1

44. Monte Carlo Neutrino Generators

44. Monte Carlo Neutrino Generators
Revised September 2021 by H. Gallagher (Tufts U.) and Y. Hayato (Kamioka Observatory, ICRR,
UTokyo; Kavli IPMU (WPI), UTokyo).
Monte Carlo neutrino generators are programs or libraries which simulate neutrino interactions
with electrons, nucleons and nuclei. In this capacity their usual task is to take an input neutrino
and nucleus and produce a set of 4-vectors for particles emerging from the interaction, which arethen input to full detector simulations. Since these generators have to simulate not only the initialinteraction of neutrinos with target particles, but re-interactions of the generated particles in the
nucleus, they contain a wide range of elementary particle and nuclear physics. Viewed more broadly,
they are the access point for neutrino experimentalists to the theory inputs needed for analysis.
Examples include cross section libraries for event rate calculations and parameter uncertainties andreweighting tools for systematic error evaluation.
Neutrino experiments typically operate in neutrino beams that are neither completely pure nor
mono-energetic. Generators are a crucial component in the convolution of beam flux, neutrino
interaction physics, and detector response that is necessary to make predictions about observable
quantities. Generated events are used to define experimental analyses, like event selection criteria, for efficiency determination and background subtraction. Similarly they are used to relate
reconstructed quantities back to true quantities. In these various capacities they are used from thedetector design stage through the extraction of physics measurements from reconstructed observable. Monte Carlo neutrino generators play unique and important roles in the experimental studyof neutrino interactions and oscillations.
There are several neutrino event generators available, such as ANIS [1], GENIE [2, 3], GiBUU[4, 5], MARLEY [6], NEGN [7], NEUT [8], NUANCE [9], the FLUKA routines NUNDIS/NUNRES [10, 11], and NuWro [12], as well as tools to facilitate cross-generator comparisons, such as
NUISANCE [13]. Historically, experiments would develop their own generators. This was often
because they were focused on a particular measurement, energy range, or target, and wanted toensure that the best physics was included for them. These ‘home-grown’ generators were often
tuned primarily or exclusively to the neutrino data most similar to the data that the experiment
would be collecting. A major advance in the field was the introduction of conference series devoted to the topic of neutrino interaction physics, NuINT (https://indico.cern.ch/event/703880/)
and NuFACT (https://indico.cern.ch/event/855372/) in particular. Event generator comparisons
have been a regular staple of the NuINT conference series from its inception, and a great dealof information on this topic can be found in the Proceedings of these meetings. These meetings
have facilitated experiment-theory discussions leading to the first generator developed by a theory
group (NuWro) [12], the extension of established nuclear interaction codes (FLUKA and GiBUU)
to include neutrino-nuclear processes [4, 5, 10, 11], and inclusion of theorists in existing generator
development teams.
These activities have led to more careful scrutiny of the crucial hadron and nuclear theory inputs
to these generators, which is evaluated in particular through comparisons to electron-scattering
data. At this point in time all simulation codes face challenges in describing the full extent of
the lepton scattering data, and the tension between incorporating the best available theory versus
obtaining the best agreement with the data plays out in a variety of ways within the field. For thefield to make progress, inclusion of state of the art theory needs to be coupled to global analyses
that correctly incorporate correlations between measurements. Given the rapid pace of new data
and the complexity of analyses, this is a significant challenge for the field in the coming years.
There are many neutrino experiments which use various sources of neutrinos, from reactors,
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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44. Monte Carlo Neutrino Generators

