∎

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
11institutetext: Department of Physics, Indian Institute of Technology
Hyderabad,
Hyderabad, 502284, Telangana, India

# Probing neutrino-nucleus interaction in DUNE and MicroBooNE

R K Pradhan e1,addr1  R Lalnuntluangae2,addr1  and A Giri e3,addr1

Abstract: The neutrino experiments utilize heavy nuclear targets to achieve
high statistics neutrino-nucleus interaction event rate, which leads to
systematic uncertainties in the oscillation parameters due to the nuclear
effects and uncertainties in the cross-section. Understanding the interaction
of neutrinos with the nucleus becomes crucial in determining the oscillation
parameters with high precision. We investigate the uncertainty in quasi-
elastic interaction due to nuclear effects by selecting exactly 1 proton, 0
pions, and any number of neutrons in the final state using DUNE and MicroBooNE
detectors, and the effects on oscillation parameters in the DUNE detector. The
kinematic method along with this selection can be used for accurate neutrino
energy reconstruction in the quasi-elastic channel where the nuclear effects
are inevitable.

##  1 Introduction

The primary goal of the current and future neutrino experiments is precisely
measuring the neutrino oscillation parameters. The oscillation probability
depends on the neutrino energy and the neutrino beam is not mono-energetic, it
spreads in a broad range of energy spectrum due to their production in the
decay of produced hadrons. So, an event-by-event neutrino energy
reconstruction is required for a better understanding of uncertainty in
neutrino oscillation. Due to high statistics requirements, neutrino
experiments use heavy nuclear targets (Argon Anderson et al. (2012); Acciarri
et al. (2014), Iron Abe et al. (2014); Tice et al. (2014), Water Abe et al.
(2017), Oxygen Abe et al. (2014), etc), which arises complications in the
reconstruction due to the complexity of the nucleus. For accurate neutrino
energy reconstruction, understanding neutrino-nucleus interactions is very
crucial.

Most Long-Baseline Neutrino Experiments (LBNE) run in the few GeV energy
regions where quasi-elastic scattering is one of the most important
interaction channels however there is a significant contribution of 2p-2h
interaction which makes the situation complicated Aguilar-Arevalo et al.
(2010); Martini et al. (2009). In this work, we focus on muon neutrino charge
current quasi-elastic (CCQE) scattering ($\nu_{\mu}\,n\rightarrow\mu^{-}\,p$).
In a nucleus, the neutron is neither free nor at rest, but it moves and is
bound inside a nuclear potential. To understand CCQE scattering, one can refer
to impulse approximation (IA) Benhar et al. (2005), in which interactions
occur on the individual nucleons. In the IA, the neutrino interacts with a
pair of nucleon inside the nucleus, or a bound nucleon followed by the final
state interaction (FSI) Dytman (2009). Due to FSI, the proton produced in
primary interaction undergoes various hadronic scattering such as (in)elastic
scattering, pion production, hadron absorption, and charge exchange which
results in knocked-out nucleons and mesons in the final state. This results in
the misidentification of non-QE events as QE events. Pion production is the
major background to the QE scattering events. In some events, these produced
pions again get absorbed leading to two nucleon knock-out, called 2p-2h
scattering. The cross-section of CCQE-like interactions measured in T2K can be
explored in Ref. Abe et al. (2016). Due to FSI, a CCQE can also produce pion,
thus can’t be classified as CCQE-like. These backgrounds lead to an
uncertainty in the neutrino energy reconstruction, which impacts the
oscillation parameters. The effects on the parameters of oscillation for
experiments such as T2K and MiniBooNE are studied here Lalakulich et al.
(2012); Coloma et al. (2014).

Monte Carlo event generators are used for the simulations of neutrino
interactions, developed with different nuclear models. They are used for
predictions and improvement of the experiments. The generators such as GENIE
Andreopoulos et al. (2010), NuWro Juszczak et al. (2006), GiBUU Buss et al.
(2012), and NEUT Hayato and Pickering (2021) are extensively used for physics
analysis. In this work, we aim to select high-purity CCQE events for DUNE
detectors Acciarri et al. (2016) using the realistic particle thresholds, and
its effects on the oscillation parameters is studied using both GENIE and
NuWro. Previous studies for LBNE using the GiBUU model can be found in Ref.
Mosel et al. (2014). The analysis methods for our event selection have been
implemented for the MicroBooNE detector Acciarri et al. (2017) using GENIE and
NuWro models.

