# Impact of Cross-Sectional Uncertainties on DUNE Sensitivity due to Nuclear
Effects.

Srishti Nagu1 111E-mail: srishtinagu19@gmail.com,Jaydip Singh1 222E-mail:
<EMAIL_ADDRESS>Jyotsna Singh1 333E-mail<EMAIL_ADDRESS>R.B. Singh1 444Email<EMAIL_ADDRESS>Department Of Physics,
University of Lucknow, Lucknow, India.1

###### Abstract

In neutrino oscillation experiments precise measurement of neutrino
oscillation parameters is of prime importance as well as a challenge. To
improve the statistics, presently running and proposed experiments are using
heavy nuclear targets. These targets introduce nuclear effects and the
quantification of these effects on neutrino oscillation parameters will be
decisive in the prediction of neutrino oscillation physics. Limited
understanding of neutrino nucleus interactions and inaccurate reconstruction
of neutrino energy causes uncertainty in the cross section. The error in the
determination of cross section which contributes to systematic error
introduces error in the neutrino mixing parameters that are determined by
these experiments. In this work we focus on the variation in the predictions
of DUNE potential, arising due to systematic uncertainties, using two
different event generators-GENIE and GiBUU. These generators have different
and independent cross-section models. To check the DUNE potential with the two
generators we have checked the sensitivity studies of DUNE for CP violation,
mass hierarchy and octant degeneracy.

##  I Introduction

The aim of studying the properties and interactions of the most elusive
particles neutrinos is to infer their true nature and explore physics beyond
the standard model. The past decades have witnessed remarkable discoveries in
the field of neutrino oscillation physics with the help of phenomenal
experiments, substantiating the existence of neutrino oscillations. Neutrino
oscillation implies the change of neutrino flavor as they travel i.e., a
neutrino which is generated with a certain flavor after traveling a certain
distance might end up having a different flavor. Neutrino oscillation
parameters that govern neutrino oscillation physics are mixing angles
$\theta_{ij}$ where $j>i$=1,2,3 ($\theta_{12},\theta_{13},\theta_{23}$), dirac
phase $\delta_{CP}$ and the magnitude of mass squared differences, $\Delta
m_{21}^{2}$ known as solar mass splitting and $\Delta m_{31}^{2}$ known as
atmospheric mass splitting. Much progress on the precise determination of
neutrino oscillation parameters has been made by achieving nearly precise
determination of the mixing angles $\theta_{12}$, $\theta_{23}$ and non-zero
value of $\theta_{13}$ theta13_1 ; theta13_2 ; theta13_3  and mass squared
differences $\Delta m_{21}^{2}$, $|\Delta m_{31}^{2}|$. The remaining unknown
parameters on the canvas of neutrino oscillation physics are- (1)the sign of
$\Delta m_{31}^{2}$ or the neutrino mass ordering. There are two possibilities
of arrangement, for the neutrino mass eigenstates $m_{i}$(i=1,2,3). One is
normal mass ordering or the normal mass hierarchy(NH) where the neutrino mass
order is- $m_{1}\ll m_{2}\ll m_{3}$ and the other is inverted mass ordering or
the inverted mass hierarchy(IH) where the neutrino mass order is-
$m_{2}\approx m_{1}\gg m_{3}$ (2) determination of the octant of
$\theta_{23}$, whether the value of $\theta_{23}$ lies in the lower octant(LO)
$0<\theta_{23}<\pi/4$ or higher octant(HO) $\pi/4<\theta_{23}<\pi/2$. This
uncertainty in the octant of $\theta_{23}$ is known as the octant degeneracy
problem which describes the incapability of an experiment to distinguish
between the values $\theta_{23}$ and $(\pi/2-\theta_{23})$ (3) determination
of the value of dirac phase $\delta_{CP}$ which can lie in the range
$-\pi<\delta_{CP}<\pi$. As we know if the value of this parameter differs from
0 or $\pi$, it would indicate CP violation in the leptonic sector. This
discovery can shed light on the origin of leptogenesis leptog  and can be a
tool to answer some of the intriguing questions like baryon asymmetry of the
universe baryog . Precise CP phase value is also required for the exact
absolute neutrino mass measurement in double beta decay experiments and also
for explaining the sterile neutrinos phenomenon jd . The global analysis 0c
shows two sets of best fit values of neutrino oscillation parameters in
1$\sigma$ and 3$\sigma$ ranges that correspond to the analysis done, with and
without Super-Kamiokande atmospheric neutrino data.

