Equation (6.6) shows that as well as measuring the number of events in each bin, the number of target nucleons, T , and the total integrated flux, φ, must also be known.
T is 5.5× 1029 nucleons, and is calculated from the known composition and mass of the components in the FGD [51].
φ is calculated from flux histograms provided by the beam group. Separate flux histograms are provided for each T2K run, and are then weighted by the good quality POT for each run, and summed to give the total flux histogram shown in Figure 6.2.
The total integrated νe flux for this analysis is 1.35× 1011 cm−2.
NEUT
NEUT provides a tool to extract predicted cross-sections on the elemental constituents of the FGD: carbon, hydrogen, oxygen, titanium, silicon and nitrogen. For each element, the tool produces histograms of the cross-section as a function of energy for each mode (CCQE, DIS etc). Bin widths of 50 MeV are chosen to match the flux histograms, and the cross-sections are evaluated at the centre of each bin. All the CC modes are then summed for each element, and the total cross-section per nucleon for the FGD is computed using
σFGD= �
e=C,H,O,Ti,Si,N
σefe
Ae
, (6.8)
where e denotes the element, σe the CC cross-section for that element, fe the fraction by mass of the FGD composed of that element, and Ae the number of nucleons per atom of that element. For reference, fe and Ae are listed for each element in Table 6.1.
Figure 6.3 shows the predicted cross-section as a function of energy for the FGD, and for carbon alone. Although the FGD is 86% carbon, there is a significant difference between the cross-section on carbon and the average cross-section on the FGD. This means that although this measurement is described as the νe CC inclusive cross-section on carbon, this is only an approximate statement.
Also shown in Figure 6.3are the flux-averaged cross-section predictions. These are calculated by multiplying the contents of the flux and cross-section histograms, and dividing by the total flux. The predicted flux-averaged CC νe cross-section on the FGD
νe cross-section measurement 133
Element fe (%) Ae
Carbon 86.1 12
Hydrogen 7.4 1
Oxygen 3.7 16
Titanium 1.7 48
Silicon 1.0 28
Nitrogen 0.1 14
Table 6.1: Composition of the FGD. fe is the fraction by mass of the FGD composed of that element. Ae is the number of nucleons in that element.
(GeV) Eν
0 1 2 3 4 5
/nucleon)2 cm-39 10× (σ CC eν
0 2 4 6 8 10 12 14 16 18
POT)21 /50MeV/102 flux (/cmeνT2K
0 2000 4000 6000 8000 10000
106
×
flux νe
T2K NEUT pred. (C) NEUT pred. (FGD)
NEUT flux-av. (C) NEUT flux-av. (FGD)
Figure 6.3: CC inclusive νe cross-section predictions from NEUT on carbon and the FGD as a whole. The curves are the cross-sections as a function of neutrino energy. The points are the flux-averaged cross-sections.
is 1.224× 10−38 cm2/nucleon. The mean neutrino energy, Emean, is used for the location of the “dot”, and the horizontal error bars follow the same convention as used for the T2K νµ CC inclusive measurement in 2013 [51]: variables Elower and Eupper are defined such that φ(Elower) = φ(Eupper) and
�Eupper
Elower
∂φ(E)
∂E dE
�∞
0
∂φ(E)
∂E dE = 90%, (6.9)
and the variances in the intervals [Elower, Emean] and [Emean, Eupper] are then calculated to obtain the errors, through
σ2− =
�Emean
Elower(E− Emean)2 ∂φ(E)∂E dE
�Emean
Elower
∂φ(E)
∂E dE (6.10)
σ2+ =
�Eupper
Emean (E− Emean)2 ∂φ(E)∂E dE
�Eupper
Emean
∂φ(E)
∂E dE , (6.11)
where σ− and σ+ are the extents of the lower and upper error bars. For the T2K νe
flux, Emean = 1.343 GeV, Elower= 0.025 GeV, Eupper = 3.175 GeV, σ−= 0.772 GeV and σ+ = 0.933 GeV.