accelerators, the atmosphere, and astrophysical sources, thereby covering a range of energies from
MeV to TeV. Much of the emphasis has been on the few-GeV region in the generators, as thisis the relevant energy range for short- and long-baseline neutrino oscillation experiments. These
generators use the impulse approximation, which treats the nucleus as a collection of independent
nucleons and the primary interaction occurs between the probe and a single nucleon, for most of theinitial interaction, and subsequently simulates the interactions of secondary particles in the nucleus
in semi-classical ways. Semi-classical hadron transport approaches are commonly used as they areable simulate a variety of nuclei in a single model, fully describe complex multi-particle final states,
and for practical considerations as these approaches are fast. However, there are several challenges
facing these simulations coming mainly from the complexity of the nuclear physics, and the needto avoid double counting in combining perturbative and non-perturbative models for the neutrinonucleon scattering processes. The overall validity of this impulse approximation-based scheme, andin particular the importance of scattering channels that involve more than one nucleon, is a crucial
question that is the topic of much current work. While generators share many common ingredients,
differences in implementation, parameter values, and approaches to avoid double counting canyield dramatically different predictions [14]. In the following sections, interaction models and theirimplementations, including final-state interactions of hadrons produced in the nuclei, are described.
In order to assure their validity, neutrino event generators are tuned and validated against awide variety of data. This means that the quality of the outputs from generators is limited by thequality of experimental data. The existing neutrino-nucleon and neutrino-nucleus scattering data
sets are not restrictive enough to eliminate many approaches. On the other hand, there are various
precise data sets, which use photon, charged lepton, and hadron probes. These data sets are extensively used to validate the generators. The results from these external data tuning exercises areimportant for experiments as they quantify the uncertainty on model parameters, needed by experiments in the evaluation of generator-related systematic errors. Electron scattering data plays animportant role in determining the vector contribution to the form-factors and structure functions,as well as in evaluating specific aspects of the nuclear model [15, 16]. Hadron scattering data isused in validating the nuclear model, in particular of interactions between hadrons produced in the
primary interaction and the residual target nucleus (final state interactions). As mentioned, tuningof neutrino-nucleon scattering and hadronization models relies heavily on the previous generationof high energy neutrino scattering and hydrogen and deuterium bubble chamber experiments. Morerecent data from the K2K, MiniBooNE, NOMAD, SciBooNE, MINOS, T2K, ArgoNEUT, MINERvA, NOvA, MicroBooNE, ICARUS, SBND and Ninja experiments has been, or will be, used
for this purpose, although caution is advisable due to possible biases that mismodeled nuclear
effects might introduce. In practice, there are often choices to be made about which models to
include in an overall simulation, and which data sets to use for tuning. The process of developing
a comprehensive tune in GENIE is discussed in Ref. [3].
However, all the recent experiments use nuclear targets and it is challenging to disentangle
the neutrino-nucleon interaction from the effects of the nuclear environment. For example, it isdifficult to separate charged current quasi-elastic scattering from the multi-nucleon quasi-elastic
like scattering or single pion production, in which a pion is absorbed in the nucleus. Sometimes,
there are tensions between the data sets from different experiments. Data on multi-pion and meson
production are limited, but are anticipated to be measured in the near future experiments. Next
generation experiments are expected to provide better detection capabilities with higher statistics
and these data will help in evaluating and validating the models implemented in the generators.
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44. Monte Carlo Neutrino Generators