The pre-requisite for precise knowledge of neutrino oscillation physics
depends on many factors amongst which precise reconstruction of neutrino
energy is of extreme importance. As we know that the neutrino oscillation
probability itself depends on the energy of the neutrinos, any incorrect
measurement of neutrino energy will be propagated to the measurements of
neutrino oscillation parameters since it causes uncertainties in the cross
section measurement and event identification. Many important long baseline
neutrino oscillation experiments use accelerator generated neutrino beams.
These neutrino beams are not monoenergetic, thus for the reconstruction of
neutrino energy complete information of final state particles is required. The
energy reconstruction of neutrinos from final state particles need careful
examination since the identification of final state particles in presence of
nuclear effects is a challenging task because the particles produced at the
initial neutrino-nucleon interaction vertex and the particles captured by the
detector can be different or not identical. Presently running and proposed
future experiments use heavy nuclear targets in order to collect large event
statistics as a a priori requirement of neutrino oscillation experiments but
at the same time, the use of heavy nuclear targets gives a boost to nuclear
effects. The high event statistics obtained, minimize the statistical error
and shifts the attention to explore a handle to control systematic errors. The
uncertainties in the determination of neutrino-nucleus cross-sections arising
due to the presence of nuclear effects are one of the most important sources
of systematic errors. It is important to investigate the precise neutrino-
nucleon interaction cross-sections in an attempt to reduce systematic errors.
The studies of interrelation between uncertainties in neutrino-nucleon cross
sections and its impact on the determination of neutrino oscillation
parameters have been explored previously in many research works crossoss1 ;
crossoss2 ; crossoss3 ; crossoss4 . To control the systematic errors, the
current knowledge of nuclear effects is still insufficient, as stated in
nuceffect1 ; nuceffect2 ; nuceffect3 ; nuceffect4 ; snaaz ; moselcross .

Here, in an attempt to capture nuclear effects, we have selected two different
simulation tools, GENIE 4  and GiBUU 5 . Both the neutrino event generators
incorporate nuclear effects in their simulation codes but differ in the
selection of nuclear models and computation of various neutrino-nucleus
interaction processes. The nucleus is a collection of nucleons and the study
of the effect of all the nucleons in neutrino- nucleus interactions is not
trivial. Different neutrino event generators which include nuclear effects in
their analysis program use different approximations to define nuclear effects.
Since the result of an experiment must be model independent, this motivated us
to perform our analysis.

The future Long-Baseline Deep Underground Neutrino Experiment(DUNE) ref0 ;
ref1 ; ref4 ; ref2 ; ref3 , as one of the third generation neutrino experiment
is in the process of being set up at United States and aims to explore the key
problems in neutrino physics i.e. the determination of neutrino mass
hierarchy, octant degeneracy, CP violation and certainly new physics. Pinning
down the systematic uncertainties in the proposed experiment will help us to
achieve these goals up to the desired sensitivity. An intense megawatt scale
muon neutrino beam produced at Fermilab will aim at two detectors: the near
detector, 575 meters from the target (fine-grained magnetic spectrometer) and
the far detector which will be a suite of four independent 10 kt Liquid Argon
Time Projection Chambers(LArTPCs), situated 1300 km from the target in the
Sanford Underground Research Facility(SURF), South Dakota. The DUNE-LBNF flux
spreads in the energy range 0.5 to 10 GeV, having an average energy of 2.5
GeV. It is composed of QE(Quasi-Elastic), RES(Resonance), DIS(Deep Inelastic
Scattering) and Coherent neutrino-nucleon interaction processes each having a
different energy dependent cross-section.

The paper is organized in the following sections: In Section II we describe
the neutrino event generators GENIE and GiBUU used in this work including a
detailed comparison of the physics incorporated in them. We outline the
simulation and experimental details in Section III, followed by a discussion
of the CP sensitivity, mass hierarchy and octant sensitivity results in
Section IV. Finally, we present our conclusions in Section V.

