First Measurement of Energy-Dependent Inclusive Muon Neutrino Charged-Current Cross Sections on Argon with the MicroBooNE Detector

We report a measurement of the energy-dependent total charged-current cross section $\sigma\left(E_\nu\right)$ for inclusive muon neutrinos scattering on argon, as well as measurements of flux-averaged differential cross sections as a function of muon energy and hadronic energy transfer ($\nu$). Data corresponding to 5.3$\times$10$^{19}$ protons on target of exposure were collected using the MicroBooNE liquid argon time projection chamber located in the Fermilab Booster Neutrino Beam with a mean neutrino energy of approximately 0.8 GeV. The mapping between the true neutrino energy $E_\nu$ and reconstructed neutrino energy $E^{rec}_\nu$ and between the energy transfer $\nu$ and reconstructed hadronic energy $E^{rec}_{had}$ are validated by comparing the data and Monte Carlo (MC) predictions. In particular, the modeling of the missing hadronic energy and its associated uncertainties are verified by a new method that compares the $E^{rec}_{had}$ distributions between data and an MC prediction after constraining the reconstructed muon kinematic distributions, energy and polar angle, to those of data. The success of this validation gives confidence that the missing energy in the MicroBooNE detector is well-modeled and underpins first-time measurements of both the total cross section $\sigma\left(E_\nu\right)$ and the differential cross section $d\sigma/d\nu$ on argon.

We report a measurement of the energy-dependent total charged-current cross section σ (Eν ) for inclusive muon neutrinos scattering on argon, as well as measurements of flux-averaged differential cross sections as a function of muon energy and hadronic energy transfer (ν). Data corresponding to 5.3×10 19 protons on target of exposure were collected using the MicroBooNE liquid argon time projection chamber located in the Fermilab Booster Neutrino Beam with a mean neutrino energy of approximately 0.8 GeV. The mapping between the true neutrino energy Eν and reconstructed neutrino energy E rec ν and between the energy transfer ν and reconstructed hadronic energy E rec had are validated by comparing the data and Monte Carlo (MC) predictions. In particular, the modeling of the missing hadronic energy and its associated uncertainties are verified by a new method that compares the E rec had distributions between data and an MC prediction after constraining the reconstructed muon kinematic distributions, energy and polar angle, to those of data. The success of this validation gives confidence that the missing energy in the MicroBooNE detector is well-modeled and underpins first-time measurements of both the total cross section σ (Eν ) and the differential cross section dσ/dν on argon.
Current and next-generation precision neutrino oscillation experiments aim to answer several critical questions in particle physics [1] by: i) searching for CP violation in the lepton sector [2,3],ii) determining the neutrino mass ordering [4], and iii) searching for light sterile neutrinos [5]. For this purpose, the SBN [6] and DUNE [7,8] experiments employ liquid argon time projection chambers (LArTPCs) [9][10][11][12], a tracking calorimeter that enables excellent neutrino flavor identification and neutrino energy (E ν ) reconstruction in the GeV energy range [13]. These experiments are designed to measure the neutrino flavor oscillations as a function of E ν , which requires a good understanding of the neutrino energy spectrum, neutrino-argon interaction cross sections [14], and LArTPC detector response. High-precision measurements of ν-Ar cross sections, particularly those related to energy reconstruction, are of paramount importance.
