Neutrino-nucleus CC0$\pi$ cross-section tuning in GENIE v3

This article summarizes the state of the art of $\nu_\mu$ and $\bar{\nu}_\mu$ CC0$\pi$ cross-section measurements on carbon and argon and discusses the relevant nuclear models, parametrizations and uncertainties in GENIE v3. The CC0$\pi$ event topology is common in experiments at a few-GeV energy range. Although its main contribution comes from quasi-elastic interactions, this topology is still not well understood. The GENIE global analysis framework is exploited to analyze CC0$\pi$ datasets from MiniBooNE, T2K and MINERvA. A partial tune for each experiment is performed, providing a common base for the discussion of tensions between datasets. The results offer an improved description of nuclear CC0$\pi$ datasets as well as data-driven uncertainties for each experiment. This work is a step towards a GENIE global tune that improves our understanding of neutrino interactions on nuclei. It follows from earlier GENIE work on the analysis of neutrino scattering datasets on hydrogen and deuterium.


I. INTRODUCTION
A major experimental program aims to measure neutrino-nucleus interactions over the few-GeV region.MiniBooNE was the first neutrino experiment to provide a double-differential flux-integrated CC0π cross-section measurement with high statistics on carbon [1].Since then T2K [2], MicroBooNE [3] and MINERνA [4] have produced a large body of measurements on different nuclei, such as carbon or argon.However, a detailed quantitative understanding of neutrino-nucleus interactions is still missing.
In order to avoid biases in cross-section measurements due to theory assumptions, neutrino experiments focus on the study of specific topologies instead of interaction processes like Quasi-ELastic (QEL) scattering.The most dominant event topol-ogy below the 1 GeV region is CC0π, which is usually defined as an event with one muon and no pions in the final state.As a consequence of the nuclear medium, different interaction processes contribute to the CC0π measurement.Neutrino Charged-Current (CC) QEL interactions are the dominant contribution to this topology inside the few-GeV energy range.Two-particles-two-holes (2p2h) contributions have been shown to be crucial for the correct description of the data at these kinematics.Adding to the complication, the Shallow-Inelastic Scattering (SIS) process non-trivially intermixes with other underlying mechanisms; this is due in part to the fact that pions produced after a CC REsonance Scattering (RES) interaction can be absorbed due to Final-State Interactions (FSI).Moreover, Deep-Inelastic Scattering (DIS) can also contribute, with an interplay existing between the description of DIS at slightly higher energies and the treatment of the Non-Resonant Background (NRB) in the SIS region.In GENIE we refer to the NRB as SIS, see Ref. [5] for details. Figure 1 summarizes the ν µ 12 C CC interaction processes and topologies of interest at the few-GeV region as a function of the neutrino energy.In addition, the flux predictions used for the crosssection measurements of MiniBooNE, MicroBooNE, T2K ND280 and MINERνA are also provided.
The GENIE Collaboration is building a global analysis of the neutrino, charged-lepton and hadronscattering data.This comprehensive analysis of the world's lepton-nuclear scattering data is being constructed in a staged manner, with recent efforts focused initially on the analysis of neutrino scattering on hydrogen and deuterium for the purpose of tuning aspects of the GENIE framework associated with the free-nucleon cross section: namely the SIS region [5] as well as tuning of hadronic multiplicities relevant for neutrino-induced hadronization models [9].The present work extends this analysis campaign to a second stage: an explicit tune of nuclear model parameters to recent nuclear data.
This work is further necessitated by outstanding discrepancies between GENIE predictions and more recent datasets, which use heavy nuclei as targets.Several neutrino collaborations, such as Micro-BooNE and MINERνA, tried to address these discrepancies by tuning GENIE against the ν µ CC0π T2K and inclusive ν µ CC MINERνA datasets, respectively [10][11][12].All these tunes simulate 2p2h interactions with the Valencia model [13].In both cases, the results suggest an enhancement of the 2p2h cross section.These tunes are not available for wider use within GENIE, and in some cases, these were performed with obsolete GENIE versions which differ substantially from the latest one.
In this paper, we describe the GENIE analysis of the available ν µ and ν µ CC0π datasets from Mini-BooNE, T2K, MINERνA and MicroBooNE.The main goal is to provide improved simulations tuned to nuclear data and quantify the major sources of uncertainties in CC0π measurements.In order to do so, new degrees of freedom are developed within the GENIE Monte Carlo (MC) event generator in order to quantify the effect of variation away from the nominal models.Most of the new degrees of freedom can be used to tune other available Comprehensive Model Configurations (CMCs) in GENIE.In this analysis we focus on the 'retuning' of the G18 10a 02 11b tune against ν µ -12 C CC0π data from MiniBooNE, T2K and MINERνA.The G18 10a 02 11b was previously tuned against free-nucleon data [5].In this paper, we refer to G18 10a 02 11b as the nominal tune.
All predictions shown in this paper are calculated using the G18 10a 02 11b tune.G18 10a 02 11b uses the Valencia model to simulate QEL and 2p2h events in the nuclear medium, while FSIs are mod- The corresponding fraction of the total ν µ -12 C events arising from each of the 0π topologies.This plot assumes a momentum threshold for protons of 450 MeV/c while the GENIE predictions are obtained with the G18 10a 02 11b tune.(Bottom) Summary of ν µ (continuous lines) and ν µ (dashed lines) normalized flux distributions for T2K ND280 at JPARC [6], MiniBooNE and MicroBooNE with the Booster Neutrino Beam (BNB) [7], and MINERνA with the Neutrino at the Main Injector (NuMI) [8].The flux predictions for neutrino and antineutrino modes are refereed to as "Forward Horn Current (FHC)" and "Reverse Horn Current (RHC)," respectively.
The CC0π topology is usually defined as a CC event with no pions in the final state, regardless of the number of protons in the event.However, the CC0π topology definition is not universal as it varies between the different published measurements as a consequence of the different detection capabilities of each experiment.In some analyses, its definition is optimized to study more exclusive final states with a specific proton multiplicity.The following nomenclature is adopted to avoid confusion for the reader: analyses requiring one or more protons in the final state are referred to as CCNp0π, where N ≥ 1.If the analysis requires exactly zero or one proton in the final state, N is the replaced by the corresponding number, i.e.CC0p0π or CC1p0π, respectively, for events with either no visible protons or precisely one.In some cases, the topology definition requires at least two protons in the final state.This is denoted as CC2p0π.We note that, in this case, CC2p0π events include the very small probability to have N> 2 final-state protons -a scenario which is challenging to isolate experimentally.When there is no requirement on the proton multiplicity, the topology is refereed to as CC0π.Fig. 1 presents the fraction of ν µ CC events as a function of the neutrino energy for different CC topologies.In this particular plot, the CC0π topology contribution is broken down into more exclusive topologies depending on the proton multiplicity.It can be concluded that CC0π events dominate the event rate for E ν < 1.5 GeV.At higher energies, the contribution from events with pions in the final state (CC other) dominates.
Tab. II lists the available CC0π and CCNp0π cross section measurements to date.The table summarizes the information of interest for the evaluation of the GENIE predictions: the target type, neutrino flux mean energy and event topology definition.The neutrino flux spectrum associated with each experiment is provided in Fig. 1 (bottom) [6][7][8].We use the same neutrino flux prediction for MiniBooNE and MicroBooNE.
The kinematic quantity column in Tab.II lists the kinematic quantities used to extract the crosssection measurements.The definition of each kinematic quantity is given in Appendix A. Some of the available measurements are double-differential or triple-differential ones.This is indicated by a comma-separated list for the kinematic quantities used in the corresponding analysis.In addition, the year of the data release and the number of bins (N Bins ) for each dataset are specified.The details on the analysis requirements for MiniBooNE, T2K ND280, MINERνA and MicroBooNE datasets as well as comparisons of the G18 10 02 11b predictions to the data are presented in Appendix B. The main observations from Appendix B are summarized in Sec.II B. For completeness, Tab.II includes measurements from SciBooNE and NOMAD which are not discussed further in this paper as their analysis strategy is limited with respect to the other measurements discussed in this work.
There is a now large body of CC0π data in the literature.This work focuses on the tuning of double differential flux-integrated CC0π and CC0p0π cross-section measurements on carbon from Mini-BooNE, T2K ND280 and MINERνA.This is sufficient for an initial study.Additional single-and triple-differential CCNp0π datasets are not considered in the first iteration of this work; these will be included in future iterations.However, some comparisons are given in this paper.
It is important to note the differences between the measurements considered in this work.These are highlighted in Appendix B. A significant difference is the treatment of uncertainties.Bin-to-bin correlation are not reported by MiniBooNE for many of their cross section measurements (including the data used in this work).In addition, flux uncertainty is given as a single normalization uncertainty of 10.7% and 17.2% for neutrino and antineutrino measurements on carbon respectively.These treatments involve approximations from modern treatments and do not fully incorporate MiniBooNE uncertainties [39].Despite the statistical limitations of this measurement, MiniBooNE's datasets are included in the analysis for a complete study of CC0π datasets on carbon.In this work, an additional normalization systematic uncertainty is added to ac-count for the missing flux correlation, as suggested by Ref. [25].

B. Dataset overview and initial considerations
The need for a tuning exercise for GENIE is clear.A few comparisons of G18 10a 02 11b against the available nuclear data are shown here.The remaining plots are in Appendix A.
We observe in Figs. 2 to 4 that CC0π and CC0p0π datasets are under-predicted, whilst the CCNp0π datasets are in quite good agreement with the G18 10a 02 11b predictions.As a consequence, a coherent global tune of CC0π and CCNp0π datasets is not possible.Hence, the analysis is mostly focused on CC0π and CC0p0π datasets.Nonetheless, understanding the tension is essential for future tuning efforts.This tension is further explored in this paper.
MiniBooNE CC0π (Fig. 2) and T2K ND280 CC0p0π data are both under-predicted at muon backward angles, where the contribution to the prediction is mostly from CCQEL events.At forward angles, where the contribution from non-CCQEL events is significant, the data are also under-predicted.The disagreement with MINERνA CC0π data are most significant in the region where 2p2h events dominate, 0.15 < p T < 0.7 GeV, see Fig. 3. Single-Transverse Kinematic Imbalance (STKI) variables [40] bring in new sensitivities and comparisons against MINERνA data are shown in Fig. 4. Non-QEL events and FSI contributions dominate the region of high δp T and δα T .These contributions are essential to describe the data.
The G18 10a 02 11b predictions as a function of the leading proton momentum show a dependency of 2p2h with W : at high proton momentum, 2p2h events with W > M ∆ = 1232 MeV/c 2 dominate, whilst the opposite is true at low momentum.This is highlighted in Fig. 4a.2p2h events contributing to the T2K ND280 CC0p0π sample (Fig. 5) have W < W Dip = 1120 MeV/c 2 .Higher multiplicity samples have a significant contribution from 2p2h events with W > W Dip .The contribution from 2p2h events with W < M N = 938 MeV/c 2 is negligible for all the analyses discussed in this paper.
For further comparisons to data, see Appendix B.

