Measurement of the charged-current electron (anti-)neutrino inclusive cross-sections at the T2K off-axis near detector ND280

The T2K neutrino beam is produced at the Japan Proton Accelerator Research Complex (J-PARC) by colliding 30 GeV protons with a graphite target. The pions and kaons produced in the target are focused by three magnetic horns and decay in flight to produce neutrinos. T2K can run with either forward horn current (FHC) or with reverse horn current (RHC) producing beams in neutrino or anti-neutrino enhanced mode, respectively.

The T2K beamline T2Kbeamline is simulated using FLUKA2011 fluka1 ; fluka2 , GEANT3 geant3 and GCALOR gcalor . The simulated yields of hadronic particles are tuned using the NA61/SHINE na61shine1 ; na61shine2 ; na61shine3 thin target measurements. The neutrino fluxes at the off-axis near detector ND280 in FHC and RHC are shown in Figure 1. The off-axis position of the near detector, from the neutrino beam direction, results in a narrow-band $\nu_{\mu}$ or $\bar{\nu}_{\mu}$ beam, however, the same does not occur with $\nu_{e}$ and $\bar{\nu}_{e}$ fluxes due to their production via three-body decays, resulting in broader $\nu_{e}$ and $\bar{\nu}_{e}$ spectra. The mean of the $\nu_{e}$ energy spectrum at ND280 is 1.28 GeV in FHC and 1.98 GeV in RHC. The mean of the $\bar{\nu}_{e}$ energy spectrum in RHC is 0.99 GeV. The total integrated $\nu_{e}$ flux at ND280 in FHC is $\Phi_{\nu_{e}}^{FHC}=\left(2.67\pm 0.24\right)\times 10^{11}$ $\rm neutrinos/cm^{2}$ and in RHC is $\Phi_{\nu_{e}}^{RHC}=\left(2.65\pm 0.21\right)\times 10^{10}$ $\rm neutrinos/cm^{2}$. The total integrated $\bar{\nu}_{e}$ flux at ND280 in RHC is $\Phi_{\bar{\nu}_{e}}^{RHC}=\left(1.00\pm 0.10\right)\times 10^{11}$ anti-neutrinos$\rm/cm^{2}$.

The $2.5^{\circ}$ off-axis near detector, ND280, is located 280 metres from the proton target. The main goal of ND280 is to constrain the neutrino flux and the interaction cross-sections. It is composed of several sub-detectors located inside a 0.2 T magnet, as depicted in Figure 2. The front part is the $\pi^{0}$ detector (P0D) ND280P0D and is optimised to measure neutrino interactions with $\pi^{0}$ production. The rear part is the tracker and it is optimised to measure charged particles produced in neutrino interactions. It consists of two Fine-Grained Detectors ND280FGD , the first of which is composed of layers of plastic scintillator (FGD1) and the second has alternating layers of plastic scintillators and water (FGD2).

The P0D, FGD1 and FGD2 provide the target mass for neutrino interactions and each is followed by a Time Projection Chamber (TPC1, TPC2 and TPC3) ND280TPC . The TPCs are filled with a gas mixture based on argon and provide excellent track reconstruction with a momentum resolution of roughly 8% for 1 GeV/c tracks. This can be combined with energy loss ($dE/dx$) measurements in order to perform PID of tracks crossing the TPCs. The measured and the expected $dE/dx$ are used to define the "pull" (the difference between the measured mean ionization and the expected one divided by the resolution) of each particle species. The TPC energy loss for negatively and positively charged tracks originating in FGD1 is shown in Figure 3. Notice the region below 200 MeV/c where the electron $dE/dx$ curve crosses with the muon and pion dE/dx curves, and the region around 1 GeV/c where the proton $dE/dx$ curve crosses with the electron $dE/dx$ curve.

The P0D and the tracker are surrounded by the lead-scintillator Electromagnetic Calorimeter (ECal) ND280ECal and a Side Muon Range Detector (SMRD) ND280SMRD . The ECal measures the energy of photon and electrons (EM energy) and provides additional PID for minimum ionizing particles (MIP), electromagnetic showers (EM) and highly ionizing stopping particles (HIP) like protons.