This paper is organized as follows. In section 2, we present the formalism for
the neutrino interactions with the nuclear target and the methods for
reconstructing the energy of the neutrino. The Monte Carlo event generators
are described in section 3. The simulation specifications of the DUNE and
MicroBooNE detectors are mentioned in section 4. We then discuss the results
from our analysis in section 5, followed by the conclusion in section 6.

##  2 Formalism

Consider a charged current neutrino interaction with a nuclear target knocking
out n nucleons and producing m mesons. These produced particles can be used
for the neutrino energy reconstruction, using the kinematic and calorimetric
methods Ankowski et al. (2015). The reconstructed energy from the generalized
kinematic approach Ankowski et al. (2015) is,

$\centering
E_{\nu}=\frac{2(nM-\epsilon_{n})E_{l}+W^{2}-(nM-\epsilon_{n})^{2}-m_{l}^{2}}{2(nM-\epsilon_{n}-E_{l}+|k_{l}|cos\theta)}\@add@centering$
(1)

The invariant hadronic mass squared is defined as,

$W^{2}=\left(\sum_{i}E_{p_{i}^{\prime}}+\sum_{j}E_{h_{j}^{\prime}}\right)^{2}-\left(\sum_{i}p_{i}^{\prime}+\sum_{j}h_{j}^{\prime}\right)^{2}$

where, $E_{p^{\prime}}$ and $p^{\prime}$ ($E_{h^{\prime}}$ and $h^{\prime}$)
are the energy, and momentum of final state nucleons (mesons) respectively. M
($m_{l}$) is the mass of the final state nucleon (lepton), $E_{l}$ and $k_{l}$
are the energy and momentum of the outgoing lepton respectively, and $\theta$
is the scattering angle, $\epsilon_{n}$ the neutron separation energy.

For a pure charge current quasi-elastic (CCQE) scattering, where the hit
nucleon (neutron) is at rest, the kinematic method for reconstructing the
neutrino energy Coloma et al. (2014),

$E_{\nu}=\frac{2(M_{n}-E_{B})E_{l}-(E_{B}^{2}-2M_{n}E_{B}+m_{l}^{2}+\Delta
M^{2})}{2(M_{n}-E_{B}-E_{l}+|k_{l}|cos\theta)}$ (2)

where, $M_{n}$ is the free neutron rest mass, $\Delta
M^{2}=M_{n}^{2}-M_{p}^{2}$, $M_{p}$ is the rest mass of the proton, and
$E_{B}$ the binding energy.

For better accuracy in the analysis, rather than assuming constant binding
energy value, we consider a distribution for the neutron excitation energy for
the nucleus Furmanski and Sobczyk (2017), for both 12C and 40Ar target. A
Gaussian distribution of separation energy E with mean $E_{\alpha}$, and
deviation $\sigma_{\alpha}$, is considered according to table 1 for 12C
Frullani and Mougey (1984), and in table 2 for 40Ar Ankowski and Sobczyk
(2008).

Subshell |  $E_{\alpha}$ (MeV) |  $\sigma_{\alpha}$ (MeV) | No. neutron $n_{\alpha}$
---|---|---|---
1$s_{1/2}$ | 40.8 | 9.1 | 2
1$p_{3/2}$ | 20.3 | 5 | 4
Table 1: Neutron Shell Structure in 12C Frullani and Mougey (1984) Subshell |  $E_{\alpha}$ (MeV) |  $\sigma_{\alpha}$ (MeV) | No. neutron $n_{\alpha}$
---|---|---|---
1$s_{1/2}$ | 62 | 6.25 | 2
1$p_{3/2}$ | 40 | 3.75 | 4
1$p_{1/2}$ | 35 | 3.75 | 2
1$d_{5/2}$ | 18 | 1.25 | 6
2$s_{1/2}$ | 13.15 | 1 | 2
1$d_{3/2}$ | 11.45 | 0.75 | 4
1$f_{7/2}$ | 5.56 | 0.75 | 2
Table 2: Neutron Shell Structure in 40Ar Ankowski and Sobczyk (2008)

The probability distribution for the separation energy is given by,

$P(E)=\frac{1}{N}\sum_{\alpha}n_{\alpha}G(E-E_{\alpha},\sigma_{\alpha})$ (3)

where the sum of neutrons

$N=\sum_{\alpha}n_{\alpha}$

and $G(E-E_{\alpha},\sigma_{\alpha})$ is the distribution function.