##  II EVENT GENERATORS: GENIE AND GiBUU

The interaction cross section($\nu$-Ar) used in this work is computed with two
neutrino event generators: GENIE(Generates Events for Neutrino Interaction
Experiments)2.12.06 4  and GiBUU(Giessen Boltzmann-Uehling-Uhlenbeck)v-2019 5
. We have considered the quasi-elastic(QE), resonance (RES) from $\Delta$
resonant decay and contribution from higher resonances, two particle-two
hole(2p2h/MEC) and deep inelastic scattering(DIS) interaction processes. The
estimated total cross-section from the two generators is further converted
into the input format of GLoBES package. Neutrino cross section as a function
of neutrino energy is shown in Figure 1. From both the generators, we observe
a difference in the value of cross sections for $\nu_{e}$ and $\nu_{\mu}$
both. Above energy 3 GeV, a difference in the cross-sections of $\nu_{e}$ and
$\nu_{\mu}$ is observed if they are computed via GiBUU whereas no such
difference is observed if they are estimated using GENIE. The difference in
the neutrino-nucleon interaction cross-sections introduced by the two
generators results in different event rates as a function of reconstructed
neutrino energy, as illustrated in Figure 2. The presented event rate is
estimated using GLoBES, by considering normal hierarchy as true hierarchy and
$\delta_{CP}=-\pi/2$.

Figure 1: Total neutrino-argon interaction cross-section per nucleon as a
function of neutrino energy by GENIE and GiBUU in the energy regime 1-10 GeV,
for different charged current processes considered in our work.

Figure 2: Left panel shows $\nu_{\mu}$ disappearance and right panel shows
$\nu_{e}$ appearance event distributions as a function of reconstructed
neutrino energy for both reference and optimized beamline designs in the
energy regime 1-10 GeV.

In this section, we will discuss the qualitative theoretical difference in the
nuclear models and the way neutrino-nucleus interaction processes are
considered in both the generators and some common approach used for the
treatment of neutrino-nucleus interaction analysis. The selected event
generators differ in the selection of nuclear models describing the neutrino-
nucleus interactions. GENIE is a ROOT 7a  based neutrino event generator
designed using object oriented methodologies and developed entirely in C++. It
is used by the majority of neutrino baseline experiments running around the
world, such as MINERvA 7b , MINOS 7c , MicroBooNE 7d , NovA 7e  and T2K 7f .
On the other hand, GiBUU is based on a coupled set of semiclassical kinetic
equations, these equations describe the dynamics of a hadronic system in phase
space and time. It is based on FORTRAN routines. GiBUU has been adopted not
only by the neutrino community but also by other research communities in
nuclear and particle physics experiments to explain the various interactions
viz.($\nu$, e, $\gamma$, A, p, $\pi$)-A, these interaction phenomena has been
tuned and explained very well by GiBUU 5 ; 7bb . The RFG model used in GENIE
is based on the model suggested by A. Bodek and J.L. Ritche ABodak , that
includes short-range nucleon-nucleon correlations while in GiBUU, RFG is
updated by adding a density dependent mean-field potential term in which all
nucleons are assumed to be bound. When simulating QE neutrino-nucleus
interaction process, RFG model refrfg  is used to describe the nuclear
structure by both the generators. The neutrino-nucleus interaction dynamics is
also described by the RFG model refrfg  under the assumptions of PWIA(Plane
Wave Impulse Approximations), further details can be found in refpwia .
Modeling of QE scattering in GENIE is according to Llewellyn Smith model
qemodel . For details on QE cross section mechanism in GiBUU, one can look
into minibooneQE ; moselQE . For nuclear density distribution both the
generators use Woods-Saxon parametrization woodsaxon . The value of axial mass
used by GiBUU is $M_{A}$= 1 GeV/c2 while GENIE uses a variable value of axial
mass between 0.99-1.2 GeV/c2. The vector form factors used by GiBUU and GENIE
are BBBA07 refGibuuVct  and BBBA05 refGenieVct  respectively.

The average energy of the neutrino beam (LBNF) for the DUNE experiment is
$\sim 2.5$ GeV and RES is the dominant process at this energy. GiBUU consists
of 13 kinds of resonance modes. The MAID analysis refGibuuMAID1 ;
refGibuuMAID2  of the electron scattering data provides vector form factors
for each of the 13 resonance modes in GiBUU. On the other hand, GENIE consists
of 16 resonance modes based on Rein Sehgal model RShegal . An additional
contribution to the total neutrino cross-section at neutrino energy less than
1 GeV contributes via processes involving two particle two hole (MEC/2p2h)
excitations. This process arises mainly from nucleon-nucleon correlations in
the initial state interactions(ISI), neutrino coupling to the 2p2h and FSI. A
debate on the importance of this process can be found in 2p2h . For simulating
DIS processes, GiBUU uses PYTHIA gibuudis  while GENIE applies the model of
Bodek and Yang bodekyang  along with Aivazis, Olness and Tung model discharm ,
particularly for DIS charm production.