The ECal EM energy is reconstructed under the hypothesis that the energy deposit is due to an electromagnetic shower. Comparing the TPC momentum with the ECal EM energy, electrons can be separated from muons and protons. The ratio of the TPC momentum over the ECal EM energy peaks at unity for electrons and at lower values for muons and protons. The ECal EM energy resolution is approximately 10% at 1 GeV.

The ECal PID is based on the longitudinal and lateral profile of ECal clusters to generate probability density functions (PDFs). These are combined for each particle type and PID variable to form a likelihood from the products of the PDFs, see ND280NueSel for details.

$R_{MIP/EM}$ is the log-likelihood ratio of the MIP and electron hypothesis and $R_{EM/HIP}$ is the log-likelihood ratio of the electron and proton hypothesis. The $R_{MIP/EM}$ for high purity control samples (90% or better) is shown in Figure 4, where the muon sample comprises cosmic muons and muons produced by neutrino interactions outside ND280 that cross the detector (through-going muons), the electron sample is formed from electron-positron pairs from photon conversions and the protons are from neutrino interactions. The ECal can provide supplementary PID to the TPC, especially in the region around 1 GeV/c where the TPC energy loss curves of electrons and protons cross. Figure 4 also shows the $R_{EM/HIP}$ for showers (classified by $R_{MIP/EM}>0$) with $p>600$ MeV/c only. Although there are some shape differences in data and simulation for $R_{MIP/EM}$ and $R_{EM/HIP}$, the data and simulation efficiencies to select electron (or positrons) and reject muons and protons are similar. The PID efficiencies in the simulation are corrected using the data control samples.

A MC event is defined as signal if the selected primary track is an electron (positron) from a CC-$\nu_{e}$ (CC-$\bar{\nu}_{e}$) interaction with the vertex inside the FGD1 fiducial volume, which has a total mass of 919.5 kg, corresponding to $\left(5.54\pm 0.04\right)\times 10^{29}$ nucleons. Backgrounds are separated into four categories: photon, muon, proton and other backgrounds. The photon background category considers events where the selected primary track is an electron or positron from a photon conversion and the true conversion point is inside the FGD1 fiducial volume. Events where the selected primary track is a muon (proton), but misidentified as electron enter the muon (proton) background category. Any other backgrounds including misidentified pions, electrons from photons converting outside of the fiducial volume but reconstructed inside the fiducial volume, electrons from $\pi^{0}$ Dalitz decay and Michel electrons go into the other background category.

The event selection for CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ events is described in the following:

1. (i)

   Only events during periods of good beam and detector quality are used. The event time has to be reconstructed within one of the eight distinct beam bunches.

2. (ii)

   The highest momentum negatively charged (leading negatively charged) FGD1-TPC track, for the CC-$\nu_{e}$ selection, or the highest momentum positively charged (leading positively charged) FGD1-TPC track, for CC-$\bar{\nu}_{e}$ selection, with a vertex in the FGD1 fiducial volume is selected. The leading positively charged track in the CC-$\bar{\nu}_{e}$ selection must also be the highest momentum track (from all negatively and positively charged tracks).

3. (iii)

   To ensure reliable PID and momentum measurements, the selected leading track is required to have at least 18 TPC hits if it enters the ECal or 36 TPC hits if it does not enter the ECal. The momentum spectra of the selected leading negatively charged and leading positively charged tracks with the minimum number of TPC hits are shown in Figure 5. Notice the large number of protons selected as the leading positively charged track in the RHC CC-$\bar{\nu}_{e}$ selection. Some data-MC discrepancies are visible in the low momentum region which contains the poorly modelled photon and other backgrounds.

   

   

   

4. (iv)

   TPC PID is applied to select electrons and remove minimum-ionizing tracks. Using the electron TPC pull, the leading track must agree with the electron TPC $dE/dx$ hypothesis. If the leading track does not enter the ECal, then additional cuts on the TPC PID are applied using the muon and pion TPC pulls. The event is rejected if the leading track agrees with the muon or pion TPC hypothesis; events around 150 MeV/c, including electrons, are rejected where the only information (TPC) is unable to distinguish them.

5. (v)

   Additional PID is applied using either the ECal EM energy or the ECal PID depending on the momentum of the leading track as it enters the ECal. To maximize the efficiency, if the leading track has $p>1$ GeV/c and is fully contained in the ECal, the reconstructed ECal EM energy is used to separate EM showers from MIPs and it is required to be larger than 1 GeV. Otherwise the ECal MIP/EM shower PID discriminator $R_{MIP/EM}$ has to agree with the EM shower PID hypothesis. Events that pass the TPC and ECal PID are shown in Figure 6. For the CC-$\bar{\nu}_{e}$ selection a complication arises since the TPC energy loss curves for positrons and protons cross around 1 GeV/c (see Figure 3) leaving a significant amount of proton background.