III. DISCUSSION OF CC0π MODEL IMPLEMENTATION IN GENIE
This section describes the parameters available to most directly influence CC0π predictions within G18 10a 02 11b.The parameters selected for this TABLE II: Summary of CC0π analyses of ν µ and ν µ interactions on nuclei.For each analysis, information on the neutrino flux mean energy, target type and event topology is provided.The kinematic quantity column specifies the list of kinematic quantities used in the cross-section measurement.Integrated cross-section measurements are denoted with a "−".All kinematic quantities are defined in Appendix.A.
The last column specifies whether the dataset is considered in the analysis.3: MINERνA ν µ CC0π double differential flux-averaged cross-section as a function of the muon longitudinal momentum, p , and transverse momentum, p T [28].The corresponding slices on p T are compared against the G18 10a 02 11b tune.The GENIE prediction is divided into different interaction modes.
analysis are optimized for the G18 10a 02 11a tune.The complete list of parameters is shown in Tab.III.The parameter ranges of interest used for the Professor parametrization are also provided.These can be grouped into five categories: CCQEL, CCSIS, CC2p2h, FSI or nuclear model parameters.
Not all the parameters from Tab. III have been included in the analysis presented in this paper.Only the parameters included in the final tune are described in this section.Other parameters of interest to tune CC0π data that have been excluded from this analysis are described in Appendix.C. The reasons for excluding these parameters are summarized in Appendix C 3.
Most of these parameters can be applied to other CMCs [5].We strive to have as many common, model-independent parameters to allow for systematic comparison between CMCs, but this is not always possible.An extension of this work to other CMC will be a subject of a future paper.FIG.4: MINERνA ν µ CCNp0π differential flux-averaged cross-section as a function of the leading-proton momentum, p p (a), and the STKI variable [29,30] δp T (b).The data are compared against the G18 10a 02 11b tune.The GENIE prediction is divided into interaction modes.

A. Charged-current quasi-elastic implementation
The QEL cross section at the free-nucleon level is parametrized with the QEL axial mass, M QEL A , and a QEL scaling factor, S QEL .Both parameters are common in the simulation of neutrino interactions on free nucleons and nuclei.M QEL A appears as the main degree of freedom in the widely-used dipole parametrization of the QEL form factor.We point out that more elaborate CMCs based on the z-expansion model [42] are now available in GENIE.In this work, preference is given to tune M QEL A as hydrogen and deuterium data provide informative priors to help constrain this parameter [5].The range for the parameters considered in the Professor parametrization but not used in the final tune is not reported.In such cases, the parameters are fixed to the corresponding nominal values (in parenthesis) in the final analysis, described in Sec.IV.The last column specifies whether the parameter is considered in the final analysis.

Parameter Nominal Value Range In Final Tune
The QEL cross section is affected by the dynamics of the nuclear medium.We include long-range nucleon-nucleon correlations in our calculations with the Random-Phase Approximation (RPA) correction [15].The main effect of the RPA correction is a suppression of the QEL cross section at low Q 2 .This correction is well supported by data and theory, but models differ in predicting its exact strength.This uncertainty is incorporated in GENIE with two parameters: one to scale the nominal QEL crosssection prediction with RPA corrections, ω RPA , and the other one to scale the QEL cross section without RPA corrections, ω No RPA .The total QEL cross section is calculated as a linear combination of the cross-section with and without RPA corrections: This parametrization can be used to scale the QEL cross section when ω RPA +ω No RPA = 1.If ω No RPA = 0, ω RPA has the exact same effect as S QEL .Therefore, S QEL is not included in the tune.One benefit of this approach is that possible scaling factors on the RPA parametrization do not alter the agreement with free-nucleon data.In addition, it reduces the analysis computing time.In Fig. 6, the CC QEL cross section as a function of the neutrino energy is shown for different combinations of ω RPA and ω No RPA .
Choosing each parameter range of interest is crucial for the correct evaluation of the post-fit uncertainties.In some cases, such as for ω No RPA , we sample negative values to allow the best-fit result to be at its physical limit of 0. In the case of the RPA parametrization, we impose the additional condition that 0.4 < ω RPA + ω No RPA < 1.6 in the sampling on the phase space so that σ QEL > 0. Fig. 7 shows the distribution of sampled parameter values for ω RPA and ω No RPA .Notice that the two limit cases are at the centre of the phase space.
It is desirable to apply priors to ω RPA and ω No RPA , as effectively, parameter combinations for which S RPA ≡ ω RPA + ω No RPA = 1 act as a scaling of the QEL cross section.Hydrogen and deuterium QEL cross-section measurements are compatible with S RPA = 1.However, nuclear effects might introduce an uncertainty in the scaling.A possible way to include this information is to consider uncorrelated priors on the sum, S RPA , and the difference, ∆ RPA ≡ ω RPA − ω No RPA , with σ S and σ ∆ being the variance associated with the priors on S RPA and ∆ RPA , respectively.In terms of ω RPA and ω No RPA , this approach includes a correlation between these parameters: the RPA correction, which we aim to constrain from data.In order to avoid strong constraints on ∆ RPA , σ ∆ = 5.Alternative parameterizations of the RPA correction uncertainty are available in the literature.Theory driven uncertainties specific for the Nieves model are estimated in Ref. [43].Alternatively, T2K [12] and MicroBooNE [10] use empirical parameterizations to characterize the uncertainty on the RPA correction.For the first iteration of this work, we opted for a simple parameterization with two parameters to reduce the computational complexity of the tune.
Our method is similar to the RPA parametrization used in the latest theory-driven MicroBooNE tune [10].The MicroBooNE Collaboration employed the GENIE ReWeight package to parametrize the RPA effect as a linear combination from the QEL cross section with the RPA correction to the QEL cross section without RPA using a single parameter limited to [0,1].We refer to this tune as µBooNE tune.Both approaches are equivalent when S RPA = 1.

B. Charged-current multi-nucleon implementation
The tuning of 2p2h models takes a central role in this work.As discussed in Sec.II, untuned GE-NIE CC0π G18 10a 02 11b predictions underesti-mate the data in regions where 2p2h events contribute.
Previous tuning attempts by other neutrino collaborations indicate a preference for a higher 2p2h cross section.The simplest approach to enhance 2p2h is to use a global scaling factor.We refer to this parameter as S 2p2h .MINERνA opted for an empirical approach where they add an extra Gaussian contribution to enhance 2p2h interactions in q 0 and q 3 .This is tuned to MINERνA CC inclusive data.This tune is known as MnvGENIE v1 tune [44,45].The µBooNE tune incorporates the 2p2h cross-section uncertainty with a linear extrapolation between the GENIE 2p2h Empirical and Valencia model to account for possible shape differences.In addition, S 2p2h is also considered.
Different GENIE 2p2h models predict a slightly different strength and shape for the 2p2h cross section [46].These differences motivated the development of a new parametrization that is able to modify the strength as well as the shape of the cross section in the q 0 -q 3 space.This is accomplished by scaling the 2p2h differential cross section a function of W : is the scaling function and d 2 σ 2p2h /dq 0 dq 3 the nominal double-differential cross section calculation.The scaling function, S(W ), depends linearly on W .In this work, the scaling function is optimized for the Valencia model which has two characteristic peaks in the q 0 -q 3 space, as it can be seen in Fig. 8.The peaks are situated at W = M N and W = M ∆ .The dip between the two peaks is at W Dip .This is implemented by imposing the following boundary conditions: The S 2p2h parameters are referred to in this work as 2p2h scaling parameters.The limits of the 2p2h phase space are defined by W P L,min and W P L,max .The upper limit is obtained by simply imposing Q 2 = 0.The lower limit is parametrized as a function of q 0 and q 3 .This is an empirical approach that breaks the intrinsic microscopic model and it is only used to explore a possible dependency of the 2p2h cross section on W .In all GENIE v3 CMCs, the 2p2h scaling parameters are set to 1.
Only three out of the five 2p2h scaling parameters are included in the tune: S 2p2h

S 2p2h
P L,max .Events with W < M N are negligible for all CC0π measurements of interest for this work, hence, S 2p2h P L,min is not included in the tune.In addition, S 2p2h Dip is also not included as the region between N and ∆ peaks is too narrow in W and the data cannot be sensitive to such parameter.In order to facilitate readability, the S 2p2h P L,max parameter is redefined as S 2p2h P L .In the particular case of T2K ND280, variations of S 2p2h ∆ and S 2p2h P L do not affect the CC0p0π predictions.This is highlighted in Fig 5, where only events with W < W Dip contribute to the 2p2h cross section prediction with no protons above the detection threshold.Therefore, these parameters are not included when tuning against T2K ND280 CC0p0π data.
The dependency of the scaling function with W for a particular set of parameters is shown in Fig. 9 (top).This particular example enhances (suppresses) the 2p2h cross-section peak in the The effect on the predictions of interest for this paper depends on the neutrino energy, proton multiplicity and proton momenta, as discussed in Sec.II.

C. Charged-current shallow-inelastic implementation
SIS events also contribute to the CC0π signal as pions can be absorbed by the nuclear medium.Therefore, SIS mismodeling impacts the interpretation of the measurements and must be considered in the tune.The parameters available in GENIE to These parameters have been previously tuned against hydrogen and deuterium data [5].This is a lesser issue for MiniBooNE and T2K ND280 CC0π data, more significant for the higher energy MINERνA data.Nuclear effects in SIS and DIS remain imperfectly understood and are therefore an important open area, both for the current study as well as future neutrino-nuclear interaction research.Nuclear-medium effects were studied for pion and electron beams [47] and found to be moderately significant.
The S RES parameter is the only SIS parameter included in the CC0π tune.NRB parameters are not included: single pion NRB parameters have a small impact on the CC0π predictions.In addition, higher multiplicity SIS/DIS contributions are negligible.In later instances, we refer to SIS/DIS contributions as DIS.

