Nuclear Transparency in Monte Carlo Neutrino Event Generators

Hadron cascade model is an essential part of Monte Carlo neutrino event generators that governs final state interactions of knocked-out nucleons and produced pions. It is shown that such model enriched with physically motivated modifications of nucleon-nucleon cross section and incorporation of nuclear correlation effects is able to reproduce experimental nuclear transparency data. Uncertainty of nucleon final state interactions effects is estimated and applied to recent neutrino-nucleus cross section measurements including an outgoing proton in the experimental signal. Conclusions are drawn on a perspective of identification of events originating from two-body current mechanism.


I. INTRODUCTION
There has been a lot of effort to understand neutrinonucleus cross sections better [1]. The motivation for this research comes from long-and short-baseline neutrino oscillation experiments and their demand to reduce systematic errors. The most critical cross section uncertainties come from nuclear effects and in particular from a two-body current (multinucleon ejection) mechanism [2]. It is important to know precisely a fraction of events originating from this mechanism because in experiments like T2K they cannot be distinguished from charge current quasi-elastic (CCQE) events on bound nucleons leading to a bias in the neutrino energy reconstruction [3]. After the MiniBooNE experiment reported an unexpectedly large value of the effective quasi-elastic axial mass [4], a parameter present in the standard parameterization of the nucleon-nucleon transition matrix element [5], a lot of theoretical [2,[6][7][8][9][10][11][12][13] and experimental [14][15][16] studies were done aiming to understand the situation. Experimental studies were focused on CC0π (sometimes called also CCQE-like) measurements with the signal defined as "no pion in the final state" [14]. It is the closest to the meaning of CCQE as it is defined for neutrino-nucleon scattering: where l is lepton flavor, n, p are neutron and proton, respectively. According to current understanding, in the impulse approximation, most of the CC0π events originate from the CCQE mechanism but there is also a significant contribution from the two-body current mechanism and pion production with consequent absorption. In the typical CC0π measurements one neglects information from final state proton(s) and there is little sensitivity to identify and measure separately contributions from the aforementioned dynamical mechanisms. MINERvA experiment made an attempt to resolve events' kinematics completely with calorimetric-type measurement of the * kajetan.niewczas@uwr.edu.pl interacting (anti-)neutrino energy [17,18]. A study done in the context of GENIE Monte Carlo (MC) generator [19] allowed to identify a kinematical region where more strength from the two-body mechanism is needed, relative to predictions of the theoretical model of Nieves et al [7,20].
Another strategy to learn about two-body current contribution is to also use information from the final state protons. ArgoNeuT experiment reported results on the distribution of relative proton angles in a two-proton sample of events [21,22]. MINERvA experiment used protons in a study of CC0π events on carbon, iron and lead targets with Q 2 (four-momentum transfer) derived from the leading proton only, as opposed to the common approach of using the muon information [23]. T2K experiment did several proton measurements and in particular those using information from transverse components of muon and proton momentum vectors [24,25]. Most recently, MINERvA did a measurement analogous to the T2K one, with the main difference coming from neutrino beam energy. MINERvA for the first time measured reconstructed neutron momentum obtaining a clear peak originating from Fermi motion [26,27].
A conclusion from the T2K and MINERvA measurements is that by combining information from both muon and proton one gets more sensitivity to multinucleon ejection mechanism. It also becomes clear that MC generators [28] used in data analysis should well describe nucleon final state interactions (FSI), typically modeled in the intranuclear cascade approach [29][30][31].
The most important test that intranuclear cascade model should pass is the ability to reproduce nuclear transparency data from electron scattering studies [32]. Nuclear transparency is defined as a probability that a knocked-out nucleon is not subject to re-interactions inside the residual nucleus. As will be seen, in the case of carbon target, used in MINERvA and T2K measurements, typical transparency values are of the order of 60%. In the vast majority of events, knocked-out nucleons interact at most once, therefore the ability to reproduce transparency value is the most important feature of cascade models.
The goal of this paper is to present a procedure of checking if neutrino MC event generators reproduce nuclear transparency data. The discussion will be done using NuWro generator [33] but can easily be repeated with other neutrino MCs. Our main conclusion will be that NuWro nucleon cascade model, after enrichment of its physical content, describes nuclear transparency very well. By comparing NuWro results with the electron transparency data we also estimated the uncertainty of nucleon mean free path as calculated in the NuWro cascade model. Our conclusion is that if it is scaled up and down by ∼ 30% one gets an error bound covering experimental uncertainties. With this estimate, we investigated how important is an impact of the FSI uncertainty for the recent T2K proton measurements. This defines a bound of experimental sensitivity in attempts to draw conclusions about two-body current mechanism. Our final conclusion is that nucleon FSI effects are controlled well enough to make a possibility of investigating details of multinucleon ejection dynamics realistic.
The paper is organized as follows. In section II we describe how nuclear transparency is defined and measured in electron scattering experiments, following the procedures described in Refs. [34,35]. In section III, NuWro generator is presented focusing on the description of the nucleon cascade model. Section IV contains the details of transparency computations in NuWro. The last two sections: V and VI contain discussion of the results and our conclusions.