   

   

   

6. (vi)

   Search for the paired FGD1-TPC electron or positron track from a potential photon conversion. The paired track must have opposite charge than the leading track, start within 5 cm from the leading track and agree with the electron TPC dE/dx hypothesis. If several paired tracks are found, the pair with the lowest invariant mass is considered since it is more likely to come from a photon conversion. Pairs with invariant mass less than 110 $\rm MeV/c^{2}$ are removed.

7. (vii)

   Veto P0D, TPC and ECal activity upstream of the vertex and remove events with additional vertices in FGD1. Events with multiple vertices more likely to come from a $\nu_{\mu}$ interaction with one or more $\pi^{0}$ in the final state.

8. (viii)

   For the CC-$\bar{\nu}_{e}$ selection, additional selection criteria are applied if the leading positively charged track has $p>600$ MeV/c, the region which is contaminated by the proton background. If the leading positively charged track produce shower activity in FGD2 then it is selected. If the leading positively charged track enters the ECal, the proton background can be removed by comparing the ECal EM energy ($E$) and the TPC momentum ($p$) using a cut $E/p>0.65$. In addition, the $R_{EM/HIP}$ shower PID discriminator has to agree with the EM shower hypothesis.

9. (ix)

   For the CC-$\bar{\nu}_{e}$ selection, if the leading positively charged track stops in FGD2, the FGD2 energy loss must not agree with the proton hypothesis.

10. (x)

    Remove external background by comparing the time stamps of the leading track between FGD1 and ECal. This cut aims to remove tracks originating in the ECal and stop in FGD1 but are mis-reconstructed with the wrong direction.

11. (xi)

    Check if the leading track is broken inside FGD1. A track is broken if it originates in FGD1 and is not reconstructed as a single track, but is broken into two or more components. In such pathological cases the leading track could originate outside the fiducial volume but mis-reconstructed within it. If the leading track follows an isolated FGD1 track then the event is removed.

Figure 7 summarises the CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ selections.

The momentum and angular (with respect to the neutrino direction) distributions of all selected CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ candidates are shown in Figures 8 and 9, respectively. These plots also show the total systematic uncertainty on the MC event yields, which is discussed in section 6. A significant data deficit is observed at low momentum ($p<600$ MeV/c) in the FHC CC-$\nu_{e}$ channel. In this region the photon background is dominant which has significant systematic uncertainties associated with the $\pi^{0}$ production. Roughly a third of the photon background comes from neutrino interactions outside of FGD1. These external backgrounds could originate from neutrino interactions on heavy targets, like iron, copper or lead with significant final state interaction systematic uncertainties. In addition, roughly another third of the photon background come from NC interactions which are poorly measured. The final third of the photon background come from CC-$\nu_{\mu}$ and CC-$\bar{\nu}_{\mu}$ interactions, usually when the muon is emitted in high angles and it is lost. In such occasions the most energetic of the other tracks is selected as the leading track. A similar data deficit is also observed in the statistically poorer RHC CC-$\bar{\nu}_{e}$ channel. In addition, an excess of events has been observed in the RHC channels at high momenta (more visible in the RHC CC-$\bar{\nu}_{e}$ channel). For the photon background produced from $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ interactions in FGD1, roughly 10% is coming from NC DIS interactions in all three selections. The relevant fraction of FGD1 CC DIS events entering the photon background is approximately 4% in CC-$\bar{\nu}_{e}$ selection, 16% in RHC CC-$\nu_{e}$ selection and 20% in FHC CC-$\nu_{e}$ selection. The differences are due to the additional selection criteria applied to CC-$\bar{\nu}_{e}$ and the presence of protons which can be selected as the leading track instead of the primary muon or background positron.