D. Discussion
The choice of tuning parameters is always complicated as these must sample the core physics dependencies with minimal correlation.In the µBooNE tune [10], only four parameters were used with an emphasis on RPA and 2p2h modeling.Although the RPA and 2p2h components are still important here, additional parameters are used to examine these aspects more fully.Since this exercise uses a broader range of neutrino energy, more parameters are needed to account for pion production.However, this contribution is small at neutrino energies ∼1 GeV and, although larger for MINERνA, we find that a single normalization parameter is sufficient to describe the CC0π data included in this study.Additional potential parameters are introduced here and discussed more fully in Appendix C.
Similarly, as we discuss in more detail in Sec.IV D, there can in principle be nonneglible correlations among the parameters associated with the nuclear models tuned in this current study and those associated with single-nucleon degrees of freedom as explored in Ref. [5].A possible approach is to fit both sets of parameters comprehensively.In the present work we concentrate on a more targeted partial tune of these nuclear parameters in order to map their relationship to the corresponding data taken on nuclear targets.This is further justified by the fact that the leading sensitivity to the nuclear parameters is provided by the nuclear data fitted here.Ultimately, however, performing nuclear tunes with frozen single-nucleon parameters can be expected to influence the resulting nuclear tune through the correlations mentioned above; systematically disentangling these correlations will require a more global comprehensive tune involving simultaneous fits of both types of data, an undertaking which will be informed by the present study with respect to model priors, methodology, and an understanding of compatibility of nuclear data sets explored in partial tunes as discussed below.
In terms of specific nuclear model choices, the nuclear binding energy is a complicated topic that we quantify through a single number in existing GE-NIE models which is independent of the momentum distribution.This is adequate for inclusive electron scattering [48].In more sophisticated treatments of semi-exclusive data, the binding energy and the missing momentum are interrelated via spectral functions [49].Any binding energy parameters are found to be highly correlated with the other parameters chosen for tuning.We choose to leave this out of the tuning procedure and show the effect of these parameters in Appendix C.
Similarly, FSI has been studied for many years and there are many disagreements about the proper treatment [50].Although this is a natural aspect of a full tune, the CC0π data are not particularly sensitive to this aspect; FSI parameters are most sensitive to CCNp0π data.A global analysis of CC0π and CCNp0π is out of the scope of this analysis and it is left for future iterations of this work.We show some interesting CCNp0π sensitivities in Appendix C.

IV. TUNING PROCEDURE
This section summarizes the tuning procedure for the analysis.The main goal is to tune GENIE against MiniBooNE, T2K ND280 and MINERνA CC0π data.

A. Construction of the GENIE prediction
In order to build the prediction associated with each dataset specified in Sec.II, we generate ν µ and ν µ CC events for the experiment target using the neutrino fluxes from Fig. 1.In this work, the events are generated with the G18 10a 02 11b tune [5].
To compute the prediction associated with the ith dataset, we generate N TOT i events.Events that do not satisfy the corresponding selection criteria specified in Sec.II are rejected.The number of accepted events in the jth bin is N i j (θ).θ is the vector of tunable parameters specified in Tab.III.
We build the corresponding n-differential fluxintegrated cross section prediction for a given set of observables, O, as where Φ i is the integrated flux for the ith dataset, ∆O i k corresponds to the jth n-dimensional bin volume for the quantities used in the differential cross section calculation, and dφ/dE ν is the expected flux at a given neutrino energy.For a target mix, the averaged cross section is evaluated by summing over the nucleus type in the target mix, T i .The ratio of a specific nucleus type with respect to the total nuclei is R Ti and σ Ti (E ν ) is the total cross section for a given nucleus type.

B. Avoiding the Peele's Pertinent Puzzle
The bin-to-bin covariance matrix provided by each experiment is considered in the evaluation of the χ 2 .The T2K ND280, MicroBooNE and MINERνA datasets have highly correlated bin-to-bin covariance matrices.Previous attempts to fit neutrino-nucleus data using the full covariance matrices result in a significant reduction of the cross section [10,51,52].These results are not surprising in highly correlated bins (ρ > 60%) in the Gaussian approximation [53].This is known as Peele's Pertinent Puzzle (PPP) [53,54].
To avoid PPP, we change our variables in order to reduce the correlation for the ith dataset using the following prescription: D i j corresponds to the ith dataset mean value at the jth bin.The jth and kth indices run over the number of bins associated with the ith dataset.This is known as Norm-Shape (NS) transformation.After the NS transformation, the integral is moved into the first bin of the ith dataset, whilst the rest describes the shape distribution.This transformation is applied to both data and predictions.
The bin-to-bin covariance associated with the ith dataset, Σ D (D) i jk , transforms as follows: where After the NS transformation the relative uncertainties are constant when the normalization changes.
The same transformation is applied to the prediction mean values and covariance.Before the NS transformation, the prediction covariance only has diagonal elements.This is not true after the NS transformation.However, the off-diagonal elements  on the prediction covariance are small and are neglected in this work.The prediction central values and errors after the NS transformation are denoted as Y i j (θ) and δY i j (θ) respectively.

C. Professor parametrization
Given that performing a multi-parameter bruteforce scan is not feasible, we use Professor [55] to parametrize the behavior of our predicted cross section and error in each bin in the NS space.We refer to this quantities as Ỹ i j (θ) and δ Ỹ i j (θ).In this particular tune, we opted for a fourth-order parametrization.This work, where originally eleven parameters were included in the analysis, requires a total of 2k event generations with θ sampled across the ranges specified in Tab.III.The accuracy of the parametrization is shown in Fig. 10.The distribution is centered at zero with a standard deviation of 0.05.This distribution is similar to previous GE-NIE tunes [5].This parametrization is used for the estimation of the best-fit values by minimizing the χ 2 .
The accuracy of the parameterization can be improved by increasing the order of the polynomial.However, an increase in the order of the polynomial is computationally expensive.For instance, a fifthorder polynomial requires 6k generations.A fourthorder polynomial is enough to describe the MC response in this work.

D. Discussion of data-driven priors
The basic structure of this tune is based on the model of separate nucleon and nucleus efforts.Although the emphasis here is on neutrino-nucleus parameters, some of the parameters of interest were already tuned to neutrino-nucleon data [5].Particularly, the G18 10a 02 11b tune with hydrogen and deuterium data provided with data-driven constraints for M QEL A and S RES [5].These parameters are crucial for the description of free-nucleon data and are strongly correlated with other aspects of the nuclear tune.This correlation was observed in the µBooNE tune, leading to best-fit results with M QE A = 1.18 ± 0.08 GeV/c 2 [10].The effect of varying M QEL A on the MINERνA ν µ CC0π prediction is shown in Fig. 11.In this work, we chose to constrain M QEL A and S RES using data driven priors from Ref. [5].The information on the parameter priors central values as well as the correlation between the two parameters out of the free-nucleon tune is included in the χ 2 minimization.The complete information on the priors is provided in Tab.IV.In this analysis we also include priors on ω RPA and ω No RPA , as discussed in Sec.III A.

E. Evaluation of the χ 2
The complete form for our χ 2 is: being i the index that runs over the N datasets considered in the fit.∆ Ỹ i j is the difference between the NS parametrization prediction and the ith dataset at the jth bin, ∆ Ỹ i j (θ) The ω ij is the weight applied to jth bin from the ith dataset.In this work, weights are used to include or exclude data from the analysis.In other words, they are either 1 or 0. The prediction errors, δ Ỹ i j (θ), are added in quadrature to Σ N S .The second term takes care of correlated priors in our fit.θ 0 and Σ θ are the central values vector and the covariance matrix of the priors for the parameters of interest.The details on the priors applied in this analysis are described in Sec.IV D.

V. TUNING RESULTS
We adopt the following naming scheme to characterise each of the partial GENIE tunes presented in this work: Here G is a capital letter that stands for GENIE, highlighting the authorship of the tunes.
xx is a number assigned to each experiment, i.e., MiniBooNE (10), T2K ND280 (20) or MINERνA (30).When using antineutrino datasets, xx is increased by one unit.For CCNp0π datasets, xx is increased by five units.Note that this is different from the standard naming scheme used for the tunes released through the GENIE platform.The standard naming convention from Ref. [5] will be used if one or more of the tunes produced in this work or future iterations is prepared for release in GENIE.
In total, six partial tunes are performed: three tunes on neutrino CC0π data, two tunes using antineutrino CC0π data and one tune using ν µ CCNp0π data.The tunes on CC0π data aim to explore avenues for improving the agreement between GENIE and data, consolidate the main elements of the GENIE CC0π tuning methodology and provide a common ground for the discussion of tensions.The tune on CCNp0π data aims to highlight tensions between CC0π and CCNp0π datasets.All of the tunes presented in this work consider carbon datasets only.Joint fits to all available data will be performed at a future iteration of this work, aiming to produce the tunes that will be publicly released through the GENIE platform.
In all CC0π tunes, the analyses are carried out using double-differential CC0π data as a function of muon kinematics.Preference is given to datasets that do not require a minimum number of protons above detection threshold in the final state.Whenever CC0π datasets are not available for a particular experiment, the tune is performed using CC0p0π datasets instead.
G18 10a 02 11b is the starting point for all these tunes and provides the nominal predictions.The corresponding names assigned to each tune prepared for the purposes of this paper are the following: G10a Tune : GENIE tune to MiniBooNE ν µ CC0π data [25].
Other measurements, including MicroBooNE ones, are used for comparisons only.Each partial tune is performed following the recipe described in Sec.IV.