44.1 Neutrino-Nucleon Scattering
Event generators typically begin with free-nucleon cross sections which are then embedded
into a nuclear physics model. The most important processes are quasi-elastic (elastic for neutral
current (NC)) scattering, resonance production, and non-resonant inelastic scattering, which makecomparable contributions for few-GeV interactions. The neutrino cross sections in this energy range
can be seen in Figure 52.1 of this Review.
44.1.1 Quasi-Elastic Scattering
The cross section for the neutrino nucleon charged current quasi-elastic scattering is describedin terms of the leptonic and hadronic weak currents, where dominant contributions to the hadronic
current come from the vector (V) and axial-vector (A) form factors. Contributions from the pseudoscalar form factor (P) are typically small for muon and electron neutrinos and are related tothe axial form factor (A) assuming partially conserved axial currents (PCAC). Owing to isospin
symmetry, the vector form factors are related to those measured by precise electron scatteringexperiments [17]. Therefore, most of the generators use parametrizations of these form factors
taken directly from the data. For the axial form factor there is no such precise experiment, andmost of the generators use a dipole form [18]. Generally, the value of axial form factor at q 2 = 0(q is the four-momentum transfer) is extracted from the polarized nucleon beta decay experiment.
However, the selection of the axial vector mass parameter depends on each generator, with values
typically around 1.00 GeV/c2 . Recently, there are several attempts to use the other functions for theaxial form factors [19, 20] and some generators have already implemented these form factors [8, 21].
44.1.2 Inelastic Production
Most generators use the prescriptions of Rein-Sehgal [22] to simulate neutrino-induced single
pion production, or updated versions that incorporate lepton mass terms and pion pole contributions [23, 24]. To obtain the cross section for a particular channel, they calculate the amplitude forthe production of each resonance multiplied by the probability for the decay of that resonance intothat particular channel. Implementation differences include the number of resonances included,
whether the amplitudes are added coherently or incoherently, the invariant mass range over which
the model is used, how non-resonant backgrounds are included, inclusion of lepton mass terms, andthe model parameter values (in particular the axial mass). In this model it is also possible to calculate the cross-sections of single photon, and η production by changing the decay probability of theresonances, which are included in some of the programs. However, it is known that discrepancies exist between the recent pion electro/photoproduction data and the results from the simulation datawith the same framework, i.e. vector part of this model. There are several attempts to overcome
this issue [25] and some of the generators started using more appropriate form factors. Recently,
there are approaches that improve the theoretical description of non-resonant contributions in theresonance region, [26–28], and which account for interference between resonant and non-resonant
amplitudes. This work is expected to be implemented in the generators soon. GiBUU and NuWro
generators do not use the Rein-Sehgal model, and instead rely directly on electro-production data
for the vector contribution and fit bubble chamber data to determine the remaining parameters forthe axial contribution [29–32]. The dynamical coupled-channel model, which has been developed
to simulate various electro- and photo-meson productions, was extended to simulate the neutrino
single pion production [33]. In this approach pion-nucleon scattering is used to fix the form of theaxial current at q 2 = 0. This model is also being implemented and expected to be available in someof the generators in future.
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444. Monte Carlo Neutrino Generators

44.1.3 Deep and Shallow Inelastic Scattering
For this process the fundamental target shifts from the nucleon to its quark constituents. Therefore, the generators construct the nucleon structure functions F2 and xF3 from parton distributions
for high Q2 (the DIS regime: W& 2 GeV/c2 and Q2 & 1 GeV2 ) to calculate the direction and momentum of the lepton. The first challenge is in extending this picture to the lower values of Q2 andW that dominate the available phase space for few-GeV interactions (the so-called ‘shallow inelastic scattering’, or SIS regime). GRV98LO parton distribution functions [34] with the corrections
proposed in [35, 36] are widely used, while others [10] implement their own modifications to theparton distributions at low Q2 . Both DIS and SIS generate hadrons but their production depends
on each generator’s implementation of a hadronization model as described in the next section.
There are various difficulties not only in the actual hadronization but the relation with the single
meson production. It is necessary to avoid double counting between the resonance and SIS/DIS
models, and all generators are different in this regard. The scheme chosen can have a significant
impact on the results of simulations at a few-GeV neutrino energies. For simulating ultra-highenergy interactions currently being studied with neutrino telescopes, fully partonic descriptions,which utilize beyond-leading-order pQCD calculations [37, 38] are the norm [39].

44.2 Hadronization Models
For hadrons produced via baryonic resonances, the underlying model amplitudes and resonancebranching fractions can be used to fully characterize the hadronic system. For shallow- and deepinelastic scattering, a hadronization model is required. Most generators use PYTHIA [40] forthis purpose, although some with modified parameters. In addition some implement their own
models to handle invariant masses that are too low for PYTHIA, typically somewhere around
2.0 GeV/c2 . Such models rely heavily on measurements of neutrino hadro-production in highresolution devices, such as bubble chambers and the CHORUS [41] and NOMAD experiments [42],to construct empirical parametrizations that reproduce the key features of the data [43, 44]. The
basic ingredients are the empirical observations that average charged particle multiplicities increase
logarithmically with the invariant mass of the hadronic system, and that the distribution of charged
particle multiplicities about this average are described by a single function (an observation known
as KNO scaling [45]). The multiplicity of the neutral particles are usually obtained by scaling
the charged particle multiplicity. Simple parametrizations to more accurately reproduce differences
observed in the forward/backward hemispheres of hadronic systems are included in GENIE, NEUT,
and NuWro.