Final state interactions(FSI), significantly modify the event distribution
captured by the detectors in the form of changed identities, topologies and
kinematics of the initially produced particles. This necessitates proper
modeling of FSI which is incorporated differently in different nuclear models
which are in use. The treatment of FSI by GENIE and GiBUU is entirely
different. GENIE simulates nuclear re-interactions using Intranuke hA and hN
as FSI models, details can be found in dytman1 ; dytman2 . The hA model is
data driven while hN incorporates theoretical descriptions. In GiBUU, FSI is
modeled by solving the semi-classical BUU equations where various particle
species are coupled via a mean field potential and collision terms. Further
differences in nuclear models, cross section models and FSI models used by the
two generators can be found in ref45 . Recent development and its
implementation in both the generators i.e. GENIE and GiBUU can be found in
refGenieMEC ; refGibuu2p2h .

##  III SIMULATION AND EXPERIMENTAL DETAILS

For the simulation of the DUNE experiment, we have considered a far detector
with a fiducial volume of 40 kton liquid argon(A=40) placed at a distance of
L=1300 km from the wideband neutrino beam source with a running time of 3.5
years, each in neutrino and antineutrino mode. The neutrino fluxes used here
correspond to the 80 GeV beam configuration 3a , with an assumed beam power of
1.07 MW for two beamline designs (1) reference design (2) optimized design. We
perform the sensitivity analysis for DUNE with both reference and optimized
beams to explore the physics potential of DUNE. The main differences between
the two beam designs include the geometry of the decay pipe and design of the
horn. Further details regarding the potential beamline designs can be found in
3b  and difference in few parameters are enlisted in Table II. For performing
the sensitivity analysis we have used the GLoBES(General Long Baseline
Experiment Simulator) 1 ; 2  package which requires cross-section, neutrino
and anti-netrino beam fluxes and detector parametrization values as input. The
cross-section input format is:
$\hat{\sigma}(E)=\sigma(E)/E[10^{-38}cm{{}^{2}}$/GeV], one can find further
details in globesmanual . The computation of binned event rates is performed
by an energy smearing algorithm which we have chosen to be a Gaussian function
of energy resolution globesmanual . The energy resolution for $\nu_{e}$ is
15$\%/\sqrt{E}$(GeV) and for $\nu_{\mu}$ is 20$\%/\sqrt{E}$(GeV) cadams . The
true values of the oscillation parameters 0c  considered in this analysis are
presented in Table I. The numerical procedure carried out to study the
sensitivities is done by calculating $\Delta\chi^{2}$ using the default
definition present in GLoBES. The oscillation analysis includes both the
$\nu_{e}(\overline{\nu_{e}})$ appearance and $\nu_{\mu}(\overline{\nu_{\mu}})$
disappearance channels and the systematics considered in our analysis are
presented in Table III. The relevant background that is considered in this
work for the muon disappearance channel is the neutral current interaction.
For the backgound of electron appearance channel we have considered
contributions from three different channels i.e. charged current interactions
of $\nu_{e}\rightarrow\nu_{e}$, misidentified CC
$\nu_{\mu}\rightarrow\nu_{\mu}$ and neutral current(NC) backgounds.