While historical accelerator-based neutrino experiments often reported E ν -dependent cross sections [15,16], recent experiments tend to limit cross-section measurements to the directly observable lepton and/or hadron kinematics [17]. This paradigm shift was triggered by concerns that quantities not directly measurable in detectors (e.g. the missing hadronic energy of the interaction from undetected neutral particles) may not be correctly modeled in simulations, which is of particular concern in a broad-band neutrino beam. In this letter, we demonstrate that the MicroBooNE tune model [18] (based on GENIE-v3 [19]) of missing energy with its associated uncertainty can be validated with inclusive muon neutrino charged-current (ν µ CC) interactions from the the MicroBooNE detector [20]. After constraining the * microboone info@fnal.gov lepton kinematics distributions of Monte Carlo (MC) to those of data, the comparison of reconstructed hadronic energy E rec had distributions between data and the updated MC prediction reveals whether the model is able to describe the relationship between the lepton kinematics and the visible hadronic energy. This procedure validates whether the missing hadronic energy is sufficiently modelled given the prior knowledge of the neutrino flux and detector effects. This new procedure enables a first measurement of the differential cross section as a function of the energy transfer to the argon dσ/dν. Together with the differential cross section as a function of the muon energy (dσ/dE µ ), the E ν -dependent cross sections are extracted. These data could be used to isolate problems for low-E ν cross sections and to reduce modelling uncertainties for the low-ν method [21][22][23] to constrain the shape of neutrino energy spectrum in future experiments.
The MicroBooNE detector is a 10.4×2.6×2.3 m 3 LArTPC. It consists of approximately 85 ton of liquid Ar in the active TPC volume for ionization charge detection, along with 32 photomultiplier tubes (PMTs) [24] for scintillation light detection. This work makes use of a data set corresponding to an exposure of 5.3×10 19 protons on target (POT) from the Booster Neutrino Beamline (BNB), which produces a neutrino flux with an estimated 93.6% ν µ purity [25] and a mean E ν of 0.8 GeV. At these energies, ν-Ar interactions are dominated by quasielastic and meson-exchange current interactions as well as resonant pion productions, and the final-state hadrons consist mostly of protons and neutrons with some charged and neutral pions. The O(1) MeV energy threshold [26] of LArTPC allows detecting these particles down to low kinetic energies.
Compared to earlier work [27], this measurement incorporates an improved TPC detector simulation and signal processing procedure [28,29], the Wire-Cell tomographic event reconstruction [30,31], and a many-to-many TPCcharge to PMT-light matching algorithm for cosmic-ray rejection [31]. In particular, the "generic neutrino detection" [32,33], which limits the cosmic-ray muon backgrounds to below 15% at over 80% ν µ CC selection efficiency, is used as a pre-selection. The ν µ CC event selection is further improved using a set of pattern recognition techniques, including i) neutrino vertex identification, ii) track/shower topology separation, iii) particle identification, and iv) particle flow reconstruction in the Wire-Cell reconstruction package [34]. Since many of the analysis details in this work including the event reconstruction, event selection, the overall model prediction along with its systematic uncertainties, and the model validation are in common with those in searching for an anomalous lowenergy excess in inclusive charged-current ν e channel that was documented in Ref. [35], they are only briefly reviewed in this letter.
First, the reconstructed neutrino vertex is required to be inside a fiducial volume, defined to be 3 cm inside the effective detector boundary [33]. Second, a set of dedicated background taggers are constructed to further reject residual muon backgrounds that entered the detector from outside based on directional information. Finally, neutral-current (NC) events are substantially reduced by requiring a reconstructed primary muon candidate to be longer than 5 cm. Some limited charged pion rejection is achieved by detecting large-angle scattering in reconstructed track trajectories. Using input variables from the background taggers, a multivariate classifier is constructed using the modern boosted decision tree (BDT) library XGBoost [36] that yields a ν µ CC selection with an estimated 92% purity and 68% efficiency [35]. In total, 11528 ν µ CC candidates are selected and used for crosssection extraction. About 1/3 of the events are fully contained (FC) and 2/3 are partially contained (PC). Here, the FC events are defined to be events with their main TPC cluster [31] fully contained within the fiducial volume [33] and PC events are mostly because of exiting muons.
