Indicadores Económicos y Financieros

5.2.1

Definition of the matrix

A more sophisticated method of performing the extrapolation is to relate the near and far fluxes using a beam matrix Bij, such that

F_ipredicted=X j

BijNjobs. (5.3)

In contrast to the ratio method, this parameterisation allows to account for the fact that a neutrino parent which could produce a neutrino of energyEi in the far detector, may produce a neutrino of different energy Ej if it decayed towards the near detector, depending on its position and momentum at the point of decay.

5.2.2

Calculation of the elements

From (5.3), it can be seen that any beam matrix such that P jBijN

MC

j =FiMC for alli, will give the correct far detector flux prediction in the case that the Monte Carlo matches the data exactly. However, we would like to have a matrix which is approximately correct even if the Monte Carlo does not exactly model the data. This can be done (following [36]) by considering, for a large sample of parents, the range of near detector neutrino energiesEncontributed to by parents which would give a neutrino of given energyEf in the far detector. If the data does not agree with the MC, these correlations can be used to estimate how the far detector flux will change for a given change in the near detector flux.

pression Bij = X parents δEf,EiδEn,Ej " Pf Pn Pn P parentsPnδEn,Ej # = P parentsδEf,EiδEn,EjPf P parentsPnδEn,Ej . (5.4)

HereEndenotes the energy that a neutrino produced by a given parent would have if directed towards the near detector, andPnthe probability that the neutrino from the decay will be directed into unit area (defined with a normal along the neutrino direction vector) located at the near detector. (Ef, Pf)denote the same quantities for the far detector. The interpretation of the expression is as follows:

1. For the matrix element Bij, we consider only contributions from the parents giving binned far and near detector energies of Ei, Ej respectively.

2. For each parent, we calculate the ratio of the probabilities to give a neutrino in the far detector and near detector. The result is the value which Bij would take if this parent were the only one in the beam.

3. To calculate the matrix element for the whole ensemble of parents, we must combine the ratios from (2), weighting the ratio for each parent by the relative contribution of that parent to the near detector flux in energy bin

Ej, i.e. the probability that a neutrino of energy Ej observed in the near detector came from this parent. This is done by taking the ratio of the neutrino flux from this parent and the total expected flux in this energy bin

Pn

P

parentsPnδ_En,Ej. In the limit of a large sample of parents then each weighting

will be infinitesimal and the sum over parents is then equivalent to an integral over the region of parent phase space in which δEf,EiδEn,Ej = 1.

5.2.3

Applicability

It can be seen analytically, by substituting (5.4) into (5.3), that the matrix is correct in the case that the data and MC are identical. If the MC does not accurately model the data, then the phase space distribution of the parents contributing to the neutrino flux in each near detector energy binEj may be different, and therefore the correlation between the contents of this energy bin and the energy bins at the far detector may change. If this is the case, the correct beam matrix describing the experiment will not be the same as that calculated from the MC. However, a change which rescales the absolute flux in each near detector energy bin, without changing the phase space distribution contributing to it, will be corrected for by the matrix. In general, the effect of a systematic error will be a combination of these two effects and will be compensated for at least partially by the matrix.

A change in the phase space distribution of the parent flux is also likely to be corrected for poorly by the ratio method, since this will affect the flux at a given energy differently for the near and far detectors. Because the beam matrix does not assume that the populations contributing to the same energy bin in the two detectors are the same (which is certainly not completely true), one would expect it to correct more effectively for systematic shifts than the ratio. We test this a priori prediction in Section 5.5, by evaluating the effects of realistic systematic errors.