††thanks: Now at Iowa State University, Ames, IA 50011, USA††thanks: Now at
High Energy Physics Department, Argonne National Lab, 9700 S Cass Ave, Lemont,
IL 60439††thanks: Now at Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA††thanks: now at Department of Physics and Astronomy,
University of Mississippi, Oxford, MS 38677††thanks: Now at Department of
Physics, Drexel University, Philadelphia, Pennsylvania 19104, USA††thanks: Now
at University of Minnesota, Minneapolis, Minnesota 55455, USA††thanks: Now at
Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221,
USA††thanks: now at Syracuse University, Syracuse, NY 13244, USA

The MINERvA Collaboration

# Measurement of Electron Neutrino and Antineutrino Cross Sections at Low
Momentum Transfer

S. Henry  Department of Physics and Astronomy, University of Rochester,
Rochester, New York 14627 USA  H. Su  Department of Physics and Astronomy,
University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA  S. Akhter
Department of Physics, Aligarh Muslim University, Aligarh, Uttar Pradesh
202002, India  Z. Ahmad Dar  Department of Physics, William & Mary,
Williamsburg, Virginia 23187, USA  Department of Physics, Aligarh Muslim
University, Aligarh, Uttar Pradesh 202002, India  V. Ansari  Department of
Physics, Aligarh Muslim University, Aligarh, Uttar Pradesh 202002, India  M.
V. Ascencio  Sección Física, Departamento de Ciencias, Pontificia Universidad
Católica del Perú, Apartado 1761, Lima, Perú  M. Sajjad Athar  Department of
Physics, Aligarh Muslim University, Aligarh, Uttar Pradesh 202002, India  A.
Bashyal  Department of Physics, Oregon State University, Corvallis, Oregon
97331, USA  M. Betancourt  Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA  J. L. Bonilla  Campus León y Campus Guanajuato,
Universidad de Guanajuato, Lascurain de Retana No. 5, Colonia Centro,
Guanajuato 36000, Guanajuato México.  A. Bravar  University of Geneva, 1211
Geneva 4, Switzerland  G.A. Díaz  Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA  Department of Physics and Astronomy, University
of Rochester, Rochester, New York 14627 USA  J. Felix  Campus León y Campus
Guanajuato, Universidad de Guanajuato, Lascurain de Retana No. 5, Colonia
Centro, Guanajuato 36000, Guanajuato México.  L. Fields  Department of Physics
and Astronomy, University of Notre Dame, Notre Dame, Indiana 46556, USA  R.
Fine  Department of Physics and Astronomy, University of Rochester, Rochester,
New York 14627 USA  P.K.Gaur  Department of Physics, Aligarh Muslim
University, Aligarh, Uttar Pradesh 202002, India  S.M. Gilligan  Department of
Physics, Oregon State University, Corvallis, Oregon 97331, USA  R. Gran
Department of Physics, University of Minnesota – Duluth, Duluth, Minnesota
55812, USA  E.Granados  Campus León y Campus Guanajuato, Universidad de
Guanajuato, Lascurain de Retana No. 5, Colonia Centro, Guanajuato 36000,
Guanajuato México.  D.A. Harris  York University, Department of Physics and
Astronomy, Toronto, Ontario, M3J 1P3 Canada  Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA  A.L. Hart  G O Jones Building, Queen
Mary University of London, 327 Mile End Road, London E1 4NS, UK  J. Kleykamp
Department of Physics and Astronomy, University of Rochester, Rochester, New
York 14627 USA  A. Klustová  The Blackett Laboratory, Imperial College London,
London SW7 2BW, United Kingdom  M. Kordosky  Department of Physics, William &
Mary, Williamsburg, Virginia 23187, USA  D. Last  Department of Physics and
Astronomy, University of Pennsylvania, Philadelphia, PA 19104  A. Lozano
Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, Urca, Rio
de Janeiro, Rio de Janeiro, 22290-180, Brazil  X.-G. Lu  Department of
Physics, University of Warwick, Coventry, CV4 7AL, UK  Oxford University,
Department of Physics, Oxford, OX1 3PJ United Kingdom  S. Manly  Department of
Physics and Astronomy, University of Rochester, Rochester, New York 14627 USA
W.A. Mann  Physics Department, Tufts University, Medford, Massachusetts 02155,
USA  C. Mauger  Department of Physics and Astronomy, University of
Pennsylvania, Philadelphia, PA 19104  K.S. McFarland  Department of Physics
and Astronomy, University of Rochester, Rochester, New York 14627 USA  M.
Mehmood  York University, Department of Physics and Astronomy, Toronto,
Ontario, M3J 1P3 Canada  B. Messerly  Department of Physics and Astronomy,
University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA  J.G. Morfín
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA  D. Naples
Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh,
Pennsylvania 15260, USA  J.K. Nelson  Department of Physics, William & Mary,
Williamsburg, Virginia 23187, USA  C. Nguyen  University of Florida,
Department of Physics, Gainesville, FL 32611  A. Olivier  Department of
Physics and Astronomy, University of Notre Dame, Notre Dame, Indiana 46556,
USA  Department of Physics and Astronomy, University of Rochester, Rochester,
New York 14627 USA  V. Paolone  Department of Physics and Astronomy,
University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA  G.N. Perdue
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA
Department of Physics and Astronomy, University of Rochester, Rochester, New
York 14627 USA  C. Pernas  Department of Physics, William & Mary,
Williamsburg, Virginia 23187, USA  K.-J. Plows  Oxford University, Department
of Physics, Oxford, OX1 3PJ United Kingdom  M.A. Ramírez  Department of
Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104
Campus León y Campus Guanajuato, Universidad de Guanajuato, Lascurain de
Retana No. 5, Colonia Centro, Guanajuato 36000, Guanajuato México.  R.D.
Ransome  Rutgers, The State University of New Jersey, Piscataway, New Jersey
08854, USA  N. Roy  York University, Department of Physics and Astronomy,
Toronto, Ontario, M3J 1P3 Canada  D. Ruterbories  Department of Physics and
Astronomy, University of Rochester, Rochester, New York 14627 USA  H.
Schellman  Department of Physics, Oregon State University, Corvallis, Oregon
97331, USA  C. J. Solano Salinas  Facultad de Ciencias Físicas, Universidad
Nacional Mayor de San Marcos, CP 15081, Lima, Perú  V.S. Syrotenko  Physics
Department, Tufts University, Medford, Massachusetts 02155, USA  E. Valencia
Campus León y Campus Guanajuato, Universidad de Guanajuato, Lascurain de
Retana No. 5, Colonia Centro, Guanajuato 36000, Guanajuato México.  Department
of Physics, William & Mary, Williamsburg, Virginia 23187, USA  N.H. Vaughan
Department of Physics, Oregon State University, Corvallis, Oregon 97331, USA
A.V. Waldron  G O Jones Building, Queen Mary University of London, 327 Mile
End Road, London E1 4NS, UK  The Blackett Laboratory, Imperial College London,
London SW7 2BW, United Kingdom  C. Wret  Department of Physics and Astronomy,
University of Rochester, Rochester, New York 14627 USA  B. Yaeggy
Departamento de Física, Universidad Técnica Federico Santa María, Avenida
España 1680 Casilla 110-V, Valparaíso, Chile  L. Zazueta  Department of
Physics, William & Mary, Williamsburg, Virginia 23187, USA

###### Abstract

Accelerator based neutrino oscillation experiments seek to measure the
relative number of electron and muon (anti-)neutrinos at different $L/E$
values. However high statistics studies of neutrino interactions are almost
exclusively measured using muon (anti-)neutrinos since the dominant flavor of
neutrinos produced by accelerator based beams are of the muon type. This work
reports new measurements of electron (anti-)neutrino interactions in
hydrocarbon, obtained by strongly suppressing backgrounds initiated by muon
flavor (anti-)neutrinos. Double differential cross sections as a function of
visible energy transfer, $E_{\text{avail}}$, and transverse momentum transfer,
$p_{T}$, or three momentum transfer, $q_{3}$ are presented.

##  I Introduction

Predictions of interactions of GeV energy (anti-)neutrinos with nuclear
targets present challenges for experiments seeking to precisely measure
neutrino flavor oscillations. Both the DUNE [1] and Hyper-Kamiokande [2]
experiments are designed to measure muon to electron neutrino flavor
transitions with uncertainties on the order one percent. Consequently accurate
predictions of detector efficiencies, backgrounds, energy reconstruction, and
cross sections, which make use of the measurements presented here, are needed.
Compounding the problem for DUNE and Hyper-Kamiokande is that their near
detectors, which are used for studying neutrino interactions, see primarily a
flux of muon neutrinos and only a small fraction of electron neutrinos.
Therefore constraints that are solely derived from measurements of muon
neutrino interactions will need to be theoretically corrected for electron
neutrinos.

