Alternative Code to Calculate NMH Sensitivity J. Brunner 16/10/2015 1

Method to Determine NMH 2 Cos(zenith) : 40 bins from -1 to 0 E ν : 40 bin in log(E) from 2 to 100 GeV 1600 pairs of true values [E ν,θ] (bin center) Oscillation probabilities from Globes

Ingredients : Honda Flux Reference: Honda paper(2006) arxiv/astro- ph/ v3 Choices made: – Frejus site – no mountain – Solar minimum – azimuth averages 3

Honda Flux : Sanity check Low energy differences due to site & season 4

Ingredients : Cross Sections From Genie, combination Oxygen, n, p Provided by Martijn 5

Detector Parameters Dependency on Zenith angle currently ignored 1-Dim parametrisation functions derived from input histograms Generally Gaussian smearing assumed 6

Angular Resolution 7 Gaussian smearing in cos(SpaceAngle) Width adjusted to approximately fit the shown distributions ( ~ √m/E ν )

Angular Resolution - 6m 8 Gaussian smearing in cos(SpaceAngle) Width adjusted to approximately fit the shown distributions ( ~ √m/E ν ) Scaling to 9m, 12m : 1 10 GeV degradation per 3m spacing increase

9 True cos(zenith) Xxxxx Parametrisation 7 GeV < E < 8 GeV Parametrisation 7 GeV < E < 8 GeV Full Simul/Reco Jannik 7 GeV < E < 8 GeV Full Simul/Reco Jannik 7 GeV < E < 8 GeV Comparison with full Simulation

10 Assume Gaussian smearing in E ν Two free parameter : mean value & width Fit for each bin in E ν Example : E reco distribution for E ν ≈ 10 GeV Energy Resolution νeνe νμνμ Fit 9.57 ± 2.55 GeV Fit ± 2.77 GeV

11 Gaussian smearing in E ν Showers width : [0] + [1]*E ν + [2]*E ν 2 Energy Resolution - 6m ν e CC ν μ CC ν τ CC ν NC

12 Gaussian smearing in E ν Showers width : [0] + [1]*E ν + [2]*E ν 2 Energy Resolution - 9m ν e CC ν μ CC ν τ CC ν NC

13 Gaussian smearing in E ν Showers width : [0] + [1]*E ν + [2]*E ν 2 Energy Resolution – 12m ν e CC ν μ CC ν τ CC ν NC

14 Use quadratic template in log 10 (E ν ) E reco = E ν ( [0] + [1]*lgE ν + [2]*lgE ν 2 ) Energy Shift – 6m ν e CC ν μ CC ν τ CC ν NC

15 Use quadratic template in log 10 (E ν ) E reco = E ν ( [0] + [1]*lgE ν + [2]*lgE ν 2 ) Energy Shift – 9m ν e CC ν μ CC ν τ CC ν NC

16 Use quadratic template in log 10 (E ν ) E reco = E ν ( [0] + [1]*lgE ν + [2]*lgE ν 2 ) Energy Shift – 12m ν e CC ν μ CC ν τ CC ν NC

17 Use quadratic template in log 10 (E ν ) E reco = E ν ( [0] + [1]*lgE ν + [2]*lgE ν 2 ) Energy Shift – Muon 6m 9m 12m

18 Use three parameter template in E ν M eff = [0]*TanH((E ν -[1])/[2]) Effective Mass – 6m ν e CC ν μ CC ν τ CC ν NC

19 Use three parameter template in E ν M eff = [0]*TanH((E ν -[1])/[2]) Effective Mass – 9m ν e CC ν μ CC ν τ CC ν NC

20 Use three parameter template in E ν M eff = [0]*TanH((E ν -[1])/[2]) Effective Mass – 12m ν e CC ν μ CC ν τ CC ν NC

21 Use three parameter template in E ν M eff = [0]*TanH((E ν -[1])/[2]) Effective Mass – ν e 6m 9m 12m HE limit does not scale ?

22 Use three parameter template in E ν M eff = [0]*TanH((E ν -[1])/[2]) Effective Mass – ν μ 6m 9m 12m HE limit does not scale ? Systematically smaller ?

