# Comparison of Validation Methods of Simulations for Final State Interactions
in Hadron Production Experiments

S. Dytman  University of Pittsburgh, Department of Physics and Astronomy,
Pittsburgh, PA 15260, USA  Y. Hayato  University of Tokyo, ICRR, Kamioka,
Japan  R. Raboanary  University of Antananarivo, Antananarivo, Madagascar
J.T. Sobczyk  University of Wrocław, Institute of Theoretical Physics,
Wrocław, Poland  J. Tena-Vidal  University of Liverpool, Liverpool, UK  N.
Vololoniaina  University of Antananarivo, Antananarivo, Madagascar

(March, 2021)

###### Abstract

Neutrino cross section and oscillation measurements depend critically on
modeling of hadronic final state interactions (FSI). Often, this is one of the
largest components of uncertainty in a measurement. This is because of the
difficulty in modeling strong interactions in nuclei in a consistent quantum-
mechanical framework. FSI models are most often validated using hadron-nucleus
data which introduces further uncertainties. The alternative is to use
transparency data where the hadron starts propagating from inside the nucleus
and the probability of interaction is measured as a function of hadron energy.
This work examines the relationship between the $\pi^{+}$ and proton total
reaction cross section and transparency from a simulation viewpoint.

##  I Introduction

A firm knowledge of neutrino interactions is required to properly assess
backgrounds and systematic errors in neutrino oscillation measurements [1, 2,
3, 4]. Oscillation experiments use pion and proton detection to varying
degrees. Some detect these hadrons within the oscillation analysis [4], but
all depend on accurately separating events by interactions based on final
state topology. Recent oscillation articles from T2K [1] and NOvA [2]
highlight this need. A recent review [5] studies the various channels required
to describe neutrino interactions and highlights the issues in the need for
precise modeling.

Event generator codes simulate a large number of neutrino-nucleus interactions
across a broad range of energies. For proper detector modeling, the entire
final state of particles is produced. These codes are then used to produce
Monte Carlo events for each experiment which are then used to identify event
categories, deduce neutrino beam energy, assess background and estimate
systematic errors. The neutrino interacts with one of more nucleons through
what is called the primary interaction. Hadrons are always emitted through
this first interaction which is governed by the weak force; they then
propagate through the residual nucleus according to final state interactions
(FSI) modeling which is governed by the strong force. At hadron kinetic
energies characteristic for long baseline neutrino oscillation experiments
(typically less than 1 GeV), the probability of hadron reinteraction is quite
large. For pions produced mostly through the $\Delta(P_{33}(1232))$ excitation
mechanism, $\sim 75\%$ of the pions interact for carbon and $\sim 80\%$ for
argon. For protons the probability to reinteract depends on its kinetic energy
and is of the order $\sim 40-50\%$. Largely because of hadronic reinteraction,
the energy transferred to nucleus in the primary interaction is shared by many
particles knocked out of the nucleus. Any measurement of the neutrino energy
by measuring energy of all particles in the final state [2] must model FSI
effects carefully. A large fraction of that energy is due to neutral particles
such as neutrons and $\pi^{0}$ which are not easily visible in today’s fine
grained detectors [6]. Since FSI significantly modifies both the composition
and energy/angle distributions of knocked-out hadrons, a proper description is
a key component of each code.

One interesting issue is that experiments use a variety of codes for their
simulations. These codes employ models of the same physics channels, but can
use either different sources or different interpretations of the same source.
As a result, measurements of neutrino oscillations can depend on these choices
and the collaborations must properly assess the associated systematic
uncertainties. One important way to understand this situation is to produce
comparisons of the codes against appropriate benchmark measurements. Many
studies of this type have been presented in a series of NuInt workshops, e.g.
[7, 8]. Results from a recent broad study that allowed increased understanding
of the issues were published in Ref. [9]. One major theme has been the use of
event generator codes to examine the relationship between measurements that
are not directly comparable.

FSI models are complex because hadrons propagate through the residual nucleus
and can interact with any of the nucleons in it with high probability. In
addition, the struck nucleon may be interacting with another nucleon at the
same time (called medium corrections or nucleon-nucleon (NN) correlations in
this work) This problem is too complicated to be solved exactly in a
consistent quantum-mechanical framework. As a result, FSI is often one of the
principal sources of uncertainty in neutrino oscillation results [2, 1]. A
class of models called intranuclear cascade (INC) have been developed over
many years. Foundations for the semi-classical treatment of FSI in terms of
intranuclear cascade model were set by Serber [10] and Metropolis [11, 12].
The propagation of hadrons is taken to be a series of independent binary
collisions that are largely independent of each other. Quantum corrections
[13, 14, 15] have be introduced and detailed agreement with a large variety of
data has been obtained using a variety of approximations. Although theoretical
arguments show that agreement of cascade models with data is largely expected
for situations when interaction probability is small, e.g. with nucleons of
kinetic energy above 200 MeV [16], this agreement apparently extends to
regions where the hadron-nucleon cross section is very large such as pion
interactions when $\Delta$ resonance excitation is dominant [17]. Without an
underlying quantum mechanical model, approximations are required. In addition,
experiments require codes which combine accuracy for most important quantities
with numerical efficiency/speed. Even though GiBUU [13] has an excellent
dynamic model with the use of hadronic transport calculations, it is difficult
to use in an experiment. Codes in use range from empirical [18] to
theoretically motivated [19, 20] models.

The basis of existing cascade codes is always free hadron-nucleon interactions
with additional nuclear corrections and medium refinements. The free hadron-
nucleon interaction has been studied extensively and partial wave amplitude
(PWA) fits to large data sets of cross sections and polarization quantities
[21] are available.

One of the important choices required is the hadron-nucleus data set that is
used to validate the models. Most often, cross sections with hadron beams on
nuclear targets have become the sole FSI validation test and the total
reaction cross section ($\sigma_{reac}$) [22, 23] is the most broad kind of
data available. Also known as the total inelastic or nonelastic cross section,
it includes all final state channels except elastic and provides a single
number for each energy that measures the total strength for important
interactions. Use of $\sigma_{reac}$ for validation is straightforward because
a large body of data is available and each generator has a subsidiary code
using the same FSI code as in the neutrino-nucleus simulation to reproduce
these data [18, 24, 23]. The alternative is to use transparency data where the
hadron is ejected from a nucleus by non-interacting probes such as an electron
[25, 26] or a neutrino [27]. The advantage of this choice is that the final
state hadron is produced in the same way as in neutrino oscillation
measurements. Experimentally, the difficulties come from tagging the FSI
hadron in an objective way. Specifically, transparency is a ratio of hadrons
in the final state with and without FSI. Measurements to date use electron
beams and either tag the struck proton [25] using the quasielastic interaction
or normalizing the experimental result with a separate calculation of the
result with no FSI [28]. The latter choice is often used for high energy
hadrons in studies of color transparency [28] where the goal is to find
differences between data and nominal calculations as a function of $Q^{2}$ as
evidence for unusual medium effects. This article is aimed at issues important
for neutrino oscillations [2, 3] and therefore focuses on particles of kinetic
energy 1 GeV and lower. There is also a large body of precise hadron-nucleus
differential cross section data. Comparison with these data will be a second
step in benchmarking MC FSI models.

The relationship between total reaction cross section and transparency is
interesting, but not often studied. Pandharipande and Pieper [26] examined the
theory input needed to describe proton transparency in the quasielastic region
[25]. They note the importance of Pauli blocking, medium effects, and short-
range nucleon-nucleon correlations. Ref. [27] was devoted to studying the
ability of the NuWro neutrino event generator [24] to describe existing proton
transparency data. They also note the importance of medium corrections and
short-range nucleon-nucleon correlations. The GiBUU neutrino-nucleus model
[29] was compared to proton transparency data with an emphasis on high
momentum transfer, $Q^{2}>$ 1 $(GeV/c)^{2}$. Isaacson, et al. [30] recently
introduced a novel FSI model based on ideas proposed in Ref. [31] and quantum
Monte Carlo (QMC) calculations by generating a sample of realistic nuclear
configurations within this framework. They examined both $\sigma_{reac}$ and
transparency with the goal to access the role of nucleon-nucleon correlations.
They report negligible impact of correlations on $\sigma_{reac}$ and a
moderate effect on transparency.

The purpose of this article is to study the FSI effects of protons and
$\pi^{+}$ in nuclei through a combined analysis of hadron beam interactions
and neutrino hadron production transparency. This study features results from
three commonly used neutrino event generators - GENIE [32], NEUT [33], and
NuWro [24]. Since authors of each code contribute to this work, detailed
comparisons are possible. Although comparisons among generators are becoming
more common [9, 34, 35, 36, 37], comparisons involving detailed understanding
of the codes are not often available. Each code is used to calculate
$\sigma_{reac}$ and transparency. Although $\sigma_{reac}$ has been measured
for many targets [22, 23], the focus here is on carbon and argon because of
their common use in neutrino oscillation experiments. This will give an
indication of the dependence on choice of nucleus, usually called
A-dependence. Transparency measurements are much less common. Some results for
protons are published for carbon and heavier elements [25, 38, 39, 40, 28],
but none for argon yet. The goal of most previous measurements is the search
for color transparency [28, 41]; therefore, most of the existing data is at
high momentum and energy transfer to the residual nucleus. For example,
existing measurements for pion production [42] are aimed in that direction and
there is then no transparency data for pions at energies needed for the
neutrino oscillations experiments. A major goal of this study is to examine
the theoretical effects that make $\sigma_{reac}$ and transparency differ and
encourage more experiments. Since no acceptance corrections are made to match
the data, these will be referred to as Monte Carlo calculations of
transparency. This choice was made to allow examination of the relationship
between the two quantities independent of any experimental details. We will
argue that within the intranuclear cascade approach there is an inherent
relation between total reaction cross section and transparency coming from the
mean free path and nucleon density. However, additional theoretical effects
can modify the relation and a combination of precise reaction cross section
and transparency measurements can be used to investigate those effects. We
notice that the relation between transparency and $\sigma_{reac}$ becomes less
intuitive in the hadron propagation approach studied in Ref. [30]. At low
values of proton kinetic energy $\sigma_{reac}$ gets large contributions from
”long-range interactions” as illustrated in Fig. 8 of [30], i.e. nucleons far
away from nucleus center and can still have non-negligible probability to
interact.