Electron-neutrino and muon-neutrino charged current cross sections differ for
two reasons. First, the tensor structure of the hadronic current and its
contraction with the lepton current yields terms that depend explicitly on the
square of the lepton mass compared to combinations of the target mass,
neutrino energy, and energy transfer [3]. These terms are largely negligible
for electron neutrinos at accelerator energies, however they may provide non-
negligible sub-leading corrections for muon neutrinos at low neutrino energies
or energy transfers. Secondly, momentum and energy transfer limits for
electron and muon neutrino interactions differ because the kinematic limits in
momentum and energy transfer are a function of lepton mass. If the energy and
three-momentum transfer from the incoming neutrino to the final state lepton
are denoted $q_{0}$ and $q_{3}$ respectively, conservation of energy and
momentum require that

$\displaystyle
E_{\nu}-\sqrt{E_{\nu}^{2}-2E_{\nu}q_{0}-m_{l}^{2}+q_{0}^{2}}<q_{3}$
$\displaystyle<E_{\nu}+\sqrt{E_{\nu}^{2}-2E_{\nu}q_{0}-m_{l}^{2}+q_{0}^{2}},$
(1)

where $E_{\nu}$ is the energy of the incoming neutrino and $m_{l}$ is the
final state lepton mass. A second constraint comes from the relationship
between initial and final state target invariant masses. If we denote the
initial invariant mass $M$ and the final mass $M+\Delta$, then this implies a
maximum three momentum transfer

$\displaystyle q_{3}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2M(2E_{\nu}+M)}\times\left(2E_{\nu}^{2}M-\Delta^{2}E_{\nu}\right.$
$\displaystyle-$ $\displaystyle\left.2\Delta
E_{\nu}M+E_{\nu}m_{l}^{2}+(E_{\nu}+M)\sqrt{\eta}\right),~{}\text{where}$
$\displaystyle\eta$ $\displaystyle\equiv$ $\displaystyle
4E_{\nu}^{2}M^{2}-4E_{\nu}M\left(\Delta(\Delta+2M)+m_{l}^{2}\right)$ (2)
$\displaystyle+$
$\displaystyle(m_{l}^{2}-\Delta^{2})\left(m_{l}^{2}-(\Delta+2M)^{2}\right).$

Expanding in $m_{l}$, we see

$\displaystyle q_{3}$ $\displaystyle\leq$
$\displaystyle\left(E_{\nu}-\frac{\Delta(\Delta+2M)}{2M}\right)$ (3)
$\displaystyle-$ $\displaystyle
m_{l}^{2}\left(\frac{E_{\nu}+M}{2E_{\nu}M-\Delta(\Delta+2M)}-\frac{1}{2M}\right)+{\cal
O}\left(m_{l}^{4}\right),$

A simple case in (3) is where $\Delta=0$, which is approximately correct for
elastic and quasielastic scattering from nucleons, gives a limit of
$q_{3}<E_{\nu}-\frac{m_{l}^{2}}{2E_{\nu}}$. Both (1) and (3) show that
increasing $m_{l}$ eliminates regions of allowed energy and momentum transfer.
Since it is the reactions of muon neutrinos that are studied with high
statistics in near detectors, the extrapolation into certain regions of energy
and momentum transfer are not well explored experimentally.

Another effect recently discussed in the literature is that of radiative
corrections which have a strong dependence on the mass of the final state
lepton [4, 5]. This cited work concludes that the effect of radiative
corrections can be precisely predicted, although those predictions are not
currently in neutrino interaction models used by experiments.

The effects of electron and muon neutrino interaction differences have been
studied within specific models of neutrino interaction cross sections on
nucleons and nuclei [6, 7, 8, 9, 10]. Such studies illuminate possible
differences between electron and muon neutrino interactions but are not
exhaustive.

With sufficient statistics, and with strong rejection and control of
backgrounds, it is possible to directly measure the interactions of electron
neutrinos and antineutrinos. This paper describes such a measurement with the
MINERvA detector [11] using the broadband NuMI [12] beam located at Fermi
National Accelerator Laboratory. To keep backgrounds low and well-controlled,
the measurement is performed at energy and momentum transfers less than $\sim
1$ GeV, much less than the incoming neutrino energy.

A complication in these measurements is that energy transfer cannot be
directly measured in a broadband neutrino beam, but instead must be inferred
from the visible recoil products in the detector. To approximate what is
measured calorimetrically, we employ a proxy used in previous MINERvA
measurements [13, 14], $i.e.$ replace $q_{0}$ in the measurement by the
quantity $E_{\text{avail}}$, defined as

$E_{\text{avail}}\equiv\sum_{\text{protons}}T_{p}+\sum_{\pi^{\pm}}T_{\pi^{\pm}}+\sum_{\pi^{0}}E_{\pi^{0}},$
(4)

where the sums are over final state particles, and $T_{X}$ indicates the
kinetic, rather than total energy $E_{X}$ of a final state particle $X$. The
weak decay products of strange, or heavier quark, baryons are included in the
sum by adding their total energies to the sum, and by subtracting (or adding)
a nucleon mass in the case of baryons (anti-baryons). For scattering from
nuclei, this quantity differs from $q_{0}$ in that it does not have the
kinetic energy of final state neutrons nor the rest mass of charged pions, and
ignores any additional excitation energy or mass differences in the final
state nuclear system.

###  I.1 Past Results on Electron Neutrino Interactions at GeV Energies

Because of the relatively small number of electron neutrinos in GeV energy
accelerator beams and high backgrounds, previous measurements are few and are
often statistics and background limited. Qualitatively, previous measurements
fall into several categories. Some measurements have measured only flux
integrated total cross sections, or total cross sections as a function of
derived neutrino energy, with relatively low statistics, from tens to a few
hundreds of events [15, 16, 17, 18]. These low statistics measurements are
unlikely to constrain electron neutrino interaction models to the level needed
by future appearance oscillation experiments. The T2K and MicroBooNE
experiments have produced measurements of flux-integrated lepton kinematics
for samples of order a hundred or few hundred events [19, 20, 21] that can be
compared to models and could be sensitive to large deviations, $>10$%.

Several measurements have additional capabilities to test models. The NOvA
experiment has made a high statistics measurement of cross sections as a
function of lepton kinematics [22], with nearly $10^{4}$ signal events with
good purity, but makes its measurements in relatively wide bins of lepton
energy and angle. MINERvA [23] and MicroBooNE [24] have measured events
without final state pions, a sample presumably dominated by single and multi-
nucleon knockout, with good purity, and with samples of order $10^{3}$ and
$10^{2}$ events, respectively. These samples complement the measurements
reported here because of their sensitivity to this exclusive and important
reaction channel.

The measurement reported in this article, by contrast to these previous
results, is high statistics with tens of thousands of signal events in each of
the $\nu_{e}$ and $\bar{\nu}_{e}$ samples. In addition it measures
correlations between lepton kinematics (as measured primarily by $p_{T}$ and
the derived three momentum transfer, $q_{3}$) and a variable which is visible
energy transfer to the final state. It is inclusive, although in the lower
visible energy and momentum transfer regions. It is dominated by single and
multi-nucleon knockout events, and reports over a wide range of momentum
transfer, up to $1.6$ GeV in transverse momentum transfer, $p_{T}$ or $1.2$
GeV in inferred $q_{3}$.

##  II MINERvA Experiment and NuMI Beam line

The MINERvA detector is a fine-grained tracking calorimeter with a fully
active solid-scintillator tracker forming the bulk of the Inner Detector (ID).
Upstream of the tracker is an area of nuclear targets – carbon, water, iron,
and lead, interleaved with tracking planes. The downstream part of the ID
contains Electromagnetic CALorimeter (ECAL) and Hadronic CALorimeter (HCAL).
The ID is surrounded by side ECAL and side HCAL. Downstream, the MINOS Near
Detector served as a muon spectrometer for MINERvA. Muon charge and momentum
measurements were provided for muons with momenta above $\sim$1.5 GeV/c.

The active detector elements are solid-scintillator strips of triangular cross
section, with a 3.3 cm base, 1.7 cm high, arranged in planes where neighboring
strips alternate orientation with respect to the beam. Charge sharing between
neighboring strips provides a spatial resolution of $\sim$3 mm. Scintillation
light due to a charged particle traversing the scintillator is collected by a
wavelength shifting fiber located at the center of each strip and routed
through clear optical fibers to M64 Hamamatsu photo-multiplier tubes (PMT).
The electrical signals from Front End Boards mounted on top of each PMT box
are readout via the data acquisition system. The detector consists of
hexagonal modules containing one or two active planes mounted on a steel
frame. The orientation of strips in the planes can be vertical (X), +60∘ (U),
or -60∘ (V). Four types of modules were built: (i) tracker modules with strip
orientations X+U, X+V; (ii) ECAL with 0.2cm lead sheets, plus planes with
strip orientations X+U or X+V; (iii) HCAL with a 2.54cm thick iron plate, and
plane with strip orientation X, U, or V; and (iv) Target modules with passive
carbon, water, iron or lead targets.