Three methods are used to reconstruct the energy of tracks and electromagnetic (EM) showers [34,35]: i) The energy of a stopped charged particle track can be estimated by its travel range using the NIST PSTAR database [37]. ii) The kinetic energy of a charged particle track can be estimated by integrating over the reconstructed energy loss per unit length dE/dx, which is calculated from the measured dQ/dx (ionization charge per unit length) using a recombination model [38]. iii) The energy of an EM shower can be estimated calorimetrically by scaling the total reconstructed charge of the EM shower with a factor of 2.50, which is derived from simulation and includes the bias in the reconstructed charge [31] and the average recombination factor. This factor is validated with the reconstructed invariant mass of the neutral pion [39]. For stopping tracks with trajectories longer than 4 cm, the range method is used to estimate the energy. For short tracks (<4 cm), tracks  exiting the detector, tracks with "wiggled" topology [34] (e.g. low-energy electrons), and muon tracks with identified δ rays, the recombination method is used to estimate its kinetic energy. The reconstructed neutrino energy E rec ν per event is estimated by summing the kinetic energies of each reconstructed (visible) final-state particle. For each reconstructed muon, charged pion, or electron candidate, its mass is added to the energy reconstruction. An average binding energy of 8.6 MeV [40] was added for each proton identified. Figure 1 shows the FC ν µ CC distribution as a function of reconstructed neutrino energy, the selection efficiency as a function of true neutrino energy, and the smearing matrix between E rec ν and E ν according to the Monte Carlo simulation. The predicted energy resolution using the MicroBooNE MC for FC ν µ CC events is ∼10% for muon energy, ∼20% for neutrino energy, and ∼30-50% for hadronic energy. The hadronic energy resolution is dominated by the missing hadronic energy and imper-fect event reconstruction. For events well reconstructed, the resolution of the reconstructed visible hadronic energy approaches ∼10%. Among all events, the average bias (towards low energy) of E rec ν for FC ν µ CC events is less than 10% for E ν < 800 MeV and increases to ∼25% at E ν = 2.5 GeV.
The total and differential cross sections are extracted using the Wiener-SVD unfolding method [41] as follows: M i is the measured number of events in bin i of the reconstructed energy space, and B i is the expected number of backgrounds. R ij = ∆ ij · F j is the overall response matrix. S j , to be extracted, is the average (differential) cross section in bin j of the true energy, weighted by the nominal ν µ neutrino flux, which is tabulated in Ref. [27]. This definition of S j with the nominal neutrino flux coincides with a recommendation from Ref. [42] in addressing a concern on the treatment of neutrino flux uncertainty. ∆ ij , the ratio between the selected number of events in reconstructed energy bin i that originate from the true energy bin j and the generated number of events in bin j, is calculated using central-value MC. This encapsulates both the smearing between reconstructed and true space and the efficiency. F j is a constant that is calculated with the POT, number of Ar nuclei, the integrated nominal ν µ flux in bin j, and the bin width (for differential cross sections only). The Wiener-SVD unfolding is performed based on a test statistics and an additional regularization constructed from a Wiener filter [41]. V is the covariance matrix on the measured number of events in the reconstructed energy bins, encoding the statistical and systematic uncertainties for both signal and background events. Statistical uncertainties on the data are calculated following the combined Neyman-Pearson procedure [43]. The covariance matrix also includes several systematic uncertainties. The neutrino flux model uncertainty (5-15%) follows the work in Ref. [27]. It includes effects from hadron production of π + , π − , K + , K − , and K 0 L , together with total, inelastic, and quasielastic cross sections of pion and nucleon re-scattering on beryllium and aluminum. In addition, modeling of the horn current distribution and calibration is included. The neutrino-argon interaction cross section model uncertainties (∼20%) are described in Ref. [18]. Particularly, the uncertainties associated with the hadronic interactions, which are important in modeling missing energy, are conservatively estimated: the proton to neutron charge exchange and the proton knockout have 50% and 20% uncertainties, respectively [44,45]. The uncertainties on the GEANT4 models [46] used to simulate secondary interactions of protons and charged pions outside the target nucleus (∼1.5%) follows Ref. [47]. These uncertainties on the flux, cross section, and GEANT4 models are estimated using a multisim technique [48] in which parameters that govern interaction models are simultaneously varied in generating hundreds of universes to construct covariance matrices.