##  II Reaction cross section vs. transparency

###  II.1 General considerations

The relationship between $\sigma_{reac}$ and transparency is a long-standing
issue in FSI calculations. This is closely related with the question about the
best way to describe protons propagating in nuclei [25]. The issue there was
the conflict between optical-model potentials and nucleon-nucleon interactions
with medium corrections and the goal of Ref. [25] was to provide data to
resolve this conflict. The work of Pandharipande and Pieper cited above [26]
showed that the NN interaction method correctly reproduced proton transparency
data [25]. Since the data for $\sigma_{reac}$ [22, 23] is much more commonly
available, studies to date for neutrino interaction event generators largely
use that information.

Here, we will explore the model dependence of both reaction cross reaction and
transparency. Although both quantities are sensitive to the mean free path of
the hadrons, transparency samples the nuclear medium more directly since they
start propagating randomly throughout the nucleus. On the other hand, many
strongly interacting particles (esp. pions at energies where the $\Delta$
(P33(1232)) resonance is important) only interact in the periphery of the
nucleus and rarely sample the interior. In reality, hadrons produced inside
the nucleus are off-shell. Since all the codes used here other than GENIE
INCL++ propagate hadrons on-shell with off-shell effects included only at
interaction points, that aspect cannot be studied here completely. INCL++ puts
the propagating particle in a potential well (both Coulomb and nuclear) that
depends on particle type and its energy. Therefore, the momentum of the
propagating particle is constant while its energy varies with position.

It is interesting to note the intrinsic differences between the two
quantities. A formation zone accounts for the possibility of a hadron
propagating for some distance in something other than a normal state, e.g. a
particle with small transverse dimension [41]. This will affect transparency
but not $\sigma_{reac}$. Medium effects are relevant because a bound nucleon
(N) has different behavior than a free nucleon due to the surrounding
particles. Medium modifications are known to be important in the propagation
of pions [43] and protons [26] where they are often handled within a
polynomial expansion in nucleon density. They will affect the two quantities
in somewhat different ways because they have a strong dependence on nuclear
density, e.g. if $\sigma_{reac}$ is largely determined by hadrons interacting
in the surface of the nucleus. Nucleon-nucleon (NN) correlations are beyond
what can be handled with typical local density approximation techniques and
are treated separately [26, 27]. These have almost no effect on
$\sigma_{reac}$, but can be significant for transparency.

Although the codes considered here are largely aimed at neutrino interactions,
transparency can in principle be measured either with electron or neutrino
beams. Since only a fraction of the codes have the capability to simulate
electron scattering events, the simulations here use electron neutrino
($\nu_{e}$) neutral current (NC) interactions. They have almost identical
kinematic properties to electron scattering interactions, using the weak
interaction instead of the electromagnetic interaction. The main purpose is to
start the propagation of hadrons at locations according the matter density
under identical residual nucleus conditions. For proton transparency, the
neutral current elastic interaction is used as the principal interaction. For
pion transparency, the neutral current resonance interaction is used.

###  II.2 Toy model

In this section we introduce a mathematical model which relates the values of
total reaction cross section with transparency. The model is valid for
arbitrary projectiles, including pions and nucleons. It contains all the
features of the basic propagation model, neglecting nucleon-nucleon
correlations and formation zone effects which have impact on the measured
transparency and can therefore be potentially considered as a tool to quantify
those effects. We also neglected the effects coming from Pauli blocking, or
more generally from the dependence of the cross section on local density.

In the toy model, a nucleus is fully characterized by a density profile
$\rho(r)$ assumed to be radially symmetric. It satisfies the normalization
condition

$\int\rho(r)d^{3}r=4\pi\int r^{2}\rho(r)dr=A$ (1)

where $A$ is the atomic number. We assume that the bulk of the nuclear density
is closed within a sphere of radius $R$. The value of $R$ is chosen so that
its increase has no significant impact on the calculated values of
transparency and $\sigma_{reac}$ (4.65 fm for carbon and 7.1 fm for argon).

####  II.2.1 Reaction cross section

We assume a uniform flux of projectiles hitting a nucleus from the outside of
the sphere of radius $R$. The goal is to calculate the probability $P_{reac}$
that an interaction happens during passage through the nucleus for a given
value of the impact parameter. A product of the average value of $\langle
P_{reac}\rangle$ with the geometric cross section $\pi R^{2}$ defines the
reaction cross section.

Technically, it is easier to calculate $1-\langle P_{reac}\rangle$, the
average probability that the projectile travels through nucleus without
interaction. Locally, the probability to move over the distance $dz$ without
interaction is ${\rm exp}{\\{-\rho\sigma dz\\}}$. In the toy model we
disregard the difference between proton and neutron local densities ($\rho$)
and local cross sections ($\sigma$).

Taking into account that for each projectile the distance travelled inside
nucleus is determined by the impact parameter $r$, we get the following
expression (see the right side of Fig. 1):

$\displaystyle\sigma_{reac}$ $\displaystyle=$ $\displaystyle\pi
R^{2}-2\pi\int_{0}^{R}dr\ r$ (2) $\displaystyle\cdot$
$\displaystyle{\rm\exp}\left\\{-\int_{-\sqrt{R^{2}-r^{2}}}^{\sqrt{R^{2}-r^{2}}}\sigma\rho\left(\sqrt{z^{2}+r^{2}}\right)dz\right\\}$

Figure 1: Computation of transparency (left) and reaction cross section
(right) in the toy model.

####  II.2.2 Transparency

In this computation a trajectory starts at a point inside nucleus selected at
random with the probability density given by $\rho(\vec{r})$.

The computations lead to the following result for transparency ($T$) (see the
left side of Fig. 1) [26]:

$\displaystyle T$
$\displaystyle=\frac{2\pi}{A}\int_{-1}^{+1}d(\cos\theta)\int_{0}^{R}dr\
r^{2}\rho(r)$ (3) $\displaystyle\cdot$ $\displaystyle{\rm
exp}\left\\{-\int_{r\cos\theta}^{\sqrt{R^{2}-r^{2}\sin^{2}\theta}}dz\sigma\tilde{\rho}\left(\sqrt{z^{2}+r^{2}\sin^{2}\theta}\right)\right\\}$

In the above formula the tilde in $\tilde{\rho}$ accounts for the fact that
the numbers of spectator nucleons are different in nucleon reaction cross
section and transparency computations. In the case of reaction cross section
it is $A$ while in the transparency it is $A-1$. Thus
$\tilde{\rho}=\frac{A-1}{A}\rho$.

####  II.2.3 Reaction cross section to transparency ratio

Absolute values of reaction cross section and transparency depend on the
microscopic cross section $\sigma$ entering Eqs. 2 and 3. The exact value of
$\sigma$ is not known but we can use information from Sects. II.2.1 and II.2.2
to eliminate $\sigma_{reac}$ and determine a function $T(\sigma_{reac})$. This
was done for three realistic density profiles - carbon, argon and iron [44].
Results are shown in Fig. 2. In the limit of reaction cross section going to
zero, the transparency approaches the value one. In the other extreme case,
when the reaction cross section approaches the maximal possible value of the
geometric cross section ($\pi R^{2}$), the transparency goes to zero. In the
intermediate region transparency is a monotone function of reaction cross
section. Since this statement does not depend on any assumption about
projectile kinetic energy, one can expect that if the projectile kinetic
energy dependence of the reaction cross section shows a local minimum, the
transparency should exhibit a local maximum and vice versa.

Figure 2: Transparency as function of reaction cross section in the toy model

####  II.2.4 Comparison with GENIE results

The toy model presented above results in a simple analytical equation which
incorporates the most basic physics input. More complicated codes can
reproduce these results with suitable simplifications. This can be studied in
GENIE because of its modular design. The toy model curve in Fig. 3 was
obtained by transforming GENIE reaction cross section results into
transparency with the toy model ratio in Fig. 2. Fig. 3 demonstrates that
stripped-down GENIE agrees well with the toy model for proton and $\pi^{+}$
transparency results in carbon. The GENIE transparency result goes to 1 below
20 MeV because a cutoff that was introduced (see Sect. III.3 for details). The
toy model and GENIE results use slightly different charge distributions, both
consistent with electron scattering data [44]. A general conclusion about the
toy model is that it can be a useful tool to investigate nuclear effects
modifying transparency but not reaction cross section. We adopt the approach
where the reaction cross section is used as input to obtain results shown in
Fig. 2.

Figure 3: Transparency for proton and $\pi^{+}$ for carbon with results using
bare GENIE hN2018 FSI model and the toy model using GENIE reaction cross
section and the transparency/$\sigma_{reac}$ ratio as explained in the text.
In the GENIE simulation, all medium corrections and formation zone effects
have been removed. The statistical error associated with GENIE predictions is
represented with a grey band.

##  III Models

Event generator codes produce events from interactions of neutrinos,
electrons, and hadrons with nuclei using similar core methods. For this
article, the focus is on propagation of protons and $\pi^{+}$ in nuclei. GENIE
hA2018 and hN2018, NuWro, and NEUT all treat the nucleus as an ensemble of
largely independent particles. These nucleons have momenta which typically
come from a local Fermi gas distribution. That means the momentum distribution
is uniform in momentum space up to a maximum value (the Fermi momentum) which
depends on position inside the nucleus:

$p_{F}(\bf{r}\rm)=\left[\frac{3}{2}\pi^{2}\rho(\bf{r}\rm)\right]^{\frac{1}{3}},$
(4)

Thus, the high momentum nucleons tend to be in the center of the nucleus. This
is a significant improvement beyond global Fermi gas models where the Fermi
momentum is independent of position. Nucleons are in a bound system; this can
be accounted for by giving the struck nucleon a shift in mass leading to an
effective mass which is smaller. Binding energy values are historically set by
fits to high energy electron quasi-elastic scattering scattered electron
energy distributions [45] where the correction is the shift of the peak
position from the value appropriate for a free nucleon. INCL++ [20, 46] has a
more sophisticated nuclear model [14] where all nucleons are placed in a
square well potential whose depth and range depend strongly on the nucleon’s
position and momentum. There are no collisions and the nucleons are
constrained to simultaneously have a momentum distribution similar to the
local Fermi gas [20] and a spatial distribution according standard values
[44]. Tuning to a variety of hadron-nucleus scattering data is also important
[20]. The result is an improved momentum distribution and binding energy
correction. Although the binding energy correction is clearly improved over
any Fermi gas value by having a natural dependence on position and momentum,
the result of their nuclear model has the highest momentum nucleons at the
surface [20] rather than in the interior.