MINER$\nu$A utilizes the intense broadband NuMI (Neutrinos at the Main
Injector) beam running at FNAL. FNAL’s Main Injector accelerates protons up to
120 GeV which are directed to a carbon target. The pions produced by the
proton interactions on the carbon target are focused by two horns and allowed
to decay in a 675 meter decay pipe. Undecayed pions are absorbed in the hadron
absorber just downstream of the decay region and the decay muons are absorbed
in the following 240 meters of rock before reaching the detector hall.
Different energy tunes are available by varying the location of target and
horns. The two main configurations are known as the Low Energy (LE) and Medium
Energy (ME) beam tunes. The results presented here are based on the ME
configuration. NuMI also allows for neutrino or anti-neutrino beam running by
sign selecting pions and kaons by setting the magnetic horn current direction.
The forward horn current (FHC) polarity produces predominantly muon neutrinos.
The reverse horn current (RHC) polarity produces predominantly muon
antineutrinos. However both FHC and RHC contain anti-neutrino and neutrino
contamination, respectively, on the order of a few percent. This cross-
contamination results from kaon decay producing neutrinos at the higher end of
the energy spectrum and from muon decay that creates neutrinos at the lower
end. The neutrino flux prediction (see Fig 1) used by MINERvA is derived from
a Geant4 simulation of the NuMI beamline which is constrained by measurements
from neutrino-electron elastic scattering[25, 26].

##  III Neutrino Interaction Simulation

Neutrino-nucleus interactions are simulated using GENIE v2.12.6. GENIE[27] is
a Monte Carlo (MC) neutrino interaction event generator which simulates
multiple neutrino-nucleus interaction channels exclusively, including the
three primary channels – charged current quasielastic (CCQE), resonant pion
production (RES), and deep inelastic scattering (DIS) - as well as subdominant
channels such as charm and coherent pion production. The quasielastic
interactions are simulated using the Llewellyn-Smith formalism[28] with BBBA05
vector form factor modeling[29], where the axial form factor uses the dipole
form with an axial mass of $M_{A}=0.99$ GeV. The Rein-Sehgal model[30] with
axial mass of $M_{A}^{RES}=1.12$ GeV is employed to simulate resonance
productions. DIS interactions are simulated using the leading order model with
the Bodek-Yang prescription[31]. In addition, ”two particle two hole” (2p2h)
interactions are simulated using the Valencia model[32, 33, 34] and coherent
pion production is simulated by the other Rein-Sehgal model[35]. The nucleon
initial states are simulated using the relativistic Fermi gas model[36] with
additional Bodek-Ritchie tail[37] while the FSI is simulated using the
INTRANUKE-hA package[38], which is a hadronic cascade model. To better
describe MINERvA data, there are tunes applied to the prediction of CCQE, RES,
and 2p2h interactions, collectively referred to as MINERvA tune v1, and
described in Ref. [39].

The coherent channel of GENIE does not simulate coherent scattering off
hydrogen atoms, e.g., diffractive pion production. However, MINERvA data from
its low energy beam showed that the contribution of the neutral current (NC)
diffractive process is sizable[40]. In order to simulate this process, the
charged current (CC) diffractive model in GENIE, which is an implementation of
the work by Rein[41], is used with two modifications to turn the CC model into
an NC model by producing a neutrino in the final state, reducing the cross
section by a factor of two for the expected CC/NC ratio.

Figure 1: Flux predictions for FHC mode (Left) and RHC mode (Right). The
contributions from all neutrino types are shown for each beam mode.

##  IV Electron Reconstruction and Identification

The reconstruction method employed in this analysis is a combination of the
MINERvA electron neutrino CCQE measurement preformed with the Low Energy (LE)
dataset[23] and the muon neutrino low-recoil analyses[13]. The electron
candidates are reconstructed using the same method as used in the LE analysis
but updated with re-tuned algorithms to cluster hits from neutrino
interactions in time and with tracking improvements.

An inclusive electron neutrino charged-current interaction sample is selected
using the following four cuts. First, events with any MINOS matched tracks are
rejected. Second, an electron candidate is constructed for each track that
originated from the most upstream vertex and is contained in the MINERvA
detector. The hits are considered as part of electron candidate if they are
inside a 7.5-degree cone region with an apex at the event vertex and axis
along the track direction or a cylindrical region of 50 mm radius extending
from the event vertex along the track direction. If there is more than a three
radiation length separation between a hit and the next downstream hit, this
upstream hit will be tagged as the most downstream hit considered part of the
electron candidate. Third, the collection of hits considered as the electron
candidate is tested by a k-nearest-neighbor classifier using three variables:
mean $dE/dx$, the fraction of energy deposited at the downstream end, and the
median shower width [42]. The classifier is trained to distinguish
electromagnetic showers from track like particles using simulated single
particle samples including electrons, muons, photons, changed pions, and
protons. Events are selected if there is at least one electron candidate
having a kNN score greater than 0.7. Lastly, the energy of the electron
candidate is measured by employing a calorimetric sum of hit energies,
corrected for passive materials. We choose the most energetic electron
candidate as the primary candidate if multiple candidates pass the threshold.

###  IV.1 Electron and Photon Separation

Additional selections are necessary because the kNN classifier is not
optimized to distinguish between electrons and photons. We use the minimal
energy deposition in a 100 mm sliding window from 25 mm to 500 mm downstream
of the event vertex (measured along track direction) of the electron candidate
as the discriminator and require the $dE/dx$ in the minimal window less than
2.4 MeV/cm (see Fig. 2) to be considered as an electron. In addition, we
reject events with multiple vertices since they are more likely to be a
neutral-current interaction.

###  IV.2 Reconstruction of Visible Calorimetric Energy

The visible calorimetric energy is calculated as described in the MINERvA muon
neutrino low-recoil measurement[13]. We assume that hits that are not included
in the electron candidate are the result of energy deposited by the hadronic
system. These hits are summed using the calibrated visible energy in each sub-
detector and corrected for passive material. We construct two hadronic energy
estimators using the visible energy in each sub-detector to estimate $q_{0}$
and $E_{\text{avail}}$ separately. The energy transfer $q_{0}$ is estimated by
summing the visible energies in all sub-detectors and applying a spline
correction to offset the bias observed by a MC study. $E_{\text{avail}}$ is
estimated (see Eq. 4) by summing up visible energies in the tracker and ECAL
and applying a flat scale factor. We construct these variables such that the
reconstructed $E_{\text{avail}}$ is more precise but less accurate while the
reconstructed $q_{0}$ is more accurate but less precise. In addition, we
estimate that 0.8% of the EM shower energy leaks out of the electron candidate
on average using a simulation study and we therefore correct the EM shower
energy leakage from the reconstructed $E_{\text{avail}}$ and $q_{0}$.

Finally, we reconstruct $q_{3}$ using lepton kinematics and reconstructed
$q_{0}$:

$\displaystyle q_{3}$ $\displaystyle=\sqrt{Q^{2}+q_{0}^{2}}$ $\displaystyle
Q^{2}$
$\displaystyle=2(E_{l}+q_{0})(E_{l}-|\vec{p_{l}}|\cos(\theta_{l}))-m_{l}^{2}$
(5)

##  V Backgrounds and Control Samples

Figure 2: RHC mean dE/dX distribution.

Figure 2 shows the distribution of the mean dE/dx quantity described above for
both data and simulation. There is a large excess of data events in the
background dominated region with mean dE/dx $>$ 2.4 MeV/cm.

This excess is similar to what was reported in MINERvA’s LE data[23] and with
the conclusion that it may be explained through diffractive pion production.
NC diffractive $\pi^{0}$ production is similar to NC coherent $\pi^{0}$
production in that both are inelastic processes where a lepton and a pion are
produced in the forward direction while leaving the struck nucleus in the
ground state. The square of the four-momentum transfer, $|t|$, must be small
to preserve the initial state of the nucleus. Since MINERvA’s tracker material
is a CH-based hydrocarbon, there is a possibility a neutrino interaction will
occur on a free proton, referred to as diffractive pion production. Since the
proton is much less massive than a carbon nucleus, the proton recoils visibly
from the momentum imparted to the target proton in the MINERvA detector. The
recoiling proton deposits its energy upstream of the “vertex” where the
$\pi^{0}$ is identified as an electron candidate. There exists a model for an
NC Diffractive scattering process in GENIE, from the work of Rein [41], that
is valid for $W>2.0$ GeV. GENIE’s implementation of the Rein model is used to
predict this background contribution.

We divide the background processes into two cases. The first case is when a
$\pi^{0}$ is the only particle produced in the neutrino interaction which is
inclusive of coherent NC pion production and NC diffractive pion production.
The second is when photons are produced from a $\pi^{0}$ and additional
particles are also produced simultaneously including NC incoherent pion
production and non-electron neutrino CC pion production. Given the 2.5 GeV
energy requirement of the electron, the NC background typically consists of
high hadronic invariant mass $W^{2}$ events defined as

$W^{2}=M^{2}+2M\nu,$ (6)

where $M$ is the nucleon mass. The exception are the NC diffractive $\pi^{0}$
and NC coherent $\pi^{0}$ cases. Neutrino-electron elastic scattering also
contributes to the signal sample at values of zero $E_{avail}$.

Figure 3: Graphic description of upstream inline energy. Upstream inline
energy is defined as the energy deposited inside a backward oriented
$7.5^{\circ}$ cone with its apex emanating from the interaction vertex.