II. NUCLEAR TRANSPARENCY
Modeling scattering processes on nuclear targets involving nucleon degrees of freedom strongly relies on the description of nucleon propagation within nuclear medium. In order to estimate the magnitude of nucleon distortion, one can introduce a measure, called nuclear transparency, defined as the probability of a struck nucleon to escape nucleus without significant re-interactions. Much attention has been brought to this subject following the hypothesis of color transparency (CT) [32]. Such phenomenon should suppress the probability of in-medium nucleon-nucleon interaction at very high energies. CT has been extensively studied in many experiments, using quasi-free A(e,e p) scattering on various nuclei, so far without definite conclusions [32].
The general idea behind the measurement of the nuclear transparency in quasi-elastic A(e,e p) reactions is to confront an experimental yield of knocked-out protons with the theoretical prediction that does not include the distortion due to FSI. In these experiments, where an electron ejects a proton p out of a nucleus A, using measured values of energy ω = E − E and momentum q = k − k transfers (E and k are initial electron energy and momentum, primed values refer to final electron), one defines missing momentum and missing energy as where T p and T A−1 = | p m | 2 /2M A−1 are the kinetic energies of the knocked-out proton and the residual nucleus, respectively. Nuclear transparency, measured for a fixed four-momentum transfer Q 2 ≡ | q| 2 − ω 2 , is defined as where Y exp and Y PWIA are proton yields of the measurement and theoretical calculation, respectively. The phase space V is restricted to the quasi-elastic region by the conditions E m 80 MeV and | p m | 300 MeV, ensuring suppression of inelastic processes. The theoretical prediction Y PWIA is calculated under a hypothesis of plain wave impulse approximation (PWIA), i.e., that the knockedout nucleon does not undergo any re-interactions. One should be aware that the aforementioned definition suffers from a model dependency as it relies on the accuracy of theoretical PWIA computations. Over the years, the following experiments have reported nuclear transparency measurements: • D.F. Geesaman, G. Garino et al. at Bates Linear Accelerator Center [36,37], • NE-18 at Stanford Linear Accelerator Center [38,39], • E91-013 in Hall C at Thomas Jefferson National Accelerator Facility [34,35], • E94-139 in Hall C at Thomas Jefferson National Accelerator Facility [40], • E97-006 in Hall C at Thomas Jefferson National Accelerator Facility [41].
The measurements were done in different kinematical setups, with outgoing protons momenta ranging from ∼ 0.5 to ∼ 5.5 GeV/c, and for various nuclear targets, with the most widely used 12 C and 56 Fe. Information about the kinematics of transparency measurements is summarized in Table I. The PWIA predictions used by experimental groups describe target proton in the independent particle shell models (IPSM). The IPSM based calculations are known to overestimate single-particle strength in exclusive reactions [44]. This discrepancy is attributed to the shells being not fully occupied due to nucleon-nucleon correlations that cannot be fully accounted for in mean-field approaches. NE-18 at SLAC was the first experiment that introduced correlation factors c A in the definition of transparency to correct for the depletion of single-particle strength outside of the phase space V : Ref. with values c A = 0.90, 0.82 for 12 C and 56 Fe, respectively. They are larger than typically used spectroscopic factors, as they come up from the integration over a specific phase space V [41]. In this paper we compare our results to transparency results as they were published by experimental groups. Our treatment of correlation factors agrees with that from Ref. [45]. It has to be emphasised that the introduction of correlation factors is a subject of an on-going debate. Some authors argue that because the experiments were conducted in the transverse kinematics, which is less sensitive to the high-value tail of the nucleon momentum distribution, the use of correlation factors is not justified [46]. Theoretical arguments suggest that perhaps soft Q 2 dependent correlation factors would be more appropriate [47], but many recent papers on the nuclear transparency simply ignore them [32]. CLAS collaboration measured nuclear transparency of protons from short-range correlated (SRC) pairs [48] and arrived at the conclusion that transparency ratios Al/C, Fe/C and Pb/C are consistent with the absence of the correlation factors in the definition Eq. 4. Similar conclusion is also supported by theoretical computations based on the Glauber theory [49][50][51] and relativistic optical potentials [49,50,52].

