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First Measurement of Muon Neutrino Charged Current Quasielastic (CCQE) Double Differential Cross Section

Using a high statistics sample of muon neutrino charged current quasielastic (CCQE) events, we report the first measurement of the double differential cross section as a function of muon energy and angle for this process. The result features reduced model dependence and supplies the most complete information on neutrino CCQE scattering to date. Measurements of the absolute cross section as a function of neutrino energy and the single differential cross section as a function of 4-momentum transfer squared are also provided, largely to facilitate comparison with prior measurements. This data is of particular use for understanding the axial-vector form factor of the nucleon as well as improving the simulation of low energy neutrino interactions on nuclear targets, which is of particular relevance for experiments searching for neutrino oscillations.


CCQE EVENT SELECTION IN MINIBOONE
The MiniBooNE 2 detector, a spherical tank filled with mineral oil, is surrounded by 1280 8" photomultiplier tubes (PMTs) to detectČerenkov light from charged particles 3 . In the 19.2µs readout window, a "subevent" is defined as a timing cluster of PMT hits. The identification of ν µ CCQE interactions relies solely on the detection of the primary muonČerenkov light (first subevent) and the associated decay electronČerenkov light (second subevent) in these events [4]: 1 : ν µ + n → µ − + p 2 : µ − → e − +ν e + ν µ .
Note this interaction results in total 3 subevents, the primary interaction and 2 muon decays resulting in an electron and a positron. Although these events can be removed from the CCQE sample by requiring only one muon decay (a total of 2 subevents), there is still a significant number of CC1π + events that contribute to the CCQE background because one of the muon decays may be missed for various reasons. Among them, π + absorption is a large effect (>40%) with large uncertainty (∼30%). Additionally, the prediction of CC1π + backgrounds in the CCQE sample rely on the Rein and Sehgal's model [5] and final state interactions (FSIs) in the NUANCE event generator [6] which are not sufficiently accurate for a precise background prediction to measure the absolute CCQE cross section.

CC1π + BACKGROUND MEASUREMENT
Because of uncertainties in the CC1π + background predictions, we instead measure the CC1π + rate in our CC1π + data and the event generator is adjusted to match. By this, the predicted kinematic distribution of CC1π + events is modified, and the systematic error of CC1π + cross section is reduced to the level of the π + absorption uncertainty.
The left plot in of Figure 1 shows the Q 2 QE distributions 4 for data and Monte Carlo (MC) of the two samples before the reweighting of CC1π + MC events. The 2-subevent sample shows good shape agreement between data and MC. NUANCE uses the relativistic Fermi gas (RFG) model [7] for CCQE interactions. In the previous work, we adjusted 2 parameters in RFG model, the effective axial mass M e f f A and Pauli blocking parameter κ, to match the shape of the Q 2 QE distribution to data [4]. Note that analysis did not consider the overall normalization of events. The 3-subevent sample shows a large data-MC disagreement in both shape and normalization. Using these samples, a simultaneous fit was performed for the shape and normalization of the 3-subevent sample, and the normalization of the 2-subevent sample. These were then used to determine the CC1π + reweighting function which is shown in the inset plot of Figure 1b (left). In order to reduce the sensitivity to the details of the shape of the 2-subevent sample, only the 0.2< Q 2 QE (GeV 2 ) <0.6 region was considered for the normalization parameter of this function. The Q 2 QE shape of the CCQE sample was fit later although it has no impact on the cross section measurements. The effect of the CCQE normalization on the 3-subevent sample was minimal since the background from CCQE in this Q 2 QE region is small as can be seen in the left plot of Figure 1b. As a final step, with the measured CC1π + background incorporated, a shape-only fit to the 2-subevent (CCQE) sample is performed in order to extract revised CCQE model parameters [4]. The normalization of the CCQE sample is then extracted from the fit described above. The Q 2 QE distributions of data from all subevent samples is shown together with the MC prediction in the right plot of Figure 1. Data-MC agreement is good in both subevent samples. A fit to the 2-subevent sample provided adjusted CCQE model parameters, M e f f A and κ. This was a "shape-only" fit, that is, the MC was normalized with an arbitrary factor to have the same integrated event count as the background subtracted data. The fit yielded,  The open star indicates the best fit point and 1 − σ contour from the previous work [4]. Two regions are shown from the previous work, the larger area indicates the total uncertainty on the results including the background uncertainty [4].
The left plot of Figure 2 shows the Q 2 QE distribution of data, MC before, and MC after the fit with all sources of error. Data and MC after the fit agree within shape errors. The right plot of Fig. 2 is the 1 − σ contour regions of this fit together with the results from the previous MiniBooNE analysis [4]. Note that the current result is consistent (to within 1 − σ ) with κ = 1. This is because the CC1π + background resulting from the procedure in this work has changed by an amount only just consistent with the error assigned on the background in the previous work. The value for κ is quite sensitive to the CC1π + background at low Q 2 QE . However, the previous and current results are consistent at the 1 − σ level.
The effect of the new M e f f A is clearly seen in 2-dimensional plots. Figure 3 shows the data-MC ratio of CCQE candidate events as a function of muon kinetic energy T µ (GeV) and muon scattering angle cos µ . Note the muon energy and muon scattering angle observables are the basis of all reconstructed kinematics variables in the ν µ CCQE channel in MiniBooNE. In the left plot, we use the world averaged nuclear parameters (M e f f A = 1.03 GeV/c 2 , κ = 1.000) [8]. As can be seen, data-MC disagreement follows auxiliary lines of equal Q 2 . This is the same tendency observed in the previous CCQE analysis in MiniBooNE [4], indicating that data-MC disagreement is more likely due to an incorrect cross section prediction (=function of Q 2 ) than an incorrect flux prediction (=function of neutrino energy). After introducing the new M e f f A and κ (M e f f A = 1.35 GeV/c 2 , κ = 1.007), Fig. 3 right plot, data-MC disagreement is reduced and we obtain an improved cross section prediction across the entire kinematic space.
Note, this modification of the CCQE cross section prediction does not affect the CCQE absolute cross section measurement, presented below. Figure. 4 shows the flux-averaged double differential cross section, d 2 σ dT µ dcos µ , for the ν µ CCQE process. The fluxaveraged total cross section, an integral of the double differential cross section (−1 < cos µ < +1 and 0 < T µ (GeV) < ∞) is 9.412 × 10 −39 cm 2 . The total normalization error on this measurement is 10.8%.