###  V.1 Background Constraints

The backgrounds are constrained by examining sidebands in the high dE/dx
region which is further subdivided into three separate regions used to
separate $\pi^{0}$ production channels. We use two variables to define the
sidebands: upstream inline energy $E_{\rm UIE}$ and extra energy $\Psi
E_{EM}$. Upstream inline energy is defined as the energy depositions inside of
a reversed $7.5^{\circ}$ cone region, as shown in Fig. 3. $E_{\rm UIE}$ is the
best discriminator between NC $\pi^{0}$ coherent and NC $\pi^{0}$ diffractive
events, allowing for the capture of recoiling proton energy upstream of the
event vertex. $\Psi E_{EM}$ is defined by the ratio of visible energy outside
of the electron cone to energy inside:

$\Psi=\frac{E_{extra}+E_{\rm UIE}}{E_{EM}}$ (7)

The division of the sidebands regions are shown in Fig. 4 and summarized as
the incoherent $\pi^{0}$ region defined by $\psi E_{EM}$ $>$ 0.5 GeV and dE/dx
$>$ 2.4 MeV/cm, the coherent region defined by $\psi E_{EM}$ $<$ 0.5 GeV,
$E_{\rm UIE}$ $<$ 10 MeV and dE/dx $>$ 2.4 MeV/cm, and the diffractive region
defined by $\psi E_{EM}$ $<$ 0.5 GeV, $E_{\rm UIE}$ $>$ 10 MeV and dE/dx $>$
2.4 MeV/cm.

Figure 4: Representation of signal and sideband regions with respective cuts. The “excess” sideband has been subdivided into low and high upstream inline energy. $E_{EM}$ (GeV) | [2.5,5) | [5,7.5) | [7.5,10) | [10,12.5) | [12.5,15) | [15,20) |
---|---|---|---|---|---|---|---
Diffractive $\pi^{0}$ | 3.385 | 7.413 | 9.535 | 15.95 | 23.21 | 9.807 |
Coherent $\pi^{0}$ | 1.970 | 2.258 | 2.936 | 2.614 | 2.018 | 5.363 |
$P_{lep}^{t}$ (GeV) | [0,0.2) | [0.2,0.4) | [0.4,0.6) | [0.6,0.8) | [0.8,1.0) | [1.0,1.2) | [1.2,1.6)
Non-coherent $\pi^{0}$ | 0.6897 | 0.6945 | 0.7659 | 0.8151 | 0.9229 | 1.014 | 1.151
Table 1: FHC scale factors applied to $\pi^{0}$ production processes $E_{EM}$ (GeV) | [2.5,5) | [5,7.5) | [7.5,10) | [10,12.5) | [12.5,15) | [15,20) |
---|---|---|---|---|---|---|---
Diffractive $\pi^{0}$ | 5.03 | 7.868 | 7.095 | 10.114 | 10.767 | 4.134 |
Coherent $\pi^{0}$ | 1.911 | 2.000 | 2.363 | 1.894 | 1.318 | 3.693 |
$P_{lep}^{t}$ (GeV) | [0,0.2) | [0.2,0.4) | [0.4,0.6) | [0.6,0.8) | [0.8,1.0) | [1.0,1.2) | [1.2,1.6)
Non-coherent $\pi^{0}$ | 1.156 | 1.074 | 1.044 | 1.083 | 1.072 | 1.198 | 1.336
Table 2: RHC scale factors applied to $\pi^{0}$ production processes

The normalization of the $\pi^{0}$ backgrounds are each fitted using
distributions in both bins of $E_{avail}$ vs $q_{3}$ and $E_{avail}$ vs
$p_{T}$ to obtain scale factors that represent the best estimate of the
normalization of data compared to the GENIE prediction. The signal
contribution is also tuned during this global fitting due to its non-
negligible contribution in the sideband regions, although that tune is not
used in the simulation compared to background subtracted signal. The fitting
process, which is done through minimizing the negative log-likelihood assuming
Poisson distribution, is done in two steps. The first global fit is done with
RHC data in $\Psi E_{EM}$ vs $p_{T}$ in bins of $E_{EM}$ which optimizes the
NC coherent and diffractive processes. The background predictions of the
coherent and diffractive $\pi^{0}$ processes are updated by applying scale
factors on an event-by-event basis. The second global fit is done in
$E_{avail}$ vs $p_{T}$ in bins of $p_{T}$ which optimizes non-coherent
$\pi^{0}$ and signal processes separately for the respective FHC and RHC
samples. The applied scale factors for the FHC and RHC analyses are found in
Tables 1 and 2 respectively. Pre and post tune distributions in the sideband
regions can be found in Figs. 5 \- 10 for RHC and FHC respectively.

Figure 5: Pre-background tuned FHC (top) and RHC (bottom) $p_{T}$ distribution
for the incoherent $\pi^{0}$ sideband of $dE/dx>$ 2.4 MeV/cm, $\psi*E_{e}>$
0.5 GeV.

Figure 6: Post-background tuned $p_{T}$ FHC (top) and RHC (bottom)
distribution for the incoherent $\pi^{0}$ sideband of $dE/dx>$ 2.4 MeV/cm,
$\psi*E_{e}>$ 0.5 GeV.

Figure 7: Pre-background tuned $p_{T}$ FHC (top) and RHC (bottom) $p_{T}$
distribution for the low UIE sideband of $dE/dx>$ 2.4 MeV/cm, $\psi*E_{e}<$
0.5 GeV, $E_{uie}<$ 10 MeV.

Figure 8: Post-background tuned $p_{T}$ FHC (top) and RHC (bottom) $p_{T}$
distribution for the low UIE sideband of $dE/dx>$ 2.4 MeV/cm, $\psi*E_{e}<$
0.5 GeV, $E_{uie}<$ 10 MeV.

Figure 9: Pre-background tuned FHC (top) and RHC (bottom) $p_{T}$ distribution
for the high UIE sideband of $dE/dx>$ 2.4 MeV/cm, $\psi*E_{e}<$ 0.5 GeV,
$E_{uie}>$ 10 MeV.

Figure 10: Post-background tuned FHC (top) and RHC (bottom) $p_{T}$
distribution for the high UIE sideband of $dE/dx>$ 2.4 MeV/cm, $\psi*E_{e}<$
0.5 GeV, $E_{uie}>$ 10 MeV. The disagreement in FHC, as a result of tension
between this region and the incoherent $\pi^{0}$ sideband, is discussed in the
text.

Figure 11: The high UIE sideband in FHC: demonstration of the tension in the
tuning, and the alternate scenarios considered for (top) the low UIE sideband,
(middle) the high UIE sideband, and (bottom) the incoherent $\pi^{0}$
sideband.

####  V.1.1 Tensions in the FHC Constraints

As noted above, the diffractive and NC coherent backgrounds are estimated
using the RHC samples because those processes are a larger fraction of the low
UIE and high UIE sidebands, respectively in the RHC beam. However, in the high
UIE and incoherent $\pi^{0}$ FHC sideband, this fit is unable to reproduce the
shape of the data as a function of $E_{avail}$, as shown in Fig. 11.
Additional tunes and a systematic uncertainty on those tunes were developed to
address this disagreement. We considered two alternate hypotheses, neither of
which describes the data well across all of the sidebands. In the first, NC
coherent and diffractive processes are allowed to have an additional
normalization in the second global FHC fit. The rationale is that the high
energy neutrino components in the two beams, above the focusing peak, are
different. This could affect the relative event rates in the FHC and RHC
samples if the cross section has a poorly modeled rate as a function of
neutrino energy. In the second hypothesis, a subset of the non-coherent
$\pi^{0}$ processes that dominates the region with the observed high UIE
disagreement ($0.2$ GeV $<E_{avail}<0.5$ GeV and $P_{lep}^{t}<1$ GeV) are
enhanced independently from other non-coherent $\pi^{0}$ processes with a
separate scale factor. We use the average of these two fits as our base
background prediction, and take the difference between the two as an
assessment of the systematic uncertainty in this procedure. Note that the
background comes from all of these contributions together, and so the
systematic underprediction of the FHC high UIE sideband coupled with the
systematic underprediction of the incoherent $\pi^{0}$ sideband do not
indicate that the background is poorly estimated.

###  V.2 Interpretation of the Coherent and Diffractive Contributions

As shown in Tables 1 and 2, the scale factors for the NC diffractive $\pi^{0}$
background process as well as the NC coherent $\pi^{0}$ process are large. We
believe the explanation for these large scale factors is that the diffractive
and coherent $\pi^{0}$ processes are not well modeled by our reference GENIE
model. We will discuss in turn the evidence that these events are in fact
single $\pi^{0}$ in nature, the relative strength of the coherent and
diffractive processes, and the reliability of the GENIE model prediction as a
function of $E_{\pi^{0}}$.

Figure 12: RHC energy outside of electron candidate cone and vertex region in
the high UIE sideband region of dE/dx $>$ 2.4 MeV/cm, $\psi*E_{e}<0.5GeV$,
$E_{uie}>$ 10 MeV.

Measurements of events from the sidebands targeting coherent and diffractive
$\pi^{0}$ production, the low UIE and high UIE sidebands respectively, do
support the hypothesis that these events have a high energy $\pi^{0}$. Since
electromagnetic cascades spread out transversely to the direction of
propagation, there is a range of energy where single-photon showers can be
distinguished from multi-photon showers based on transverse size. Median
shower width, or median transverse width, provides the extent to which an
electromagnetic cascade spreads transversely to its direction of propagation.