III. NUWRO
NuWro [33] is a neutrino Monte Carlo generator that has been developed at the University of Wroc law since 2005. It covers neutrino energy range from ∼ 100 MeV to ∼ 100 GeV. It has been used in numerous experimental studies showing a good agreement with various cross section measurements.
For neutrino-nucleon scattering NuWro uses three interaction "modes": CCQE (or elastic for neutral current reaction), RES covering a region of invariant hadronic mass W < 1.6 GeV and DIS describing the region W ≥ 1.6 GeV. In the case of neutrino-nucleus scattering, impulse approximation is typically assumed with interactions occuring on bound and moving nucleons. A variety of options to describe such nucleons are available starting from global and local Fermi gas (LFG) models up to hole spectral function (SF) [53] with the lepton affecting FSI effects included [54], and density and momentum dependent potential [55]. The description of scattering off nuclear targets is completed with interactions mediated by meson exchange currents (MEC) and with coherent pion production (COH).
For the purpose of this study the most important NuWro ingredient is the intranuclear cascade model.

A. NuWro cascade model
The model describes in-medium propagation of pions and nucleons. The basic scheme is taken from the seminal papers by Metropolis et al. [29,30] but relevant physics ingredients are new. The MC sampling is based on the standard formula expressing a probability for a particle to propagate over a distance ∆x with no re-interaction: where λ = (ρσ) −1 is the mean free path calculated locally, expressed in terms of nuclear density ρ and an effective interaction cross section σ. In actual computations we distinguish proton/neutron densities and proton-proton/neutron cross sections. A step of ∆x = 0.2 fm was checked to be sufficient to grasp the structure of a nuclear density profile. The computations in this paper were done with the NuWro version 19.02 [56] that contains several improvements, with respect to NuWro 18.02, developed for the purpose of this study. This version uses a custom fit to the experimental free nucleon-nucleon cross sections, both elastic and inelastic, that aimed to improve the agreement with the current PDG dataset [28]. The fraction of single-pion production within inelastic interactions was adjusted to follow the fits of Ref. [57]. Moreover, the center-of-momentum (COM) frame angular distributions for the elastic scattering were updated using the parametrization of Ref. [58].
The in-medium modification of the elastic cross sections is modeled using the results of Pandharipande-Pieper study [59], where the two main effects come from Pauli blocking and in-medium nucleon effective mass. The Pauli blocking is included on the event-by-event basis, what is easy in MC simulations. We checked that NuWro cascade performance reproduces the results from Ref. [59] with a sufficient accuracy. For the inelastic nucleon-nucleon scattering we adopt a phenomenological in-medium cross section (σ * NN ) parametrization [60]: where η = 0.2, and ρ, ρ 0 are local and saturation nuclear densities, respectively. Following the experiences of Refs. [45,51,59], we included effects coming from nucleon-nucleon short-range correlations. In general, the density that enters the mean free path in Eq. 6 is assumed to be the one of nuclear matter at point r 2 , as experienced by a propagating nucleon known to be in a position r 1 . It can be expressed in terms of one-(ρ [1] A ) and two-body (ρ [2] A ) densities as ρ [1] eff ( r 2 | r 1 ) = ρ [2] A ( r 1 , r 2 ) ρ normalized to the number of remaining nucleons d 3 r 2 ρ [1] eff ( r 2 | r 1 ) = A − 1. We introduce correlation ef-fects through a following substitution ρ [1] eff,IPSM ( r 2 | r 1 ) = ρ where g(| r 21 |) is the nucleus-dependent pair distribution function [59] and N (| r 2 |) is introduced to keep the global normalization condition.
For the choice of g(| r 21 |), we rely on distributions of nucleon-nucleon distances obtained in ab initio computations for light nuclei, including carbon [61,62]. For heavier nuclei, including iron, we approximate g(| r 21 |) by the ab initio-calculated infinite nuclear matter distributions g inf (ρ avg , | r 21 |) of Ref. [59], evaluated at average nuclear density. In our computations we include effects coming from distinct shapes of g(| r 21 |) for nucleon pairs of the different isospin configurations, and accordingly we define effective densities following the scheme of Eq. 9.
The discussion of the influence of the aforementioned NuWro cascade model modifications on the results of this paper can be found in Sec. V B.