A. Discussion of partial CC0π tune results
Each tune's best-fit parameter values and the χ 2 calculated with the Professor parametrization at the best-fit point are summarized in Tab.V.The nominal and best-fit predictions are shown in Figures 12 to 18. Tab.VI provides the χ 2 values computed with each tune's GENIE prediction and corresponding dataset.In this case, the χ 2 values are calculated with the NS transformation with the GENIE predictions.Notice that the χ 2 values from Tab. VI are different to the ones provided in Tab.V.This is a consequence of the Professor parametrization not being exact.
It is observed that the description of the data after the tune improved substantially.For instance, the agreement with MINERνA ν µ CC0π before the tune is χ 2 Nominal = 626/144 DoF.After the tune, χ 2 Nominal = 151/144 DoF.This is mainly a consequence of an improvement in the overall normalization for each partial tune.
All carbon tunes show similar trends; whilst the tunes are in good agreement with the priors on M QEL A and S RES , the other parameters differ from the nominal parameter values.There is also a clear preference for QEL with RPA corrections.In addition, the tunes prefer a higher QEL, i.e. ω RPA + ω RPA > 1, and 2p2h cross section.Finally, the different tunes suggest an underlying energy dependence on the 2p2h cross section strength and shape: the G10a, G11a and G20a tunes enhance (suppress) the Valencia 2p2h cross section at the nucleon (∆) region.Alternatively, the G30a and G31a tunes enhance the cross section at the nucleon and ∆ region, with S 2p2h ∆ > S 2p2h N .A hint of an underlying energy dependence was also observed in Ref. [56].
The enhancement of the QEL cross section is crucial for the description of MiniBooNE CC0π data at cos θ µ < 0. Particularly, the G10a and G11a tunes suggest an increase of the QEL cross section of about 20%.Similar QEL scalings have been observed by MicroBooNE [10] and recent Lattice Quantum-ChromoDynamics (QCD) calculations [57].The increase (decrease) of the S 2p2h N (S 2p2h ∆ and S 2p2h P L ) is also crucial to correctly describe MiniBooNE ν µ and ν µ CC0π data.
The G20a tune also offers a better description of T2K ND280 CC0p0π data.This tune suggests a scaling of S 2p2h N = 1.7 ± 0.3, compatible with the results presented by MicroBooNE [10].In this particular case, the scaling of QEL is around 10%.The post-fit value of ω No RPA , although negative, is compatible with zero.This result is physical as ω RPA + ω No RPA > 0, hence the total cross-section is positive.This scenario can be avoided by reducing the ω No RPA range to [0,1.5].However, the param-    [25].The comparisons are restricted to the 0.2 < T µ < 1.0 GeV phase space.The predictions are computed using the parameters specified in Tab.V.The total χ 2 associated with this dataset before and after the tune are reported in Tab.VI.  [26].The comparisons are restricted to the 0.2 < T µ < 1.0 GeV phase space.The predictions are computed using the parameters specified in Tab.V.The total χ 2 associated with this dataset before and after the tune are reported in Tab.VI. differential data [27].The comparisons are restricted to the −1.0 < cos θ µ < 0.94 phase space.The predictions are computed using the parameters specified in Tab.V.The total χ 2 associated with this dataset before and after the tune are reported in Tab.VI.  [26].The comparisons are restricted to the 0 < p T < 1.5 GeV/c phase space.The predictions are computed using the parameters specified in Tab.V.The total χ 2 associated with this dataset before (after) the tune reported in Tab.VI.G30a tunes against MINERνA ν µ CC0p0π single-differential data [28].The predictions are computed using the parameters specified in Tab.V. eter range is not reduced further to allow a valid estimation of the error on ω No RPA .
Before the tune, the G18 10a 02 11b prediction under-predicted MINERνA CC0π data in the phasespace regions where 2p2h events dominate (0.15 < p T < 0.7 GeV/c).The results suggest that an enhancement of QEL, as well as 2p2h, improves the agreement with data.In fact, the G30a and G31a tunes provide with a better description of ν µ CC0π and ν µ CC0p0π data respectively.The improvement in the normalization of the cross section is reflected in the post-fit χ 2 values from Tab. VI.The same is true for the cross section as a function of the reconstructed neutrino energy, Fig. 18, and singledifferential cross section data, Figs. 17 and 19.Both tunes over-predict the data at very low Q 2 QEL .

B. Tension between CC0π partial tunes
Tensions between datasets can be explored by comparing the different tunes.Figure 20 compares the G10a, G20a and G30a predictions against Mini-BooNE ν µ CC0π data.Even though the normalization of the three tunes is similar, differences in the predicted cross-section shape exist.The G10a tune is the only one out of the three that successfully describes the shape of the data, as it can be seen in Fig. 20 (left).The other tunes underestimate the cross-section at backward muon angles.In addition, the G30a Tune over-predicts the cross section at forward angles as a consequence of the enhancement of the 2p2h cross section at the ∆-region.All tunes G31a tunes against MINERνA ν µ CC0p0π single-differential data [26].The predictions are computed using the parameters specified in Tab.V. overestimate the cross section at forward muon angles and low muon kinetic energies, as demonstrated in Fig. 20 (right).
The G31a tune is in clear tension with all the rest, including partial tunes performed with MINERνA neutrino data.In comparison with the rest of the tunes, the G31a tune prefers higher QEL and 2p2h cross sections.This leads to the over-prediction of all the other datasets.The comparison of G30a and G31a against MINERνA and MiniBooNE ν µ CC0π data are shown in Fig. 21.The effect of this tension on the χ 2 is reported in Tab.VI.
The tension between the G31a tune and the rest can have different origins.A possibility is that the model does not fully characterize the difference between neutrino and anti-neutrino fluxes.This is investigated comparing the G20a tune to the T2K WAGASCI antineutrino data.Both the T2K WA-GASCI and T2K ND280 analysis explore the ν µ and G31a (b) against MINERνA ν µ and ν µ CC0p0π integrated cross section data [26,28].The predictions are computed using the parameters specified in Tab.V.
CC0p0π topology, but these are exposed to different neutrino fluxes (see Appendix.B 2).The impact of the G20a tune to these predictions is shown in Fig. 22.It is observed that the G20a tune has little impact on the T2K WAGASCI predictions.This indicates than an additional neutrino/antineutrino modeling uncertainty should be considered in a global tune of neutrino and antineutrino data.Another possible source of uncertainty is the different topology definition for MINERνA's ν µ dataset, with requires no visible protons above T p = 120 MeV for the antineutrino sample.The proton multiplicity uncertainty is explored further in Sec.V C.

C. Tensions between νµ CC0π and νµ CCNp0π datasets
T2K ND280, MINERνA and MicroBooNE are the only experiments that released cross-section measurements for different proton multiplicities.As discussed in Sec.II B, CC0π and CC0p0π datasets are under-predicted, whilst the CCNp0π datasets are slightly over-predicted by the nominal G18 10a 02 11b prediction.This modeling limitation is also observed in Ref. [27,56,58].
After the partial tunes using ν µ CC0π data, the agreement with CCNp0π data deteriorates.This is highlighted in Tab.VII, which summarizes the postfit χ 2 values associated with CCNp0π datasets.In all cases, the χ 2 computed with each partial tune prediction increases with respect to the χ 2 computed with the nominal G18 10a 02 11b tune.All G10a, G20a and G30a tunes overpredict ν µ CCNp0π data.Figure 23 shows a comparison of the partial tune predictions against different single-differential CCNp0π cross-section measurements from MINERνA. Figure 23 shows that none of the available tunes can describe the peak at low δp T and that all partial tunes overestimate the cross section at low proton momentum and forward angles.The same observations are made when comparing the tunes against T2K ND280 and MicroBooNE CCNp0π data, see Figs. 24 and 25, respectively.
To further explore this tension, an additional tune is performed using the MINERνA ν µ CCNp0π dataset as a function of the proton angle.Following the naming scheme described at the beginning of Sec.V, this tune is referred to as G35a.The best-fit results are listed in Tab.V.
The G35a tune suggests a significant reduction of the QEL cross section.In addition, the tune suppresses the Valencia cross-section peak prediction at W = M N and shifts the ∆ peak to W > M ∆ .This result contradicts the rest of the partial tunes presented in this article, reinforcing the fact that there is a strong tension between CC0π and CCNp0π datasets.The summary of χ 2 is reported in Tab.VI and Tab.VII.
An important observation is that the G35a tune also improves the agreement with Micro-BooNE CCNp0π data, suggesting that a possible A-dependency on the parameters does not play an important role.
The tension between CC0π and CCNp0π datasets needs to be resolved before attempting a global tune of CC0π data that can describe all data available to date.Some modelling aspects that may contribute to this tension are investigated in Sec.VI.  [25].The predictions are computed using the parameters specified in Tab.V. TABLE VII: Summary of χ 2 values associated the CCNp0π datasets specified in each row.The χ 2 values are calculated using the NS method for seven different tunes: G18 10a 02 11b, G10a, G11a, G20a, G30a and G31a.The values highlighted in bold correspond to the best-fit χ 2 G35a for the partial tune using the specified dataset.This section offers an insight into possible modeling implementations that may contribute to the tension between CC0π and CCNp0π datasets and explores avenues of accommodating both within future joint tunes.None of the uncertainties described in this section has a big impact on CC0π datasets.and G31a tunes against (a) MINERνA [28] and (b) MiniBooNE [25] ν µ CC0p0π double-differential cross-section data.The predictions are computed using the parameters specified in Tab.V.

Nuclear model variations
The nuclear model determines the momentum and binding energy of the hit nucleon.In GENIE, three nuclear models are available: Relativistic Fermi Gas (RFG), Local Fermi Gas (LFG) and Correlated Fermi Gas (CFG) [14].By default, G18 10a 02 11b uses the LFG.
The nuclear model choice affects the CCNp0π predictions.Figure 26 shows the impact of the underlying nuclear model against CCNp0π singledifferential cross-section measurements as a function of δp T .Differences between the models are significant for the cross-section peak prediction at low p T .The RFG model is the only one out of the three that predicts the MINERνA data below the maximum.However, it still over-predicts the cross section at the peak.Alternatively, the CFG model successfully predicts the peak normalization.This is reflected in the χ 2 CFG , reported in Tab.VIII.
The main characteristic of the RFG and the CFG implementations in GENIE is that nucleons can have a momentum above the Fermi momentum in its ground state.This tail in the momentum distribution is a consequence of nucleon correlations in the nuclear medium.As a consequence of including those effects in the nuclear model, the description of the tail of the δp T distribution improves.This study suggests that using a more elaborate nuclear model is key to describe CCNp0π measurements.
The differences between the three GENIE nuclear model predictions are not enough to explain the discrepancy between CC0π and CCNp0π data: all models predict a higher cross section for processes with protons in the final state concerning those with no protons in the final state.This is highlighted in Fig. 27.This can be caused by FSI or initial state effects.For example, more sophisticated nuclear models based on spectral functions were found to better describe CC0π and CCNp0π data, suggesting that a better nuclear model might be key to resolve the tension [51].FIG.23: Comparison of the G18 10a 02 11b, G10a, G20a and G30a tunes against MINERνA ν µ CCNp0π single-differential cross-section data as a function of (a) δp T , (b) α p , (c) p p or (d) θ p .In order to ease the readability of these plots, no statistical errors are shown.The predictions are computed using the parameters specified in Tab.V.