Figure 12 shows the post background tuned energy outside the electron
candidate cone and the vertex region, referred to as the extra energy
($E_{extra}$). Most events from the high and low UIE sidebands populate the
first few bins and are well described. From these distributions, it is
apparent that the event has little non-shower activity. Additionally, the post
background tuned inline-upstream energy cone distribution, shown in Fig. 13,
indicates that the shape of the diffractive and coherent processes agree with
what we would expect from energy upstream of the event vertex.

Figure 13: RHC upstream inline energy in the high UIE sideband region of dE/dx
$>$ 2.4 MeV/cm, $\psi*E_{e}<0.5GeV$, $E_{uie}>$ 10 MeV.

We note that the relative rate of the diffractive reaction, with high upstream
inline energy, and the coherent events, with low upstream inline energy is
consistent with naive scaling arguments which would suggest a dependence on
the atomic number, $A$, somewhere between $A^{1/3}$ and $A^{2/3}$ between
carbon and hydrogen. This suggests that the large difference between the two
in the GENIE model, approximately a factor of $30$, is incorrect.

On the subject of the $\pi^{0}$ energy dependence of the scale factors, a
separate MINERvA analysis studying neutrino-induced coherent $\pi^{+}$
production on different targets [43] concluded that the Rein-Sehgal and PCAC-
based Belkov-Kopeliovich (B-K) models do not accurately describe the angular
dependence on $\theta_{\pi}$, the energy-dependence on $E_{\pi}$, or the
A-dependence. The fact that this is also seen in the charged current analog
reaction makes the energy dependent scale factors needed in this analysis more
plausible.

Figure 14: Post-background tuned FHC (top) and RHC (bottom) $p_{T}$
distribution for the Signal region of $dE/dx<$ 2.4 MeV/cm.

###  V.3 Background Subtracted Signal Distributions

Figure 14 shows the lepton $p_{T}$ distribution for the signal region for both
FHC and RHC samples. The FHC sample has approximately 46,700 selected events
with a total estimated background of 24,600 events. The RHC sample has
approximately 28,300 selected events and a total estimated background of 8,000
events.

##  VI Separation of $\nu_{e}$ and $\bar{\nu}_{e}$ Events

While $\bar{\nu}_{e}$ and $\nu_{e}$ events are indistinguishable in data, the
MC simulation provides a prediction for the $\bar{\nu}_{e}$ contribution to
the FHC sample and the $\nu_{e}$ contribution to the RHC sample. To correct
for the contamination from these events in the respective samples, we form an
estimator based on FHC data and the MC simulation that gives a prediction of
the $\nu_{e}$ background found in the RHC sample and vice versa. The procedure
is identical for the two measurements, so we will describe the procedure to
correct the RHC measurement.

In this procedure, a corrected FHC sample is used to replace the MC prediction
for the $\nu_{e}$ event rate in the RHC sample. First a ratio of RHC/FHC
$\nu_{e}$ events distributed in true neutrino energy is formed, seen in Fig.
15 $\nu_{e}$, as a function of true neutrino energy.

Figure 15: Ratio of RHC/FHC $\nu_{e}$ in True Neutrino Energy

This ratio must be applied to correct the FHC $\nu_{e}$ events to make a
prediction for them in RHC. To apply the correction to the data, a neutrino
energy estimator is developed out of the reconstructed available energy and
the reconstructed electron energy,

$\displaystyle E_{est}=E_{e}+E_{avail}$ (8)

The accuracy of the formed energy estimator to predict the true neutrino
energy for the samples in this analysis is shown in Fig. 16.

Figure 16: Neutrino energy estimator vs true neutrino energy in RHC.

The energy estimator value is then used to correct events on an event-by-event
based by the ratio fo the fluxes in two beams shown in Fig. 15.

A complication is that in the data, there are background contributions to the
RHC samples, as well as contributions from $\bar{\nu}_{e}$. The simulation is
used to predict the initial $\bar{\nu}_{e}$ background to the RHC sample, and
the other backgrounds are predicted as described above with the tunes to the
control samples. Each of these contributions is weighted on an event by event
basis by the energy estimator from the reconstruction, whether the source is
data or simulation. After this weighting, the RHC $\nu_{e}$ prediction is
formed by taking the corrected FHC data, and subtracting the corrected
$\nu_{e}$ background prediction and the corrected other sources of background.
Because the flux correction is made event-by-event, these samples can be used
to predict the background in the measured reconstucted variables. This
procedure is iterated once, replacing the initial MC prediction of
$\bar{\nu}_{e}$ events with the data corrected version from this procedure.
This background is less than a few percent in most bins, and largest in high
$p_{T}$ bins with high $E_{avail}$ in RHC or low $E_{avail}$ in FHC.

Figure 17: The scaled RHC (FHC) sample prediction (orange) for the $\nu_{e}$
electron antineutrino ($\bar{\nu}_{e}$ electron neutrino) background compared
to the predicted simulated sample and the observation in the data with all
backgrounds subtracted for $p_{T}<1.6$ GeV/c. The top plot shows the effect on
the $\nu_{e}$ sample, and the bottom the effect on the $\bar{\nu}_{e}$ sample.

##  VII Measurement of Differential Cross Sections

Calculation of the flux-integrated differential cross section per nucleon for
kinematic variable $x$, in bins of i, is measured by the following equation:

$(\frac{d\sigma}{dx})_{i}=\frac{\sum_{j}U_{ij}(N_{j}^{data}-N_{j}^{bkg})}{\epsilon_{i}T\Phi(\Delta
x)_{i}}$ (9)

where $(\frac{d\sigma}{dx})_{i}$ is the differential cross section as function
of $x$ at bin $i$, $U_{ij}$ is the unfolding matrix, $N_{j}^{data}$ is the
measured number of events in bin $j$ of reconstructed variable $x$,
$N_{j}^{bkg}$ is the predicted number of background events in bin $j$,
$\epsilon_{i}$ is estimated acceptance at bin $i$, $T$ is number of nucleon
targets, $\Phi$ is integrated neutrino flux (or integrated anti-neutrino
flux), and $(\Delta x)_{i}$ is bin width normalization of bin $i$.

The double differential cross sections $d^{2}\sigma/dE_{avail}dq_{3}$ and
$d^{2}\sigma/dE_{avail}dp_{T}$ are calculated using the selected number of
events and subtracting the number of background events predicted by the
simulation. An iterative unfolding approach using the D’Agostini[44, 45]
method as implemented in RooUnfold[46] was used to correct the background-
subtracted distributions for resolution effects. Several unfolding studies
were carried out using randomly thrown pseudodata samples generated from
alternate physics models. The number of unfolding iterations is determined
through $\chi^{2}$ values calculated by comparing the unfolded pseudodata with
the truth pseudodata. The alternate physics models used in the studies include
modification to the 2p2h enhancement to include a reweight giving nn/pp or np
pairs additonal strength and a modification of RPA suppression affecting the
low $Q^{2}$ or high $Q^{2}$ regions. The result of the unfolding studies for
both the RHC and FHC analyses indicate that a different number of unfolding
iterations are required for the two different distributions based on the
different minimum $\chi^{2}$ values. It is decided that $E_{avail}$ vs $p_{T}$
will be unfolded with 10 iterations and $E_{\text{avail}}$ vs $q_{3}$ unfolded
with 15 iterations.

The number of events after unfolding is then divided by the efficiency. In
this analysis, a few bins on the edge of the phase space contain low
statistics, leading to large statistical uncertainty in the efficiency
estimation. To account for this, the efficiency in the low statistic bins are
estimated by the average of adjacent bins. The RHC efficiency is found in
Figs. 18 and 19 for the $E_{avail}$ vs $q_{3}$ and $E_{avail}$ vs $p_{T}$
distributions respectively. In both cases, the efficiency decreases at higher
$E_{avail}$ values. This is most likely because it is more difficult for the
tracking algorithm to reconstruct a proper electron candidate track at higher
$E_{avail}$ due to the greater amount of hadronic activity overlapping with EM
showers. The equivalent FHC efficiencies are found in Figs. 20 and 21. The
inefficiency for high $E_{avail}$ events is due to the overlapping of EM
showers and hadronic activity.

Figure 18: RHC Efficiency for $E_{avail}$ vs $q_{3}$ distribution. Figure 19:
RHC Efficiency for $E_{avail}$ vs $p_{T}$ distribution. Figure 20: FHC
Efficiency for $E_{avail}$ vs $q_{3}$ distribution. Figure 21: FHC Efficiency
for $E_{avail}$ vs $p_{T}$ distribution.

The normalization factors include 3.234$\times 10^{30}$ nucleon targets and
the flux integral from 0 to 100 GeV for a total integrated flux value of
$\Phi=2.34\times 10^{12}$ $\bar{\nu}/cm^{2}$ for the anti-neutrino analysis
and $\Phi=6.7\pm 0.2\times 10^{11}$ $\nu/cm^{2}$ for the neutrino analysis.
The double differential cross sections $d^{2}\sigma/dE_{avail}dq_{3}$ and
$d^{2}\sigma/dE_{avail}dp_{T}$ are found in Figs. 22 and 23 for the RHC
analysis and Figs. 24 and 25 for the FHC analysis.