Flux-averaged double differential cross section
The kinematic quantities, T µ and cos µ , have been corrected for detector resolution effects only. This result is the most model-independent measurement of this process possible with the MiniBooNE detector. No cuts on the recoil nucleons are used to define this process. The neutrino flux is an absolute prediction and was not adjusted based on measured processes in the MiniBooNE detector.   In addition to the experimental result, Figure 5 also shows the prediction for the CCQE process from the NUANCE simulation with three different variations of parameters in the underlying RFG model. The predictions are fluxaveraged and absolutely normalized. The RFG model is plotted with both the world-averaged CCQE parameters (M A = 1.03 GeV,κ = 1.000) and with the CCQE parameters extracted from this analysis (M A = 1.35 GeV, κ = 1.007). The model using the world-averaged CCQE parameters underpredicts the measured values significantly (by ≈ 30%). The model using the CCQE parameters extracted from the shape fit to the MiniBooNE CCQE data are within ≈ 10% of the data, consistent within the normalization uncertainty of ≈ 10%. The prediction with the CCQE parameters from this analysis scaled by 1.10 is also plotted and is in good agreement with the data.

Flux-unfolded total cross section
The flux-unfolded total cross section (σ [E QE,RFG ν ]) as a function of estimated neutrino energy E QE,RFG ν is shown in Figure 6. The quantity E QE,RFG ν is a model-dependent estimate of the neutrino energy obtained after correcting for both detector and nuclear model resolution effects. These results depend on the details of the nuclear model used for the calculation. The dependence is only weak in the peak of the flux distribution but becomes strong at E ν < 0.5 GeV and E ν > 1.0 GeV, in the "tails" of the flux distribution.
In Figure 6 data are compared with the NUANCE implementation of the RFG model with the world averaged parameter values (M e f f A = 1.03 GeV, κ = 1.000), and the parameters extracted from this work (M e f f A = 1.35 GeV, κ = 1.007). These are absolute predictions from the model -they are not scaled in any way. The measurement is ∼20% higher than the RFG model prediction with world average parameter values at the flux peak (700 − 800 MeV). The prediction with the RFG parameter values extracted from the shape-only fit to MiniBooNE CCQE data reproduces the data significantly better, to within 1σ for every point over the entire measured energy range. Figure 6(b) shows the CCQE results from the LSND [9] and NOMAD [10] experiments. It is interesting to note that NOMAD results are better described with the world-average M e f f A and κ values. 10 × MiniBooNE data with shape error MiniBooNE data with total error =1.000 κ =1.03 GeV,