3}$ | $1.46947$ |  $0.566,2.38\times 10^{-3}$(IH) | $(15)$
$\bar{\nu}_{\mu}$ QE (Kine) (IH) (Eff=100%) | $0.476$ | $2.48\times 10^{-3}$ | $1.0685$ |  | $(15)$
Table 3: Summary of best-fit values and corresponding $\chi^{2}$ values of the
oscillation parameters along with global best-fit values for antineutrino for
different scenarios studied in this work.

The highlights and new findings of this work can be summarised as - using flux
and realistic detector specifications of NO$\nu$A in this study, and in all
cases that we investigated, the calorimetric method (except for detector
efficiency 31.2% and 33.9%) produces more precision than the kinematic one, in
the parameter sensitivity analysis, for both neutrino and antineutrino and
both the mass hierarchies. It is noted that realistic detector effects enhance
the uncertainty in the sensitivity analysis in both calorimetric and kinematic
methods, while better efficiency of the detector reduces it. Less smearing is
observed in migration matrices at ND when the calorimetric method is used
(Fig. 3 and 4), than when the kinematic method, which is a clear indication
that the former method is more efficient for energy reconstruction. It is
observed that at ND, the kinematic method shows more bias in smearing than the
calorimetric method when nuclear effects are included (Fig. 3). In sensitivity
analysis too, this feature is seen. The dependence of event distribution at ND
with randomly chosen BE of target Carbon was shown in Fig. 7, and no
significant dependence on BE is observed. When realistic detector
specifications are not included, the kinematic method is affected more by the
nuclear models (and it depends on the correct understanding of neutrino-
nucleus interactions), while the calorimetric method is less biased by them
(Fig. 8 and 9). A large contribution to uncertainties due to multi-nucleon
enhancement (QE+RPA and 2p2h) both in neutrino and antineutrino cases is
observed (Fig. 10 and 11). This contribution is significantly large for the
kinematic method than for the calorimetric method, as an ideal detector set-up
was used in Figs. 10 and 11. With the inclusion of realistic detector setup
details in these two methods (with nuclear effects), it is found that the
calorimetric method shows more bias for efficiency 31.2% and 33.9%
(uncertainty increases, e.g. see Figs. 12-15) in sensitivity contours, as
compared to the kinematic method. We did this analysis for both the mass
hierarchies as well as for incoming neutrino and antineutrino beams. We note
that the quality and quantity of the above biases are almost the same in the
two MH for neutrino, and uncertainty is generally less in neutrino than in
antineutrino. Through our results in Fig 16, we showed that CP and MH
sensitivity have some dependence on energy reconstruction methods, and hence
these should be included in such analyses of experimental data carefully.

Though our results are in qualitative agreement with those of a similar work
Ankowski:2015jya , they are an addition of more detailed and up-to-date
results to those of Ankowski:2015jya  giving a deeper insight - as we used
realistic detector specifications of NO$\nu$A LBL experiment, used a latest
complete model for nuclear interactions Deka:2021qnw  along with long-range
RPA corrections and multi-nucleon enhancement, used latest global fit values
of known neutrino oscillation parameters, compared the effect of two different
values of detector efficiencies, showed migration matrices in details and did
the whole analysis for both the mass hierarchies and for both neutrino and
antineutrino. These results therefore can be useful for pinpointing neutrino
mass hierarchy as well as the measurement of the leptonic CP phase. As the
calorimetric method is seen to be less sensitive to nuclear models, thus it
appears to be an effective way of minimizing systematic uncertainties in
oscillation experiments, though with a careful inclusion of nuclear effects.
Of course, one needs to improve the performance of detectors too, for more
precise extraction of unknown neutrino oscillation parameters. The results
presented in this work can be used as a guiding tool for a careful inclusion
of all these effects into neutrino event generators and nuclear models for
oscillation parameter sensitivity analysis particularly at NO$\nu$A and in
general at all LBL experiments, including leptonic CP phase and MH
measurement.

##  VIII Acknowledgments

KB and PD acknowledge support from FIST grant SR/FST/PSI-213/2016(C) dtd.
24/10/2017(Govt. of India) in upgrading the computer laboratory of the Physics
department of GU where part of this work was done. KB thanks DST-SERB, Govt.
of India for a research project EMR/2014/000296 (2015-2019) during which some
initial part of this work was done.

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