Figure 22: A decomposition of the RHC cross section result into contributing
interaction types in $E_{avail}$ vs $q_{3}$ on a y log scale. The y axis is on
a log scale truncated at $10^{-2}$ to enable a better view of the tail end of
the cross section. Figure 23: A decomposition of the RHC cross section result
into contributing interaction types in $E_{avail}$ vs $p_{T}$ on a y log
scale. The y axis is on a log scale truncated at $10^{-2}$ to enable a better
view of the tail end of the cross section. Figure 24: A decomposition of the
FHC cross section result into contributing interaction types in $E_{avail}$ vs
$q_{3}$. Figure 25: A decomposition of the FHC cross section result into
contributing interaction types in $E_{avail}$ vs $p_{T}$.

###  VII.1 Systematic Uncertainties

Statistical uncertainties dominate systematic uncertainties in nearly every
bin of these measurements. Uncertainties on the measured cross sections can be
categorized into four major groups: flux, detector model, interaction model,
and MINERvA tunes. The breakdown of the RHC fractional systematic uncertainty
for $d^{2}\sigma/dE_{avail}dq_{3}$ is shown in Fig. 26 and
$d^{2}\sigma/dE_{avail}dp_{T}$ in Fig. 27. The equivalent plots for FHC are
shown in Figs. 28 and 29 respectively.

The uncertainties related to the flux can be broken down into two major
categories: focusing uncertainties associated with all components related to
the NuMI beam and hadron production uncertainties related to the uncertainty
of hadron production from the proton beam incident on the graphite target. The
flux uncertainty is fairly constant with $E_{avail}$ and $q_{3}$/$p_{T}$,
around 4.7%.

The detector model uncertainties consist of the uncertainties pertaining to
the simulation of particle propagation through the detector, particle and
kinematic reconstruction and the particle response of the detector. The
detector model uncertainty can be broken into two groups: hadronic energy and
electron reconstruction. A systematic uncertainty is assessed on the
correction for leakage of electron energy outside of the electron cone. The
energy leakage outside the cone leads to an overestimation of the available
energy. The energy leakage was estimated to be 0.8% of the electron energy. We
estimate the energy leakage by simulating electron initiated showers with
various energies and angles. By comparing this simulation to our sample of
neutrino-electron elastic scattering ($\nu+e\rightarrow\nu+e$) events, we
conclude that the simulation underestimates the energy leakage by $5\pm 2$
MeV, and the $2$ MeV uncertainty from this study is the assigned systematic
uncertainty. The leading uncertainty in $q_{3}$ bins with the highest
$E_{avail}$ is the leakage uncertainty. The highest $p_{T}$ bin shows large
systematic error values, similar to the GENIE error summary, due to the low
number of events in that bin. The leakage uncertainty is the leading
systematic uncertainty for the lower $p_{T}$ bins.

The interaction model uncertainties encompass GENIE interaction model
uncertainties as well as GENIE final state interaction uncertainties. For the
RHC analysis, the leading systematic uncertainty for most bins is the axial
mass $M_{A}$ resonance production (MaRES) which adjusts the $M_{A}$ in the
Rein-Sehgal cross section, affecting the shape and normalization. This next
leading systematic is the $M_{V}$ resonance production (MvRES), which adjusts
the axial vector mass $M_{V}$ in the Rein-Sehgal cross section, and the
charged current resonance normalizaion (NormCCRES) that implements changes the
normalization of CC Rein-Sehgal cross section. As shown in Figs. 22 and 23,
the CC resonant pion production has a large contribution to the cross section
measurement. Since the GENIE MaRES and MvRES parameters control resonant pion
production it is not surprising that they are among the leading contributors
to the uncertainty. For the RHC analysis, the probability for elastic
scattering of nucleons while conserving the total rescattering probability
(FrInelas_N), contributes to the first bin of $q_{3}$ and highest $E_{avail}$
bin with less contribution for higher $q_{3}$ bins. Most likely this would
include a neutron losing a large amount of energy in a collision, resulting
with a proton in the final state.

The systematic error breakdown for the MnvTunes shows the low $Q^{2}$ tune
affects the highest bin of $E_{avail}$ for a given $q_{3}$ and $p_{T}$. This
is expected because these are the regions in which the process is most
dominant. The low recoil 2p2h tune has a larger systematic uncertainty for
values of $q_{3}$ compared to $p_{T}$. The shifting of the 2p2h model impacts
the $E_{avail}$ distribution and the effects are seen more easily in $q_{3}$
due to the model dependency.

Figure 26: RHC: Total cross section error summary broken down into four major
subgroups for $E_{avail}$ vs $q_{3}$ Figure 27: RHC: Total cross section error
summary broken down into four major subgroups for $E_{avail}$ vs $p_{T}$
Figure 28: FHC: Total cross section error summary broken down into four major
subgroups for $E_{avail}$ vs $q_{3}$ Figure 29: FHC: Total cross section error
summary broken down into four major subgroups for $E_{avail}$ vs $p_{T}$

###  VII.2 Discussion and Interpretation of Anti-neutrino and Neutrino Results

The measurement shows more rate than predicted for the anti-neutrino analysis
in the first bin of $E_{avail}$ in the cross section both for $E_{avail}$ vs
$q_{3}$ in Fig. 22 and for $E_{avail}$ vs $p_{T}$ as seen in Fig. 23. The
events predicted to populate the first bin of available energy tend to be
events where the final state is neutral, typically final state neutrons, and
in the first bin of $E_{avail}$ $\sim$ 60-90% of the model prediction is
charged current quasielastic events. The quasielastic events are expected to
be the dominant contributor to the first $E_{avail}$ bin because, in the
absence of final state interactions, there is only a lepton and neutron in the
final state.

Looking more closely at the $q_{3}$ bin of 0.4-0.6 GeV in Fig. 22, there is a
population of inelastic events that leak into the first bin of $E_{avail}$. It
is possible that some type of inelastic events with mostly neutrons in the
final state is not being correctly simulated. It also could involve events
where the final state pion doesn’t have much energy and is absorbed within the
nucleus, resulting in only final state neutrons. The last proposal to explain
the high cross section in the first $E_{avail}$ bin is that the MC simulation
predicts too many quasielastic events at higher values of $E_{avail}$.
Increasing the population of quasielastic MC events near zero $E_{avail}$
would improve this prediction.

In contrast to the anti-neutrino results there is a deficit of data events
over the simulated prediction for the neutrino analysis found in the first bin
of $E_{avail}$ seen in the cross section plots for both $E_{avail}$ vs $q_{3}$
as shown in Fig. 24 and $E_{avail}$ vs $p_{T}$ (Fig. 25) in lowest respective
bins.

###  VII.3 Comparison to Muon Neutrino and Antineutrino Measurements

These results can be compared with MINERvA’s measurement of the analogous
samples from muon neutrinos and antineutrinos. However, there are differences
in the measurements that make a direct comparison challenging. In particular,
there all of the measurements of muon neutrino and antineutrino processes are
made with neutrino spectra which are substantially different than the ones
measured in these results.

The $\bar{\nu}_{e}$ cross section result would be most appropriately compared
with MINERvA’s low recoil LE $\bar{\nu}_{\mu}$ result [47]. In addition to the
flux differences, there are also selection differences between the two
analyses, so the signal definitions are not identical. The $\bar{\nu}_{\mu}$
result requires a lepton momentum of greater than 1.5 GeV, while the
$\bar{\nu}_{e}$ analysis requires lepton energy greater than 2.5 GeV to
eliminate a large $\pi^{0}$ background at low electron energy. In addition,
the $\bar{\nu}_{e}$ analysis has no scattering angle requirements while the
$\bar{\nu}_{\mu}$ analysis requires the lepton scattering angle to be less
than 20 degrees due to the difficulty of reconstructing high angle muons. The
analyses are also reported using different binning. Table 3 shows a binning
comparison between the ME and LE results for $E_{avail}$ and Table 4 shows the
comparison for $q_{3}$.

ME $\bar{\nu}_{e}$ |  LE $\bar{\nu}_{\mu}$
---|---
$0.0<E_{avail}(GeV)<0.04$ | $0.0<E_{avail}(GeV)<0.03$
$0.04<E_{avail}(GeV)<0.08$ | $0.03<E_{avail}(GeV)<0.07$
$0.8<E_{avail}(GeV)<0.12$ | $0.07<E_{avail}(GeV)<0.17$
$0.12<E_{avail}(GeV)<0.16$ | $0.17<E_{avail}(GeV)<0.27$
$0.16<E_{avail}(GeV)<0.24$ | $0.27<E_{avail}(GeV)<0.35$
$0.24<E_{avail}(GeV)<0.32$ | $0.35<E_{avail}(GeV)<0.5$
$0.32<E_{avail}(GeV)<0.4$ |
$0.4<E_{avail}(GeV)<0.5$ |
Table 3: Comparison between the $E_{avail}$ binning used for the low recoil ME $\bar{\nu}_{e}$ analysis (left) and the low recoil LE $\bar{\nu}_{\mu}$ analysis (right). The $\bar{\nu}_{e}$ binning is truncated at 0.5 GeV for the comparison to LE but the results are reported up to 1.2 GeV. ME $\bar{\nu}_{e}$ |  LE $\bar{\nu}_{\mu}$
---|---
$0.0<q_{3}(GeV)<0.2$ | $0.0<q_{3}(GeV)<0.2$
$0.2<q_{3}(GeV)<0.4$ | $0.2<q_{3}(GeV)<0.3$
$0.4<q_{3}(GeV)<0.6$ | $0.3<q_{3}(GeV)<0.4$
$0.6<q_{3}(GeV)<0.8$ | $0.4<q_{3}(GeV)<0.5$
$0.8<q_{3}(GeV)<1.0$ | $0.5<q_{3}(GeV)<0.6$
$1.0<q_{3}(GeV)<1.2$ | $0.6<q_{3}(GeV)<0.8$
Table 4: Comparison between the $q_{3}$ binning used for the low recoil ME
$\bar{\nu}_{e}$ analysis (left) and the low recoil LE $\bar{\nu}_{\mu}$
analysis (right).

