Global Analysis with the event-by-event extrapolation

1. Global Analysis with the event-by-event extrapolation Peter Litchfield • I will describe how the event-by-event extrapolation method would work for a global fit including • CC - events • CC + events • NC events • Anti-fiducial events • Rock events • Except for the rock events this is all working now.

2. Event-by-event Method • The method is based on the beam MC which relates near detector events to far detector events from the same parent decay • Predicted far detector spectra are built up from individual near detector MC events • The following stages are carried out • Correction of the MC using near detector data • Extrapolation to the far detector using beam MC truth events plus any physics (oscillation parameters etc). • Selection of far detector MC truth events with the same physics as the near detector event and construction of predicted far detector reconstructed distributions. • Parameter fitting and error determination

3. My weights SKZP weights Run 2 MC cc events MC correction • The beam MC is corrected on an event by event basis by weighting each event by the appropriate element of the reconstructed E v Eshw matrix of • E is signed by the measured sign of the  • Events with no reconstructed  have their own column • All events are used, with only good beam and detector cuts (and a minimum of data cleaning cuts) • Global since , and nc events are weighted separately • Equivalent to the SKZP fit • Does it work? Event weight

4. Extrapolation to the far detector • The extrapolation is done by the beam MC. • Each near detector MC event has • The ratio of the probabilities that the neutrino from this beam particle decay hit the near detector to that it hit the far detector (just decay physics and geometry) • The energy that a neutrino from this decay would have when it hit the far detector (decay physics) • From these one can calculate truth far detector distributions from the near detector MC • Any physics that happens on the way, standard oscillations, separate oscillations, sterile  oscillations, etc can be added as weights to the event. • Since all near MC events are extrapolated the analysis is still global

5. Far detector predictions • We now have a near detector MC event, weighted by the correction weight and any physics weights, and a corrected far detector energy. • Events from the far detector MC are selected that have the same truth physics (CC/NC, QEL/RES/DIS) and the same truth energy and y • The distribution of any reconstructed quantity for these events can be produced, normalised to one near MC event (unweighted) • Any cuts, selections, etc can be applied to these events yeilding, for example, CC and NC samples,  and samples, fiducial and anti-fiducial events, separation into resolution bins, etc, etc. • These distributions are summed for all the near detector MC events with relative pot and fiducial volume weights to get the correct final normalisation

6. Rock,  and e events • Distributions for event types that are not on the standard far detector MC can be produced in the same way • Events are selected from Rock MC,  ore far MC files and their reconstructed parameter distributions calculated, weighted in addition for cross-section differences. • Again all cuts and selections can be applied and the results summed with the standard MC distributions • We now have a global prediction for all Far detector reconstructed events • We can analyse any selection independently or all together as a global analysis

7. Run 1,2A Predicted Distributions No oscillations Selected NC events Selected CC events NC with reconstructed track NC with no track

8. Predicted Distributions m2=0.00238, sin22=1.0, 13=0 Selected NC events Selected CC events NC with reconstructed track NC with no track

9. Comparison with Beam Matrix • Agreement is not bad but there is a systematic difference • Beam peak is narrower in my extrapolation • No definitive explanation of the difference at the moment • My guess is that it is to do with events being mis-reconstructed out of the beam peak in the near detector • I extrapolate truth so it does not transmit to the far detector prediction

10. Fitting Procedure • In principle one can use whatever fitting procedure one wants to compare far data with far prediction • I use methods which fit well with the extrapolation method • “Unbinned” Extended Maximum Likelihood Grid Search • Feldman-Cousins error analysis • Extended maximum likelihood method described in DocDb-5109 • Unbinned data is compared with a binned probability density given by the far detector prediction to produce a likelihood • The low statistics data is unbinned • The probability distribution comes from the MC and near detector data which have high statistics and can be binned in fine bins. • Likelihoods from many small selections can be summed • Calculation is quick and fits well with the F-C analysis

11. Error Analysis • I do not like the nuisance parameter fits • Systematic errors are not Gaussian, therefore they do not give correct coverage • One has to be very careful with MINUIT when near physical boundaries where derivatives are not continuous • I don’t buy the argument that somehow one is going to improve one’s fit by fitting for the systematics • They are complicated to do with many systematics • So I use F-C to calculate allowed regions, described in DocDB-5109 • Better coverage • Can use my far detector MC event library to produce fake experiments • Systematic effects easy to introduce in the fake experiments • Unbinned likelihood fitting quick

12. Summary • A system exists for doing a global fit (except for rock events), the only reason I haven’t done one is that the data was in the closed box • As individual far detector event predictions are available, addition of rock and anti-fiducial events and any desired separations, e.g. bins of resolution, are easily added • I use a simple and rigorous fitting procedure which can easily accommodate different data sets, from a global fit to E v Eshw for all events, to data divided by event type and/or resolution and/or any other desired separation