B. NuWro as a tool in transparency studies
Using MC event generators one can define the "MC transparency" as a fraction of events with no re-interactions at all. However, experimentally one cannot distinguish these events from those with "soft" FSI. Because of that, in order to make a comparison reliable, we go through all the steps of the experimental procedures to extract transparency.
NuWro keeps information about particles before and after FSI. This is exactly what is needed in the computation of nuclear transparency. Particles after FSI correspond to those that are detected in experiments. Particles before FSI correspond to theoretical computations in PWIA.
NuWro does not yet have complete electron scattering module, hence in this study we decided to use neutral current (NC) ν e interactions on bound proton targets. In doing so, we collect samples of NC events with exactly the same (electron mass is negligible) kinematics as in the transparency electron scattering experiments. In both electron and neutrino cases, a radial distribution of interaction points inside the nucleus is the same and given by the nucleus density profile.
A main challenge is to reproduce experimental situations with complete information on the kinematics and applied cuts. For every kinematical setup we ran a simulation with the neutrino beam energy equal to E e . Then, the energy E e and the in-plane angle θ e for the outgoing electron/neutrino were fixed around the central value of the spectrometer. Analogically, the momentum p p and the in-plane angle θ p for the knocked-out proton were fixed. As in all the experiments the electron and proton spectrometers are set in-plane, the out-of-plane angles were fixed to the same value φ e = φ p . The exclusive cross section formula is symmetric with respect to the rotation of the system, hence only the relative out-of-plane angle between the electron and proton plays a role, here set to φ e p = 0. All of the variables E e , θ e , p p , θ p , φ e p were fixed with the accuracy provided by the spectrometers' energetic or angular acceptance, namely ∆E e , ∆θ e , ∆p p , ∆θ p , ∆φ e p = ∆φ e + ∆φ p . On the top of those cuts, additional conditions were imposed using the information about the variables E m , | p m |. The beam energies and central spectrometers values for every setup can be found in Tab. I, while the acceptances and the cuts on missing variables are put into Tab. II.
In order to establish a proper framework for comparing nuclear transparency results with experiment, we tested different ways of modeling the initial nuclear state in NuWro. The SF-and LFG-based simulations were compared with exclusive properties of the knocked-out protons that were reported by the E91-013 experiment at JLAB. As can be seen in Fig. 1, SF in NuWro is able to accurately reproduce a measured shape of the angular distributions of knocked-out protons. The angular dependence of transparency reproduces a general flat shape  that can be seen in Fig. 2. of Ref. [34] with sufficient precision. On the other hand, the angular distributions of the measured yield of protons for the LFG-based simulations is peaked too strongly around the central value, what leads to the overestimation of the proton transparency. Due to its simplicity, LFG model fails to properly predict the exclusive kinematics, what is a prerequisite in reliable nuclear transparency studies. We conclude that only NuWro simulations that uses SF as the model for the initial nuclear state can give reliable results in comparison with exclusive electron scattering experiments. Unfortunately, such conclusion imposes a limitation on the nuclear targets that can be simulated, as the hole spectral functions are available only for a limited number of nuclei making impossible an estimate of the A-dependence of nuclear transparency in NuWro. The only targets that can be compared with the transparency measurements are 12 C and 56 Fe.