Lastly, the two analyses took different approaches in unfolding. The
$\bar{\nu}_{e}$ unfolds using coarse binning and a large number of iterations
and the $\bar{\nu}_{\mu}$ analysis unfolds using fine binning and a small
number of iterations. This is due in large part to the difference in
observables. The ME $\bar{\nu}_{e}$ analysis has to account for the energy
leakage outside the electron cone and into the available energy. Overall, the
LE $\bar{\nu}_{\mu}$ has a much better energy resolution compared to the
$\bar{\nu}_{e}$ analysis. With the consideration of the differences between
the two analyses, the cross section result for the LE $\bar{\nu}_{\mu}$ is
shown in Fig. 30 and the relevant cross section bins for the $\bar{\nu}_{e}$
are shown in Fig. 31.

Figure 30: $d^{2}\sigma/dE_{avail}dq_{3}$ cross section per nucleon compared
to the model with RPA and tune 2p2h components. Figure from Ref. [47] Figure
31: The $\bar{\nu}_{e}$ cross section result truncated at $0.8$ GeV on logy
scale in $q_{3}$ for comparison to the LE result. Figure 32: Comparison of
published MINERvA $\nu_{\mu}$ ME measurements with the present ME $\nu_{e}$
results.

There are similar features between the two results. As expected in the cross
section model prediction, both cross section results have quasielastic events
as the dominant contributor in the first bin of $E_{avail}$. There is a
population of 2p2h events for values of low $E_{avail}$. The delta resonance
becomes the dominant process at the higher values of $E_{avail}$ in both
predicted cross sections. There is a noticeable difference between the data
results for values of $\sim E_{avail}>0.2$ GeV. The reference MC prediction
for the $\bar{\nu}_{\mu}$ cross section consistently exceeds the data while
the $\bar{\nu}_{e}$ prediction falls below the data.

Both cross section results contain many events that have no available energy.
This creates a sharp peak at zero $E_{avail}$ followed by a cross section that
falls slowly compared to the size of the peak in the first $E_{avail}$ bin.
Therefore, to compare the first two $E_{avail}$ bins for both results we
assume that the peak at zero $E_{avail}$ is a Kronecker delta function-like
peak and the remaining cross section distribution is flat. To determine the
magnitude of the delta function we subtract the second $E_{avail}$ bin from
the first, or the flat distribution from the peak, leaving us with a
$d\sigma/dE_{avail}$ value. We multiply $d\sigma/dE_{avail}$ by the bin width
so that we end up with a cross section that is differential in each $q_{3}$
bin. This process is repeated for each result’s data and MC values. The
resultant bin combination for the two samples are 0.0 $<E_{avail}$(GeV) $<$
0.08 for the $\bar{\nu}_{e}$ cross section result and 0.0 $<E_{avail}$(GeV)
$<$ 0.07 for the $\bar{\nu}_{\mu}$ cross section result. Tables 6 and 7
summarize the results. Tables 8 and 9 are the correlation matrices for the
reported bins with the correlation matrix ordering defined in Table 5.

The conclusion drawn from the comparison between the data/MC peak at zero is
that the ME $\bar{\nu}_{e}$ result is consistent with the LE $\bar{\nu}_{\mu}$
result in all $q_{3}$ bins except the first. The ME $\bar{\nu}_{e}$ result has
a significant enhancement over the simulation.

Similarly, we can in addition compare the $\nu_{e}$ cross section result with
MINERvA’s low recoil ME $\nu_{\mu}$ result[14]. As with the comparison above
for the $\bar{\nu}_{e}$, there are significant differences between the
$\nu_{e}$ and $\nu_{\mu}$ analyses including the flux and features of the
reconstruction. The cross section results of ($\nu_{\mu}$ ME and $\nu_{e}$)
are compared in Fig. 32. We conclude the $\nu_{e}$ result is qualitatively
consistent with the $\nu_{\mu}$ results, except in the lowest $q_{3}$ and
$E_{avail}$ bin where there is some indication of a difference. In this bin
the measured $\nu_{e}$ cross section is smaller than the $\nu_{\mu}$ cross
section, albeit with large uncertainties.

|  0.0 $<q_{3}(GeV)<$ 0.2 |  0.2 $<q_{3}(GeV)<$ 0.4
---|---|---
Estimated peak at zero | 1 | 3
Subtraction size from first bin | 2 | 4
|  0.4 $<q_{3}(GeV)<$ 0.6 |  0.6 $<q_{3}(GeV)<$ 0.8
Estimated peak at zero | 5 | 7
Subtraction size from first bin | 6 | 8
Table 5: Correlation matrix ordering for both results. Values in the table refer to bin numbers. Note that the correlation matrix value for the data/MC peak is equivalent to the estimated peak at zero value. |  0.0 $<q_{3}(GeV)<$ 0.2 | Diagonal Unc.
---|---|---
Estimated data peak at zero | 0.99 | 0.20
Data subtraction size from first bin | 0.015 | 0.005
Estimated MC peak at zero | 0.52 | n/a
MC subtraction size from first bin | 0.01 | n/a
Data/MC peak at zero | 1.90 | 0.38
|  0.2 $<q_{3}(GeV)<$ 0.4 | Diagonal Unc.
Estimated data peak at zero | 2.18 | 0.35
Data subtraction size from first bin | 0.36 | 0.09
Estimated MC peak at zero | 1.87 | n/a
MC subtraction size from first bin | 0.29 | n/a
Data/MC peak at zero | 1.17 | 0.19
|  0.4 $<q_{3}(GeV)<$ 0.6 | Diagonal Unc.
Estimated data peak at zero | 3.12 | 0.45
Data subtraction size from first bin | 0.67 | 0.13
Estimated MC peak at zero | 2.43 | n/a
MC subtraction size from first bin | 0.79 | n/a
Data/MC peak at zero | 1.53 | 0.18
|  0.6 $<q_{3}(GeV)<$ 0.8 | Diagonal Unc.
Estimated data peak at zero | 3.24 | 0.40
Data subtraction size from first bin | 0.51 | 0.08
Estimated MC peak at zero | 2.05 | n/a
MC subtraction size from first bin | 0.52 | n/a
Data/MC peak at zero | 1.58 | 0.19
Table 6: Summary of results for ME $\bar{\nu}_{e}$. |  0.0 $<q_{3}(GeV)<$ 0.2 | Diagonal Unc.
---|---|---
Estimated data peak at zero | 0.73 | 0.078
Data subtraction size from first bin | 0.00 | 0.000
Estimated MC peak at zero | 0.9 | n/a
MC subtraction size from first bin | 0.08 | n/a
Data/MC peak at zero | 0.74 | 0.08
|  0.2 $<q_{3}(GeV)<$ 0.4 | Diagonal Unc.
Estimated data peak at zero | 3.1 | 0.33
Data subtraction size from first bin | 0.38 | 0.06
Estimated MC peak at zero | 2.55 | n/a
MC subtraction size from first bin | 0.36 | n/a
Data/MC peak at zero | 1.15 | 0.12
|  0.4 $<q_{3}(GeV)<$ 0.6 | Diagonal Unc.
Estimated data peak at zero | 4.1 | 0.51
Data subtraction size from first bin | 0.66 | 0.12
Estimated MC peak at zero | 3.46 | n/a
MC subtraction size from first bin | 0.51 | n/a
Data/MC peak at zero | 1.16 | 0.14
|  0.6 $<q_{3}(GeV)<$ 0.8 | Diagonal Unc.
Estimated data peak at zero | 3.5 | 0.42
Data subtraction size from first bin | 0.44 | 0.06
Estimated MC peak at zero | 2.91 | n/a
MC subtraction size from first bin | 0.26 | n/a
Data/MC peak at zero | 1.20 | 0.15
Table 7: Summary of results for LE $\bar{\nu}_{\mu}$. 1.00 | 0.47 | 0.16 | -0.03 | 0.36 | 0.04 | 0.36 | 0.15
---|---|---|---|---|---|---|---
0.47 | 1.00 | -0.06 | 0.51 | 0.07 | 0.27 | 0.13 | 0.04
0.16 | -0.06 | 1.00 | -0.32 | 0.49 | -0.30 | 0.51 | -0.21
-0.03 | 0.51 | -0.32 | 1.00 | -0.30 | 0.60 | -0.30 | 0.42
0.36 | 0.07 | 0.49 | -0.30 | 1.00 | -0.26 | 0.53 | -0.26
0.04 | 0.27 | -0.30 | 0.60 | -0.26 | 1.00 | -0.30 | 0.72
0.36 | 0.13 | 0.51 | -0.30 | 0.53 | -0.30 | 1.00 | -0.14
0.15 | 0.04 | -0.21 | 0.42 | -0.26 | 0.72 | -0.14 | 1.00
Table 8: Correlation matrix for ME $\bar{\nu}_{e}$ result for relevant bins. 1.00 | 0.00 | 0.82 | 0.37 | 0.67 | 0.38 | 0.56 | 0.42
---|---|---|---|---|---|---|---
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
0.82 | 0.00 | 1.00 | 0.19 | 0.83 | 0.30 | 0.76 | 0.38
0.37 | 0.00 | 0.19 | 1.00 | 0.21 | 0.75 | 0.01 | 0.70
0.67 | 0.00 | 0.83 | 0.21 | 1.00 | 0.32 | 0.86 | 0.38
0.38 | 0.00 | 0.30 | 0.75 | 0.32 | 1.00 | 0.10 | 0.85
0.56 | 0.00 | 0.76 | 0.01 | 0.86 | 0.10 | 1.00 | 0.24
0.42 | 0.00 | 0.38 | 0.70 | 0.38 | 0.85 | 0.24 | 1.00
Table 9: Correlation matrix for the LE $\bar{\nu}_{\mu}$ result for relevant
bins.