The detector response uncertainty follows the work in Ref. [49], considering the effects of variations in the TPC waveform, light yield and propagation, space charge effect [50,51], and ionization recombination model. For each source, the same set of MC interactions are resimulated through the detector response simulation with a 1σ change to the corresponding detector model parameter. The differences in the selected number of events between the modified and original simulations are used to construct a covariance matrix with a bootstrapping [52] procedure. The uncertainty of modeling the "dirt" events that originate outside the cryostat follows the work in Ref. [35]. The statistical uncertainty of the Monte-Carlo sample is treated using the methods described in Ref. [53]. The uncertainties on the POT (2% based on in-situ proton flux measurements [25]) and the number of target nuclei (∼1%) are also included.
Given Eq. (1), the uncertainties on the neutrino flux, GEANT4 model, detector model, and POT enter through B i and the numerator of ∆ ij . "Dirt" uncertainties enter through B i . In comparison, the cross section uncertainty enters through B i and both numerator and denominator of ∆ ij . Although the uncertainty on the predicted inclusive cross section is ∼20%, it is reduced to ∼5% because of the cancellation between numerator and denominator of ∆ ij .
A prior condition of using the Wiener-SVD unfolding method to extract cross sections is that the data must be well-described by the overall model prediction within its uncertainties. In Fig. 1 and Fig. 2, data and simulation are shown for key reconstructed kinematic variables including i) neutrino energy E rec ν , ii) muon energy E rec µ , iii) cosine of muon polar angle cos θ rec µ , and iv) hadronic energy E rec had . The compatibility between the data and prediction is demonstrated quantitatively by decent χ 2 /ndf values (ndf is the number of degrees of freedom) with corresponding p-values larger than 0.05 considering full uncertainties using the Pearson χ 2 [54]. To examine different components of systematic uncertainties, we further utilize the conditional covariance matrix formalism [55] to adjust the model prediction and reduce its uncertainties by applying constraints from data. Figure 2a) shows the comparison of the E rec µ distribution for PC ν µ CC in data to that of the model prediction after applying constraints from the FC E rec µ events. While the uncertainties are largely reduced, there is only a small change to χ 2 /ndf. The data and constrained model agree within uncertainties, verifying the modeling of the invisible energy of muons outside the active detector volume for PC events. Figure 2b) shows the comparison of the cos θ rec µ distribution for both FC and PC ν µ CC candidates in data with the model prediction after applying a constraint from the E rec µ distributions of the same set of ν µ CC can- didate events. Compared to the previous case, the correlated statistical uncertainties between the cos θ rec µ distributions and the E rec µ distributions are estimated with a bootstrapping procedure. While the uncertainties are significantly reduced after applying the constraint, the change to χ 2 /ndf is small, showing well-modeled muon kinematics.
We will examine the modeling of the mapping between the reconstructed energy of the hadronic system E rec had and the energy transfer to the argon nucleus ν = E ν −E µ after taking into account the muon results. The mapping of E rec had to ν (or E rec ν to true E ν ) relies on the overall cross section model to correct for the missing energy going into undetected neutrons, low-energy photons and other particles below the detection threshold. To validate the model, we examine the E rec had distribution for the FC ν µ CC candidates in data with that of the model prediction after applying constraints from two one-dimensional distributions in muon kinematics: E rec µ and cos θ rec µ in Fig. 2c). After applying constraints, the uncertainties on the model prediction for E rec had are significantly reduced because of the cancellation of common systematic uncertainties, such as neutrino flux. At the lowest energies, it reduces from 20% to 5%. Nevertheless, the χ 2 /ndf values still yields p-value above 0.5, indicating that the model describes the relationship between E rec had and E rec µ well within its uncertainty. In particular, the difference between the data and the model prediction in the first three bins of E rec had is significantly reduced after applying the constraints. This test further validates that the modeling of the missing hadronic energy can describe data within its associated uncertainty. We note the conditional covariance matrix formalism, which is used to update the MC predictions and their uncertainties given the data constraints (more details can be found in Ref. [35]), is only used in validating the overall model, and is not used in extracting cross sections through the unfolding procedure. With fake data, we show that the χ 2 /ndf has a significant increase with a shift of ∼15% in the hadronic energy fraction allocated to protons (mimicking a variation of the proton-inelastic cross section), and this procedure is also able to distinguish between two GENIE models (see Supplemental Material [56]). In addition, the model validation procedure is shown to be much more sensitive to detect an insufficient input model compared to the extracted cross sections.