IV. RESULTS
In Fig. 2 the transparency results for carbon and iron are shown together with data points from several experiments. In experimental papers, transparency is discussed as a function of Q 2 but this variable can be translated into average proton momentum. The transparency curve has a characteristic shape reproduced in all theoretical computations: a saturation at larger values of proton momentum and a decline in the region of ∼ 1 GeV/c. Saturation can be explained by a roughly constant value of total free nucleon-nucleon cross sections for larger values of the incident nucleon momentum. A region of transparency decline comes from a complicated interplay of various nuclear effects and is the most difficult to model.
NuWro results for carbon reproduce the transparency data quite well. For application in neutrino physics, the most important region is that of low nucleon momentum, starting from ∼ 500 MeV/c, which is a detection threshold in experiments like T2K and MINERvA. We can see that the value of the first available experimental point, from Ref. [37], is reproduced well but then the decline of NuWro transparency is not steep enough. Predictions from our model are slightly above the data in the saturation region. For the iron target, the same shape of the transparency curve can be seen. Small differences, including data overshooting at low momenta, can be attributed to nucleon-nucleon correlation effects being introduced in more approximate way with respect to carbon, see the discussion in III A. In general, the agreement with the data points is good.

A. Model uncertainties
As discussed in the Introduction, nucleon FSI effects contribute to the background in all attempts to measure multinucleon ejection contribution to the inclusive cross section. Thus, it is not enough to have good qualitative agreement with the transparency data but also it is important to estimate uncertainty inherent in the NuWro nucleon FSI model. Our approach is to assess an uncertainty of nucleon mean free path as calculated inside NuWro. We tried to obtain a 1σ error bound by demanding that 2/3 of experimental points together with experimental errors are entirely inside the bound. In order to achieve that we multiply mean free path calculated within NuWro by a constant overall scaling factor. The result is shown in Fig. 3. The upper and lower dashed curves were obtained by scaling up and down central mean free paths by 30%. A discussion of possible sources of uncertainty in NuWro FSI model will be presented in Sec. V B.

B. Application I: Single transverse variables
As the first application, we discuss T2K measurements of single transverse variables [25]. These variables, in CC0π analyses, are defined in the following way where k corresponds to the outgoing lepton and index T denotes the transverse projection w.r.t. the beam direction. NuWro results are obtained with the SF model known to produce better results then LFG [25]. Due to many adjustments in the FSI model, the results obtained with NuWro 19.02 differ notably from the ones of older versions of NuWro. The most significant effect is an increase of normalization. This does not change much the values of χ 2 for the SF-based results, but makes the χ 2 values larger for the LFG-based ones. In Fig. 4, we show how much of uncertainty comes from possible NuWro FSI mismodeling. We see that applying a global decrease of the cascade mean free paths by 30% decreases the normalization of the results. We checked that this does not lead to a significant change of the calculated values of χ 2 . On the other hand, making mean free paths 30% larger causes a slight increase of the value of χ 2 . A general conclusion is that for single transverse variables, the error coming form FSI strength seems to be well under control.

C. Application II: Proton multiplicities
An observable that is potentially very sensitive to nucleon FSI effects is a distribution of number of reconstructed protons. The dominant contribution to the experimental signal comes from CCQE events. Thanks to FSIs, there is a fraction of events with more than one proton, otherwise such events would be impossible. Another impact of FSI is that due to rescattering some protons loose kinetic energy dropping down below the detection threshold, what results in events with no detected protons. In general, the FSI net effect is mostly a migration of events from N=1 to N=0.
In Fig. 5, we show a comparison of NuWro predictions with the T2K data from Ref. [25]. We see that the uncertainty coming from unknown strength of FSI is not large. Here, larger nucleon mean free paths results in increasing proton multiplicities. The data shape suggests that FSI strength should be set at the biggest value acceptable by  the nuclear transparency data. An impact of FSI on the distribution is smaller than expected. It is because the experimental proton acceptance cuts eliminate most of events with FSI.