For oscillation analysis, the survival probability of muon neutrino can be
calculated by using the following expression Zyla et al. (2020),

$P_{\nu_{\mu}\rightarrow\nu_{\mu}}\approx
1-sin^{2}2\theta_{\mu\mu}\,sin^{2}\frac{\Delta m^{2}_{\mu\mu}L}{4E_{\nu}}$ (4)

with

$sin^{2}\theta_{\mu\mu}=cos^{2}\theta_{13}\,sin^{2}\theta_{23}$
$\begin{split}\Delta m^{2}_{\mu\mu}=&sin^{2}\theta_{12}\Delta
m_{31}^{2}+cos^{2}\theta_{12}\Delta m_{32}^{2}\\\
&+cos\delta_{CP}\,sin\theta_{13}\,sin2\theta_{12}\,tan\theta_{23}\Delta
m_{21}^{2}\end{split}$

The values of the oscillation parameters, $\Delta m^{2}$ and $\theta$ are
taken from Ref. Bishai et al. (2012). The distance of the DUNE far detector is
$L\approx$ 1300 km far from the near detector.

Figure 1: Unoscillated muon neutrino flux for DUNE Near detector and
MicroBooNE (left) and oscillated flux for DUNE Far detector (right).

##  3 Monte Carlo Event Generator

In this work, two MC generators, GENIE and NuWro are used for simulating the
neutrino-nucleus interactions. Both GENIE and NuWro are used by accelerator-
based neutrino experiments. The Fermi Gas Model Bodek and Ritchie (1981) is
implemented to define the impact of the nuclear environment considering
nucleon-nucleon correlation effects Bodek and Ritchie (1981) in GENIE, and the
spectral function Benhar et al. (1989) in NuWro.

The default model for QE scattering in GENIE is the Llewellyn-Smith (LS) model
Llewellyn Smith (1972). The new models for QE scattering and MEC/2p-2h
processes in GENIE are the Valencia QE model Nieves et al. (2004); Gran et al.
(2013) and SuSAv2 Dolan et al. (2020); Megias et al. (2016). The Valencia
model is based on the Local Fermi Gas model with Coulomb correction effects
and Random Phase Approximation (RPA). In contrast, the SuSAv2 is based on the
Relativistic Mean Field (RMF) theory. In NuWro, the QE interactions are
described by the LS model with options for vector and dipole axial vector form
factors. GENIE implements the axial mass $M_{A}$ extending from 0.99 to 1.2
GeV/$c^{2}$, whereas NuWro confines this parameter in a range of 0.94-1.03
GeV/$c^{2}$. In this work, the value of the CCQE axial mass is considered as
$\approx 0.96\,GeV/c^{2}$ in both GENIE and NuWro, and the vector form factors
used are BBA07 Bodek et al. (2008) and BBA05 Bradford et al. (2006)
respectively. GENIE has 4 FSI models: hA, hN, INCL++, and Geant4 Alvarez-Ruso
et al. (2021), while in NuWro, the FSI is described by the cascade model based
on the algorithm by Metropolis et al. Metropolis et al. (1958). Empirical MEC
model Katori (2015) is also available for MEC/2p-2h scattering in GENIE, and
transverse enhancement model Bodek et al. (2011) and Marteau-Martini model
Martini et al. (2009) are also available in NuWro for 2p-2h scattering.

The baryonic resonances are modeled using the Berger-Sehgal (BS) model Berger
and Sehgal (2007) in GENIE. NuWro uses the Rein-Sehgal (RS) model Rein and
Sehgal (1981) for $\Delta$(1232) resonance, and the Adler-Rarita-Schwinger
model Hernandez et al. (2008) for higher resonances.Other available resonance
models in GENIE are Kuzmin-Lyubushkin-Naumov (KLN) Kuzmin et al. (2004) and
the RS model.

In the inelastic region (DIS), both the generators use the Bodek-Yang Model
Bodek and Yang (2002) for the cross-section calculation, and utilize PYTHIA
Sjostrand et al. (2006) to produce hadronic final states with different values
of PYTHIA parameters such as invariant hadronic mass $W$. GENIE employs it to
$W$ = 1.8 or 2.0 GeV with KNO scaling Koba et al. (1972) while NuWro is to 1.6
GeV. The invariant hadronic mass threshold used in this work is $W$ = 1.9 GeV
in GENIE and 1.6 GeV in NuWro.

