Extrapolation Techniques

1. Extrapolation Techniques • Four different techniques have been used to extrapolate near detector data to the far detector to predict the neutrino energy spectra. • Far/Near – simplest, used for the nc paper • Beam fits – modify the beam parameters to fit the near detector • Matrix method – translates the near beam to the far by multiplying matrices, used for the cc analysis • Event by event method – Uses each near MC event, weighted to agree with near detector data, to construct a far detector prediction

2. The Beam MC • Generates proton interactions in the target (and surrounds) • Tracks secondary particles (,K,) through the target, the magnetic horns and the decay pipe • Generates a decay at a given position • Calculates the probability that the neutrino from the decay will hit the near and far detectors and the energy of such a neutrino • For 2-body decays (→) this is easy since the decay is uniform in the  center of mass. Doing a Lorentz transform to the CM one can calculate the angle subtended by the detector and thus the probability • For multi-body decays it is more complicated but can be done assuming a decay distribution • The decay neutrinos are used to generate events in the near and far detectors. By keeping account of the relative probabilities correctly normalised near and far distributions can be produced

3. The far/near method • Relies on the MC to transform the near detector data in reconstructed energy bin i to the far detector • To oscillate the spectrum the true energy is required and thus the distribution of true energies of the events in the reconstructed energy bin • The sum is done using a 2-D histogram of measured v true energies • In principle beam, cross-section and detector differences are taken into account in the MC • Robust but introduces a lot of smearing, particularly for nc events

4. Beam fits • The beam MC generates particles with distributions in pt and pz • The beam fits attempt to refine these distributions by fitting the near detector cc energy distributions for many (9) different configurations of target and horn parameters. The pt and pz distributions are distorted by applying fitting parameters. • But neutrino cross-sections and detector parameters are also have errors and thus need to be included in the fits • Up to 16 beam parameters and 10 other parameters to be fitted • Significant improvement to near detector distributions for reasonable distortions of the beam and other parameters • Then the refined beam parameters are used in the MC to predict the far detector energy distribution • Fits the near detector data OK but lots of parameters, are they all correct?

5. NuMu beam matrix Matrix method • More sophisticated version of the F/N method • Uses the beam MC to generate a matrix of far detector event numbers corresponding to one near detector event as a function of near truth energy v far truth energy • BUT: the matrix applies to the truth quantities and getting to the truth from the measured quantities is complicated. • Involves multiplying togther nine separate matrices, smearing and statistics

6. ND Data: Selected, reconstructed energy spectrum Multiplicative purity correction Matrix correction: reco→true energy True energy neutrino flux passing through ND fiducial volume Divide out cross-sections Multiplicative efficiency correction True energy neutrino flux passing through FD fiducial volume Multiplicatively apply efficiencies Apply cross-sections Oscillate if appropriate FD prediction: Selected, reconstructed energy spectrum Matrix correction: true→reco energy Additively apply impurities Matrix method Beam matrix

7. Event-by-event method • Also uses the beam MC to generate far detector distributions but does it on an event by event basis using weights rather than by matrices • The near MC is corrected as a function of reconstructed E and Eshw by the ratio of reconstructed data to reconstructed MC • E is signed by the sign of the reconstructed charge. Events with no  have their own column • NC and + events are incorporated naturally • Both beam flux and cross-sections are corrected together

8. Event-by-event method • Each near MC event is used to generate a far detector distribution. • Events are selected in the far detector with the same truth parameters, nc/cc, initial and final state, projected E and y • Their reconstructed E and Eshw are histogrammed and normalised to one event • The distribution is weighted with • Near detector correction weight • Ratio of far/near probabilities (including fiducial mass, pot count) • The oscillation probability • Some small correction weights • The distributions are summed for all the near MC events to produce far detector predictions to be compared with the far data • Notice near data only compared to near MC, far data to far MC

9. What to use for a universal analysis? • For the CC analysis all methods give similar results • Backgrounds are small, detectors are similar • The matrix method, which in principle was the most sophisticated at the time of the first paper, has been used until now. • For the NC analysis the F/N method is used for the paper • The matrix method has a problem that for NC events the correlation between measured E and true E is small • The CC matrix can be used but then the correction matrices are different for the near and far data, increased systematics • The analysis is planning on using the matrix method with separate matrices for  and • For the rock events again the CC matrix can be used to generate the flux in the rock but again there is no cancellation of near and far efficiencies

10. Each near detector MC event is extrapolated individually with its truth • No separation in the near detector into CC/NC, / , differences in near/far reconstruction and selection efficiencies automatically taken into account. • Far detector CC/NC, / predicted distributions generated automatically. They can be fitted separately or together. • Rock event predictions can be generated by selecting events with the same truth parameters from the rock MC. • Predictions for oscillations to , e, s can be generated by selecting from appropriate MC samples and applying the appropriate oscillation probabilities to the near detector . Event-by-event method advantages

11. My code will currently analyse all contained events, NC/CC, / , inside/outside the fiducial volume • To include rock events, needs a PAN of the rock MC plus the incorporation of the rock events in the same way the  events are currently done • Needs an improved systematics analysis, Feldman-Cousins? • MC which generates near and far MC events from the same beam decay? • John Marshall at Cambridge wrote a completely independent version of the method and it was checked that it gave consistent results. It only analysed CC events and it is not clear what its future is now he has left. • The collaboration correctly insists on cross-checks. It would not be bad for somebody else to implement the method. There are a number of wrinkles that could be done differently and my code is not very beautiful or C++’ed. • I may not be around much longer, it would be nice if somebody could take the method over. What exists and is needed?