Most of the efficiency loss is observed at low momentum since the electron and muon/pion $dE/dx$ energy loss curves cross around 150 MeV/c (see Figure 3). In addition, high angle tracks that do not enter the TPC are not selected and the events are lost. Another important source of efficiency loss is due to electron shower or bremsstrahlung in FGD1. As a result the primary electron track does not enter the TPC or another track is selected as the leading track. As estimated from the MC, 35-45% of the signal electrons or positrons are lost because the primary electron track does not enter the TPC. The efficiency loss is larger in the FHC CC-$\nu_{e}$ channel since the electron momentum spectrum is softer and at higher angles. The true vs reconstructed momentum and angular distributions in the MC for the selected signal electrons and positrons are shown in Figure 10. The effect of bremsstrahlung is visible as the reconstructed momentum spectrum is biased towards lower momenta. A summary of efficiencies and purities is shown in Table 1.

The muon mis-identification probability (probability of a muon to be mis-identified as an electron after applying the PID) was studied in previous T2K publications ND280NueSel with very good agreement between data and MC. Similarly, the proton mis-identification probability is important for the CC-$\bar{\nu}_{e}$ selection. A high-purity independent sample of protons has the same PID criteria as the CC-$\bar{\nu}_{e}$ selection applied and the number of protons that survive is checked. An independent control sample that can be used is the FHC CC-$\bar{\nu}_{e}$ selection. This channel has a tiny signal contribution and a much larger proton background and it is not used in the cross-section measurements. Before applying the proton rejection cuts (viii) and (ix), approximately 94% of the leading tracks selected with $p>600$ MeV/c and not entering the ECal are protons. The measured proton mis-identification probability is the fraction of protons that survive from these independent proton enriched samples and is $\left(4.6\pm 0.8\right)$% for the data compared to $\left(5.0\pm 0.3\right)$% in the MC. The errors are statistical only. The proton purity is lower in the case where the leading track enters the ECal and is approximately 70% with the rest to be mostly positrons. Due to the relatively low proton purity of this sample, only an approximate proton mis-identification probability can be measured in this case, $\left(9.4\pm 0.1\right)$% in the data compared to $\left(11.9\pm 0.05\right)$% in the MC.

The CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ selections in the MC are repeated using GENIE (2.8.0) instead of NEUT (5.3.2) MC. There are some differences between these two neutrino generators, see section 3 and for more details the description in ND280numucc4pi . One of the most important is that the neutrino multi-nucleon interaction simulations are turned-off in this version of GENIE. Efficiencies and purities for NEUT and GENIE agree quite well. Compared to the selected events, both NEUT and GENIE predictions disagree with data at low momenta with the FHC CC-$\nu_{e}$. The prediction of the photon background in particular is similar in both neutrino generators. Tables 1 and 2 summarize the event selections using NEUT and GENIE.

The steps to select photon candidates are:

1. (i)

   Only events during periods of good beam and detector quality are used. The event time has to be reconstructed within one of the eight distinct beam bunches.

2. (ii)

   The highest momentum negatively charged or highest momentum positively charged FGD1-TPC track (leading track) with a vertex in the FGD1 fiducial volume is selected.

3. (iii)

   The leading track must be compatible with the electron TPC $dE/dx$ hypothesis. If the leading track enters the ECal, and has momentum $p>1$ GeV/c and ECal energy E, then $E/p>0.5$ is required in order to clean up the high momentum tail.

4. (iv)

   Require a second track with opposite charge to the leading track, also compatible with the electron TPC $dE/dx$ hypothesis and with a starting position within 5 cm from the primary track.

5. (v)

   The invariant mass calculated from the leading and paired tracks must be less than 55 $\rm MeV/c^{2}$. The distributions of the invariant mass of the selected $e^{-}e^{+}$ pairs are shown in Figure 11. The invariant mass cut is very effective to remove backgrounds from misidentified muons, protons and electrons from CC-$\nu_{e}$ interactions.

   

   

   

6. (vi)

   Although the photon selection at this stage is very pure, it is contaminated by external photons (photons from neutrino interactions outside FGD1). To remove external photons the same veto cuts used in the CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ selections are applied.

The signal and background categories are the same as for the CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ selections. The momentum and angular distributions of the selected photon candidates are shown in Figures 12 and 13, respectively. The systematic uncertainties on the MC event yields are also shown in these plots, see section 6 for details. A MC excess below 300 MeV/c is visible. In the angular distributions a significant MC excess is observed at high angles in the FHC CC-$\nu_{e}$ selection but not in the photon control selection (Figures 9 and 13).