Table 1: Oscillation parameters considered in our work Parameter | Best Fit Value | 3$\sigma$ Range
---|---|---
$\theta_{12}$ | 0.590 | -
$\theta_{13}$ | 0.151 | -
$\theta_{23}$(NH) | 0.867 | 0.703 - 0.914
$\theta_{23}$(IH) | 0.870 | 0.710 - 0.917
$\delta_{CP}$ | 0 | $-\pi-+\pi$
$\Delta m^{2}_{21}$ | 7.39e-5 eV2 | -
$\Delta m^{2}_{31}$(NH) | 2.525e-3 eV2 | +2.427 $\rightarrow$ +2.625
$\Delta m^{2}_{31}$(IH) | -2.512e-3 eV2 | −2.611 $\rightarrow$ −2.412
Table 2: A comparison of beamline parameters taken from CDR for Reference Design flux and Optimized Design flux are as follows 3b . These parameters are used in our analysis. Parameter | Reference Beamline Design | Optimized Beamline Design
---|---|---
Proton Beam Power | 1.07 MW | 1.07 MW
Energy of Proton Beam | 80 GeV | 80 GeV
Horn Style | NuMI-style | Genetic Optimization
Horn Current | 230 kA | 297 kA
Diameter of Decay Pipe | 4 m | 4 m
Length of Decay Pipe | 204 m | 241 m
Table 3: Systematic uncertainties for signal and backgound channels used for both the reference and optimized beamline designs. Channel | Signal | Background
---|---|---
$\nu_{\mu}(\overline{\nu_{\mu}})\rightarrow\nu_{e}(\overline{\nu_{e}})$ | 5$\%$ | 10$\%$
$\nu_{\mu}(\overline{\nu_{e}})\rightarrow\nu_{\mu}(\overline{\nu_{e}})$ | 5$\%$ | 10$\%$

##  IV SENSITIVITY STUDIES FOR DUNE

In this section, we have tried to explore the impact of cross-sectional
uncertainties (shortcomings in the theoretical aspect of nuclear physics
models as implied in the generators) on the three major goals DUNE aims to
resolve i.e. (1)CP phase violation (2)mass ordering (3)octant ambiguity.
Several attempts have been made to perform the sensitivity analysis for long
baseline experiments like DUNE, for example, determination of octant
sensitivity and $\delta_{CP}$ sensitivity have been performed in sens1 ; sens2
; sens3  while a complete sensitivity analysis for other experiments like T2K
7f , T2HK t2hk , NO$\nu$A 7e  along with DUNE have been performed in sens4 ;
sens5 .

###  IV.1 CP VIOLATION SENSITIVITY

In order to observe CP violation, the value of the CP phase must be different
from CP conserving values i.e. 0 or $\pm\pi$. Since we do not know the true
value of $\delta_{CP}$, the analysis is performed by scanning all the possible
true values of $\delta_{CP}$, over the entire range $-\pi<\delta_{CP}<+\pi$
and comparing them with the CP conserving values. Our test parameters are
$\delta_{CP},\theta_{23}$ and $|\Delta m^{2}_{31}|$. While performing our
analysis we have marginalized over the test parameters in 3$\sigma$ range, as
mentioned in Table I. To calculate the CP violation sensitivity we perform
calculations as-

Figure 3: CP sensitivity measurement as a function of true value of
$\delta_{CP}$ for NH(left panel) and IH(right panel) by GENIE(blue lines) and
GiBUU(red lines). Reference and optimized designs are represented by solid and
dashed lines respectively.

$\Delta\chi^{2}_{0}=\chi^{2}(\delta_{CP}=0)-\chi^{2}_{true}$ (1)
$\Delta\chi^{2}_{\pi}=\chi^{2}(\delta_{CP}=\pi)-\chi^{2}_{true}$ (2)
$\Delta\chi^{2}=min(\Delta\chi^{2}_{0},\Delta\chi^{2}_{\pi})$ (3)

Qualitative handle on the measurement of CP violation is obtained by using,
$\sigma=\sqrt{\Delta\chi^{2}}$ and is illustrated in Figure 3. Left panel of
Figure 3 shows CP sensitivity when normal hierarchy is considered as true
hierarchy. In this analysis 1$\sigma$ variation is observed in CP sensitivity
results at $\delta_{CP}\sim 0.5/\pi$ in the range $0<\delta_{CP}/\pi<1$ for
the results obtained by GENIE and GiBUU for the DUNE experiment. This gives us
a hint that the physics analysis for CP sensitivity from these two generators
will have a difference in the results by 1$\sigma$ if mass hierarchy is found
to be normal.

This 1$\sigma$ difference in the CP sensitivity is observed for both the
reference and optimized beam designs as reflected in Figure 3 left panel. The
right panel of Figure 3 shows CP sensitivity when inverted hierarchy is
considered as true hierarchy. The CP sensitivity results with the two
different generators show a variation of more than 1$\sigma$ in the range
$0<\delta_{CP}/\pi<1$ around 0.5/$\pi$ for both reference and optimized beam
designs. In the negative half range($-1<\delta_{CP}/\pi<0$) of the CP values,
the variation between GENIE and GiBUU predictions for reference and optimized
beam designs is seen to be lesser than 1$\sigma$ for both normal and inverted
hierarchy cases.