Nucleon Final State Interaction model variations
Mismodeling of nucleon FSI can cause migration between CC0p0π and CCNp0π samples [58].Ref. [59] suggests increasing the nucleon mean-free path in cascade models might improve the agreement with CCNp0π data from T2K ND280 and MINERνA.This possibility is explored here.The effect of the mean-free path implementation is validated against proton transparency data for carbon.
A crucial test for FSI models is to be able to reproduce nuclear transparency data from electron scattering experiments.Transparency is defined as the probability for the knocked-out nucleon to not undergo FSIs in the nuclear environment and it can be measured using electrons or neutrinos.In transparency measurements, the final-state nucleon is produced inside the nucleus.This feature is common with neutrino experiments, making transparency data extremely valuable to characterize and test FSI modeling uncertainties.Unfortunately, nuclear transparency measurements are scarce.Few data points on proton transparency on carbon as a function of the proton momentum are available in Ref. [60][61][62][63].
Transparency can be easily calculated within MC event generators as a ratio between the distribution of final-state protons which did or did not rescatter while leaving the nuclear environment.Ref. [58] provided the first direct comparison of transparency calculations using a neutrino event generator.This analysis took into account the experimental acceptances of the electron scattering experiments in the transparency definition.Such an analysis could be replicated in GENIE; however, it is out of the scope of the present work.To be able to compare GENIE's G20a and G30a tunes against T2K ND280 ν µ CCNp0π single-differential cross-section data as a function of (a) δp T or (b) δα T [27].In order to ease the readability of these plots, no statistical errors are shown.The predictions are computed using the parameters specified in Tab.V.
transparency calculations with data, we scale the GENIE predictions by the ratio between the transparency prediction from Ref. [58] with and without acceptance cuts.This approach was used in Ref. [50].
The effect on proton transparency calculations for carbon when varying the mean-free-path is shown in Fig. 29.The red and blue bands show the effect on the predictions when scaling up and down the nucleon mean-free path by 10% and 30% respectively.The 10% variation describes the data points with proton kinetic energies above 600 MeV within the 1σ error bound.The 30% variation covers all the data available.Fig. 29 suggests that at most a 30% variation of the mean-free-path is feasible for low-momentum protons.This is also supported by Ref. [50], where the authors observed a strong model dependency at low proton kinetic energies.Another article [56] finds that low momentum protons have a small re-scattering probability [56], reinforcing the  [32].In order to ease the readability of these plots, no statistical errors are shown.The predictions are computed using the parameters specified in Tab.V.
need of a more realistic model for the nuclear ground state.
The impact on the T2K ND280 cross section of variations in the nucleon mean-free path is shown in Fig. 28 as a function of proton multiplicity.It is observed that a higher nucleon mean-free path results in an increase of the proton multiplicity.A higher cross-section is predicted for events with no protons above detection threshold when reducing the meanfree path.However, variations of the mean-free path are not enough to explain the observed tension.
Another possible line of study would be to determine whether more elaborate FSI models can resolve the tension.The hA and hN FSI models are build on a simplistic view of the nuclear environment.More complex approaches offer an improved description of CCNp0π data [27,64,65].Such a study is out of the

VII. CONCLUSIONS
This article describes the first neutrino-nucleus cross-section tuning effort within the GENIE Collaboration.The goal of this work is to tune GENIE against CC0π data and quantify the major sources of CC0π modelling uncertainties.In total, five partial tunes using double-differential flux-integrated ν µ or ν µ CC0π cross-section measurements on carbon as a function of the outgoing muon kinematics.Each tune is performed with data from either MiniBooNE, T2K ND280 or MINERνA following the same analysis procedure.Even though these experiments all This tune confronted a number of new challenges with respect to previous GENIE free nucleon tuning efforts.This led to important changes in the GENIE tuning software.In particular, modern nuclear data provide the full correlation between the data release bins due to systematic uncertainties.In order to incorporate this information in the analysis, the definition of the χ 2 is modified to avoid the Peele's Pertinent Puzzle, which leads to nonphysical normalization factors.This analysis considers a total of seven parameters which attempt to capture the basic features of the component interactions -QEL, 2p2h, and RES as implemented in the G18 10a 02 11b model set.Some of the parameters used in this work affect the simulation of neutrino interactions on free nucleon.We chose to constrain these parameters with correlated priors coming from previous GE-NIE tunes to bubble chamber data [5].In addition, new parametrizations that encapsulate possible nuclear uncertainties were developed, using the Valencia model [13] for the QEL and 2p2h processes as a basis for choosing parameters.These affect the strength of the RPA correction for CCQEL calculations in a nuclear environment as well as the strength and shape of the 2p2h cross section -these are the main topic of this work.Other relevant CC0π parameterizations that affect the nuclear or FSI models are discussed.These are found to be highly correlated with other aspects of the tune or not too sensitive to the CC0π data considered in the tune.For these reasons, these are not included in the tunes presented here.
All tunes present a common trend: the QEL and 2p2h cross sections are enhanced and there is a preference for the QEL cross section with RPA corrections.In addition, the tune results are in agreement with the priors imposed on free nucleon parameters.Despite similarities, a clear energy dependence is observed for the 2p2h cross-section shape: Cross Section [10

Graph
FIG. 29: Comparison of proton transparency in carbon against the available electron scattering data [60][61][62][63].The GENIE prediction (black) is calculated using the G18 10a 02 11b.The error bands correspond to the expected uncertainty when varying the nucleon mean free path by 10% (red) and 30% (grey).The predictions have been corrected according to experiments acceptance effects as determined in Ref. [58].
the MiniBooNE and T2K tunes enhance the 2p2h cross section at the nucleon region, W = M N , while suppressing it at the ∆ region, W = M ∆ .Alternatively, both MINERνA tunes enhance the cross section in both regions, with an even higher scaling factor at the ∆ region.This suggests a dependence of the CC0π cross section on the neutrino energy which is manifested in this work as a change in the shape and strength of the 2p2h cross section.
Tensions between the various CC0π partial tunes and existing CCNp0π data from T2K ND280, MINERνA and MicroBooNE exist.These results were shown in Table VI and are a key result of this work.Parameters that give good agreement with one data set give poor χ 2 values for the other data sets.Since MiniBooNE and T2K are at very similar energies, this indicates tensions between the data sets of the two experiments.In all cases, the tunes over-predict CCNp0π data.This tension was further investigated with a dedicated tune, which is performed using MINERνA ν µ CCNp0π data.The result suggests that a reduction of the QEL and 2p2h cross sections would improve the agreement with all CCNp0π data, contradicting the CC0π partial tunes.This tune also improves the agreement with MicroBooNE CCNp0π data, suggesting that a possible A-dependence on the tune parameters is small, as was indicated in the MicroBooNE tune [10].The disagreement with CCNp0π data is further explored in this paper, highlighting the importance of using a more realistic nuclear model and possible changes to FSI models to describe existing CCNp0π data.Tensions between neutrino and anti-neutrino tunes are also observed, suggesting the need of an additional modelling uncertainty.The observed tensions must be addressed before attempting to perform a global tune with all the available data.

VIII. ACKNOWLEDGEMENTS
We would like to thank Andy Buckley (University of Glasgow, UK) and Holger Schultz (Institute of Particle Physics Phenomenology, University of Durham, UK) for their support interfacing the Professor tool with the software products that underpin the GENIE global analyses.We would like to thank the CC-IN2P3 Computing Center, as well as the Particle Physics Department at Rutherford Appleton Laboratory for providing computing resources and for their support.This work, as well as the ongoing development of several other GE-NIE physics tunes was enabled through a PhD studentship funded by STFC through LIV.DAT, the Liverpool Big Data Science Centre for Doctoral Training (project reference: 2021488).The initial conceptual and prototyping work for the development of the GENIE / Professor interfaces, as well as for the development of the GENIE global analyses framework that, currently, underpins several analyses, was supported in part through an Associateship Award by the Institute of Particle Physics Phenomenology, University of Durham.We thank: Stephen Dolan and Jaafar Chakrani for providing the method to correct for Peelle's Persistent Paradox; Daniel Ruterbories and Stephen Dolan for guidance to implement MINERνA's and T2K's analysis requirements in GENIE Comparisons; Xianguo Lu for discussion and comments on the paper.
Appendix A: Kinematic quantities of interest for CC0π measurements Differential neutrino cross-section measurements are given as a function of different kinematic quantities.In this paper, these kinematic quantities are classified into direct, inferred with an underlying process hypothesis or inferred without an underlying process hypothesis.

Direct
Kinematic quantities that can be measured by the detector are classified as direct.For instance, an example of direct quantity would be the muon momentum, p µ , or angle with respect to the beam-line axis, θ µ .In some cases, the muon kinetic energy, T µ , is used instead.All cross-section measurements specified in Tab.II released data as a function of the muon kinematics.Depending on the detector capabilities, direct quantities can also be related to proton kinematics.We refer to proton momentum and angle as p p and θ p respectively.
Muon direct quantities depend strongly on the neutrino energy and these are less sensitive to nuclear effects.This motivated the recent efforts on the study of more exclusive topologies that allow measurement of the cross section as a function of the outgoing-proton kinematics [27,29,31,32].These depend weekly on the neutrino energy and are significantly altered by nuclear effects [40].
In some cases, the differential cross-section measurements are presented as a function of the reconstructed direct quantities.Kinematic quantities in the reconstructed space are denoted with a 'reco' superscript.For instance, p reco µ stands for reconstructed muon momentum.

Inferred kinematic with an underlying process hypothesis
This category includes measurements that rely on the reconstruction of neutrino properties assuming a specific interaction type.For instance, the kinematics of a CC0π event can be reconstructed under the hypothesis that the initial nucleon was at rest and that there is no inelastic production of mesons in the final state (QEL hypothesis).Under this hypothesis, the reconstructed neutrino energy (E QEL ν ) and squared four-momentum transferred (Q 2 QEL ) are: where M i (M f ) is the initial (final) nucleon mass, M µ is the muon mass, and E µ is the muon energy.
For the neutrino analysis, M i = M n and M f = M p , whereas M i = M p and M f = M n for the antineutrino case.The binding energy, E b , depends on the target type.Its specific value is provided in each analysis.
The main disadvantage of using these quantities is that the underlying hypothesis is uncertain.The presence of the nuclear environment complicates the characterization of event topologies: no single-event topology is produced only by a single underlying process.This is highlighted by the T2K ND280 results on inferred kinematics [27].In their analysis, they reconstruct the energy and momentum of the outgoing proton assuming a QEL interaction.Instead of presenting the cross-section measurement as a function of the inferred with kinematic with an underlying process quantities, they used the difference between the direct and the inferred one.In particular, T2K ND280 explored this quantity for the proton kinematics: These are referred to as proton inferred kinematics quantities.Here, the superscript indicates whether the kinematic quantity is direct (i.e p direct p ) or inferred (i.e.p QEL p ) .The reconstructed proton energy and momentum under the QEL hypothesis are: These kinematic quantities can be used to highlight nuclear effects in CC0π measurements, as the quan-tities defined in Eq.A3 deviate from zero when nuclear effects are present.