23 Use three parameter template in E ν P track = [0] + [1]*E ν **[2] Particle ID – 6m ν e CC ν μ CC ν τ CC ν NC

24 Use three parameter template in E ν P track = [0] + [1]*E ν **[2] Particle ID – 9m ν e CC ν μ CC ν τ CC ν NC

25 Use three parameter template in E ν P track = [0] + [1]*E ν **[2] Particle ID – 12m ν e CC ν μ CC ν τ CC ν NC

26 Use three parameter template in E ν P track = [0] + [1]*E ν **[2] Particle ID – ν e 6m 9m 12m

27 Use three parameter template in E ν P track = [0] + [1]*E ν **[2] Particle ID – ν μ 6m 9m 12m

Method to Determine NMH 28 Start with 1600 pairs of [E ν,θ] Apply Resolution matrices – NE reco [i] = Σ M E [i][j] NE true [j] – Nθ reco [i] = Σ M θ [i][j][k] NEθ true [j][k] Apply Particle ID Below (IH-NH)/√ NH for 1 year NH  IH |Δm 2 31 |  |Δm 2 31 | - 2Δm 2 21 |Δm 2 32 |  |Δm 2 32 |

Method to Determine NMH 29 Start with 1600 pairs of [E ν,θ] Apply Resolution matrices – NE reco [i] = Σ M E [i][j] NE true [j] – Nθ reco [i] = Σ M θ [i][j][k] NEθ true [j][k] Apply Particle ID Below (IH-NH)/√ NH for 1 year NH  IH |Δm 2 31 |  |Δm 2 31 | - Δm 2 21 |Δm 2 32 |  |Δm 2 32 | +Δm 2 21

Event Rate in ORCA – 6m Events per year per GeV One example bin in cosθ (width 0.1 at ~45 0 ) Numbers for full angular range No Resolutions, no PID ν μ CC 21,700 ν e CC 17,300 ν τ CC 2,500 NC 5,300 NH  IH |Δm 2 31 |  |Δm 2 31 | - Δm 2 21 |Δm 2 32 |  |Δm 2 32 | +Δm

Event Rate in ORCA – 9m Events per year per GeV One example bin in cosθ (width 0.1 at ~45 0 ) Numbers for full angular range No Resolutions, no PID ν μ CC 24,800 ν e CC 17,300 ν τ CC 3,100 NC 5,300 NH  IH |Δm 2 31 |  |Δm 2 31 | - Δm 2 21 |Δm 2 32 |  |Δm 2 32 | +Δm

Event Rate in ORCA – 12m Events per year per GeV One example bin in cosθ (width 0.1 at ~45 0 ) Numbers for full angular range No Resolutions, no PID ν μ CC 23,800 ν e CC 14,500 ν τ CC 3,000 NC 4,600 NH  IH |Δm 2 31 |  |Δm 2 31 | - Δm 2 21 |Δm 2 32 |  |Δm 2 32 | +Δm

Event Rate in ORCA – 6m ν μ CC 21,700 ν e CC 17,300 ν τ CC 2,500 NC 5,300 Events per year per GeV (Flux, CrossSection, M eff ) One example bin in cosθ (width 0.1 at ~45 0 ) Numbers for full angular range Resolutions added, no PID 33

Event Rate in ORCA - 9m ν μ CC 24,800 ν e CC 17,300 ν τ CC 3,100 NC 5,300 Events per year per GeV (Flux, CrossSection, M eff ) One example bin in cosθ (width 0.1 at ~45 0 ) Numbers for full angular range Resolutions added, no PID 34

Event Rate in ORCA – 12m ν μ CC 23,800 ν e CC 14,500 ν τ CC 3,000 NC 4,600 Events per year per GeV (Flux, CrossSection, M eff ) One example bin in cosθ (width 0.1 at ~45 0 ) Numbers for full angular range Resolutions added, no PID 35

Event Rate in ORCA – 6m Events per year per GeV (Flux, CrossSection, M eff ) One example bin in cosθ (width 0.1 at ~45 0 ) PID added ; CP phase : 0,90,180, Tracks 28,200 Cascades 18,700

Event Rate in ORCA – 9m Events per year per GeV (Flux, CrossSection, M eff ) One example bin in cosθ (width 0.1 at ~45 0 ) PID added ; CP phase : 0,90,180, Tracks 30,600 Cascades 20,000

Event Rate in ORCA – 12m Events per year per GeV (Flux, CrossSection, M eff ) One example bin in cosθ (width 0.1 at ~45 0 ) PID added ; CP phase : 0,90,180, Tracks 26,600 Cascades 19,300