GENIE hA2018 and hN2018, NuWro, and NEUT all use a semi-classical approach
where the hadron is assumed to be free and move in straight lines as it
propagates. The governing equation then comes from an evaluation of mean free
path $\tilde{\lambda}$ which depends on position within the nucleus and the
total cross section for interacting with nucleons. Hadrons propagating through
a nucleus are moved in steps in position. The probability of traveling a
distance $\lambda$ (small enough to satisfy an assumption of constant nuclear
density $\rho$) without interaction is:

$P(\lambda)=e^{-\lambda/\tilde{\lambda}}$ (5)

where

$\tilde{\lambda}=\left(\sigma_{p}\rho_{p}(r)+\sigma_{n}\rho_{n}(r)\right)^{-1}$
(6)

and $\sigma_{p}$ ($\sigma_{n}$) is the total cross section for interaction on
either a proton ($p$) or neutron ($n$). Pauli blocking can either be included
in average by reducing values of $\sigma_{p/n}$ or can be checked for each
microscopic interaction. To satisfy the assumption of (approximately) constant
density, the particles are moved by a small step with typical values from 0.05
fm (GENIE) to $0.2$ fm (NEUT and NuWro). Tests show that a range of step
values give no visible impact on the final results but smaller steps make the
computations slower. Using the probability calculated in Eq. 5 at each step it
is decided if an interaction is going to happen. Although the GENIE INCL++
model is different in a variety of ways from the other approaches, it is still
based on the mean free path concept for the energies considered here. For
INCL++, the entire hadron-residual system changes through time steps and
interactions occur when the distance of separation is less than
$\sqrt{\sigma_{hN}/\pi}$ fm [20]. A different method of propagating hadrons is
proposed in Ref. [30]. A configuration of nucleon positions is generated with
the QMC method where all nucleons are interacting with realistic potentials.
Hadrons to be tested (protons in their work) propagate as point-like on-shell
particles in small time steps. Interactions can occur if there are target
nucleons inside a cylinder with symmetry axis along the propagating nucleon
momentum vector. The cylinder’s transverse size is the same as the INCL++
sphere of closest approach. In order for the interaction to be generated, hit
nucleon initial momenta are sampled from local/global Fermi gas model density
information and final momenta are checked for the Pauli blocking. This method
of propagating nucleons and generating interactions becomes very different
from what is presented here at low values of nucleon kinetic energy when the
nucleon-nucleon cross section is large.

For transparency computations in this work the hadron starting points are
generated according to a density distribution as determined from elastic
electron scattering. To make sure that all propagating particles are deposited
in the nucleus the same way, identical interactions are chosen for all the
generators – neutral current elastic (NCEL) interaction to produce a
propagating proton and the neutral current resonance (NCRES) or deep inelastic
scattering (NCDIS) interaction for $\pi^{+}$. The choice could be either
electron or neutrino beams because each has essentially no interaction with
the nucleus until the principal interaction. In this case, we chose neutrino
interactions because not all participating codes have working electron
capabilities. We also choose neutral current (NC) $\nu_{e}$ interactions to
avoid issues with an improperly prepared residual nucleus. For example, a
$\nu_{\mu}$ charged current (CC) interaction would have both an improper
residual nucleus because of the conversion of a neutron to a proton and an
energy imbalance due to the emission of a muon which has significant mass. It
is important to note that the transparency simulations here do not reproduce
the requirements of the measurements [25, 38, 39]. Therefore, these results
should have a qualifying label such as Monte Carlo transparency and will only
be indirectly compared with data in Sect. IV.

Nuclear effects can be included in a semiclassical model in a number of ways.
Various medium effects are added in accord with external models and data. They
typically use a local density approximation.

There are two interesting effects that modify theoretical results for
transparency but not for reaction cross section. The first is formation
zone/time. Formation time is also included in Ref. [30]. The second effect is
nucleon-nucleon correlations. They are studied in detail in Ref. [30] and
implemented in an approximate way in NuWro.

To summarize, nuclear effects are all handled in similar ways by the various
codes. However, different implementation choices have been made. They are
summarized in Table 1 for protons and Table 2 for pions. These effects vary in
size among the codes and will be studied in detail in Sec. V.

Table 1: Brief summary of model components for protons in NCEL interactions that influence $\sigma_{reac}$ and transparency in different ways.  Generator | Pauli | medium | NN | formation
---|---|---|---|---
| blocking | effects | correlations | zone
GENIE hA2018 | none | Ref. [26] | none | none
GENIE hN2018 | none | Ref. [26] | none | none
GENIE INCL++ | yes | Ref. [46] | none | none
NuWro 19.02 | yes | Ref. [26] | Ref. [26] | none
NEUT v5.4.0.1 | yes | none | none | none
Table 2: Brief summary of model components for pions in NCRES interactions that influence $\sigma_{reac}$ and transparency in different ways.  Generator | Pauli | medium | formation
---|---|---|---
| blocking | effects | zone
GENIE hA2018 | none | none | none
GENIE hN2018 | none | Ref. [19] | none
GENIE INCL++ | yes | Ref. [46] | none
NuWro 19.02 | yes | Ref. [19] | none
NEUT v5.4.0.1 | yes | Ref. [19] | Ref. [47]

###  III.1 NEUT

Like the other generators, pion and nucleon FSI are simulated using a custom
semiclassical INC model. The nuclear density function is of Woods-Saxon type.
The mean free paths for low momentum pion interactions in the Delta region are
calculated using the prescriptions by Salcedo et al. [19]. These authors
proposed a model of $\Delta$ propagation in a finite nucleus. The model
includes a modification of $\Delta$ self-energy due to medium effects via the
local density approximation. This was an appropriate way to reproduce the
pion-nucleus data available at the time, and was aimed for Monte Carlo
simulation. The main advantage is an accurate description of pion cross
sections for a wide range of nuclei. This approach was first adopted by NEUT
and later included in NuWro and GENIE. The mean free paths for pions with
momentum above 500 MeV/c), are obtained using the free pion-nucleon scattering
data. In the calculation of the mean free paths for the low momentum pions,
the Local Fermi Gas model is used and thus Fermi motion of the nucleon in the
nucleus was taken into account. Pauli blocking was implemented by requiring
the nucleon momentum after the interaction to be larger than the local Fermi
surface momentum ($p_{F}(r)$) (see Eq. 4).

In order to improve the match of the simulation to pion-nucleus total cross
section data, energy independent normalization factors were introduced for
each pion-nucleon interaction type, e.g. charge exchange or absorption. These
factors were then fit to various pion-nucleus scattering data sets for pion
energies up to 2 GeV. [23]

In the determination of the kinematics of the scattered pions, the results of
phase shift analysis obtained from $\pi-N$ scattering experiments [48] are
used. Although the mean free paths are determined according to Salcedo-Oset,
the outgoing kinematics were modified according to medium corrections as
suggested by Seki et al.[49]. Also, Pauli blocking effect is taken into
account to be consistent with the mean free path.

FSI for nucleons uses the differential cross-sections of nucleon scattering in
nucleus from the work by Bertini et al. [50], which was based on various
experimental results. Elastic scattering and one or two $\Delta(1232)$
production processes are included. In order to simulate the $\Delta(1232)$
production, a simple isobar production model [51] is used. The produced pions
are tracked using the pion re-scattering code from the point of generation.
The treatments of the binding energy (effective nucleon mass) for nucleon
scattering is quite different than either pion scattering in NEUT or the other
generators. In simulating the nucleon FSI in NEUT, the effective nucleon mass
${M_{N}}^{eff}$ of the bound nucleon is set to be

${M_{N}}^{eff}=\sqrt{({M_{N}}^{free}-8MeV/c^{2})^{2}-(p_{F}^{surf})^{2}},$ (7)

where $\rm{M_{N}}^{free}$ is the on-shell nucleon mass and $p_{F}^{surf}$ is
the local Fermi surface momentum (see Eq. 4), respectively. While tracking the
nucleon in the nucleus, an interaction with the other nucleon happens only
when total energy is larger than 2 $\times M_{N}^{free}$. This largely
suppresses the low energy interactions of nucleons as shown in the later
comparisons, in Figs. 5 and 6. In simulating hadron production processes such
as single meson or multi-pion production and deep inelastic scattering, a
formation zone effect based on SKAT data is included in NEUT [52, 47]. The
production point of the hadrons for those interactions are shifted using the
formation length ($L_{FZ}$), where

$L_{FZ}=p/\mu^{2},$ (8)

p is the momentum of the hadron and $\mu=0.08$(GeV/c2). The actual size of the
shift is determined as $L_{FZ}\times(-\log({\rm rand}[0,1])$, where rand[0,1]
is a positive random number up to 1. Because of the geometrical effects, this
makes the distribution of the hadron production locations shift to the outer
region of the nucleus as shown in Fig. 4. The density of the surrounding
region is lower than the central part and thus, the interaction probability of
those particles becomes smaller.

Figure 4: Effects of the formation zone in NEUT. The plot shows the
interaction position (distance from the nucleus center) distributions of the
neutrino (blue) and the starting position of the pion (green) coming from the
neutrino interaction. The peak at 6 fm is at the outer edge of the Carbon
nucleus in the simulation. Because of the formation zone effect, the
production position of pion is shifted to the region of lower density.

###  III.2 NuWro

In NuWro the expected distance travelled by hadrons is calculated to be:

$\lambda=-\tilde{\lambda}\cdot\log(\mbox{rand}[0,1])$ (9)

with $\tilde{\lambda}$ defined in Eq. 6. An interaction is generated at the
distance $\lambda$ if its value calculated in Eq. 9 satisfies $\lambda<0.2$
fm. If instead $\lambda\geq 0.2$ fm the particle is moved by a distance $0.2$
fm along its trajectory.