44.3 Nuclear Physics
The nuclear physics relevant to neutrino-nucleus scattering at few-GeV energies is complicated,
involving Fermi motion, nuclear binding, Pauli blocking, in-medium modifications of form factors
and hadronization, intranuclear rescattering of hadrons, and many-body scattering mechanisms
including long- and short-range nucleon-nucleon correlations.
44.3.1 Treatment of nuclear effects
In order to obtain the cross-section off nucleons in the nucleus, it is necessary to take into accountvarious in-medium effects. Most of the models used for neutrino-nuclear scattering kinematics weredeveloped in the context of few-GeV inclusive electron scattering, by experiments going back nearly
50 years. The basic models employed in event generators rely on impulse approximation schemes,
the most simple of which is the Relativistic Fermi Gas Model. The most common implementations
have been the Smith-Moniz [46] and Bodek-Ritchie [47] models. However, the results from neutrinonucleus scattering experiments in 2000 and afterwards, beginning with K2K, MiniBooNE haveshown large discrepancies from the naive expectation from the models. Most striking differences
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544. Monte Carlo Neutrino Generators

are a suppression of forward going muons (low Q2 ), a high Q2 enhancement in the event rate,
and an overall larger than expected number of observed events. In order to reproduce the data,
the quasi-elastic axial mass was used as the effective parameter and increased by roughly 20%
from the nominal values obtained by an earlier generation of bubble chamber experiments usinghydrogen or deuterium [18]. This inconsistency between nucleon and nucleus targets suggeststhat the simple nuclear model is not appropriate in describing the data. Moreover, these simple
Fermi-Gas models are not expected to describe the kinematical distributions of final state nucleons.
Therefore, several generators have to implement better models, such as local Fermi-Gas model ormore sophisticated models. Within the electron scattering community, the analogous calculations
have for decades relied on spectral functions, which incorporate information about nucleon momenta
and binding energies in the impulse approximation scheme [15]. Therefore, most of the generators
have implemented the realistic spectral functions in their latest releases.
Actually, the discrepancies in small q2 could not be solved alone by just introducing the local
Fermi-gas model nor spectral function models. This implies that the additional medium correction
effects are needed to be taken into account. One of the implemented solutions is the local Fermi-Gas
model with medium correction calculated using the random phase approximation, which is known togive large suppression in small q2 [48,49]. Although, the fundamental parts of the models are same,actual implementations are quite different between the generators. Especially, the constructionsof the final state hadron kinematics are quite different. Especially for the quasi-elastic scattering
case, treatment of the nucleon masses in the nucleus, the binding and the separation energies are
sometimes quite different [50]. Recently, Super-Scaling model with relativistic mean field theory
effects (SuSAv2) [51,52] is implemented in some of the generators. Indications of the suppression insmall q2 in the single pion productions were also observed in K2K and MiniBooNE and it is clearlyobserved at the MINERνA and NOνA experiments. However, the models to explain the suppression
have not been implemented, other than the phenomenological ones. Also, q2 dependence of thecross-section is largely model and parameter dependent. Further studies of newly implemented
single pion models may give new insights. The cause of the discrepancy of small q2 seems to be
identified but the issue of the observed interaction rates are not solved. This implies that there
must be some interaction channels which are missing and not considered in the generators.
These led to a revisitation of the role played by excitation of multiple particle-hole states in the
nucleus, and the experimental study of these scattering channels is an area of intense experimental
interest [53]. The contribution of these scattering processes is an extremely active area of theoreticalresearch as well, with significant implications for generators and analyses [54]. Several approaches,ranging from strictly phenomenological descriptions to full theoretical calculations, have recently
been incorporated into generators [55–57]. One example of a phenomenological approach utilizes
an Effective Spectral Function [58] and a Transverse Enhancement Model [59], which together
encapsulate information derived from electron scattering experiments at relevant kinematics. The
microscopic model of Nieves and collaborators is now available in GENIE and NEUT [60, 61].
SuSAv2 model [52] also has capability to simulate this multi-nucleon quasi-elastic like interaction
and is also implemented in GENIE.
One of the challenges in incorporating full theoretical models of these processes is that they are
typically slow, so generators have developed new approaches whereby much of the computation isdone offline, and the generators simply read in the hadronic tensor components. This allows for a
full prediction of the lepton kinematics, however the ability to simulate the hadronic component ofthese multinucleon states then relies on separate models. The other challenge is that the theoreticalmodels are not designed to describe exclusive final states. These approaches, while making simulations computationally tractable, neglect correlations between the lepton and hadron kinematics.
Also, there are limitations of the model itself to describe some of the kinematic regions. However,
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644. Monte Carlo Neutrino Generators