Figure 2: The probability density function (PDF) of reconstructed neutrino
energy for Argon in DUNE near (left) and far (right) detector using GENIE
(solid) and NuWro (dotted). Figure 3: The ratio of true and reconstructed
energy PDF as a function of neutrino energy for Argon in DUNE near (left) and
far (right) detectors with FSI (blue) and without FSI (red) using GENIE
(solid) and NuWro (dotted).

##  4 Simulation Details

In this paper, nuclear effects on energy reconstruction are analyzed for the
DUNE near and far detector and the MicroBooNE detector. DUNE mainly uses Argon
as the target material. The proposed DUNE near detector HpgTPC Duffy (2019)
consists of Argon and Methane (CH4), and liquid Argon as the target material
in the far detector. MicroBooNE also uses liquid Argon targets. CC $\nu_{\mu}$
interaction are simulated for Argon and Carbon target using the DUNE Abi et
al. (2021) and MicroBooNE $\nu_{\mu}$ flux Fleming (2012) as shown in Fig. 1.
The GENIE v3.01.00 (tune G18_10a_02_11a) and NuWro-21.09.2 are used for the
simulation. We consider realistic simulations and include the Quasi-elastic
(QE), Meson exchange (MEC/2p-2h), Resonance (RES), and Deep Inelastic
Scattering (DIS) in neutrino mode. In the RES channel, we consider only the
first resonance region, $\Delta$ or $P_{33}(1232)$. For the simulation, we
have used the Nieves model for QE, the Berger Sehgal model for the resonance,
and the CCQE axis mass ($M_{A}$) is considered to be 0.96 GeV/$c^{2}$. For
FSI, the hA model is employed in GENIE, and the Oset model in NuWro. To study
the FSI effect, we have also simulated the interaction events by switching
ON/OFF the FSI in both MC generators.

Figure 4: The PDF of reconstructed neutrino energy for Carbon (left) and Argon
(right) in MicroBooNE detector using GENIE (solid) and NuWro (dotted). Figure
5: The PDF ratio of true and reconstructed energy as a function of neutrino
energy for Carbon (left) and Argon (right) in MicroBooNE detectors with FSI
(blue) and without FSI (red) using GENIE (solid) and NuWro (dotted).

Ideally, CCQE interaction can be identified as 1 proton and 0 pions in the
final state. But it could be possible that the pion produced due to FSI gets
absorbed inside the nucleus knocking out nucleons. We consider a selection of
exactly 1 proton, 0 pions, and any number of neutrons in the final state Mosel
et al. (2014) for the CCQE interaction. The unobserved neutrons in the
detector are a consequence of the nuclear effects. Due to limitations in the
sensitivity of detectors, all the produced particles cannot be detected. We
have applied the detector cuts for both the near and far detectors. The
detection thresholds for the DUNE ND are, the total energy cut for muon is 226
MeV Hewes et al. (2021) ($\approx$ 200 MeV momentum), and a minimum kinetic
energy threshold of 3 MeV for proton Hamacher-Baumann et al. (2020). The
neutrons in the detector have a threshold of kinetic energy from 50MeV to 700
MeV Hewes et al. (2021). In case of FD, muons have a detector threshold of 30
MeV kinetic energy, and 50 MeV for protons and neutrons Acciarri et al.
(2016). A study on the proton kinetic energy threshold of 20 MeV with 80%
efficiency can be found in Ref. Singh et al. (2020). In the MicroBooNE
detector, the leading proton has a threshold momentum of 300 MeV/c ($\approx$
47 MeV kinetic energy ), and the momentum threshold for muon is 100 MeV/c
Fleming (2012). The selection (1 proton + 0 pions + X neutrons) with the
corresponding detector thresholds for the DUNE and MicroBooNE detectors are
applied to reconstruct the neutrino energy for the CCQE.

