Extrapolation (the Beam Matrix)

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5.3 Extrapolation (the Beam Matrix)

The MINOS experiment was designed with two detectors in order to reduce the effect of systematic uncertainties. Many systematics, such as the neutrino flux, the neutrino cross section, and the modeling of the hadronic energy, affect both detectors in the same way, allowing them to effectively

“cancel out.” For example, imagine the actual neutrino flux were 10% higher than simulated. With only one detector those 10% more events at the Far Detector would change the apparent oscillation probability. With two detectors, however, that flux increase would increase the number of events at the Near Detector as well. So, if we predict the Far Detector based on the Near Detector that 10%

increase would be expected and thus not affect the measured oscillation probability.

There is an important caveat though. The example above relied on the change in the flux having

θf To far detector

Decay Pipe

!

⁺

!

⁺_(soft)

(stiff)

θn Target

ND

Figure 1: A diagram of neutrino parents in the NuMI decay pipe, illustrating the different solid angles subtended by the near and far detectors at the parent decay point. (Figure taken from [10].)

Figure 2: Simulated true neutrino energy spectra in the near (left) and far (right) detectors. Parents responsible for producing neutrinos in the hatched near detector energy bins produce a different neutrino energy distribution in the far detector, shown by the corresponding hatching. (Figure taken from [4].)

Figure 2 illustrates the effect of parent decay kinematics on the near and far detector energy spectra. It highlights how parents responsible for certain near detector neutrino energies produce a different far detector neutrino en- ergy distribution. The effect is most pronounced for higher energy parents (and therefore higher energy neutrinos) which typically travel further along the decay pipe before decaying: the neutrinos are then produced closer to the near detector, allowing a wider range of contributing decay angles. (This effect outweighs the increased Lorentz boost of the higher energy parents, which narrows the range of decay angles in the laboratory frame.) The effect primarily lowers the typical energy of a neutrino at the near detector in com- parison to the far as the parents are travelling predominantly towards the far

5

Figure 5.10: Diagram of the neutrino parents in the NuMI decay pipe. A parent will typically have a wide range of neutrino decay angles that reach the Near Detector and a very narrow range that will reach the Far Detector.

the same effect at both detectors. This comes from an implicit assumption that both detectors see an identical flux. However, for MINOS this is not quite true due to the kinematics of the decays that produce the neutrinos. For a given parent, the energy of the daughter neutrino in the parent’s rest frame is fixed:

Eν^∗= m²p−m²µ

2mp (5.8)

where mp is the mass of the parent (p=π^±, K^±), mµ is the mass of the muon and the neutrino mass is negligible. However, the energy of the neutrino in the lab frame depends on the relative angle,θ, between the parent’s direction of travel and the neutrino’s:

Eν= Eν^∗

γp(1−βpcosθ) (5.9)

where γp is the parent’s Lorentz factor and βp is its velocity. The flux is also a function of angle.

While the parent emits the neutrino isotropically in its rest frame, once boosted to the lab frame the flux becomes angle-dependent:

dN

dcosθ = 1

2γ²p(1−βpcosθ)² (5.10)

where againγpis the parent’s Lorentz factor andβpis its velocity. The derivations of these formulas can be found in Appendix A.

The Far Detector is sufficiently distant so that for a given parent, there is only one narrow range of angles that will produce a neutrino that will reach the Far Detector, uniquely determining the neutrino energy from that parent. The Near Detector, however, covers a much wider solid angle since it is significantly closer to the end of the decay pipe. Consequently, this same parent can produce neutrinos at a range of energies in the Near Detector (see Figure 5.10). The effect on the spectrum can be seen in Figure 5.11: neutrinos of a particular energy in the Near Detector (shaded regions on the left) correspond to a range of parent energies with different decay angles which will produce a range of energies at the Far Detector (shaded regions on the right). The Far Detector distribution

84 Antineutrinos in a Neutrino Beam

/ GeV True E

ν

0 5 10 15 20 25 30

Events / GeV

3

10

0 0.5 1 1.5 2

Near detector Monte Carlo MINOS preliminary

/ GeV True E

ν

0 5 10 15 20 25 30

Events / GeV

3

10

0 0.5 1 1.5

Far detector Monte Carlo MINOS preliminary

Figure 5.11: The relationship between the energies of ¯νµ events observed in the Near Detector with those observed in the Far Detector. The colored regions on the left and right show the differing neutrino energy distributions in the detectors for neutrinos coming from the same parents.