###  IV.2 MASS HIERARCHY SENSITIVITY

Determination of mass hierarchy is one of the most crucial problems in
neutrino physics i.e. the quest of finding the true nature of neutrino mass
ordering, is normal or inverted. Mass hierarchy sensitivity is calculated by
assuming normal(inverted) hierarchy as true hierarchy and comparing it with
inverted(normal) hierarchy by using equation 4(5). So we set opposite
hierarchies in true and test values. Figure 4 shows the mass hierarchy
sensitivity for both the normal hierarchy(left panel) and inverted
hierarchy(right panel) cases. The $\Delta\chi^{2}$ quantity for mass hierarchy
sensitivity, is calculated as below-

$\Delta\chi^{2}_{MH}=\chi^{2}_{IH}-\chi^{2}_{NH}$ (4)
$\Delta\chi^{2}_{MH}=\chi^{2}_{NH}-\chi^{2}_{IH}$ (5)

Figure 4: Mass hierarchy sensitivity measurement as a function of true value
of $\delta_{CP}$ for NH(left panel) and IH(right panel) by GENIE(blue lines)
and GiBUU(red lines). Reference and optimized designs are represented by solid
and dashed lines respectively.

For the true normal hierarchy case presented in the left panel of Figure 4, we
observe a variation of more than 2$\sigma$ for reference beam and more than
3$\sigma$ for optimized beam predictions made by GENIE and GiBUU in the range,
$-1<\delta_{CP}/\pi<0$ at $\sim-0.5/\pi$ while in the range of positive
$\delta_{CP}$ values i.e. $0<\delta_{CP}/\pi<1$ we observe a variation of
around 1$\sigma$ at $\sim 0.5/\pi$ between the GENIE and GiBUU predictions,
for both the beams. This analysis illustrates that there will be a difference
in the sensitivity prediction of DUNE if they are made using two different
generators(i.e. GENIE and GiBUU). The dependence of physics results on the
selection of the generators will restrict the physics analysis goals.

Figure 4(right panel) shows the mass hierarchy sensitivity study when inverted
hierarchy is considered to be true hierarchy. In this analysis, we observe a
difference of around 5$\sigma$ for reference beams and around 6$\sigma$ for
optimized beams in the sensitivity results performed by GENIE and GiBUU. This
difference is observed in the range $0<\delta_{CP}/\pi<1$ around
$\delta_{CP}\sim 0.5/\pi$. In the range $-1<\delta_{CP}/\pi<0$, we observe a
variation of around 3$\sigma$ for reference beams and more than 3$\sigma$ for
optimized beams at $\delta_{CP}\sim-0.5/\pi$ by GENIE and GiBUU.

###  IV.3 OCTANT SENSITIVITY

Figure 5: Octant sensitivity measurement as a function of true value of
$\delta_{CP}$ for NH-LO(top left), NH-HO(top right), IH-LO(bottom left) and
IH-HO(bottom right) by GENIE(blue lines) and GiBUU(red lines) Reference and
optimized designs are represented by solid and dashed lines respectively.

It is not yet confirmed that the atmospheric mixing angle $\theta_{23}$ lies
in the lower octant($0<\theta_{23}<\pi/4$)-LO or in the higher
octant($\pi/4<\theta_{23}<\pi/2$)-HO with $\pi/4$ as its maximal value. The
prime problem in resolving the octant degeneracy is the appearance of several
disconnected regions in the multi-dimensional neutrino oscillation parameter
space. This makes it difficult to pinpoint the exact or the true solution for
a given set of true values, known as parameter degeneracy. While performing
the octant sensitivity calculations in the lower(higher) octant, the test
value of $\theta_{23}$ is varied in the lower(higher) octant range. The true
value of $\theta_{23}$ in lower octant is 0.703 and in the higher octant it is
0.867 whereas the range of test values for $\theta_{23}$ in LO is [0.785,
0.961] and for HO is [0.609, 0.785]. For octant sensitivity, the metric
$\Delta\chi^{2}$ is defined as-

$\Delta\chi^{2}_{octant}=|\chi^{2}_{\theta_{23}^{test}>\pi/4}-\chi^{2}_{\theta_{23}^{true}<\pi/4}|$
(6)