FSI effects are described by a custom made semiclassical intranuclear cascade
(INC) model [24, 27]. The essential ingredient of the model are hadron-nucleon
microscopic cross sections. For pions, in the kinetic energy range
${T_{k}=85-350\text{ MeV}}$, NuWro, like NEUT, uses the results of microscopic
calculation from Ref. [53] implemented in the region of $T_{k}\leq 350$ MeV as
described in Ref. [19]. The model parameters are tabulated as functions of
nuclear density in intervals of $0.1\rho(r)/\rho_{0}$ and can be used for
arbitrary nuclei once its density profile is known. $\rho(r)$, $\rho_{0}$ are
local and saturation nuclear densities, respectively. The parameters are
interpolated linearly to cover the whole range of nuclear density. Unlike in
NEUT, no adjustment of the parameter values to the experimental data for pion
absorption and reaction cross sections was done.

Since the model in Ref. [19] is restricted to the energies up to the
$\Delta(1232)$ region only, it is supplemented with a phenomenological model
for kinetic energies above 350 MeV. Free pion-nucleon cross sections are taken
from the available experimental data.

For nucleons NuWro uses as a starting point a custom fit to the experimental
free nucleon-nucleon cross sections, both elastic and inelastic. The fraction
of single-pion production within inelastic interactions was adjusted to follow
the fits of Ref. [54].

The in-medium modification of the elastic nucleon-nucleon cross sections is
modeled using the results of the Pandharipande and Pieper study [26], where
two main effects come from Pauli blocking and in-medium nucleon effective
mass. The Pauli blocking is included on the event-by-event basis.

For inelastic nucleon-nucleon scattering NuWro adopts a phenomenological in-
medium microscopic cross section ($\sigma^{\ast}_{\mathrm{NN}}$)
parameterization [55]:

$\sigma^{\ast}_{\mathrm{NN}}=\left(1-\eta\frac{\rho(r)}{\rho_{0}}\right)\sigma^{\mathrm{free}}_{\mathrm{NN}},$
(10)

where $\eta=0.2$.

Following the guidance of Refs. [26, 56, 57], the NuWro cascade model includes
correlation effects coming from nucleon-nucleon correlations. The main effect
comes from short-range correlations. The density that enters the mean free
path formula is (for the details see Ref. [27]):

$\begin{split}\rho^{[1]}_{\mathrm{eff,IPSM}}(\vec{r}_{2}|\vec{r}_{1})=\rho^{[1]}_{\mathrm{A-1}}(\vec{r}_{2})g(|\vec{r}_{21}|)N(|\vec{r}_{1}|),\end{split}$
(11)

normalized to the number of spectator nucleons

$\int\mathrm{d}^{3}\vec{r}_{2}\
\rho^{[1]}_{\mathrm{eff}}(\vec{r}_{2}|\vec{r}_{1})=A-1.$ (12)

$g(|\vec{r}_{21}|)$ is the nucleus-dependent pair distribution function [26]
and $N(|\vec{r}_{1}|)$ is introduced to keep the global normalization
condition. For the choice of $g(|\vec{r}_{21}|)$, we rely on distributions of
nucleon-nucleon distances obtained in ab initio computations for light nuclei,
including carbon [58, 59]. For heavier nuclei including iron, we approximate
$g(|\vec{r}_{21}|)$ by the ab initio-calculated infinite nuclear matter
distributions $g_{\mathrm{inf}}(\rho_{\mathrm{avg}},|\vec{r}_{21}|)$ of Ref.
[26], evaluated at average nuclear density. The model includes effects coming
from different shapes of $g(|\vec{r}_{21}|)$ for nucleon pairs of the distinct
isospin configurations.

For both NuWro and GENIE, formation time/zone effects are included in DIS
events, therefore not included in the main results of this study. A somewhat
similar effect, originating from finite $\Delta$ life-time can also be applied
in resonance events. NuWro has this as an option that is not used in this
study.

###  III.3 GENIE

GENIE allows many configurations, each of which contains a different set of
models. In addition, many of the model sets have configurable parameters
inside; all simulations here come from the version 3.0.6 public release with
the exception of INCL++ which is in v3.2.0. Each model set has a complete set
of core interactions containing the most compatible models. For example, the
G18_10a_02_11a model set has the local Fermi gas nucleon momentum distribution
for all interaction models in it. For this work a single GENIE model set was
chosen to deposit a proton (NCEL) or $\pi^{+}$ (NCRES or NCDIS) within the
nucleus as discussed above. In version 3 of GENIE, there are multiple FSI
models that can be used in conjunction with each set of primary interactions.
Three of them are used for the present study. hA has been the traditional
default model and hN was added in v3.111 The names are derived from the scope
of each model. hN is a traditional hadron-nucleon INC model and hA is a custom
model which has a more empirical characterization of the effect of multiple
hadron-nucleon interactions. Both hA2018 and hN2018 have significant
modifications which make them more complete than previous versions. In
particular, hA2018 has deleted the hadron-nucleus elastic scattering process
which has had the effect of significantly increasing the transparency while
leaving $\sigma_{reac}$ largely unchanged. Both use the same semi-classical
stepping method according to mean free path as described above with a step
size of 0.05 fm. Neither use Pauli blocking to suppress interactions at lower
energies. No formation zone effects are employed in any of these FSI codes.
All use a modified Gaussian spatial nucleon density function which is very
similar to the Woods-Saxon function used by NuWro and NEUT.

The hN FSI model has very similar features to what is in NuWro and NEUT. It
uses the Salcedo-Oset [19] model for pions with kinetic energy below 350 MeV.
This model has nuclear medium modifications. Above 350 MeV, unmodified free
$\pi N$ interactions [21] are used. Nucleon interactions are also based on
nucleon-nucleon cross sections which are based on the phase shift fits of the
GWU group [21] to a variety of data. Nucleon FSI models use the Pandharipande-
Pieper medium modifications [26] at all energies. The hN model uses an energy
cutoff below which low energy hadrons don’t propagate; the cutoff depends on
the nucleus as $A^{0.2}$.

The hA model is more empirical. The main differences with respect to hN come
after a final state interaction is chosen. The hA model has a single
interaction which is heavily based on data instead of the multiple
interactions of cascade models such as hN, NEUT, and NuWro employ. It uses the
same mean free path values as hN with the exception of the pion-nucleon cross
section. Although it uses the Pandharipande-Pieper model for nucleons, the
pion interaction has no medium modifications.

The INCL++ model [20, 46] has been recently added. Its development has been
completely independent of the other codes discussed here and is being used in
production codes for the first time. The original goal of this model was a
global capability to describe nucleon-nucleus interactions at energies
spanning the range studied here. Understanding the spallation process was a
major goal and it is one of the best existing codes [60]. All particles are
tracked as a function of time. The interplay of radial and momentum dependence
of the nucleon distribution is carefully considered [14]. The propagating
hadron and all nucleons in the residual nucleus are in a square well potential
whose width depends on location. The depth and radius were originally tuned to
spallation data, but the radius was tuned to single nucleon emission data in a
recent publication [20]. By choosing constant well depth, the accounting for
energy and momentum is much simpler. The INCL++ nuclear model is therefore
more sophisticated than the other codes used in this study and automatically
has medium effects on both nucleons and pions. An extra effect in INCL++ is
the effective binding of the propagating hadron that decreases the kinetic
energy. Tracking of higher energy particles is similar to the other approaches
here with a more sophisticated nuclear model. After a stopping time (7.8 x
10-22s for argon), propagation is handed over to a pre-equilibrium/compound
nucleus model, ABLA07 [61]. Although this is important for low energy nucleons
in the final state, it is unlikely to have a strong effect on the results in
this work. The $\Delta$ is a separate particle that propagates independently
with competing interactions and decay possibilities. The other codes don’t
have an explicit $\Delta$, but the propagating pion has effects of both the
$\pi\rightarrow\Delta$ transition and medium effects according to Ref. [62].

##  IV Comparisons

We now present a series of comparisons between total reaction cross sections
and transparency for protons and $\pi^{+}$ interacting with carbon
(specifically ${}^{12}C$) and argon (${}^{40}Ar$) targets. The goal is to
identify similarities and differences in results obtained with NEUT, NuWro and
GENIE using codes that a user could employ.

Fig. 5 provides a broad look at proton-carbon total reaction cross section and
transparency up to 1 GeV kinetic energy. Transparency data is not shown
because the results are Monte Carlo values in that experimental acceptances
are not taken into account (details are discussed later in this section). No
results of the calculations are shown below 20 MeV because most models are not
aimed at energies that low. It is notable that there is separation by proton
energy in $\sigma_{reac}$, larger energies ($T_{p}>\sim 200$ MeV) where the
reaction cross section calculations tend to agree with each other and the data
together with a lower energy region where data is less certain and the
calculations diverge as well. For the higher energy region, the total cross
section calculations are slowly rising and transparency calculations are
slowly falling. This relationship was first discussed in Sect. II.2 and agrees
with the predictions there. Although the NuWro value for $\sigma_{reac}$
agrees with the other calculations, the transparency prediction is above the
others as NN short range correlations influence transparency but not reaction
cross section (see Sect. V).

Fig. 6 focuses on the lower proton energies so that details from Fig. 5 become
apparent. The $\sigma_{reac}$ data [22] has some scatter because much of it
comes from publications from 1960’s and 1970’s. Overall, the data can be
understood as a rise as the energy decreases to account for the increasing
$pN$ cross section and the influence of compound nuclear processes. The cross
section peaks at $\sim$30 MeV and decreases at lower energies due to Pauli
blocking, other in-medium effects, and Coulomb repulsion. The peaking is seen
in NuWro and GENIE-INCL++ in good agreement with the data. INCL++ has both
compound nuclear processes and Coulomb effects, but both are absent in NuWro.
Since both INCL++ and NuWro have Pauli blocking, this seems to be the most
important contributor. GENIE-hA2018 continues to increase to lower energies
while GENIE-hN2018 follows the same trend but is cut off at 20 MeV. Neither
have Pauli blocking in the code versions used here and the hN code compensates
for this with the empirical cutoff, as explained in Sec. III.3. The NEUT
simulation starts to fall off at about 80 MeV for both carbon and argon
because of the different treatment of binding energy and Pauli-blocking as
explained in the Sec. III.1. The corresponding transparency calculations for
GENIE-hN2018 and NEUT have a rapid rise in transparency where $\sigma_{reac}$
is rapidly decreasing as discussed in Sec. II.2. The peak in $\sigma_{reac}$
at about 40 MeV corresponds to a dip in transparency for NuWro and GENIE-
INCL++.