Inferred without an underlying process hypothesis
This category includes those kinematical quantities which are inferred from direct ones but do not assume a specific underlying interaction process.An example of interest for this work is the Single-Transverse Kinematic Imbalance (STKI) variables [40].STKI provide direct constraints on nuclear effects that, in some cases, have a weak dependence on the neutrino energy.STKI quantities are inferred from the muon and primary state hadron kinematics and only detectors capable of measuring low energy hadrons can provide such information.So far, only T2K ND280 and MINERνA have released single-differential fluxintegrated cross-section measurements as a function of these quantities [27,29,30].
The transverse momentum imbalance, δp T , is defined as the sum of the transverse muon and proton momentum: As the neutrino travels in the longitudinal direction, the transverse muon momentum is related to the transverse momentum transfer as p µ T = −q T .The angle between δp T and −p µ T is known as boosting angle, δα T : The deflection of the nucleon with respect to q T is measured with the δφ T angle: A more recent study investigates the CC0π crosssection dependency on the muon-proton momentum imbalances parallel (δp T y ) and longitudinal (δp T x ) to the momentum transfer in the transverse plane [30].These quantities are mathematically defined as: given the Cartesian coordinate system defined with respect to the neutrino and muon kinematics.The neutrino direction is given by pν .All these quantities define what experiments refer to as STKI variables. ( FIG. 30: Graphical definition of the STKI variables in a ν µ CCQEL neutrino interaction on a nuclear target.The incoming neutrino, represented as a dashed arrow, interacts with a free nucleon at rest (a) or with a bound nucleon subject to Fermi motion (b).The outgoing muon (proton) is represented in blue (red).The transverse plane is represented in grey.The incoming neutrino is perpendicular to the transverse plane.Nuclear effects distortion the free-nucleon picture (a) creating an imbalance between the muon and nucleon transverse momentum (b).The STKI variables that define this imbalance are highlighted in orange.
A graphical representation of the definition of the STKI variables for a neutrino interaction with and without nuclear effects is shown in Fig. 31.When the interaction occurs with a static free nucleon, i.e. no nuclear effects, p µ T = −p p T , δp T = 0 and δφ T = 0, see Fig. 30a.However, this picture is modified by Fermi motion, nucleon correlations, non-QEL interactions and FSI.If FSI effects and nucleon correlations are neglected, δp coincides with the initial nucleon momentum p Ni .Moreover, δα T is uniform due to the isotropic nature of the Fermi motion.FSI effects smear the δp T distribution and modify the shape of the δα T distribution.In GE-NIE, the hA FSI model enhances the cross section at δp T > 0.2 GeV/c and δα T ∼ 180 • ; see Fig. 31.This region is refereed to as high-transverse kinematic imbalance region.
Ref. [40] demonstrated that the δp T and δα T dependence on the neutrino energy is smaller than possible uncertainties due to FSI modeling.The δφ T variable has a stronger dependence on the neutrino energy as it scales with δp T /p µ T : at higher neutrino energies, the distribution at small angles becomes narrower.The dependency of the STKI variables in GENIE with the neutrino energy is shown in Fig. 32.Changes in the neutrino energy affect mostly the tail of the δp T distribution and the δα T distribution at backward angles.This section offers with comparisons of GENIE against all CC0π and CCNp0π data available from MiniBooNE, T2K, MINERνA and MicroBooNE.The corresponding GENIE predictions are obtained by replicating the analysis within GENIE: neutrino interaction events are simulated for each experiment given the neutrino flux, target material, and analysis cuts.The normalized neutrino flux spectra is reported in Fig. 1.With this information, the GENIE prediction for the corresponding differential flux-integrated cross section is evaluated.
The format of all the comparisons with data reported in this appendix is common: the data and differential cross-section prediction are represented in black.In addition, the contribution from different interaction models is shown for CCRES, CC2p2h and CCDIS/SIS.The contribution to the G18 10a 02 11b predictions from CCDIS/SIS events is really small at the neutrino energies considered in this work.For this reason, the contribution is grouped into a single category (DIS).The 2p2h contribution is divided further into four categories that depend on the event invariant mass, W .The W regions are: • W > M ∆ The data error bars include statistical and systematic uncertainties.The errors on the x-axis represent the bin width used in the original analysis.

MiniBooNE CC0π cross-section measurement
The MiniBooNE experiment studies neutrinos produced with the BNB [7].MiniBooNE published the first-high statistics ν µ and ν µ CC0π flux-integrated double differential cross-section measurement on carbon, at E ν ∼ 800 MeV and E ν ∼ 500 MeV respectively [25,26].The fluxunfolded total cross section, σ E QEL ν , and the flux-integrated single differential cross section as a function of the squared four-momentum transferred, dσ/dQ 2 QEL , were also reported.Both MiniBooNE analyses study CC0π events with a muon in the final state and no pions.The signal topology of a muon in the detector is described in two sub-events: the first one associated with the primary Cherenkov light from the muon, and the second one, produced by the Cherenkov light from the Michel electron, which is produced in the muon decay.This requirement provides a sample of mostly CC events, as neutral-current events only have one sub-event.
Positively charged pions produced in the detector leave a distinct signature in the detector, as the π + decays immediately into a muon and a muon neutrino.The Cherenkov light from the π + contributes to the total light of the primary muon.This process can be distinguished from a CCQEL interaction as the muon produced from the pion decay will also decay into a Michel electron (three sub-events).Negatively charged pions are absorbed by the nuclear environment and contribute to the CC0π topology.In the GENIE predictions, pion production events are removed by requiring no pions in the final state.
Recoil protons also emit scintillation light.However, such scintillation light signal produced is either indistinguishable from the muon signal or its momentum below the Cherenkov threshold.For this reason, no requirements based on the recoil proton are considered in the MiniBooNE analyses.
The analysis considers further model-dependent cuts to correct for backgrounds and extract the CCQEL cross-section from the CC0π sample.In the original publication, these are referred to as irreducible backgrounds.An example of irreducible background is CC1π events that were not removed by the cut on the pion subevent topology or pion production events in which the pion is absorbed.This is corrected using a MC simulation tuned to ν µ CC1π MiniBooNE data.Information on ν µ CC1π + sample is used to characterize this background and correct for single-pion events which were not removed by the CC0π selection criteria in the neutrino and antineutrino analyses.This procedure is one of the main limitations of this dataset as it incorporates strong biases in the reported measurement.The contribution to the cross-section measurement from irreducible backgrounds is also reported, allowing the comparison against CC0π data.
The quality of the MiniBooNE CC0π data release is poor in comparison with the rest.The MiniBooNE collaboration provided measurements in bins of T µ and cos θ µ , but did not provide the bin-to-bin covariances for either of the two measurements.Instead, they quoted a normalization systematic uncertainty of ∼ 10.7% (17.2%) for the neutrino (antineutrino) measurement.As suggested by Ref. [25], this error is added as a systematic in our database, effectively including a correlation between the bins.
In Figs. 2 and 33, the flux-integrated double differential ν µ and ν µ CC0π cross section data as a function of p µ and T µ are compared against GENIE.The main observation is that the GENIE tune underpredicts the data.In particular, the G18 10a 02 11b disagreement with the data are more significant at backward angles, where the cross-section is determined by CCQEL events only.The disagreement is also observed at forward angles, where there is a significant contribution from non-QEL events.