We present the results for reference and optimized beam designs for four
combinations of true hierarchy and octant configurations viz. NH-LO, NH-HO,
IH-LO and IH-HO presented in Figure 5, top and bottom panels respectively. For
NH-LO case (top left panel of Figure 5), in the negative range of
$\delta_{CP}$ values at $\sim-0.5/\pi$ there is a difference of around
2$\sigma$ between sensitivity results by GENIE and GiBUU and a negligible
difference is seen at $\sim 0.5/\pi$ in the positive half range of
$\delta_{CP}$ values. For NH-HO case(top right panel of Figure 5), the octant
sensitivity predictions for reference and optimized beam designs vary by less
than 2$\sigma$ for $\delta_{CP}$ value around 0.5/$\pi$ in the positive range
of $\delta_{CP}$ i.e. $0<\delta_{CP}/\pi<1$. A variation of less than
1$\sigma$ is observed in the negative half range of $\delta_{CP}$ values i.e.
$-1<\delta_{CP}/\pi<0$, around $-0.5/\pi$ for both reference and optimized
beam designs. For IH-LO case(bottom left panel of Figure 5), there is a
substantial difference of around 3$\sigma$ between GENIE and GiBUU predictions
for reference and optimized beams in the negative half range of $\delta_{CP}$
values, particularly around -0.5/$\pi$ while a difference of around 1$\sigma$
is observed in the positive half range of $\delta_{CP}$ values between GENIE
and GiBUU predictions. For IH-HO case(bottom right panel of Figure 5), a
difference of less than $\sim 3\sigma$ is observed around $0.5/\pi$ and a
variation of less than 1$\sigma$ is observed between GENIE and GiBUU
sensitivity predictions in the negative half range of $\delta_{CP}$ values.

##  V Summary and Conclusion

Construction of a nuclear model requires the precise combination of
information about the energy dependence of all exclusive cross-sections and
nuclear effects. The simulated results depend on this nuclear model where
ignorance of theoretical uncertainties cost inaccuracy. In order to evaluate
results from the neutrino oscillation experiments, we use event generators,
which are build upon these nuclear models. They predict the neutrino-nucleon
event rates for a particular nuclear target along with the topology of final
state particles which is critical for oscillation analysis. Due to the use of
heavy targets, nuclear effects play an important role in the prediction of
neutrino oscillation physics. Since nuclear effects are not well understood,
thus different generators use different approximations to accommodate the
nuclear effects giving rise to different results.

Our analysis of nuclear effects in the present work reflects the extent of
ambiguity in the physics of the selected neutrino event generators. The
inbuilt theoretical models in the event generators used in this work i.e.
GENIE and GiBUU must be revisited so that the results are independent of the
selection of the generators. Since the DUNE flux peaks at 2.5 GeV and at this
energy the resonance process is the most dominant process, hence it should be
tackled properly. Neutrinos having sufficient energy, strike the nucleon in
the nucleus and produce baryonic resonances $N^{*}$, which further quickly
decays to a nucleon and a single pion. These are classified as resonant single
pion production processes. At lower energies, resonance process is dominated
by $\Delta$(1232) production. Contribution from CC $\pi$ production creates
complication in neutrino energy determination for disappearance experiments.
Thus accurate modeling of nuclear effects in the pion production process is
necessary and to mention the largest background contribution to
$\nu_{\mu}\rightarrow\nu_{e}$ oscillation channel comes from NC $\pi^{0}$
production.

There is a need to determine cross-section for various nuclei and upgrade the
present cross-section data as there is insufficient cross-section data
available to understand the existing nuclear effects. The nuclear
structure(mass number, atomic number) of every target nucleus is different and
poses a non-negligible complication in precise calculation of neutrino-nucleon
cross-section.

The upcoming promising experiment DUNE is designed to answer CP sensitivity,
mass hierarchy, octant sensitivity and new physics. To check the potential of
DUNE, using any of the present generators, at the initial step needs a deep
understanding and study of the generator itself. Any uncertainty in the
generators will be propagated to the results. Here, DUNE potential is studied
using two different generators: GiBUU and GENIE. The analysis performed by
different generators for sensitivity studies of DUNE is not the same, as can
be seen in our analysis(from Figures 3,4 and 5). The variation of few sigmas
is observed in the results of two generators. These uncertainties need to be
monitored and addressed while stating the DUNE potential. In our work, only
cross-sectional uncertainty is taken into account. In a future work, we will
consider uncertainties arising from final state interactions while checking
the DUNE potential.