The detailed correspondences are also interesting. GENIE-INCL++ has the most
complete nuclear model and has the best agreement with the $\sigma_{reac}$
data. While GENIE-INCL++ is always below NuWro in transparency in Fig. 6,
NuWro is sometimes larger and sometimes smaller for $\sigma_{reac}$. Even
though GENIE-hA2018 has a minimal set of nuclear corrections, it tends to
agree with GENIE-INCL++ for energies larger than about 30 MeV.

Figure 5: Total reaction cross section and transparency for proton-carbon.
Available data for $\sigma_{reac}$ [22] is shown along with calculations from
GENIE, NuWro, and NEUT. The transparency results are Monte Carlo, i.e. with no
acceptance corrections.

Figure 6: Total reaction cross section and transparency for proton-carbon,
same as Fig. 5 except for an expanded scale to show details. Available data is
shown along with calculations from GENIE, NuWro, and NEUT.

While there is a significant body of data across the full range for the total
reaction cross section, the data for transparency is scarce. No data is shown
in Fig. 5 because a computation of acceptance correction factors to match the
transparency data from electron beams is beyond this study. Fig. 7 shows a
partial accounting of this effect. NuWro made transparency calculations with
and without the experimental acceptance effects [27]. The ratio is used to
estimate the impact of acceptances on the other model results by using the
ratio of acceptance-corrected to Monte Carlo transparencies as an energy-
dependent scale factor that is applied to all the calculations in Fig. 7. We
see that these estimated acceptance corrections put all calculations in
reasonable agreement with the data [25, 63, 38] with the exception of NuWro.
The effect of short range NN correlations has increased the transparency
calculation so that it is now above the data [38, 39, 40]. The lowest energy
points are of particular interest because of the sensitivity to nuclear
effects. Although the calculations match the measurement of transparency for
protons of kinetic energy $180$ MeV [25], they are somewhat above the newer
data points at kinetic energy $\sim$ 350 MeV [39, 40]. None of the
calculations reported here show such a steep rise.

Figure 7: Transparency for proton-carbon where the calculations have been
corrected according to acceptance effects as determined in Ref. [27].
Available data [25, 39, 38, 40], is shown along with calculations from GENIE,
NuWro, and NEUT.

Plots comparing $\sigma_{reac}$ and transparency are shown for $\pi^{+}$ and
carbon ${}^{12}C$ target in Fig. 8. Here, the dominant feature is a peak
corresponding to excitation of the $\Delta$(P33(1232)) resonance at kinetic
energy of about 165 MeV. This corresponds to a dip in the transparency
results.

As was seen for protons, values of $\sigma_{reac}$ for high energy pions (here
larger than about 400 MeV) have reasonable agreement among the calculations,
slightly underestimating a few existing experimental points. However, the
spread of the simulations for transparency is much larger for $\pi^{+}$ than
for protons. This is due to the extra effects of formation zone (NEUT) and
differing treatments of the higher mass resonances (NuWro, GENIE INCL++). The
effect of resonances at masses above the $\Delta$ is seen for GENIE hA and hN,
but not for the others. If a precise measurement could be made, these features
could be tested.

Treatments of the $\Delta$ resonance in nuclei have been studied with pion
[43] and electromagnetic [64] probes. They typically find small shifts and
increases in width for nuclei. The codes studied here have minimal corrections
to the $\nu\rightarrow\Delta$ vertex and only INCL++ treats the $\Delta$ as a
propagating particle. Due to differing treatments of the interactions and
nuclear models, the variations among the simulations are significant for the
$\Delta$ peak in both $\sigma_{reac}$ and transparency. At the same time, the
hN, hA, and NuWro results are close together for $\sigma_{reac}$ and
transparency for kinetic energies below roughly 300 MeV. Since hN and NuWro
share usage of the medium corrections of Salcedo-Oset [19] and hA doesn’t have
that effect, this results implies that the medium corrections in FSI aren’t
very important. (See Sect. V for more detail.) It is interesting that although
INCL++ is above the other simulations for $\sigma_{reac}$ at the peak, the
prediction for transparency is shifted with respect to the others. All
propagating particles are in a mean field potential in INCL++ [46] which
depends on the kinetic energy and position. This potential includes both
nuclear and Coulomb contributions and is not in any of the other codes. As a
result, the energy of the $\pi^{+}$ is shifted and the dip in transparency
moves to lower energy. It is also notable that NEUT is in excellent agreement
for $\sigma_{reac}$ because the $\pi$N cross sections were fit to it [23].
Although the NEUT result is above hN, hA, and NuWro for $\sigma_{reac}$, it is
also above the other results for transparency. This is due to the formation
zone effect (see Sect. V).

Figure 8: Total reaction cross section and transparency for $\pi^{+}$-carbon.
Available data [23] are shown along with calculations from GENIE, NuWro, and
NEUT.

To study atomic mass (A) dependence, calculations are repeated for the argon
target with results shown in Figs. 9, 10, and 11. These calculations can be
directly compared with result for a carbon target in Figs. 5, 6, and 8 above.
The importance of nuclear medium effects can be expected to increase as the
size of the nucleus increases. However, the gross features of each model are
unchanged with this significant increase in nuclear mass. Although many basic
nuclear effects scale linearly with A, other detailed effects such as FSI
scale as $A^{2/3}$ and NN correlations can vary significantly for small
changes in A. Since reasonable agreement was obtained with iron (A=56)
transparency data for NuWro in Ref. [27], no strong dependence on nucleus is
expected. We chose argon as a second target because of its importance in
neutrino oscillation experiments. Since there is no data for this nucleus,
only a comparison among the simulations is possible. Pauli blocking is a
bigger effect for protons in argon. This and other nuclear effects make the
spread of curves somewhat more pronounced. Medium effects make the $\Delta$
$\sigma_{reac}$ peak wider for pions with a corresponding effect in
transparency. The tentative conclusion is that A dependence is not significant
or the models fail to account properly for it.

Figure 9: Subject of plots is same as in Fig. 5 but for the proton-argon
interaction. Since there is no data available, only Monte Carlo calculations
from GENIE, NuWro, and NEUT are shown.

Figure 10: Subject of plots is same as in Fig. 9 except for an expanded scale
to show details. Since there is no data available, only Monte Carlo
calculations from GENIE, NuWro, and NEUT are shown.

Figure 11: Total reaction cross section and transparency, same as Fig. 8
except for $\pi^{+}$-argon. Since there is no data available, only Monte Carlo
calculations from GENIE, NuWro, and NEUT are shown.

##  V Discussion

It is clear that there are important differences in the FSI codes analyzed
here. In addition, there is never a guarantee that implementation has been
done in the same way. The goal of this section is to explore each code in more
detail to see effects of particular components which modify reaction cross
section and transparency results. These effects include medium corrections,
nucleon-nucleon correlations, and the formation zone.

The modular structure of GENIE allows study of many theoretical components.
Fig. 12 shows the impact of the Salcedo-Oset [19] medium effects on $\pi^{+}$
total reaction cross section and transparency for carbon. When these effects
are taken away, no nuclear effects remain and the $\pi$N cross sections are
the free values. In fact, the structures in the results come from the
underlying $\pi$N cross sections. Although the authors give the range of model
viability as 80-350 MeV as the range of applicability, GENIE also allowed it
to work at pion energies smaller than 80 MeV because the effects are small
there. A small discontinuity at 350 MeV can be seen in both results. Since
there is no difference in calculation for kinetic energy larger than 350 MeV,
the curves at those energies in Fig. 12 can only differ from using samples
with different random number sets or binning. In the GENIE adaptation, the
Salcedo-Oset modification of microscopic cross section subtracts about 10%
from the total reaction cross section at the cross section peak
(T${}_{\pi}\sim$ 180 MeV) and adds about 15% to the transparency.

Figure 12: Total reaction cross section and transparency for pion-carbon
calculated in GENIE hN2018. This illustrates the importance of the Salcedo-
Oset medium effects [19] in these quantities. ”No medium corrections” means a
simulation with free $\pi$-N cross sections.

Medium effects for nucleon-nucleon interactions were explored by Pandharipande
and Pieper [26], also using the local density approximation. GENIE implemented
this as a set of look-up tables as a function of nucleon energy and nuclear
density for a variety of nuclei. There is a small dependence on nucleus and
that is handled with a linear interpolation between tables.

Results for protons and carbon are shown in Fig. 13. As was seen for pion
interactions, the largest effect is found at lower kinetic energy where the
interaction cross section is large and nuclear effects are important. Here,
the effect is a decrease in the total reaction cross section and an increase
in transparency. The effect grows as the energy decreases, similar result was
produced for NuWro in Fig. 5 in Ref. [27]. Here, we extend the effect to lower
energy and it becomes as large as 80%.

Figure 13: Reaction cross section and transparency for proton-carbon in GENIE
hN2018. Show dependence of Pandharipande-Pieper medium effects [26] in
$\sigma_{reac}$ and transparency.

In Fig. 14 we show the effect of nucleon-nucleon short range correlations on
the transparency results in NuWro, as described in Sec. III.2. We see that the
effect is to increase the transparency by 10-15% in the whole range of proton
kinetic energies. The origin of the effect is illustrated in Fig. 4 from Ref.
[30]. Because of nucleon-nucleon correlations, the probability of having
another nucleon in a sphere of radius $\sim 0.8$ fm around any nucleon is
strongly suppressed. Interactions typically occur in the central region of
nucleus with higher density. Due to correlation effects there, nucleons are
more likely to leave this region avoiding any reinteraction. Correlations do
not affect reaction cross section where only the single nucleon density is
relevant in the adopted approach.

Figure 14: Transparency for proton-carbon in NuWro. Show dependence of short-
range correlation effects [27].