T2K CC0π cross-section measurements
The Tokai-to-Kamioka (T2K) experiment is an accelerator-based long-baseline experiment that studies neutrino oscillations.Neutrinos are generated at the Japan Proton Accelerator Research Complex (J-PARC) facility [6].The target for the neutrino beam is 280 m away from the T2K near detectors [2]: INGRID, WAGASCI and ND280.The T2K ND280 detector is used to measure neutrino interactions on carbon at E ν ∼ 600 MeV.The WA-GASCI module was recently added to the T2K ND facility and it measures neutrino interactions at 0.86 GeV.Details on the detector setup can be found in Ref. [27,37].Most measurements described here use the detector central tracker region, composed of three time projection chambers (TPC) and two finegrained detectors (FGD1 and FDG2).The FGDs are the target mass and are also used to track charged particles.Carbon measurements use the FGD1 as the target mass.The central region is surrounded by an electromagnetic calorimeter (ECal), which is contained within a magnet.This setup allows measuring the particle charge and momentum.This in-formation, together with energy deposition, is used to identify charged particles.
The first double-differential ν µ CC0π measurement provided by T2K ND280 was released back in 2015 [41].This measurement is surpassed by Ref. [27], which considers improved constraints on systematic uncertainties.Ref. [27] provides additional measurements including double-and tripledifferential measurements for different proton multiplicities as well as two CCNp0π single-differential cross-section measurements as a function of STKI and proton inferred kinematics quantities.
All measurements from Ref. [41] require one muon and no pions in the final state, regardless of the number of nucleons in the event.Any event must contain at least one track in the TPC, which must be either a muon or a proton.If it is a proton, they look for a muon-like track in the FDG1 or ECal.Other events with tracks that are not consistent with the muon-like or proton-like signature are rejected.Events with low-momentum charged or neutral pions are removed by requiring no Michel electrons or photons.At the MC level, this is implemented by removing events with pions or photons in the final state, respectively.
The selected sample is divided further depending on the number of protons above the detection threshold of 500 MeV/c: no protons (CC0p0π), one proton (CC1p0π) or more than one visible proton (CC2p0π) in the final state.The CC0p0π and CC1p0π are double-and triple-differential cross-section measurements as a function of the muon and muon and proton kinematics respectively.The total CC2p0π cross-section is also reported.The STKI and proton inferred kinematics are obtained with the CCNp0π sample: they require the presence of at least one visible proton (p p > 500 MeV/c).
Efficiency corrections for ν µ CCNp0π events can be model dependent.To avoid this, different kinematical restrictions are considered for each analysis, selecting regions in which the efficiency is flat or well understood.These are specified in Tab.IX.Events with more than one proton are reconstructed using the information from the highest energy one, which has to satisfy the kinematical limits of Tab.IX.The samples are not corrected for events with protons below the detection threshold or any of the kinematical cuts considered in the analysis.The same cuts are applied at the generator level when evaluating the GENIE predictions.
The GENIE comparison against the ν µ CC0p0π double-differential cross section are presented in Fig. 34.The main contribution to the CC0p0π topology comes from CCQEL events.The second contribution is from CC2p2h events with M N < W < W Dip .The contribution from 2p2h events with This disagreement in the overall normalization is also observed in Fig. 5, which compares GENIE against the cross-section as a function of the proton multiplicity.This observation conflicts with ν µ CC1p0π data, which is not under-predicted.There are some outstanding differences between the GE-NIE predictions for CC0p0π and CC1p0π data.Whilst the total contribution from 2p2h events is similar, the main 2p2h contribution comes from 2p2h events with W > W Dip .In addition, the fraction from RES events is higher with respect to the CC0p0π one.MINERνA extracted several CC0π and CCNp0π measurements using the NuMI low-energy flux [28][29][30]38].A CC0π measurement using the NuMI medium energy flux is also available [33].This review focuses on the CC0π and CCNp0π measurements obtained with the low-energy flux.
The exact target mixture is composed of carbon (88.51%), hydrogen (8.18%), oxygen (2.5%), titanium (0.47%), chlorine (0.2%), aluminium (0.07%), and silicon (0.07%).In the calculation of the GE-NIE predictions, only the three most abundant targets are considered.The relative mass abundances are renormalized to take this approximation into account.This simplifies the computing power and has a negligible effect on our predictions.
a. MINERνA νµ and νµ CC0π cross-section measurement MINERνA reported the CC0π differential fluxintegrated cross-section as a function of muon momentum in the transverse (T ) and longitudinal ( ) direction relative to the neutrino beam [28,38].The differential cross-section as a function of E QEL ν and Q 2 QEL are reported as well [28].The neutrino energy and the momentum transferred are reconstructed under the QEL hypothesis, described in Sec.A 2. The binding energy used to reconstruct E QEL ν according to Eq. A1 in their neutrino and antineutrino analysis is E b = 34 MeV and E b = 30 MeV respectively.
The ν µ CC0π topology is defined as an event with one muon, µ − , any number of protons and neutrons, any photons below nuclear de-excitation energies, E γ ≤ 10 MeV, no mesons and no heavy or excited baryons in the final state.The MINERνA detector is not able to measure the muon charge as it does not have a magnetic field.For this reason, muons are identified by looking for tracks that have a match with the MINOS detector, which is used to determine the muon momentum and charge.Because of geometric acceptance, both analyses require θ µ < 20 • .Events containing low-energy photons are accepted as they can arise from nuclear deexcitation.Pions are removed by applying a cut on the recoil energy, E recoil ≤ 500 MeV, defined as the activity that is not coming from a muon or any tracked protons.E recoil is corrected for the calorimetric detector response [38].The recoil energy does not include energy deposited at less than 150 mm from the neutrino vertex as it could be due to proton absorption nearby the vertex.Moreover, events with Michel electrons are removed, as they assume The ν µ CC0p0π topology [38] is similar to the ν µ CC0π one, with some differences.Due to the nature of this interaction, the muon must be positively charged.Moreover, the analysis requires there are no visible protons in the final state, i.e. protons with kinetic energy above 120 MeV.Finally, mesons are removed using the information on the recoil energy deposited outside the vertex region only.
The GENIE prediction is evaluated with MC events that satisfy the criteria specified above with few exceptions: the removal of events with mesons in the final state is based on true information only.Baryons, are short living and decayed into mesons using the GENIE particle decayer.The requirements on the removal energy are not implemented in our MC analysis either as the data was already corrected for this effect.GENIE comparisons against the doubledifferential ν µ CC0π measurement is shown in Figs. 3 and 38 for ν µ CC0p0π data.For both ν µ and ν µ data, the G18 10a 02 11b underestimates the data.This is true especially in the phase-space regions in which 2p2h events dominate.In high p T regions, where the contribution of 2p2h events is negligible, the agreement improves.This can be seen for the 0.85 < p T < 2.5 GeV/c slices in Fig. 3. Consequently, the reconstructed neutrino energy is also under-predicted, as observed in Fig. 37. FIG.39: MINERνA ν µ CCNp0π differential flux-averaged cross-section as a function of STKI variables [29,30].The data are compared against the G18 10a 02 11b tune.The GENIE prediction is divided into interaction modes.The notation for histograms is the same as in Fig. 35.
uated 500 m away from the BNB beam at Fermilab [3,7].LArTPC detectors use complex software algorithms to reconstruct the neutrino event topology with excellent spatial resolution in the detector [69][70][71].For instance, MicroBooNE can reconstruct proton tracks of 2 cm with a ∼ 26% efficiency [32].Different Particle IDentification (PID) algorithms, based on the characteristic signal of each particle in the detector, allow the identification of proton and µ/π candidates, but these methods fail to distinguish between muons and pions.MicroBooNE provides the first high-statistics cross-section measurements on argon: ν µ CC inclusive [72], ν µ CC1p0π [31], ν µ CCNp0π [32], and ν µ CC π 0 production [73].The detector is situated 500 m away from the BNB beam at Fermilab [7].In this section, we focus on the description of the CCNp0π measurement [32], given that the CC1p0π measurement [31] is a subsample of the CCNp0π one.
The CCNp0π analysis presents a total of five sin-gle differential flux-integrated cross-section measurements.The single differential cross sections are given in terms of the muon momentum (p µ ), muon angle (θ µ ), leading proton momentum (p p ), leading proton angle (θ p ), and the angle between the muon and the leading proton (θ µp ).The CCNp0π topology is defined as an event with one muon, at least one visible proton, any number of neutrons and no pions in the final state.In the analysis, the muon candidate is the longest track which is not identified as a proton.Other tracks in the event must be compatible with the proton PID hypothesis.In order to guarantee at least a 5% efficiency in the momentum reconstruction, they require the muon (proton) to have a momentum of at least 100 MeV/c (300 MeV/c).In addition, the leading proton candidate is must have a reconstructed momentum of less than 1.2 GeV/c.This cut avoids regions of the phase-space in which the proton candidate length is greater than the muon one.These analysis criteria removes events with pions below 30 MeV/c, which are not reconstructed.No corrections are applied to remove events with protons or pions below the detection threshold.The same requirements are applied to the corresponding MC predictions.
The differential cross-section measurements were not unfolded to true muon momentum and muon angle.Instead, the results are presented in terms of the reconstructed quantities.The smearing matrices that convert from the reconstructed to the truth quantities are provided in the data release and are used for the evaluation of the GENIE predictions in the reconstructed space [32].This method is known as forward folding.
Figure 40 presents the comparison between the MicroBooNE data and the GENIE predictions.The nominal agreement for the G18 10a 02 11b tune is reasonably good, except for the bin at highest cos θ reco µ , which is largely over-predicted.The contribution of non-QEL interactions increases at forward muon and proton angles, see Figs. 40b and  40d.The G18 10a 02 11b dependency on 2p2h events at different W with the proton momenta is re-encountered.

Appendix C: Additional Nuclear Uncertainties
Here, we explore modeling aspects that were not included in the tuning exercise.

Nuclear model implementation
Uncertainties in the nuclear model affect the dynamics of the outgoing muon and nucleon after a QEL or a 2p2h interaction.
In the Valencia model implementation in GENIE, the differential cross section is evaluated at an effective energy transfer q0 , which takes into account the nucleon removal energy.The implementation in the QEL and 2p2h processes is slightly different.The effective energy transfer q0 used in the Valencia QEL model implementation is: E Ni is the energy of the off-shell initial nucleon, which is bound with a binding energy E b .E p is the energy of the initial nucleon on-shell with a momentum p, E p = M 2 N + p 2 .E N f is the energy of the nucleon produced after the QEL interaction, which is on-shell.In other words, the effective energy transfer is reduced relative to the ordinary one by the amount of energy needed to put the initial nucleon on the mass shell.The binding energy and initial nucleon momentum are determined by the corresponding nuclear model.In this work, for QEL in-teractions we refer to q0 as qQEL 0 .Notice that qQEL 0 depends on the event kinematics.
In the Valencia 2p2h model implementation, the effective energy transfer is calculated as: q0 = q 0 −q 2p2h shift , where q 2p2h shift ≡ M (A Z+1 )−M (A Z ).
In this case, q 2p2h shift is independent of the event kinematics.For a carbon target, q 2p2h shift ( 12 C) = 16.8MeV, whilst q 2p2h shift ( 40 Ar) = 0.99 MeV for argon.Shifts on q0 are effective modifications of the binding energy in the nuclear model.It is possible to apply relative shift to q0 for both QEL and 2p2h calculations by modifying q QEL 0 and q 2p2h shift .This modification translates as: shift → q 2p2h shift (1 + f 2p2h ) f QEL and f 2p2h are two dimensionless parameters.In the GENIE v3 version, both parameters default to 0. Both f QEL and f 2p2h parameters are included in the initial iteration of this analysis.
Ref. [74] suggests that shifts on qQEL 0 (q 2p2h shift ) of 0 − 20 MeV (0 − 40 MeV) for QEL (2p2h) are in reasonable agreement with electro-scattering data.The effect of such variations on the 2p2h cross-section prediction is shown in Fig. 41.The biggest variation is observed on dσ/dQ 2 for both QEL and 2p2h.For the 2p2h cross section, this systematic shifts peaks position in W .