The purity of the photon control samples is approximately 80% when selecting electrons and 85% when selecting positrons. A significant fraction of the selected photon candidates is classified in the other background category where the photons are coming from a true conversion point outside the FGD1 fiducial volume, but are mis-reconstructed inside of it. Including these events in the photon category definition increases the purity to approximately 90%. The rest of the other background contributes (5 - 6)% in the photon control samples and comes from $\pi^{0}$ Dalitz decay, general mis-reconstructions like broken tracks and accidental matching when at least one of the two tracks selected in the pair is not electron or positron. The signal leakage (CC-$\nu_{e}$ or CC-$\bar{\nu}_{e}$) in the photon control samples is around (3 - 4)% when the selected leading track is an electron. The leakage is otherwise negligible when the selected leading track is a positron. The muon background entering the photon control samples is less than 1% in all of the cases.

When selecting electrons as the leading track in the photon control selections, approximately 40% of the photon candidates come from external photons, approximately 30% come from NC interactions in FGD1 and the other 30% come from CC interactions in FGD1. When selecting positrons as the leading track in the photon control selections the contributions are slightly different. Approximately 45% of the photon candidates come from external photons, approximately 35% come from NC interactions in FGD1 and 20% come from CC interactions in FGD1. Often the event is rejected if the selected highest positively charged momentum track is a proton. However, since the protons are invisible when selecting negatively charged tracks the same event could be selected when searching for the highest momentum negatively charged track. This explains the difference in the number of photon candidates in RHC when the leading track selected is the electron or the positron.

Although the photon control samples are of high purity they have some differences compared to the photon background entering the CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ selections. The main reason is that the photon control selection requires both the electron and positron to be reconstructed in the TPC, while the photon background is mostly related to events where either the electron or positron is lost, usually when it is not very energetic or emitted at high angles. As a result, the photon background consists mostly of highly asymmetric events where most of the energy of the photon goes into one of two electrons. For high angle events it is predominantly due to one of the two electrons being lost, resulting in more high angle photon background in the CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ selections.

This angular dependence will introduce different external photons to the photon background and the photon control selection. Most of the external photons entering the photon control samples come from neutrino interactions in the P0D or in the aluminium frame of TPC1. For the photon background, however, a significant population of external photons are also from neutrino interactions in the ECals. The photons mostly come from $\pi^{0}$ decays and Table 3 shows the different contributions to the photon background and the photon control selections from CC and NC interactions and from external photons. Despite the differences discussed, the origin of the photon background entering the CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ selections and the photon control selections is similar. This provides confidence that the photon control samples can be used to constrain the photon background in the signal channels. Additional simulation studies are also performed to check for shape variations in momentum and angle in the photon selections and in the photon background in the signal selections. These studies include the variation of the relevant fraction of CC/NC photon events by a factor of 2, weighting the nominal MC by varying the Delta resonance width by $\pm 1\sigma$ and varying the external photon background between (40 - 75)% based on the target material the neutrino interaction occurred. In all the cases the effect on the momentum and angular shapes is found to be very small.

A summary of systematic uncertainties on signal and background MC event yields for the CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ selections is shown in Table 5. The systematic uncertainties on signal yields are dominated by the flux (8 – 10%) and cross-section modelling (13 – 14%). The larger cross-section systematic uncertainties come from the large uncertainties considered on the quasi-elastic axial mass $M_{A}^{QEL}$ and multi-nucleon interactions, each contributing (6.5 – 8.5)% to the total cross-section systematic uncertainty. Detector systematic uncertainties on signal yields are (2 – 4)% with the most important being the TPC PID and TPC-ECal matching efficiencies. For CC-$\bar{\nu}_{e}$, the ECal PID and FGD2 shower efficiency, which are related to the proton background rejection, are also important. For an inclusive CC selection, final state interaction systematic uncertainties on signal yields are small. They are only considered if a charged pion, after final state interactions, becomes more or less energetic than the primary electron or when there is a $\pi^{0}$ involved as the secondary electrons can be more or less energetic than the primary electron. The total systematic uncertainty on the signal yields is approximately (16 – 17)% in all the channels.