gets wider at higher energies since higher energy parents tend to get further down the decay pipe (i.e. closer to the Near Detector) before decaying, enhancing the solid-angle effect (and outweighing the increased Lorentz boost which tends to narrow the outgoing neutrino energy distribution). Since the larger angles tend to have lower energies, events at the Near Detector tend to shift downward into the peak, leading to a more peaked spectrum at the Near Detector than at the Far Detector.

In order to get a correct prediction of the Far Detector given the Near Detector spectrum, the Monte Carlo is used to create a “beam matrix” that relates the Near Detector spectrum to the Far Detector spectrum.⁴ The rows each correspond to a bin of Far Detector true neutrino energy and the columns each correspond to a bin of Near Detector true neutrino energy. Each whole column is effectively the Far Detector neutrino energy spectrum that would be produced by the collection of parents that produced neutrinos at the Near Detector energy to which that column corresponds.

Each column is normalized to one Near-Detector neutrino so that by matrix multiplying with the Near Detector spectrum the Far Detector spectrum is obtained. The matrix for antineutrinos in the neutrino-mode beam is illustrated in Figure 5.12.

The matrix is populated by taking many simulated neutrino parents and forcing them to decay towards both the Near Detector⁵and the Far Detector. The corresponding neutrino energies and the probabilities of those decay directions are calculated. The matrix is then filled in the element defined by the two detector energies with a weight defined by the probability of that parent producing a neutrino at each detector. Once the full matrix is filled each column is normalized to a single Near Detector neutrino as described above.

Since the matrix is based on simulation, it assumes that the Near and Far Detector spectra are in

4I played a key role in adapting the extrapolation, designed for the neutrino analysis, to antineutrinos.

5Locations within the Near Detector are selected randomly in three dimensions, the same procedure used in generating the Near Detector simulation. This more accurately samples the range of decay angles that reach the Near Detector.

energy / GeV ν

µ

Near detector

0 10 20 30

energy / GeV

µ

ν Far detector

0 10 20 30

10-12

10-11

10-10

10-9

10-8

10-7

10-6

Low energy beamMINOS preliminary Monte Carlo

Figure 5.12: The beam matrix for ¯νµ’s in the neutrino-mode beam. Each cell relates a Far Detector energy bin to a Near Detector one. The content of each cell represents the mean number of ¯νµ events expected in the Far Detector for one event in the Near Detector. This distribution is treated as a matrix to relate the energies measured in the Near Detector to those expected in the Far Detector.

true neutrino energy, are perfectly pure, and that events are selected perfectly. In reality, of course, there are backgrounds and selection efficiencies, and only the reconstructed visible energy of each event is available. Each of these effects (purity, efficiency, visible energy) is corrected using the full detector Monte Carlo, both at the Near Detector and at the Far Detector. Backgrounds and efficiency are corrected by multiplying or dividing the appropriate histogram. The reconstructed visible energy is converted to corrected (‘true’) neutrino energy by matrix-multiplying a two-dimensional histogram which relates reconstructed and true energies in the Monte Carlo. This way the energy spectrum of selected antineutrinos in the Near Detector can be transformed into the neutrino flux×cross section in neutrino energy at the Near Detector. This Near Detector flux×cross section is multiplied by the beam matrix to produce a corresponding Far Detector flux ×cross section in neutrino energy.

The corrections applied for the Near Detector, but now based on the Far Detector simulation, are then applied in reverse: the spectrum is converted back to reconstructed energy by multiplying with a true-to-visible matrix, the spectrum is reduced by multiplying the selection efficiency, and data-based background predictions are added, finally resulting in a prediction of the Far Detector spectrum given the spectrum observed in the Near Detector without any oscillations. The prediction can be produced for any choice of oscillation parameters by applying the oscillation probability to the unoscillated Far Detector flux×cross section before the other corrections are applied.

86 Antineutrinos in a Neutrino Beam