##  VI Acknowledgment

We would like to thank Prof. Raj Gandhi for various suggestions regarding this
work. SN and JD would like to thank Prof. U. Mosel for many helpful
discussions regarding GiBUU related issues and to Newton Nath, Suprabh
Prakash, Debajyoti Dutta and Mehedi Masud for their help in GLoBES. This work
is supported by the Department of Physics, Lucknow University. Financially it
is supported by the Government of India, DST Project no-SR/MF/PS02/2013,
Department of Physics, University of Lucknow.

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Here we present the DUNE sensitivity plots for CP and octant sensitivity in
the case of unknown hierarchy. These are represented by figures A1 and B1
respectively.

##  Appendix A CP VIOLATION SENSITIVITY

The CP sensitivity for DUNE is checked for unknown mass hierarchy. The
sensitivity plots are illustrated in Figure A1, where left panel is for normal
hierarchy and right panel is for inverted hierarchy.

Left panel of Figure A1 shows CP sensitivity when normal hierarchy is
considered as true hierarchy. In the positive half range of $\delta_{CP}$ i.e.
$0<\delta_{CP}/\pi<1$, at $\delta_{CP}\sim 0.5/\pi$, a variation of more than
1$\sigma$ in the prediction of CP sensitivity by the two generators is
observed.

The right panel of Figure A1 shows CP sensitivity when inverted hierarchy is
considered as true hierarchy. The CP sensitivity results with the two
different generators show a variation of more than 1$\sigma$ in the range
$0<\delta_{CP}/\pi<1$ around 0.5/$\pi$ for both reference and optimized beam
designs. In the negative half range($-1<\delta_{CP}/\pi<0$) of the CP values,
the variation between GENIE and GiBUU predictions for reference and optimized
beam designs is seen to be less than 1$\sigma$ for both normal and inverted
hierarchy cases.

Figure 6: CP sensitivity measurement for unknown hierarchy as a function of
true value of $\delta_{CP}$ for NH(left panel) and IH(right panel) by
GENIE(blue lines) and GiBUU(red lines). Reference and optimized designs are
represented by solid and dashed lines respectively.

##  Appendix B OCTANT SENSITIVITY

Figure 7: Octant sensitivity measurement for unknown hierarchy as a function
of true value of $\delta_{CP}$ for NH-LO(top left), NH-HO(top right), IH-
LO(bottom left) and IH-HO(bottom right) by GENIE(blue lines) and GiBUU(red
lines) Reference and optimized designs are represented by solid and dashed
lines respectively.

The octant sensitivity for DUNE is checked for unknown mass hierarchy. The
sensitivity plots for reference and optimized beam designs for four
combinations of true hierarchy and octant configurations viz. NH-LO, NH-HO,
IH-LO and IH-HO are presented in Figure B1, top and bottom panels
respectively.

For NH-LO case (top left panel of Figure B1), in the negative range of
$\delta_{CP}$ values at $\sim-0.5/\pi$ there is a difference of around
2$\sigma$ between sensitivity predictions by GENIE and GiBUU but an
appreciable difference of $\sim 4\sigma$ is observed at $\sim 0.5/\pi$ in the
positive half range of $\delta_{CP}$ values.

For NH-HO case(top right panel of Figure B1), the octant sensitivity
predictions for reference and optimized beam designs vary by less than
2$\sigma$ for $\delta_{CP}$ value around 0.5/$\pi$ in the positive range of
$\delta_{CP}$ i.e. $0<\delta_{CP}/\pi<1$. A variation of less than 1$\sigma$
is observed in the negative half range of $\delta_{CP}$ values i.e.
$-1<\delta_{CP}/\pi<0$, around $-0.5/\pi$ for both reference and optimized
beam designs.

For IH-LO case(bottom left panel of Figure B1), there is a substantial
difference of around 3$\sigma$ between GENIE and GiBUU predictions for
reference and optimized beams in the negative half range of $\delta_{CP}$
values, particularly around -0.5/$\pi$ while a difference of around 1$\sigma$
is observed in the positive half range of $\delta_{CP}$ values between GENIE
and GiBUU predictions.

For IH-HO case(bottom right panel of Figure B1), a difference of less than
$\sim 3\sigma$ is observed around $0.5/\pi$ and a variation of less than
1$\sigma$ is observed between GENIE and GiBUU sensitivity predictions in the
negative half range of $\delta_{CP}$ values.