There is no simple way in NEUT to calculate the flux-averaged differential cross-section as a function of the interaction kinematics. Instead, the ND280 Monte Carlo files are used. The number of generated events in each bin are counted, and divided by the number of target nucleons and the total simulated flux. As a cross-check to validate the method, the total flux-averaged cross-section is computed from the ND280 MC events. This gives a cross-section of 1.231× 10−38 cm2/nucleon, compared to the 1.224× 10−38 cm2/nucleon found using NEUT directly. This small discrepancy can be explained by the fact that the direct NEUT prediction was calculated by summing the contents of bins of width 50 MeV. Although the flux histogram is an average over the 50 MeV range, the cross-section is just evaluated at the central value. As the flux and cross-section can vary rapidly across each bin, the true flux-averaged cross-section of the bin is not necessarily the same as that found by multiplying the two histograms, and as the difference between the calculations is small, it shows that there is no problem with the method used to calculate the cross-sections from the ND280 MC files.
One final subtle point relates to applying the detector systematics. The nominal set of detector systematics affects which events enter each sample, and the weight of those events. These weights just reflect the detector efficiency, rather than the physics that
νe cross-section measurement 135
is being simulated. To keep the same predicted νe cross-section, events entering the νe
sample with a detector systematic weight wd must also be counted as events that are missed with a weight 1− wd. Applying this procedure means that the same predicted νe cross-sections are found when turning detector systematics off and when setting them to their nominal values.
GENIE
GENIE is the alternative neutrino interaction generator used by T2K, and is also used by many other neutrino experiments. Comparing the T2K data to the GENIE prediction is of more relevance to the neutrino community as a whole than comparing to NEUT, which is mostly used by the T2K and Super-Kamiokande collaborations.
Generating cross-section predictions for GENIE proceeds in a similar way to NEUT.
Cross-section splines directly from the generator are used to compute the total CC flux-averaged cross-section, and differential cross-section predictions are found from ND280 MC files. As a cross-check, the total cross-section from the MC files is compared to the direct prediction from the generator.
The GENIE cross-section splines give a total predicted CC νe cross-section on the material composition of the FGD of 1.072× 10−38 cm2/nucleon.
GENIE MC files are used to compute the differential cross-sections. Fewer GENIE files are available than NEUT MC files, and a total of 28.04× 1020 simulated POT is available, just 4.75 times the data POT. More POT would be desirable, but as the GENIE MC is only used to draw predicted cross-sections (and not used as part of the analysis) it is less critical that there are large statistics. As with the NEUT files, true νe CC interactions that occur in the FGD FV are selected, and a total flux-averaged cross-section of 1.083× 10−38 cm2/nucleon is found from the GENIE MC files. As was the case for NEUT, the prediction from the MC files is slightly different to the prediction from the cross-section splines (1.083× 10−38cm2/nucleon rather than 1.072× 10−38 cm2/nucleon).
Predictions
The flux-averaged predictions for the NEUT nominal, BANFF pre-fit, BANFF post-fit and GENIE nominal models are shown in Figure 6.4 (see Section 5.2.1 for details of the BANFF fit). Predictions for both the full electron kinematic phase-space and the
(GeV)
> 550MeV && cos(
pe
Figure 6.4: Flux-averaged cross-section predictions from ND280 MC. The left plot shows the prediction for the full electron kinematic phase-space; the right plot shows the restricted phase-space. The overflow and underflow bins (represented by > and
< markers) are normalised to the bin width shown on the plot.
Model σCCνe (×10−39 cm2/nucleon)
Full phase-space Restricted phase-space
NEUT 12.31 7.38
BANFF pre-fit 13.49 8.01
BANFF post-fit 11.85 6.89
GENIE 10.83 6.41
Table 6.2: Flux-averaged cross-section predictions from the NEUT and GENIE interaction generators. The restricted phase-space is defined by events with pe > 550 MeV/c and cos(θe) > 0.72.
restricted phase-space are shown. The restricted phase-space corresponds to events with pe > 550 MeV/c and cos(θe) > 0.72. Note that the BANFF post-fit re-weighting changes the shape of the νe flux as well as the cross-section parameters, which can be seen in the shifted mean energy and horizontal error bars. The predictions are also tabulated in Table 6.2. The differential cross-section predictions are shown in Figure 6.5.
νe cross-section measurement 137
> 550MeV && cos(
pe
> 550MeV && cos(
pe
Figure 6.5: Flux-averaged differential cross-sections in pe (top) and cos(θe) (middle) and Q2 (bottom) from ND280 MC. The left column shows the prediction for the full electron kinematic space; the right column shows the restricted phase-space. Underflow and overflow bins are marked by < and > respectively, and are normalised to the width shown on the plot.