The systematic uncertainties on the MC background event yields are separated into photon background and all other backgrounds. The total systematic uncertainties on the MC photon background event yields are approximately (23 – 26)% in all channels and are dominated by the cross-section and external systematic uncertainties. Cross-section systematic uncertainties (16 – 19)% are dominated by the charged-current and neutral-current resonant and DIS production models. The flux systematic uncertainties are around 8% and the final state interaction systematic uncertainties are (1.5 – 3.0)%. Detector systematic uncertainties are (3 – 6)%, with TPC PID, FGD1 and ECal time resolutions, TPC-ECal matching efficiency and pion secondary interactions being the most important. Approximately a third of the photon background comes from neutrino interactions outside FGD1, either in other sub-detectors or outside the ND280 and the majority of these events populate the low momentum and/or high angle regions.

The systematic uncertainties on the other backgrounds MC event yields vary from (19 – 33)% since different sources of backgrounds contribute to each channel. The biggest difference comes from the external background which dominates the systematic uncertainties on the other background event yields and is different in each channel since the neutrino flux is different. Flux systematic uncertainties are around 8% and the cross-section systematic uncertainties are around (11 – 12)%. Detector systematic uncertainties are (4.0 – 6.5)%, which are larger than the corresponding detector systematic uncertainties for signal and photon background event yields.

The systematic uncertainties on the photon control samples are roughly (20 – 23)% and are summarised in Table 6. The dominant sources are coming from the external background and cross-section modelling.

Systematic uncertainties are included in the fit as nuisance parameters and are calculated as $\pm 1\sigma$ variations of the nominal samples in the signal and photon control samples, $S_{ij}(\vec{\theta})$, $B_{ij}^{\gamma}(\vec{\theta})$, $B_{ij}^{other}(\vec{\theta})$, $B_{il;PC}^{\gamma}(\vec{\theta})$ and $B_{il;PC}^{other}(\vec{\theta})$. These variations can either change the normalisation or produce bin-dependent shape differences or have a correlated effect on shape and normalisation. For each variation (or set of variations) a nuisance parameter is used to interpolate between the $\pm 1\sigma$ uncertainties with a Gaussian constraint. Systematic uncertainties that are common between samples or channels are fully correlated in the fit. A summary of the nuisance parameters included in the fit is shown in Table 7.

Variations from the cross-section uncertainties are calculated by varying each cross-section parameter by $\pm 1\sigma$ and changing the nominal samples. Some of the cross-section uncertainties may produce asymmetric variations and these are considered in the fit. Variations related to the final state interaction systematic uncertainties, including their correlations, are estimated following the methodology described in T2KOscLong .

Variations from the flux uncertainties are calculated using the full beam covariance taking into account all the correlations between the neutrino beams, neutrino flavours and energy bins.

Variations of the nominal samples arising from the detector, pile-up and external background systematics are evaluated using MC simulations varying the systematics to change the number of events in each reconstructed bin. Three nuisance parameters are used for the three pile-up systematics in each beam mode (FHC or RHC). Four nuisance parameters in each beam mode are used to describe the external background systematic uncertainties. The external backgrounds are separated based on their origin (ND280 or sand interactions), their background category (photon or other backgrounds) and their beam mode (FHC or RHC).

MC statistical uncertainties, describing the finite size of the simulated events in each sample, are also included as nuisance parameters in the fit following the Barlow-Beeston barlowbeeston approach considering one nuisance parameter per channel and bin.

The choice of the binning depends on a number of factors, some of the most important are: sufficient number of signal events in each bin, isolation of the backgrounds in specific $p-\theta$ regions, event migration due to bremsstrahlung and flat efficiency corrections.

The first criterion for the binning choice is to not consider high angle events ($\theta>45^{\circ}$) since the acceptance due to detector effects is almost zero. In addition, the photon background is large and the statistics in the photon control channels is poor. The high angle regions and the low momentum ($p<300$ MeV/c) bins are background enriched and are kept in the fit as background validation regions. Approximately 75% of the external photon background is contained in the low momentum and high angle regions. The signal contribution in the low momentum bins ($p<300$ MeV/c) is tiny and, to help the fit performance, it is kept constant in the fit. Two angular regions are considered to better describe the middle and forward angular cases. The momentum bins are identical in each angular region.

The momentum bins are optimised to minimize the effect of bremsstrahlung. Since bremsstrahlung is not a detector smearing effect, but a physics effect depending on the initial electron kinematics and the material propagated, special requirements are considered to minimize this effect.