Final state interaction implementation
Final-state interactions (FSI) are crucial for modeling nuclear cross sections as they affect the event topology and kinematics of an event.There are different models available in GENIE to simulate FSI [14,50].In particular, G18 10a 02 11b models FSI with the INTRANUKE hA model [22].
INTRANUKE hA is an empirical model that considers a single interaction which is based on hadronnucleus data [50].In particular, pion-nucleus data are used to determine the inelastic (Inel), absorption (Abs), charge-exchange (CEx) and pion production (πProd) fractions (f i ).The fractions depend on the pion kinetic energy and the nuclear atomic number.These fractions satisfy that i f π ± i = 1 (unitarity condition), where i is an index that runs over the available processes aforementioned.
Two parameters are introduced to be able to modify the f π ± Abs and f π 0 Abs while preserving unitarity: The other fractions are also modified as a consequence of this scaling.Notice that variations of S π ± Abs do not scale f π ± Abs linearly.Similarly, a scaling parameter is introduced to scale the charged pion mean-free path.This is referred to as S π ± MFP .The same approach can be applied to other processes and to nucleon fractions.
Figure 42 shows the dependence of each hA fraction as a function of the pion kinetic energy (T π ) for carbon and argon targets.The FSI fractions and their uncertainty are extracted from fits to hadronnucleus scattering data [50,75].The uncertainty associated with f π ± Abs is 15%.Variations of the FSI parameters considered in this work result in the migration of CC1π events into the CC0π sample.The effect on the prediction depends on the topology definition.For CC0π samples, it mostly affects the overall normalization of the cross-section.The measurement most sensitive to this variation is the ν µ CCNp0π MINERνA differential cross section as a function of δα T , see Fig. 43.A decrease in S π ± Abs reduces the cross section at δα T ∼ 180 • .In addition, this model variation also affects the slope of the distribution.
In this tune, only parameters related to charged pion absorption are included: S π ± Abs and S π ± MFP .Pion inelastic fractions are not relevant at the energies of interest for this work.Nucleon FSI parameters are relevant for the study of exclusive cross-section measurements with protons in the final state.Ideally, to perform a global tune with CC0π and CCNp0π data, nucleon FSI parameters must be considered in the analysis.Including these parameters in the analysis substantially increases the computing time.In addi- FIG.41: Flux-integrated differential ν 12 µ C CC2p2h cross section dependence with W , Q 2 or p µ .Events are generated with the G18 10a 02 11b tune and the NuMI ν µ low energy flux [8].The top (bottom) three plots show the CCQEL (CC2p2h) differential cross section as a function of W , Q 2 or p µ .The black prediction corresponds to the GENIE v3 case, where no shifts on qQEL 0 and q 2p2h shift are considered.The variations considered for the f QEL and f 2p2h parameters correspond to an absolute shift to qQEL 0 and q 2p2h shift of 20 MeV for QEL interactions and of 40 MeV for 2p2h interactions.tion, it is desirable to first understand the tensions between CC0π and CCNp0π measurements.Therefore, it is therefore convenient to reduce the complexity of the analysis and focus on CC0π datasets only.Nucleon FSI parameters will be included in future iterations of this work.

Final choice of parameters for the CC0π tune
A series of preliminary tunes were performed using different priors or parameter sets.The goal of this study is to determine which parameters to include in the final tune.
Nuclear effects in the QEL cross section are tweaked with the RPA parametrization.
Freenucleon cross-section data suggests that the QEL cross section should not be scaled.This condition can be incorporated in our analysis by imposing a more restrictive prior on S RPA of σ S = 0.01.Tunes performed using this prior result in worse goodness of fit, suggesting that a less restrictive prior on the sum is desired to improve the agreement with the data.This motivated our choice for a prior on the sum of σ S = 0.2, as described in Sec.III A. FSI interactions are important to describe CC0π measurements.Additional parameters must be consistent with previous data [50], making this tricky.The results of test cases with FSI suggest that variations of these parameter that respect pion-nucleus scattering data do not have a big impact on the tune results.Consequently, these parameters are not included in the final analysis.
Various choices were made to get a more representative result.Although full coverage in parameters can be sought, that is not always possible or desirable.For this study, Professor allows a large parameter set which don't have to be ReWeight variables.A study of the f QEL and f 2p2h parameters in the tune In some cases, these correlations lead to unphysical values for f QEL and f 2p2h .For this reason, these parameters are excluded from the analysis.The final-parameter set used in this work is summarized in Tab.III.

FIG. 1 :
FIG.1:(Top) Summary of contributions from each interaction process to the CC ν µ cross section on12 C as a function of neutrino energy, E ν .(Middle)The corresponding fraction of the total ν µ -12 C events arising from each of the 0π topologies.This plot assumes a momentum threshold for protons of 450 MeV/c while the GENIE predictions are obtained with the G18 10a 02 11b tune.(Bottom) Summary of ν µ (continuous lines) and ν µ (dashed lines) normalized flux distributions for T2K ND280 at JPARC[6], MiniBooNE and MicroBooNE with the Booster Neutrino Beam (BNB)[7], and MINERνA with the Neutrino at the Main Injector (NuMI)[8].The flux predictions for neutrino and antineutrino modes are refereed to as "Forward Horn Current (FHC)" and "Reverse Horn Current (RHC)," respectively.

FIG. 5 :
FIG.5: T2K ND280 flux-averaged ν µ CCNp0π differential cross section as a function of the proton multiplicity[41].The data are compared against the G18 10a 02 11b tune.The GENIE prediction is divided into different interaction modes.

FIG. 6 :
FIG. 6: Impact of the RPA parametrization on the CCQEL cross section.The G18 10a 02 11b prediction is shown in black.The other predictions are obtained with the same tune while changing the RPA weight values.(a) Total CCQEL cross section for 12 C.(b) Flux-integrated differential cross section as a function of Q 2 .The prediction is obtained with the NuMI flux in low-energy mode.

FIG. 9 :; 2 .
FIG. 9: Graphic representation of the 2p2h scaling as a function of W .On the top, the default parametrization (dashed blue) and an example scaling function (green) are shown.The highlighted dashed vertical lines correspond to the tunable scaling parameters for W = W min P L , M N , W Dip , M ∆ , and W max P L .The bottom figure shows the Valencia 2p2h flux-integrated cross section as a function of W for the G18 10a 02 11b tune in blue, and the same prediction scaled with the example scaling function in green.This plot is obtained simulating ν µ interactions on 12 C with the NuMI ν µ low energy configuration [8].

FIG. 10 :
FIG.10: Fractional difference between true MC predictions in the NS space calculated with a given θ parameter set.

5 QELFIG. 11 :
FIG. 11: Impact of M QEL A variations on MINERνA CC0π flux-integrated differential cross section predictions as a function of p T .The red line corresponds to the GENIE prediction computed with the M QEL A best-fit value from the µBooNE tune [10].No other parameters are modified from their nominal values.

FIG. 12 :
FIG.12: Comparison of the G18 10a 02 11b and G10a tunes against MiniBooNE ν µ CC0π double differential data[25].The comparisons are restricted to the 0.2 < T µ < 1.0 GeV phase space.The predictions are computed using the parameters specified in Tab.V.The total χ 2 associated with this dataset before and after the tune are reported in Tab.VI.

FIG. 13 :
FIG.13: Comparison of the G18 10a 02 11b and G11a tunes against MiniBooNE ν µ CC0π double differential data[26].The comparisons are restricted to the 0.2 < T µ < 1.0 GeV phase space.The predictions are computed using the parameters specified in Tab.V.The total χ 2 associated with this dataset before and after the tune are reported in Tab.VI.

FIG. 14 :
FIG.14:Comparison of the G18 10a 02 11b and G20a Tune against T2K ND280 ν µ CC0p0π double differential data[27].The comparisons are restricted to the −1.0 < cos θ µ < 0.94 phase space.The predictions are computed using the parameters specified in Tab.V.The total χ 2 associated with this dataset before and after the tune are reported in Tab.VI.

FIG. 15 : 10 [FIG. 16 :
FIG.15:Comparison of the G18 10a 02 11b and G30a tunes against MINERνA ν µ CC0π double differential data[28].The comparisons are restricted to the 0.25 < p T < 0.85 GeV/c phase space.The predictions are computed using the parameters specified in Tab.V.The total χ 2 associated with this dataset before and after the tune are reported in Tab.VI.

FIG. 24 :
FIG.24:Comparison of the G18 10a 02 11b, G10a, G20a and G30a tunes against T2K ND280 ν µ CCNp0π single-differential cross-section data as a function of (a) δp T or (b) δα T[27].In order to ease the readability of these plots, no statistical errors are shown.The predictions are computed using the parameters specified in Tab.V.

FIG. 31 :
FIG. 31: Probability density function of δp T and δα T for the G18 10a 02 11b tune with (a) and without (b) FSI.Both predictions are obtained simulating ν µ CCQEL interactions only on 12 C at 1 GeV with the G18 10a 02 11b tune.

FIG. 32 :
FIG. 32: Probability density function of STKI variables.The predictions are obtained simulating ν µ CCQEL interactions only on 12 C with the G18 10a 02 11b tune at different neutrino energies.

FIG. 37 :
FIG.37: MINERνA ν µ CC0π flux-averaged cross-section as a function of the reconstructed neutrino energy, E QEL ν[28].The data are compared against the G18 10a 02 11b tune.The notation for the histogram is the same as in Fig.35.

FIG. 40 :
FIG.40: MicroBooNE ν µ CCNp0π flux-averaged differential cross section on40 Ar as a function of muon and proton kinematics.The GENIE prediction is obtained with the G18 10a 02 11b tune.The nominal prediction is divided into interaction modes.

FIG. 42 :
FIG.42: hA FSI pion fractions for (a)12 C and (b)40 Ar as a function of the pion kinetic energy.The error bands represent the fraction variation when applying a S π ± Abs = 1.2 on the pion absorption fraction, which corresponds to a variation of ∼ 15%for the pion absorption fraction on carbon at T π = 200 MeV.

FIG. 43 :
FIG. 43: Impact of S π ±Abs on MINERνA CCNp0π flux-integrated differential cross section predictions as a function of δα T .

TABLE IV :
[5]ors (a) and covariance matrix (b) for M QEL A and S RES obtained to the free-nucleon tune from Ref.[5].

TABLE V :
Best-fit parameter values for the different partial tunes.Parameter values within parenthesis are kept fixed during the fit.The χ 2 values are calculated with the Professor parametrization, in accordance to Eq. 1.

TABLE VI :
Summary of χ 2 values associated the CC0π datasets specified in each row.The χ 2 values are calculated using the NS method for seven different GENIE predictions: G18 10a 02 11b, G10a, G11a, G20a, G30a, G31a and G35a.The values highlighted in bold correspond to the best-fit χ 2 for the partial tune predictions.

TABLE VIII :
Summary of χ 2 values associated with the CCNp0π datasets specified in each row.The χ 2 values are calculated using the NS method for three GENIE predictions.The GENIE predictions are calculated with the G18 10a 02 11b tune.Each prediction uses a different nuclear model: RFG, LFG or CFG.

TABLE IX :
[27]e-space restrictions for the T2K ND280 analyses from Ref.[27].The proton cuts are only applied to the highest energy proton.