###### Acknowledgements.

This document was prepared by members of the MINERvA Collaboration using the
resources of the Fermi National Accelerator Laboratory (Fermilab), a U.S.
Department of Energy, Office of Science, HEP User Facility. Fermilab is
managed by Fermi Research Alliance, LLC (FRA), acting under Contract No. DE-
AC02-07CH11359. These resources included support for the MINERvA construction
project, and support for construction also was granted by the United States
National Science Foundation under Award No. PHY-0619727 and by the University
of Rochester. Support for participating scientists was provided by NSF and DOE
(USA); by CAPES and CNPq (Brazil); by CoNaCyT (Mexico); by ANID PIA / APOYO
AFB180002, CONICYT PIA ACT1413, and Fondecyt 3170845 and 11130133 (Chile); by
CONCYTEC (Consejo Nacional de Ciencia, Tecnología e Innovación Tecnológica),
DGI-PUCP (Dirección de Gestión de la Investigación - Pontificia Universidad
Católica del Peru), and VRI-UNI (Vice-Rectorate for Research of National
University of Engineering) (Peru); NCN Opus Grant No. 2016/21/B/ST2/01092
(Poland); by Science and Technology Facilities Council (UK); by EU Horizon
2020 Marie Skłodowska-Curie Action; by a Cottrell Postdoctoral Fellowship from
the Research Corporation for Scientific Advancement; by an Imperial College
London President’s PhD Scholarship. D. Ruterbories gratefully acknowledges
support from a Cottrell Postdoctoral Fellowship, Research Corporation for
Scientific Advancement award number 27467 and National Science Foundation
Award CHE2039044. S.M. Henry’s work was supported in part by the National
Science Foundation Graduate Research Fellowship under Grant No. DGE-1419118.
We thank the MINOS Collaboration for use of its near detector data. Finally,
we thank the staff of Fermilab for support of the beam line, the detector, and
computing infrastructure.

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# Comment on ”Anisotropic peridynamics for homogenized microstructured
materials” by Vito Diana, Andrea Bacigalupo, Marco Lepidi and Luigi Gambarotta
[Comput. Methods Appl. Mech. Engrg. 392 (2022) 114704]

Marcin Maździarz<EMAIL_ADDRESS>Institute of Fundamental Technological
Research Polish Academy of Sciences, Warsaw, Poland

###### keywords:

Continuum-molecular model, Anisotropic pair potential, Cauchy material ,
Poisson’s ratio

††journal: Computer Methods in Applied Mechanics and Engineering

Recently, Vito Diana, Andrea Bacigalupo, Marco Lepidi and Luigi Gambarotta [1]
proposed an anisotropic continuum-molecular model based on pair potentials for
describing the linear behavior of periodic heterogeneous Cauchy materials in
the framework of peridynamic theory.

Unfortunately, the proposed model for an isotropic material may have zero or
even negative shear stiffness and this will be the case for _Silicene_ or
_Germanene_ , for example. In addition, 2D symmetries were confused with 3D
symmetries, and hence the limits for the Poisson’s ratio.

In Ref.[1, Eq.38] two elastic quadratic potentials of pairwise axial
(${w}_{s}$) and shearing deformation measures (${w}_{\gamma}$) in the special
case of isotropic materials reduce to:

$\displaystyle{w}_{s}=\frac{1}{2}s^{2}\left|\mathbf{\xi}\right|k_{n}=6\pi{h}{\delta}^{3}s^{2}\left|\mathbf{\xi}\right|\left(C_{1111}-C_{1212}\right).$
(1)

$\displaystyle{w}_{\gamma}=\frac{1}{2}{\gamma}^{2}\left|\mathbf{\xi}\right|k_{t}=6\pi{h}{\delta}^{3}{\gamma}^{2}\left|\mathbf{\xi}\right|\left(3C_{1212}-C_{1111}\right).$
(2)

The elasticity tensor for an ordinary 2D material with isotropic symmetry has
a representation in the Voigt notation:

$\displaystyle\centering{C}_{ijkl}\rightarrow\left[\begin{array}[]{ccc}{{C}_{1111}}&{{C}_{1122}}&0\\\
{{C}_{1122}}&{{C}_{1111}}&0\\\ 0&0&C_{1212}\\\
\end{array}\right],\@add@centering$ (6)

where $C_{1212}=\dfrac{C_{1111}-C_{1122}}{2}$, hence the number of independent
components reduces here to 2. We can therefore write this as:

$\displaystyle\centering{C}_{ijkl}\rightarrow\left[\begin{array}[]{ccc}{{C}_{1111}}&{{C}_{1122}}&0\\\
{{C}_{1122}}&{{C}_{1111}}&0\\\ 0&0&\dfrac{C_{1111}-C_{1122}}{2}\\\
\end{array}\right]=\left[\begin{array}[]{ccc}{{C}_{1111}}&{{C}_{1111}-2{C}_{1212}}&0\\\
{{C}_{1111}-2{C}_{1212}}&{{C}_{1111}}&0\\\ 0&0&C_{1212}\\\
\end{array}\right].\@add@centering$ (13)

The mechanical stability condition requires the elasticity tensor $C_{ijkl}$
to be positive definite, see [2]. This imposes conditions on the elastic
constants and for any isotropic two-dimensional material (2D) means:

$\displaystyle C_{1212}>0,$ (14) $\displaystyle C_{1111}>C_{1212}.$ (15)

We see that the condition 15 implies positivity of _w s_ in Eq.1 but for
${C_{1111}}>3C_{1212}$ _w γ_ in Eq.2 is negative, which is not acceptable as
the stiffness cannot be negative.

This is not some contrived, theoretical case, e.g. for $Silicene$
$C_{1111}$=68.3 N/m, $C_{1212}$=22.5 N/m, what means that
$\left(3C_{1212}-C_{1111}\right)$ = -0.8 N/m, see [3].

In Ref.[1, Section 5.1, P.16] is written: ” _… leads to the effective fourth
order elasticity tensor $\mathbb{C}$ having the elasticities of the two-
dimensional cubic symmetry …_”. No such symmetry exists for 2D materials,
there is isotropy, tetragonal, orthotropy and anisotropy, see [2].

In Ref.[1, Section 5.1, P.17] is written: ” _… representative extreme values
$\nu_{a}$ $\approx$ 1/2 (e.g. 0.49), …_”. For any isotropic two-dimensional
material (2D) the correct limits for the Poisson’s ratio, resulting from the
positive definiteness of the stiffness tensor, are $-1{\leq}{\nu}{\leq}1$, see
[4].

## References

  * [1] V. Diana, A. Bacigalupo, M. Lepidi, L. Gambarotta, Anisotropic peridynamics for homogenized microstructured materials, Computer Methods in Applied Mechanics and Engineering 392 (2022) 114704.  doi:10.1016/j.cma.2022.114704.
  * [2] M. Maździarz, Comment on ‘The Computational 2D Materials Database: high-throughput modeling and discovery of atomically thin crystals’, 2D Materials 6 (4) (2019) 048001.  doi:10.1088/2053-1583/ab2ef3.
  * [3] Z.-Q. Wang, T.-Y. Lü, H.-Q. Wang, Y. P. Feng, J.-C. Zheng, Review of borophene and its potential applications, Frontiers of Physics 14 (3) (2019) 33403\.  doi:10.1007/s11467-019-0884-5.
  * [4] K. W. Wojciechowski, Remarks on “Poisson Ratio beyond the Limits of the Elasticity Theory”, Journal of the Physical Society of Japan 72 (7) (2003) 1819–1820.  doi:10.1143/JPSJ.72.1819.