As described in Sect. III.1, NEUT uses a formation zone for pions produced in
nuclei. The formation zone effect is similar to that of correlations in that
interactions are suppressed by giving the particle a region where it won’t
interact. This effect increases the $\pi^{+}$ transparency for a wide range of
energies as shown in Fig. 15. This large effect could be easily tested in a
pion electro-production experiment.

Figure 15: Effects with and without the formation zone in NEUT for $\pi^{+}$\-
carbon. The plot shows the pion transparency in simulations of neutrino single
pion production.

Generator codes are developed by independent groups which sometimes make
different choices. Therefore, a direct comparison is both interesting and
perhaps hard to interpret. In Figs. 16 and 17, we attempt to make a direct
comparison between the generators of transparency for proton-carbon and
$\pi^{+}$-carbon, respectively, using the same theoretical inputs. For Fig.
16, the NuWro NN-correlations are removed. That result is then directly
compatible with the hN result from Fig. 5. These are compared with the NEUT
result from Fig. 5 which doesn’t have medium corrections like the others. The
results for proton-carbon nicely continue the themes previously discussed.
Above $\sim$200 MeV, the calculations are in good agreement showing a weak
model dependence there. NuWro and GENIE hN are in agreement indicating very
similar implementation of the core propagation model and medium corrections
[26]. However, NEUT doesn’t have the medium corrections and this is shown to
be a significant effect in Fig. 13. The agreement between NEUT and the others
in the range of 200-400 MeV must be caused by some additional difference such
as the choice of nucleon-nucleon interactions. At low energies, the curves
diverge according the approximations made.

With the NEUT formation zone removed, the comparison for $\pi^{+}$-carbon in
Fig. 17 becomes more straightforward in that all calculations have the basic
propagation model together with medium modifications [19]. However, the
interpretation is not as simple as for proton-carbon because GENIE hN is not
in good agreement with NuWro and NEUT. At kinetic energies above $\sim$400
MeV, the effects involved should come from the basic propagation model,
nuclear densities and pion-nucleon cross sections. NuWro and NEUT agree with
each other and GENIE hN has a different shape and larger magnitude.
Preliminary explorations indicate that the pion-nucleon cross sections
employed are different according to data base used and how pion absorption is
treated. However, a full explanation and fix is beyond the scope of this
article. On the other hand, the calculations agree on the depth of the dip due
to the $\Delta$ resonance with moderate differences in the resonance width. At
the lowest energies, GENIE hN predicts a larger transparency than the others
due to lack of Pauli blocking. The discrepancies in this plot need further
study.

An additional study was done to examine the disagreements in Fig. 17 and make
a comparison with data. To study the transparency of higher energy pions, a
new simulation was done with $\nu_{e}$-carbon NCDIS events. This gives a
sample of pions up to 4 GeV. The result for the Monte Carlo transparency is
shown in Fig. 18 where the simulations are compared with data from Ref. [42].
All calculations are in qualitative agreement with the data. Without properly
accounting for experimental conditions, no detailed test can be made.

Each curve in Fig. 18 has the standard pion formation length [52, 47] removed.
Since inclusion of this effect produced a transparency that approached 1 in
this energy range, this is an indication that the formation length derived
from neutrino data [47] is not needed to describe data. With the models shown
in Fig. 18, the main sensitivity is to the pion-nucleon total cross sections.
Although each code has a somewhat different strategy, the results are similar.
Therefore, there is no insight into the disagreements seen in Fig. 17.

Figure 16: Comparison of generator predictions of transparency for proton-
carbon.  Figure 17: Comparison of generator predictions of transparency for
$\pi^{+}$-carbon.  Figure 18: Comparison of generator predictions of
transparency for $\pi^{+}$-carbon for kinetic energies above 1 GeV with data
from Ref. [42]. Here, simulations using the NCDIS interaction without
formation zone effects are shown.

The issues in validating FSI codes are strongly dependent on probe energy.

  * •

Protons (pions) above 200 (300) MeV kinetic energy have simple or no nuclear
corrections as long as formation zone effects can be ignored. Either
$\sigma_{reac}$ or transparency as a validation goal is equally correct.

  * •

Low energy protons (similar effects will apply to low energy neutrons) have
the most sensitivity to nuclear effects. Some of the codes in this study make
severe approximations for these particles and can give untrustworthy results.
For example, Pauli blocking is significant at lower energies and its
implementation varies widely among the codes. It is notable that INCL++ has
the best nuclear model and has the best agreement with proton $\sigma_{reac}$
data. Although all simulations have good agreement with the lowest energy
transparency data at 180 MeV [25], this misses the region with the most
sensitivity. The importance of this finding will vary among experiments.
Experiments that run at low neutrino energy [4] or that need significantly
improved accuracy [3] will be most affected. With the threshold for proton
detection down to 47 MeV in the MicroBooNE liquid argon detector results [36],
examination of these effects is becoming possible.

  * •

Pions of any charge at energies where the $\Delta$ resonance is important are
difficult for all the codes studied here. This is well known [5, 65] and
improvements in them are necessary. Although all codes have the effect of the
propagating $\Delta$, INCL++ has the additional off-shell effects of the
propagating pion which produces a noticeable effect. Some of the codes are
more than two standard deviations from the $\sigma_{reac}$ data. Interesting
sensitivity to effects in the pion transparency await data for testing.

No significant A dependence is seen here. The comparisons for pC (Figs. 5 and
6) and pAr (Figs. 9 and 10) look very similar. This is because the quantities
studied here seem to be mainly sensitive to general features that can be well-
described by the models that are typically employed in these codes. Although
that suggests data for light nuclei are as good as heavier nuclei data for
validating models, that could be due to the simplified nuclear treatment in
the codes or the narrowness of the data studied.

The variations among the simulations are approximately equal in size for
$\sigma_{reac}$ and transparency in many cases. For example, the fractional
variation among NEUT, hN, hA, NuWro, and INCL++ results in Fig. 6 is very
similar for both quantities with protons having kinetic energy less than 200
MeV. The same relationship can be observed for high energy hadrons and for the
$\Delta$ region for pions. In particular, the NEUT result for pion
$\sigma_{reac}$ has been adjusted by tuning the underlying $\pi$N cross
sections to get better agreement with data than is seen for the other codes.
This adjustment has a similar fractional effect in both $\sigma_{reac}$ and
transparency. However, choices made in development of each code produce
interesting exceptions. One deviation to this observation comes in Fig. 8
where NuWro is further from the GENIE calculations for transparency than
$\sigma_{reac}$ at $\sim$400 MeV pion kinetic energy. Another is the
relationship between INCL++ and all the other calculations in Fig. 8. Although
INCL++ has a larger value of $\sigma_{reac}$ than the others, the distribution
of transparency is shifted to low $T_{\pi}$ due to the way INCL++ shifts the
energy of the hadron in the mean field potential.

Proper description of protons with less than 100 MeV kinetic energy requires
many nuclear effects. Codes that include Pauli blocking (NEUT, NuWro and GENIE
INCL++) are qualitatively different than those that don’t (GENIE hA and hN).
GENIE hN chose to stop the propagation of low energy nucleons to avoid
difficulties coming from the neglect of Pauli blocking. In NEUT, events
producing low energy nucleon interactions are suppressed due to Pauli blocking
with different results than the other codes. Data for low energy hadrons (both
cross section and transparency, particularly for protons) should be updated
and/or improved.

Medium corrections are included for pions (GENIE hN and INCL, NuWro, and NEUT
according to Ref. [19]) and for protons (GENIE hN and hA, and NuWro according
to Ref. [26]). It is interesting that all codes chose the same models for pion
and proton medium corrections. Various comparisons have shown that at least
the pion medium corrections have been implemented with an almost identical
strategy for each code. INCL++ took a different strategy in including most
nuclear effects within an overall potential for all particles and gets similar
results to the other codes with the notable exception of low energy nucleons
and pions when the $\Delta$ excitation is important.

##  VI Conclusions

The main motivation for this study comes from the needs of neutrino
oscillation experiments. It is well established that detection of final state
hadrons improves the quality of neutrino energy reconstruction [2, 66]. In
addition, studies of final state hadrons allows for better understanding of
dynamics of primary interactions inside nuclei. This has an indirect effect on
oscillation experiments that don’t depend on hadron reconstruction but must
also accurately separate events by interaction [67]. Both Cerenkov [67] and
scintillator experiments [2] report significant sensitivity to FSI effects.
Detection of low energy protons and pions is a high priority for liquid argon
detectors [3, 4] for both neutrino oscillation and cross section measurements.

$\sigma_{reac}$ and transparency provide two independent ways to validate FSI
models for Monte Carlo event generators in a general way. We have studied the
relationship between them for three of the most commonly used neutrino event
generators - GENIE [32], NEUT [33], and NuWro [24]. While $\sigma_{reac}$ is
the total inelastic cross section as measured directly with hadron beams,
transparency is the probability of escaping with no interaction when the
hadron is deposited inside the nucleus which must be measured with production
experiments. Although transparency measurements can come from either
electromagnetic or neutrino beams, monoenergetic electron beams have been much
more practical. However, the application to neutrino oscillations is of great
importance. These two quantities measure different aspects of the propagation
of hadrons in nuclei in a general way and direct comparisons show they have
different dependencies on the underlying physics. Detailed theoretical studies
of these effects are unusual [27, 30] and less often with codes that commonly
simulate experiments. Results for $\pi^{+}$ and proton propagation are studied
for carbon and argon targets here. These targets are of great interest because
many existing neutrino oscillation measurements using them are in progress or
planned.

All the codes studied here use INC approximations which are based on the idea
that hadron-nucleon interactions in the medium are isolated and similar to
what would happen for free nucleons. Various effects are then employed to
simulate the nuclear environment. We find that each code implements a unique
set of effects with varying effects on the results. These codes are all in
heavy use in neutrino experiments and are regularly validated against a
variety of data with $\sigma_{reac}$ being the most general data employed. It
is interesting that the nuclear effects are less significant for
$\sigma_{reac}$ because transparency measurements depend much more on the
nuclear environment around the location of the primary interaction. Nuclear
effects such as medium corrections, short range correlations, and formation
zone have measurable influences on transparency predictions. At the same time,
it must be noted that transparency measures properties closer to what is
needed in neutrino oscillation experiments.