A. MC transparency
In the MC approach, as mentioned in III B, a natural way of studying the nuclear transparency is to follow individual cascaded protons and check whether they interact at all. However, such definition might not catch particular aspects of the situation that are important from the experimental perspective and is expected to underestimate the final result. A refinement of the naive MC transparency definition is to take into account a finite angular acceptance of spectrometers, and therefore, allow protons to softly interact without a significant direction change, e.g., ∆θ p = 5 • . The value of 5 • approximately coincides with an angle that expands a solid angle in experimental acceptances, see Tab. II. In Fig. 6, the results for carbon, using different transparency definitions, are shown. One can see that while the "no interactions" definition is too strict, the softer definition "∆θ p = 5 • " works quite well, especially in the saturation region. However, it is unable to reproduce the first, most important for us, experimental point at p p 625 GeV/c. Knowing this behavior, the definition "∆θ p = 5 • " can be used for less exhausting cascade checks.

B. Cascade model ingredients
In order to understand sources of uncertainties in our model, we present an impact of its various ingredients on the predicted transparency. In Fig. 7, we show results obtained with: • a bare cascade model with free nucleon-nucleon cross sections, including projectile binding energy and target nucleon Fermi motion, • a model that on top of the bare model includes Pauli blocking (labeled "+ Pauli blocking"), • a model that additionally includes in-medium nucleon-nucleon cross section effects (labeled "+ in-medium effects"), • the full model that includes nucleon-nucleon correlation effects (labeled "+ correlations").
We can see that in the the region of proton momenta below 1 GeV/c all the theoretical ingredients of the model are relevant while for larger values of the momenta correlation effects play the most important role.
The basic observation for the model closest to the most naive "bare cascade" is that it underpredicts the experimentally measured transparency by a large amount. The proton momentum dependence of the corresponding curve reflects the momentum dependence of free protonproton/neutron cross sections. The effect of Pauli blocking is significant for lower momenta and slowly disappears with increasing proton momentum. Although it might not seem to be intuitive that the impact of Pauli blocking extends up to p p 2.5 GeV/c, for larger elastic scattering energies, the COM angular distributions get more and more forward/backward peaked what leads to kinematics that are prone to be Pauli blocked. As emphasised in Sec. III A, the in-medium nucleon-nucleon cross section modifications are modeled differently for elastic and inelastic interactions. This is reflected in nuclear transparency, where the modification of elastic cross sections increases with lowering proton momentum, while the inelastic part has a constant behaviour. The effect of the nuclear correlations strongly depends on average mean free paths in a given energy region. The free nucleon-nucleon cross section is higher in the saturation region, and therefore, the mean free paths are lower and the effect of correlations is more pronounced.
In general, all of the more sophisticated physical ingredients move the predicted transparency always in one direction, making it larger.

VI. CONCLUSIONS
NuWro 19.02 features an improved nucleon cascade model that, using proper comparison tools, is able to reproduce nuclear transparency data, in particular in the energy region which is crucial in the context of neutrino-nucleus scattering physics. The study presented in this paper shows that a cascade model should be enriched with many additional effects, such as, nucleon correlations, on top of a bare model with free nucleon-nucleon cross sections.
For the purpose of neutrino scattering physics, we estimated a 1σ error on the nucleon mean free paths in NuWro 19.02 with a result of 30%. This result was applied to recent T2K data that are potentially sensitive to nucleon FSI giving an uncertainty which suggests that FSI modeling in under control and that there must be other sources of data/MC disagreement that is still seen in NuWro results. There is a solid foundation for using these datasets in future research of multinucleon ejection contributions and especially a very uncertain hadronic part of its modeling [3].
All of the results obtained in this paper can be easily reproduced with any other neutrino MC generator, such as, NEUT or GENIE. The outcome of this work should allow to further reduce systematic errors in the modeling of neutrino-nucleus scattering, and moreover, open a door for future analyses of more involved exclusive interaction channels.