1. Aart Heijboer ● ORCA * catania workshop sep 2009 1 statistical power of mass hierarchy measurement (with ORCA) Aart Heijboer, Nikhef

2. Aart Heijboer ● ORCA * catania workshop sep 2009 2 Computing oscillations probabilities 2 3 4 5 6 F = U V 6 V 5 V 4 V 3 V 2 V 1 U -1 The product of transition matrices computed for a matter of constant density r i. Oscillation probability given by P ab = |F ab | 2 with F the transition matrix is the flavour basis several options to compute exp in practice Cayleigh-Hamilton formalism (hep-ph/9910546) Diagonalize H power series for exp. Cross-checked with published plots and 'Globes' package. all methods agree

3. Aart Heijboer ● ORCA * catania workshop sep 2009 3 note allowed ranges are not identical for NH and IH → taken into account in the following plots assumed zero taken from most recent global-data fit Arxiv:1205.5254 Inputs for oscillations probabilities

4. Aart Heijboer ● ORCA * catania workshop sep 2009 4 Oscillation probability (m → m) for straight up beware: there are also plots around which accidentally exaggerate the NH/IH difference by comparing the central value or NH with a IH curve at the ~1 sigma edge of the allowed global fit. NH/IH difference largely degenerate with change in Dm 2 large

5. Aart Heijboer ● ORCA * catania workshop sep 2009 5 Other zenith angles... effect is fairly large, but... almost opposite for anti-neutrinos will be washed out by E- and angle resolutions next step: compute realistic event rates

6. Aart Heijboer ● ORCA * catania workshop sep 2009 6 Interaction rates : ingredients cross-sections neutrino fluxes (Bartol), include n e cross-sections from integrating DIS formula (approximates to ~20% genie and measurements)

7. Aart Heijboer ● ORCA * catania workshop sep 2009 7 Interaction rates

8. Aart Heijboer ● ORCA * catania workshop sep 2009 8 Adding detector effects This is where the guess-work starts; all results depend crucially on these assumptions assume zenith angle muon is measured perfectly. resolution comes for angle between n and m Neutrino energy resolution: assume 25% requires reconstruction of muon and hadronic shower Acceptance: neutrino vertex inside instrumented volume require 15 hits (educated guess: need direction, E-resolution, rejection of atm. muons.). geant4 sim with full photon tracking assumed detector: (100m) 3 =1Mton 6x6 strings 20 oms/string

9. Aart Heijboer ● ORCA * catania workshop sep 2009 9 Effect of detector on oscillogram: NH

10. Aart Heijboer ● ORCA * catania workshop sep 2009 10 Effect of detector on oscillogram: IH

11. Aart Heijboer ● ORCA * catania workshop sep 2009 11 difference between NH(central values from global fit) and IH (central values from global fit) after accounting for assumed detector resolutions: maximum rate difference between NH and IH ~10% to be compared to other uncertainties (e.g. on mixing angles and masses → next slide).

12. Aart Heijboer ● ORCA * catania workshop sep 2009 12 effect of parameter uncertainties : NH each plot compares central values with +1 sigma variation in each parameter → cannot be neglected: need to deal with these (nuisance) parameters

13. Aart Heijboer ● ORCA * catania workshop sep 2009 13 (Optimal) analysis to distinguish NH and IH Optimal observable to distinguish between NH and IH hypotheses = Maximum likelihood rato So this means, for each pseudo-experiment (data): 1. assume NH and find by maximizing (this involves computing many smeared-oscillograms for NH) 2. do the same for IH 3. compute The likehood contains a gaussian constraint representing the current knowledge from the global fit Dm 2 21 and q 21 are fixed in the fit to gain speed

14. Aart Heijboer ● ORCA * catania workshop sep 2009 14 Results of parameter fit on (NH) peudo-experiments true value drawn from global-fit-allowed range fitted value ORCA can improve the current uncertainty for Dm 2 large & q 23 already with 1 year of data s(q 13 ) still dominated by other data (via Gaussian constraint in the likelihood) after 10 Mtonyr → ORCA not very sensitive to it.

15. Aart Heijboer ● ORCA * catania workshop sep 2009 15 likelihood ratio distribution......for toy experiments in which the true hierarchy is normal or inverted. remember: results depend crucially on assumptions on resolution and detector layour and acceptance

16. Aart Heijboer ● ORCA * catania workshop sep 2009 16 likelihood ratio distribution for toy experiments in which the true hierarchy is normal or inverted. remember: results depend crucially on assumptions on resolution and detector layour and acceptance. expressed in sigma's, separation between NH and IH = 3 sigma with 10 Mton x year.

17. Aart Heijboer ● ORCA * catania workshop sep 2009 17 Conclusions / thoughts Full toy analysis set up, including oscillation fit good sensitivity to Dm 2 large & q 23 before we can do MH Determining MH is not an easy measurement For assumed detector performance, need 10 Mton x year for 3 sigma Of course, assumptions could be pessimistic Still several sources of systematic to be accounted for (earth density, rate normalization, cross-sections,....) Determining the mass hierarchy with this type of detector requires Large instrumented volume (1 Mton is not enough ) When we have full simulation and reconstruction, we should investigate sparse (or variable-density ) detectors ( But demands on (energy) reconstruction quality are very high )