The basis for the complex simulation codes studied here is very simple – a
mean free path using nuclear density and free hadron-nucleon cross sections.
This was explored in Sect. II.2 where formulas are provided to calculate
either quantity. The main result is a single relationship between
$\sigma_{reac}$ and transparency for each nucleus independent of the probe.
The connection to running codes was demonstrated through agreement with a
stripped-down version of GENIE hN. These formulas could be used as the
baseline for the study of nuclear medium effects in transparency experiments.

The transparency calculations done here do not account properly for the
acceptance of the measurements. We therefore refer to them as Monte Carlo
transparency. The acceptance was taken into account in Ref. [27] and we find
that the effect is approximately a uniform enhancement of $\sim$15%
independent of energy. Applying this as a correction factor, all the existing
codes are in reasonable agreement with the proton-carbon transparency data.

One important finding of this work is that the uncertainty in the
understanding of these quantities depends very strongly on energy. At lower
energies, nuclear effects are more significant. In fact, the picture of a
localized propagating proton with a large wavelength is very questionable. At
large kinetic energy (above roughly 200 (300) MeV) for protons (pions), the
mean free path is large compared to the intranuclear spacing and all the codes
give similar results for both quantities. Nuclear effects can also apply at
higher energy. For example, the addition of NN correlations in NuWro makes it
$\sim$10% larger than the others for high energy protons and the formation
zone for high energy pions in NEUT produces an increased transparency for high
energy pions.

There are two kinematic regions where nuclear effects are particularly
important. In most cases, the effects come from treatments of the nuclear
medium which have been imported from theoretical work describing data from a
variety of probes beyond neutrinos. These are most important at low kinetic
energies and predominantly affect transparency. Proper description of protons
with less than 100 MeV kinetic energy requires many nuclear effects, including
Pauli blocking, medium corrections, compound nucleus mechanisms, and Coulomb
effects. From the comparisons here, it seems that the pN cross section
attenuated by Pauli blocking is, together with medium corrections, the most
important effect. In addition, there is significant sensitivity to nuclear
effects in the way the $\Delta$ ($P_{33}$(1232)) resonance is treated. The
shift and change in width in the $\Delta$ transparency dip is different in
INCL++ than the others. Both include the effect of $\Delta$ interactions with
the nuclear medium. This effect was already seen for other probes [43] and
makes the pion transparency an interesting experimental study. An additional
effect in INCL++ comes from including binding effects on the each propagating
hadron. The net effect is of opposite sign for the two methods.

The variations among the simulations are approximately equal size for
$\sigma_{reac}$ and transparency with extra uncertainty for the latter due to
correlations and formation zone effects. That observation has important
consequences for the choices in FSI model validation. It is interesting that
our results are somewhat different than the recent calculations of Isaacson,
et al. [30]. They have different methods to simulate interactions and a new
calculation of NN correlations. They also show a calculation which uses
methods similar to what is done here. The result is surprisingly different
than what is shown in this paper. Both of these differences need to be
explored further in a dedicated study.

The uncertainty of results is expected to grow with the size of the target
nucleus because the hadrons have a longer path length for interactions. Since
there is no data for argon, only the spread in Monte Carlo transparencies is
available to test this hypothesis here. There is no significant difference in
the spread of values for the codes studied here between carbon (A=12) and
argon (A=40). Additional studies of heavier nuclei data with these same codes
for $\sigma_{reac}$ [23, 18, 24] and transparency [27] show good agreement.
Another possibility is that there is some A-dependent effect that is missing
in the models presented here.

Comparisons among event generator codes are of growing importance [9]. In this
work, a complex quantity such as transparency is studied. One desire from such
a comparison is to assess the relative accuracy of the codes. It is clear that
almost any FSI code can work for kinetic energies about $\sim$300 MeV where
the nuclear corrections are small. The treatment of low energy protons
involves significant corrections due to nuclear effects which get very large
for kinetic energies below $\sim$50 MeV. Although GENIE-INCL++ has the most
complete set of effects, the other codes have reasonable behavior down to
$\sim$100 MeV. Below that value, NEUT, hA, and hN diverge from the data at
different values of energy. For pions, the main sensitivity is to the handling
of the $\Delta$(1232) resonance. The differences between GENIE-INCL++ and the
other GENIE models and NuWro are interesting and new experimental tests could
point to improved treatment of the $\Delta$ resonance in nuclei. Although NEUT
has excellent agreement with $\sigma_{reac}$ due to tuning of the underlying
$\pi$N cross sections, other codes have poorer agreement with the same data.
Agreement of NEUT with transparency data is not guaranteed with this tuning of
$\sigma_{reac}$. NEUT has a significant effect due to pion formation zone that
can be tested in experiments. Even though hA is the simplest FSI model, it
agrees well with the more sophisticated calculations in most situations.

Figures 16 and 17 are the closest code comparisons that can be made in this
study. Here, comparable simulations from GENIE hN2018, NEUT, and NuWro are
collected for examination. For protons, both GENIE and NuWro have medium
corrections but NEUT does not. In general, the different treatments of low
energy protons is readily apparent. The lack of Pauli blocking in GENIE hN is
seen at the lowest energies in both figures. However, the difference between
it and the other codes for pions between 400 MeV and 1 GeV is surprising and
deserves further study. Fig. 18 extends the comparison for pion-carbon
transparency to kinetic energy up to 4 GeV, testing codes in a kinematic
region where nuclear effects should be less important. All calculations are in
rough agreement with data [42]. However, these calculations are made without
the normally applied formation zone [52, 47] and disagreement with these data
is significant for calculations with this effect. This suggests that the
inclusion of formation zones as implemented are inconsistent with electron
scattering transparency data and should not be used.

Another important finding is that $\sigma_{reac}$ and transparency are related
in a basic way for the codes studied here, i.e. features in each quantity can
be related to each other. For example, the peak at $\sim$30 MeV in
$\sigma_{reac}$ corresponds to a dip in transparency of comparable size. The
relative importance of $\sigma_{reac}$ and transparency as a way to validate
Monte Carlo FSI codes can only be assessed in a preliminary way because of the
lack of transparency data. One good way to estimate uncertainty is to examine
the range of results from reasonable codes. We can first note that the range
of results presented here is very similar for the two quantities, perhaps a
little larger for transparency because of the added sensitivity to nuclear
corrections. Further, the spread of results for protons at energies above
$\sim$200 MeV is much smaller than at lower energies, where nuclear effects
due to Pauli blocking and medium effects cause additional uncertainties in the
calculations. Pions above $\sim$300 MeV are sensitive to formation zone
effects and sensitive to the detailed treatment of the $\Delta$ resonance at
lower energies. It should be noted that the only code to accurately reproduce
$\sigma_{reac}$ over the entire energy range considered here is NEUT and that
is because the $\pi$N cross sections were fit to it.

The relationship to experiment is somewhat clouded by lack of transparency
data with the same quality as the $\sigma_{reac}$ data. At the same time, the
large body of low energy proton-nucleus data is too inconsistent. Although the
transparency data for protons on carbon is significantly more sparse than for
$\sigma_{reac}$, there is no data for the argon target. Therefore, there is a
strong need for more proton transparency data at all energies. There is recent
data for pion $\sigma_{reac}$ [23], but no data for pion transparency in the
energy range studied here. This is the most pressing need for data uncovered
in this study.

At present, all FSI validations for the codes studied here use $\sigma_{reac}$
as a primary tool. While no significant errors are made this way as long as
extra effects such as formation zone and NN correlations are externally
verified, higher quality transparency data would shift emphasis toward it.
Ideally, a carefully defined balance of the two observables would be used for
validation purposes. These data provide a test of hadron interaction
probability and, perhaps surprisingly, the nuclear corrections. A full
validation must include more detailed hadron interaction data such as process-
dependent total cross sections [23] and inclusive double differential cross
sections [18].

Independent of the existence of data, the relationship of simulation of the
two quantities can be studied. We find that transparency is more sensitive to
medium effects which have a strong effect on when a hadron first interacts
when deposited into the nucleus. Therefore, an interesting finding is that
transparency measurements provide a way to study these nuclear effects in
isolation through the energy dependence.

Finally, we can make suggestions for future measurements. The quality of low
energy pion and proton beam data should be improved to allow better tuning of
models where the sensitivity to nuclear effects is large. Proton transparency
measurements shown here are either tagged (where the coincident electron is
detected for each event [25]) or untagged (relative to calculations without
the effect [38, 39]). Untagged methods are also used for studies of pion
production [28] which use pions of higher energy than are examined here. We
feel that the tagged measurements have less model dependence for the energy
region studied here where the quasielastic interaction is strong. Therefore,
new measurements for proton kinetic energies in this energy range similar to
Ref. [25] would be valuable to disentangle the nuclear effects. The lack of
any pion transparency data for kinetic energies below 1 GeV is notable.
However, a large acceptance detector such as CLAS at Jefferson laboratory
could make the measurements by either method. The dominant interaction
producing a $\pi^{-}$ would be through a $\Delta^{-}$ resonance which decays
to $\pi^{-}$n. The tagging would require both the scattered electron and the
coincident ejected neutron. For pions of kinetic energy larger than about 350
MeV, this tagging could still be used although the resonances have a smaller
decay branch to to $\pi$N.

###### Acknowledgements.

The authors thank the US Department of Energy for support under contract DE-
SC0007914. JTS was partially supported by Polish Ministry of Science and
Higher Education Grant DIR/WK/2017/05. YH was supported by by MEXT/JSPS
KAKENHI Grant Number 18H05536.