the generators need to simulate final all the state particles and thus, several assumptions are madeby authors of the generators from time to time.
Also, it is known from photo and electro-nuclear scattering that the Delta width is affected by
Pauli blocking and collisional broadening. These effects are included in some, but not all generators.
The remnant nucleus after the scattering is usually in an excited state. Therefore, the remnant
may emit gamma-rays, additional nucleons or an alpha particle during the de-excitation process.
This de-excitation gamma-ray is used in searching for proton decay in water Cherenkov detectors.
Some generators have implemented this process. NEUT implemented the de-excitation process but
only for Oxygen. Recently, GENIE has de-excitation processes for various nuclei.
When scattering from a nucleus, coherent scattering of various kinds is possible. Most simulations incorporate, at least, coherent neutral and charged single pion production. While theinteraction rate for these interactions is typically around a percent of the total yield, the unique
kinematic features of these events can make them potential backgrounds for oscillation searches.
Implemented in Monte Carlo are several PCAC-based methods [62, 63]. Microscopic models, valid
at lower neutrino energies [64–67], have also been implemented in several generators. One commonly used model by Rein and Sehgal [62] predicts a cross section for charged-current coherent pion
production that is much larger than what is observed by K2K, MINERvA and T2K. However, thecross-section is sensitive to the pion cross-section used in the model as parameters and improved
models with lepton mass correction [63] give better agreement with the recent data. This improved
model is implemented in most of the generators. In addition to coherent pion production, some generators can simulate coherent single photon production [68] and coherent elastic neutrino-nucleus
scattering.
44.3.2 Hadron Production in Nuclei
Neutrino pion production is one of the dominant neutrino scattering mechanisms in the few-GeV
region and the pions produced in the nucleus have quite large interaction cross sections. Therefore,
pion intranuclear scattering can have sizable effects on the results of simulations at these energies.
Also, recent neutrino oscillation experiments use the observed information of protons and pions
more extensively, in addition to the charged leptons. Therefore, most generators implement thisphysics through an intranuclear cascade simulation. In generators which utilize cascade models,a hadron, which has been formed in the nucleus, is moved step by step until it interacts with anucleon or escapes from the nucleus. The probabilities of each interaction in nucleus are usually
given as the mean free paths and used to determine whether the hadron has interacted or not. If
the hadron is found to have interacted, appropriate final states are selected. Usually, absorption,
elastic, charge exchange, and inelastic scatterings including particle productions are simulated as
intranuclear interactions. The determination method of the kinematics for the final state particles
heavily depends on the generators but most of them use experimentally validated models to simulate
hadron interactions in nucleus. No two intranuclear cascade simulations implemented in neutrino
event generators are the same [69]. In all cases hadrons propagate from an interaction vertex chosen
based on the density distribution of the target nucleus. In determining the generated position of
the hadrons in nucleus, the concept of the formation length is sometimes employed. Based on thisidea, the hadronization process is not instantaneous and it takes some time before generating thehadrons [12]. The basis for formation times are measurements at relatively high energy and Q2 ,and most generators that employ the concept do not apply them to resonance interactions.
GiBUU does not employ an intranuclear cascade simulation, instead, it utilizes a semi-classicaltransport model in coupled channels that describes the space-time evolution of a many body systemin the presence of potentials and a collision term [4]. This approach assures consistency between
nuclear effects in the initial state, such as Fermi motion, Pauli blocking, hadron self-energies [70],31st May, 2024

744. Monte Carlo Neutrino Generatorsand modified cross sections, and the final state, such as particle re-interactions, since the two arederived from the same model. This model has been previously used to describe a wide variety of
nuclear interaction data. Similarly, the hadronic simulation of the NUNDIS/NUNRES programs
are handled by the well-established FLUKA hadronic simulation package [10]. Recently, GENIE
included the interface to the external hadronic simulation package INCL++ [71].
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