Event Rate in ORCA Events per year per GeV (Flux, CrossSection, M eff ) One example bin in cosθ (width 0.1 at ~45 0 ) PID added ; Zoom : CP phase : 0,90,180,270 39

NMH Sensitivity Calculation Calculate Event rates W for cascade/track[40,40] for «true» parameters (including hierarchy choice) Fit «wrong» hierarchy event rates – all free parameters also fitted Minimizer : MIGRAD within Minuit2 from Root Significance from χ 2 = Σ (NH-IH) 2 /NH No pseudo-experiments (« Asimov-Set ») Sanity check : Fitting «true» hierarchy yields χ 2 = 0 40

Fit Parameters Fixed oscillation parameters : θ 12,θ 13,Δm 21 Fitted oscillation parameters : θ 23,Δm 31, [ δ CP ] Fitted Nuisance parameter – Individual normalisations for tracks, cascades, NC 6 parameters, no priors, i.e. no constraints Fit starting points at «nominal» values Convergence after ~200 calls (~10min) 41

Sensitivity to Neutrino Mass Hierarchy 42 True value for θ 23 varied, different δ CP conditions Dashed : δ CP = 0 not fitted Solid thin : δ CP = 0 fitted ; thick δ CP = 180 fitted Initial value for θ 23 : true value

Sensitivity to Neutrino Mass Hierarchy 43 Introduce scan of starting values for θ 23 Look at fitted values for θ 23 NH true – IH fitted : always second octant IH true – NH fitted : always first octant

Sensitivity to Neutrino Mass Hierarchy 44 Add nuisance parameters: spectral index, ratio ν/ν Small decrease of sensitivity Compatible with result from Martijn

Sensitivity to Neutrino Mass Hierarchy 45 Add nuisance parameter: spectral index, ratio ν/ν Simplified resolution functions : identical for ν/ν Find systematic shift towards anti-neutrino 10-30% lower neutrino rate, 20-60% higher Antinu  need prior here ?? Martijn : spread of 4% Inconsistent with above ?

46 Add nuisance parameter: spectral index, ratio ν/ν Channel dependent resolution function Ratio ν/ν much more stable  need prior here ?? No !! Martijn : spread of 4% Consistent with above !! Sensitivity to Neutrino Mass Hierarchy

47 Add nuisance parameter: spectral index, ratio ν/ν Gaussian Prior with 10% width added Minor effect

Dependency on δ CP and θ Plot presented at ICRC (Martijn) Tendency visible but scrambled due to limited statistics

Dependency on δ CP and θ Same plot (for true NH) Symmetric pattern around δ CP = Highest sensitivity for δ CP = 0, and large θ 23

Dependency on δ CP and θ Same plot (for true IH) Symmetric pattern around δ CP = Highest sensitivity for δ CP = 0, and θ 23 close to 45 0

51 Normalisations behave well without priors Jitter compatible with values from Martijn Martijn : 2.0% Martijn : 11.0% Nuisance Parameters Total NormNC Norm

52 Spectral Index & Track/Cascade ratio ok Jitter compatible with values from Martijn Martijn : 1.2% Spectral Index Track - Cascade Martijn : 0.5% Nuisance Parameters

53 ΔM 2 very stable for IH δ CP always around even for true value 0 δ CP Δm 2 32 Oscillation Parameters

54 What happens if PID changes ? Reminder : Present values Effect of modified detector performance

55 Assume improvement of PID of muon-neutrinos at 10 GeV from 75% to 85% Below : Assumed performance and new results Moderate gain of 0.2σ Effect of modified detector performance

56 What happens if energy resolution improves from ~20-25% to 10-15% ? (all channels) Reminder : Present values Effect of modified detector performance

57 What happens if energy resolution improves from ~20-25% to 10-15% ? (all channels) Below : Assumed performance and new results Average gain of 0.5σ in 3 years Effect of modified detector performance

58 NH true, 3 years data taking Effect of vertical spacing 6m 9m 12m

59 IH true, 3 years data taking Effect of vertical spacing 6m 9m 12m

Summary & Further Plans New Sensitivity code fully operational – based on parametrized detector response – Asimov sets (no pseudo-experiments) Results from pseudo-experiment study reproduced Further studies – Nuisance parameters on non-nominal values – Study more systematic effects (energy scale) – Uncertainty of resolution functions – Effect of Bjorken-Y 60