The (anti-)correlations between the momentum bins introduced by bremsstrahlung are studied with MC simulations requiring them to be less than 50%. If the chosen momentum binning fails this requirement, the momentum bins are expanded to reduce the migration of events due to bremsstrahlung and the MC simulations are repeated. Due to the large momentum bins chosen in this analysis, the effect of bremsstrahlung can be efficiently handled in the fit.

Signal efficiencies are also a significant factor for the binning choice as they should be flat to avoid model dependencies. The efficiencies in the two angular regions in each signal channel are shown in Figure 15 and are relatively flat with some small fluctuations observed between NEUT and GENIE and in the low statistics bins. Although the cross-section measurements are calculated in one dimension, fitting in $p-\theta$ is important to check for model dependencies due to the efficiency corrections. After the total statistical and systematic uncertainties are applied on signal efficiencies, the efficiency errors are artificially inflated to cover differences between NEUT and GENIE and variations between momentum bins. The efficiencies with statistical, systematic and inflation uncertainties are shown in Figure 16.

The binning choice for each signal channel is shown in Table 8. In total there are 9 free parameters controlling the photon background (one for each angular region) and 17 free parameters controlling the signal (one for each bin in the table, except for the six lowest momentum bins which are kept constant in the fit since the number of signal events is negligible).

The total cross-sections in the measured phase-space ($p>300$ MeV/c and $\theta\leq 45^{\circ}$) using NEUT and GENIE MC are shown in Table 9. The results are compatible with the NEUT and GENIE predictions, although larger cross-sections are measured in RHC, but with large statistical uncertainties.

Using the NUISANCE framework nuisance , the fit results are compared to cross-section predictions from recent neutrino generator models in NEUT (5.4.0), GENIE (2.12.10) and also from NuWro (19.02) nuwro . NEUT 5.4.0 uses a local Fermi gas (instead of spectral function). Other interaction modelling and final state interactions are similar to NEUT 5.3.2 (detailed in section 3). GENIE 2.12.10 interaction modelling is similar to 2.8.0 (detailed in section 3), with the "empirical MEC" model for the description of multi-nucleon interactions enabled. NuWro simulates the CC quasi-elastic process with the Llewellyn-Smith model with axial mass value of 1.03 $\rm GeV/c^{2}$. The nuclear model is simulated using the relativistic Fermi gas including random phase approximation corrections rpa . Multi-nucleon interactions are simulated similar to NEUT using the model from Nieves2p2h . For pion production a single $\Delta$-model by Adler-Rarita-Schwinger adler is used for the hadronic mass W $<$ 1.6 $\rm GeV/c^{2}$ with axial mass value of 0.94 $\rm GeV/c^{2}$. A smooth transition to DIS processes is made for W between 1.3 and 1.6 $\rm GeV/c^{2}$. The total cross section is based on the Bodek and Yang approach BodekYangNeut . Similar to NEUT, final state interactions are simulated using a semi-classical cascade model.

The comparisons of the data to NEUT 5.4.0, GENIE 2.12.10 and NuWro 19.02 are shown in Figure 19. A $\chi^{2}$ between the data measurements and each neutrino generator model predictions is defined as

where $k$ is either $p$ or $\cos(\theta)$, $\frac{d\sigma_{i}^{meas.}}{dk_{i}}$ is the differential cross-section measurement in bin $i$, $\frac{d\sigma_{i}^{model}}{dk_{i}}$ is the differential cross-section model prediction in bin $i$ and $V_{ij}$ is the covariance matrix as defined in equation 3 and shown in Figure 18. The $\chi^{2}$ is measured for each neutrino generator individually and is summarised in Table 10. NEUT 5.4.0 has the lowest $\chi^{2}$ compared to our data. GENIE 2.12.10 has a slightly larger $\chi^{2}$. The $\chi^{2}$ for NuWro 19.02 is significantly larger. The $\chi^{2}$ is also calculated individually for the single differential cross-sections $d\sigma/dp$ and $d\sigma/d\cos(\theta)$. A reduced covariance is used considering only the momentum or $\cos(\theta)$ part of the full covariance in Figure 18. In these cases the $\chi^{2}$, in both momentum and $\cos(\theta)$ measurements, are smaller and similar for all neutrino generators. This highlights the importance of using the combined cross-section measurements in momentum and $\cos(\theta)$ when doing model comparisons, rather than using each cross-section measurement in momentum or $\cos(\theta)$ individually.