##  5 Results

The event distributions of neutrino energy for Argon in both ND and FD are
shown in Fig. 2. For the ND, the reconstructed energy using the kinematic
method (green) is shifted by $\approx$ 0.4 GeV towards lower energies,
compared to the true energy distribution (black). The distribution at FD after
oscillation is given on the right panel (Fig. 2). It can be observed clearly
that without the selection, the distribution is distorted with flattening of
the minima around 2.4 GeV. The event rate is significantly low at the second
maxima at 1.4 GeV. When the selection (1 proton + 0 pions + X neutrons) is
considered along with the DUNE ND and FD detector cuts, the reconstructed
energy distribution (blue) agrees with the true energy for both ND and FD. The
shift of reconstructed energy is reduced to less than 100 MeV compared to true
neutrino energy. The results using GENIE are represented as solid lines.
Similarly, NuWro (dotted) results show the same effects as GENIE. For
evaluating the uncertainty in the reconstruction, we consider the event ratio
distribution of true energy with the reconstructed energy which is shown in
Fig.3. To further quantify the effect of FSI, we consider the ratio with FSI
and without FSI. For the ND, the ratio is close to 1 but in the lower energy
region ($<$ 1GeV), the reconstructed energy differs from the true energy due
to nuclear effects. As a result of the nuclear effects, 91 % of events are
true CCQE, and the remaining events are mostly contributed by 2p-2h and DIS,
out of the selected events. In the context of far detector, the reconstructed
energy disagrees with the true energy in the maxima and minima regions as
there is a downward shift in the reconstructed events. When the FSI is
switched off, it can be seen that the ratio is closer to unity in the lower
energy range, which indicates that the effect of the FSI contributes to the
uncertainty in the reconstruction. The probability distribution of neutrino
energy for Carbon and Argon in the MicroBooNE detector is given in Fig. 4.
There is a shift of $\approx$ 0.2 GeV in the reconstructed energy without the
selection i.e. the kinematic method. However, the reconstructed energy agrees
with true energy when the selection along with detector thresholds are
considered. Out of the selected events, 96% of events are from true CCQE
contributions. From the ratio of true neutrino energy to reconstructed energy
in Fig. 5, the selection criteria can be considered to significantly identify
the CCQE interaction, and the uncertainty could be due to the imapact of final
state interactions.

Figure 6: The muon neutrino survival probability in DUNE as a function of true
energy without any selection (black) and the reconstructed energy for QE with
FSI (blue) and without FSI (red) using GENIE (solid) and NuWro (dotted)
considering the selection with detector cuts.

The effects on the neutrino oscillation parameters using the selection (1
proton + 0 pions + X neutrons) are studied. The survival probability of muon
neutrinos is shown in Fig. 6. The true survival probability is calculated
using equation 4, and the same parameters are used from Bishai et al. (2012)
as a function of true neutrino energy. Assuming the experimental analysis, the
survival probability is calculated by taking the ratio of oscillated (FD) and
unoscillated (ND) event distributions. In Fig. 6, the black curve shows the
survival probability as a function of true energy, the blue curve represents
the survival probability as a function of the reconstructed energy by
considering the selection with detector cuts and the FSI effects, and without
the FSI is shown by the red curve. We can observe that there is a significant
influence in the absolute values, both at the first maximum and first minimum,
while the positions of maximum and minimum are less affected when we consider
the selections. One can see the effect of FSI in the lower energy range and at
the first maxima. This indicates a notable discrepancy in the mixing angle,
while the impact on $\Delta m^{2}$ is comparatively negligible. The same
effect has been observed for T2K by Coloma et al. Coloma and Huber (2013) as
well.

##  6 Conclusion

The purity of QE interactions selection for the MicroBooNE detector, and the
DUNE near and far detectors and their effect on oscillation parameters has
been studied using the Monte Carlo neutrino event generators GENIE and NuWro.
The physics potential studies on LBNE have shown that an energy resolution of
100 MeV is required to differentiate between different physics properties
Adams et al. (2013). The selection of 1 proton, 0 pions, and X neutrons for
CCQE shows a weighty discrepancy between the reconstructed and the true
neutrino energy approximately by 100 MeV which is the required resolution.
This shift of around 100 MeV is observed in both the ND and FD for both
generators. The uncertainty in reconstruction due to the nuclear effects
notably affects the mixing angle, with comparatively lesser effects on $\Delta
m^{2}$. Also, the ratio of true and reconstructed energy distribution in the
high-energy regions indicates the viability of this method in higher-energy
experiments.

## Acknowledgements

R K Pradhan acknowledges the DST-INSPIRE grant (2022/IF220293) for financial
support. R Lalnuntluanga thanked the Council of Scientific & Industrial
Research (file number: 09/1001(0054)/2019-EMR-I) for the financial grant. R
Lalnuntluanga and A Giri credited the grant support of the Department of
Science and Technology (SR/MF/PS-01/2016-IITH/G).

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