With the overall model validated, the total and differential cross sections per nucleon are extracted. The binning of the unfolded results is chosen by considering the energy resolution and the number of samples in the true space. Considering both FC and PC samples, the total cross section divided by the bin-center neutrino energy is shown as a function of neutrino energy in Fig. 3a), where the bin-center is calculated as the flux-weighted average neutrino energy. Excluding the PC sample does not change the overall behaviour of the cross sections, but increases their uncertainties for neutrino energy above 1.2 GeV modestly. Besides the nominal cross-section   [58], and GiBUU 2019.08 [59] after applying the Wiener filter are quantitatively compared with the measurement through calculating χ 2 /ndf with the uncertainty covariance matrix obtained from the unfolding procedure. Note that these comparisons only incorporate the central predictions from various generators without their theoretical uncertainties, which are particularly important in constructing predictions in analysis. The central predictions of GENIE v3 and NuWro are disfavored compared to the other three. Particularly, the "Micro-BooNE MC" (tuned GENIE-v3 model [18]) has better agreement than GENIE v3.0.6, given the tuned GENIE-v3 model is constructed by fitting T2K data [60] in a similar energy range. Figure 3b) and c) show the flux-averaged differential cross sections as a function of muon energy (dσ/dE µ ) and energy transfer to the argon nucleus (dσ/dν). The same set of model predictions are compared to these measurements. The model comparison of dσ/dE µ shows a shape agreement with most models, although the normalization predictions differ. The central predictions of GENIE v3 and NuWro are more disfavored. The model predictions in dσ/dν show large variations, particularly in the low energy transfer (ν) region, where the shape difference contributes considerably to the χ 2 /ndf given the correlations in the uncertainty covariance matrix. The central prediction from GiBUU has the best agreement with data in the low ν region, but is systematically lower than data at high ν region, which could be originated from an underestimation of the cross sections in the nucleon resonance region beyond ∆. Considering all three cross-section results, the GiBUU prediction has the best agreement with acceptable χ 2 /ndf values, while the performance of the NEUT prediction is comparable. The central predictions of the other three models show larger disagreement.
In summary, we present a measurement of cross section as a function of the neutrino energy based on data from a broad-band neutrino beam. We report the nominal-flux weighted total inclusive ν µ CC cross sections σ (E ν ), and the nominal flux-averaged differential cross sections as a function of muon energy dσ/dE µ and energy transfer dσ/dν using the Wiener-SVD unfolding method [41]. A new procedure based on the conditional covariance matrix formalism [55] and the bootstrapping method [52] is used to validate the model of missing energies, which enables the first measurement of dσ/dν on argon and significantly adds value to the measurement of the total cross section as function of neutrino energy σ (E ν ). These results provide a detailed way to compare data and calculations beyond what is possible with existing flux-averaged total cross section results. With additional accumulated data statistics (up to 1.2×10 21 POT from BNB) in the MicroBooNE detector, additional neutrino cross-section measurements are expected that will lead to further model development and generator improvements for neutrino scattering in argon. the Royal Society (United Kingdom); and The European Union's Horizon 2020 Marie Sklodowska-Curie Actions. Additional support for the laser calibration system and cosmic ray tagger was provided by the Albert Einstein Center for Fundamental Physics, Bern, Switzerland. We also acknowledge the contributions of technical and scientific staff to the design, construction, and operation of the MicroBooNE detector as well as the contributions of past collaborators to the development of MicroBooNE analyses, without whom this work would not have been possible.