## References

  * Abe et al. [2018] K. Abe et al. (T2K Collaboration), Phys. Rev. Lett. 121, 171802 (2018).
  * Acero et al. [2019] M. A. Acero et al. (NOvA Collaboration), Phys. Rev. Lett. 123, 151803 (2019).
  * Abi et al. [2020] B. Abi et al. (DUNE Collaboration) (2020), eprint arXiv:2002.03005.
  * Antonello et al. [2015] M. Antonello et al. (MicroBooNE, LAr1-ND, ICARUS-WA104) (2015), eprint arXiv:1503.01520.
  * Alvarez-Ruso et al. [2018] L. Alvarez-Ruso et al. (NuSTEC), Prog. Part. Nucl. Phys. 100, 1 (2018).
  * Elkins et al. [2019] M. Elkins et al. (MINERvA), Phys. Rev. D 100, 052002 (2019), eprint 1901.04892.
  * Da Motta et al. [2015] H. Da Motta, J. G. Morfin, and M. Sakuda, eds., _Proceedings, 8th International Workshop on Neutrino-Nucleus Interactions in the Few GeV Region (NuInt 12): Rio de Janeiro, Brazil, October 22-27, 2012_ , vol. 1663 (AIP, 2015).
  * Boyd et al. [2009] S. Boyd, S. Dytman, E. Hernandez, J. Sobczyk, and R. Tacik, AIP Conf. Proc. 1189, 60 (2009).
  * Betancourt et al. [2018] M. Betancourt et al., Phys. Rept. 773-774, 1 (2018), eprint 1805.07378.
  * Serber [1947] R. Serber, Phys. Rev. 72, 1114 (1947).
  * Metropolis et al. [1958a] N. Metropolis, R. Bivins, M. Storm, A. Turkevich, J. M. Miller, and G. Friedlander, Phys. Rev. 110, 185 (1958a).
  * Metropolis et al. [1958b] N. Metropolis, R. Bivins, M. Storm, J. M. Miller, G. Friedlander, and A. Turkevich, Phys. Rev. 110, 204 (1958b).
  * Buss et al. [2012] O. Buss, T. Gaitanos, K. Gallmeister, H. van Hees, M. Kaskulov, O. Lalakulich, A. B. Larionov, T. Leitner, J. Weil, and U. Mosel, Phys. Rept. 512, 1 (2012).
  * Cugnon et al. [2016] J. Cugnon, A. Boudard, J. C. David, S. Leray, and D. Mancusi, Eur. Phys. J. Plus 131, 169 (2016).
  * Ferrari et al. [2005] A. Ferrari, P. R. Sala, A. Fasso, and J. Ranft, _FLUKA: A multi-particle transport code (Program version 2005), preprint CERN-2005-010_ (2005).
  * Yariv et al. [2007] Y. Yariv et al., in _ND 2007_ (EDP Sciences, 2007), 296.
  * Ginocchio [1978] J. N. Ginocchio, Phys. Rev. C 17, 195 (1978).
  * S. A. Dytman and A. S. Meyer [2011] S. A. Dytman and A. S. Meyer, AIP Conf. Proc. 1405, 213 (2011).
  * Salcedo et al. [1988a] L. L. Salcedo, E. Oset, M. J. Vicente-Vacas, and C. Garcia-Recio, Nucl. Phys. A 484, 557 (1988a).
  * Mancusi et al. [2015] D. Mancusi, A. Boudard, J. Carbonell, J. Cugnon, J.-C. David, and S. Leray, Phys. Rev. C 91, 034602 (2015), eprint 1411.4562.
  * [21] _Said partial-wave analysis facility_ , http://gwdac.phys.gwu.edu/.
  * Carlson [1996] R. Carlson, Atom. Data Nucl. Data Tabl. 63, 93 (1996).
  * Pinzon Guerra et al. [2019] E. Pinzon Guerra et al., Phys. Rev. D 99, 052007 (2019), eprint 1812.06912.
  * Golan et al. [2012] T. Golan, C. Juszczak, and J. T. Sobczyk, Phys. Rev. C 86, 015505 (2012).
  * Garino et al. [1992] G. Garino et al., Phys. Rev. C 45, 780 (1992).
  * Pandharipande and Pieper [1992] V. R. Pandharipande and S. C. Pieper, Phys. Rev. C45, 791 (1992).
  * Niewczas and Sobczyk [2019] K. Niewczas and J. T. Sobczyk, Phys. Rev. C 100, 015505 (2019), eprint 1902.05618.
  * Dutta et al. [2013] D. Dutta, K. Hafidi, and M. Strikman, Prog. Part. Nucl. Phys. 69, 1 (2013), eprint 1211.2826.
  * Leitner et al. [2009] T. Leitner, O. Buss, and U. Mosel, Acta Phys. Polon. B 40, 2585 (2009), eprint 0905.1644.
  * Isaacson et al. [2021] J. Isaacson, W. I. Jay, A. Lovato, P. A. N. Machado, and N. Rocco, Phys. Rev. C 103, 015502 (2021), eprint 2007.15570.
  * Benhar [2003] O. Benhar, in _2nd International Workshop on Neutrino-Nucleus Interactions in the Few GeV Region_ (2003), eprint nucl-th/0307061.
  * Andreopoulos et al. [2010] C. Andreopoulos et al., Nucl. Instrum. Meth. A614, 87 (2010), eprint 0905.2517.
  * Hayato [2009] Y. Hayato, Acta Physica Polonica B 40, 2477 (2009), URL http://th-www.if.uj.edu.pl/acta/vol40/abs/v40p2477.htm.
  * Stowell et al. [2017] P. Stowell et al., JINST 12, P01016 (2017), eprint 1612.07393.
  * Le et al. [2019] T. Le et al. (MINERvA), Phys. Rev. D 100, 052008 (2019), eprint 1906.08300.
  * Abratenko et al. [2020] P. Abratenko et al. (MicroBooNE), Phys. Rev. D 102, 112013 (2020), eprint 2010.02390.
  * Abe et al. [2020] K. Abe et al. (T2K), Phys. Rev. D 101, 112004 (2020), eprint 2004.05434.
  * O’Neill et al. [1995] T. O’Neill et al., Phys. Lett. B 351, 87 (1995), eprint hep-ph/9408260.
  * Dutta et al. [2003] D. Dutta et al. (JLab E91013), Phys. Rev. C 68, 064603 (2003), eprint nucl-ex/0303011.
  * Rohe et al. [2005] D. Rohe et al. (E97-006), Phys. Rev. C 72, 054602 (2005), eprint nucl-ex/0506007.
  * Brodsky and Mueller [1988] S. J. Brodsky and A. H. Mueller, Phys. Lett. B 206, 685 (1988).
  * Qian et al. [2010] X. Qian et al., Phys. Rev. C 81, 055209 (2010), eprint 0908.1616.
  * Freedman et al. [1982] R. Freedman, G. Miller, and E. Henley, Nucl. Phys. A 389, 457 (1982).
  * De Jager et al. [1974] C. W. De Jager, H. De Vries, and C. De Vries, Atom. Data Nucl. Data Tabl. 14, 479 (1974), [Erratum: Atom.Data Nucl.Data Tabl. 16, 580–580 (1975)].
  * Moniz et al. [1971] E. J. Moniz, I. Sick, R. R. Whitney, J. R. Ficenec, R. D. Kephart, and W. P. Trower, Phys. Rev. Lett. 26, 445 (1971).
  * Boudard et al. [2013] A. Boudard, J. Cugnon, J.-C. David, S. Leray, and D. Mancusi, Phys. Rev. C 87, 014606 (2013), eprint 1210.3498.
  * Ammosov [2001] V. Ammosov, in _NuINT01 workshop_ (KEK, Tsukuba, Japan, 2001).
  * Rowe et al. [1978] G. Rowe, M. Salomon, and R. H. Landau, Phys. Rev. C 18, 584 (1978).
  * Seki and Masutani [1983] R. Seki and K. Masutani, Phys. Rev. C 27, 2799 (1983).
  * Bertini [1972] H. Bertini, Phys. Rev. C 6, 631 (1972).
  * Lindenbaum and Sternheimer [1957] S. Lindenbaum and R. Sternheimer, Phys. Rev. 105, 1874 (1957).
  * Baranov et al. [1984] D. S. Baranov et al., Yad. Fiz. 40, 1454 (1984).
  * Oset and Salcedo [1987] E. Oset and L. L. Salcedo, Nucl. Phys. A468, 631 (1987).
  * Bystricky et al. [1987] J. Bystricky, P. La France, F. Lehar, F. Perrot, T. Siemiarczuk, and P. Winternitz, J. Phys. France 48, 1901 (1987).
  * Klakow et al. [1993] D. Klakow, G. Welke, and W. Bauer, Phys. Rev. C48, 1982 (1993), eprint nucl-th/9304008.
  * Benhar [2006] O. Benhar, Nucl. Phys. Proc. Suppl. 159, 168 (2006), eprint hep-ph/0602108.
  * Cosyn and Ryckebusch [2013] W. Cosyn and J. Ryckebusch, Phys. Rev. C87, 064608 (2013), eprint 1301.1904.
  * [58] _Vmc calculations of two-nucleon densities_ , https://www.phy.anl.gov/theory/research/density2/.
  * Carlson et al. [2015] J. Carlson, S. Gandolfi, F. Pederiva, S. C. Pieper, R. Schiavilla, K. E. Schmidt, and R. B. Wiringa, Rev. Mod. Phys. 87, 1067 (2015), eprint 1412.3081.
  * [60] _Iaea benchmark of spallation models_ , http://www-nds.iaea.org/spallations/.
  * Kelic et al. [2009] A. Kelic, M. V. Ricciardi, and K.-H. Schmidt, in _Joint ICTP-IAEA Advanced Workshop on Model Codes for Spallation Reactions_ (2009), eprint 0906.4193.
  * Salcedo et al. [1988b] L. Salcedo et al., Nucl. Phys. A484, 557 (1988b).
  * Abbott et al. [1998] D. Abbott et al., Phys. Rev. Lett. 80, 5072 (1998).
  * Drechsel et al. [1999] D. Drechsel, L. Tiator, S. S. Kamalov, and S. N. Yang, Nucl. Phys. A 660, 423 (1999), eprint nucl-th/9906019.
  * Aliaga et al. [2020] L. Aliaga et al. (NuSTEC), in _NuSTEC Workshop on Neutrino-Nucleus Pion Production in the Resonance Region_ (2020), eprint 2011.07166.
  * Furmanski and Sobczyk [2017] A. P. Furmanski and J. T. Sobczyk, Phys. Rev. C 95, 065501 (2017), eprint 1609.03530.
  * Abe et al. [2015] K. Abe et al. (T2K Collaboration), Phys. Rev. D 91, 072010 (2015), eprint 1502.01550, URL http://link.aps.org/doi/10.1103/PhysRevD.91.072010.