GENIE models and global fits of neutrino scattering data

Transcription

1 GENIE models and global fits of neutrino scattering data Marco Roda - mroda@liverpool.ac.uk on behalf of GENIE collaboration University of Liverpool 7 September 17 NUFACT 17 Uppsala 1/49

2 Generators Neutrino MC Generators: A Theory/Experiment Interface Access the flux distorsion due to oscillation Every observable is a convolution of flux, interaction physics and detector effects Connect truth and observables Event topologies and kinematics Model dependencies Good Generators Support oscillation analyses uncertainty validation tune the physics models Tuning proved to be difficult So far no results Several MC Generators in use: GENIE, GiBUU, NuWro, NEUT /49

3 Role of generators Roles of MC generators in Oscillation Physics Comparing data and models You cannot study oscillations without fully understood models Validity region Highlight tensions Feedback for experiments Drive the format of cross section releases Hint toward key measurements 3/49

4 Role of generators Roles of MC generators in Oscillation Physics Comparing data and models You cannot study oscillations without fully understood models Validity region Highlight tensions Feedback for experiments Drive the format of cross section releases Hint toward key measurements Global fits Generator is the ideal place for global fits We control the model implementation Constraints on Cross Section for oscillation analysis Neutrino Cross sections priors Eventually based on data What generators can do depends on the available datasets 3/49

5 Dataset Evolving datasets - Old datasets / GeV] cm -38 CC [1 ν µ , GENIE Functions of E ν Not flux-integrated Only statistical errors 1 E ν ANL_1FT, BEBC, BEBC,5 BNL_7FT, CCFR, CHARM, FNAL_15FT,1 Gargamelle,1 IHEP_ITEP, IHEP_JINR, SKAT, [GeV] Ignore nuclear effects ANL_1FT,4 BEBC, BEBC,8 BNL_7FT,4 CCFRR, CHARM,4 FNAL_15FT, Gargamelle,1 IHEP_ITEP, MINOS, SciBooNE, trunk:g_a:numu_freenuc trunk:g16_a:numu_freenuc Poor statistical interpretation Poor model discrimination power 4/49

6 Dataset Evolving datasets - Present datasets Functions of experimental observables flux-integrated Usually differential cross-sections 1D, D Organized by topology, not process Higher statistics More statistically robust Fermilab Neutrino seminar by Mikael Kuusela - 17/4/13 cm /GeV/n] -38 [1 T µ / µ Cosθ CC π)/ 1 Cosθ µ [.9; 1 ] Phys. Rev. D81, 95 (1) σ(ν µ miniboone_nuccqe_1 T µ [GeV] 5/49

7 Dataset Evolving datasets - Present datasets Functions of experimental observables flux-integrated Usually differential cross-sections 1D, D Organized by topology, not process Higher statistics More statistically robust Fermilab Neutrino seminar by Mikael Kuusela - 17/4/13 cm /GeV/n] -38 [1 T µ / µ Cosθ CC π)/ 1 Cosθ µ [.9; 1 ] Phys. Rev. D81, 95 (1) µ Sometimes incomplete Helped the development of new models p/h σ(ν miniboone_nuccqe_1 T µ [GeV] 5/49

8 Dataset Future of datasets - a personal view One big covariance matrix per experiment Correlation between datasets Differential cross sections, dim >? No data releases with this format in SBND we are thinking about a solution? It is usually a big effort but... We finally have a way to use these datsets Statistically coherent Complete error analysis 6/49

9 GENIE - GENIE Collaboration Luis Alvarez Ruso 8, Costas Andreopoulos,5, Chris Barry, Francis Bench, Steve Dennis, Steve Dytman 3, Hugh Gallagher 7, Tomasz Golan 1,4, Robert Hatcher 1, Libo Jiang 3, Rhiannon Jones, Anselmo Meregaglia 6, Donna Naples 3, Gabriel Perdue 1, Marco Roda, Jeremy Wolcott 7, Julia Yarba 1 [ Faculty, Postdocs, PhD students] 1 - Fermi National Accelerator Laboratory, - University of Liverpool, 3 - University of Pittsburgh, 4 - University of Wroclaw, 5 - STFC Rutherford Appleton Laboratory, 6 - IPHC Strasbourg, 7 - Tufts University, 8 - Valencia University Core GENIE mission 1... provide a state-of-the-art neutrino MC generator for the world experimental neutrino community... simulate all processes for all neutrino species and nuclear targets, from MeV to PeV energy scales 3... perform global fits to neutrino, charged-lepton and hadron scattering data and provide global neutrino interaction model tunes 7/49

10 GENIE status and prospects GENIE Version.1.6 CCQE models Llewellyn Smith Nieves, Amaro and Valverde RES Rein-Sehgal Berger-Sehgal Kuzmin-Lyubushkin- Naumov MEC models Empirical Nieves Simo Vacas Nuclear Models Relativistic Fermi Gas Local Fermi Gas Effective Spectral Functions Single Kaon Λ production COH Rein-Sehgal Berger-Sehgal Alvarez Ruso FSI - Intranuke Full Intra-Nuclear cascade Schematic based on Hadron-nucleus data Only one Comprehensive Model Configuration (CMC) Default tune has not changed 8/49

11 GENIE status and prospects GENIE Version 3 Comprehensive Model Configurations Self-consistent collections of primary process models Tune names are supposed to become commonly used Help cooperation between collaborations single command-line flag --tune G16_a Complete characterization against public data Willing to host configuration provided by experiments Tunes for each CMC will also be available A step closer toward GENIE core mission graphics by grafiche.testi@gmail.com 9/49

12 GENIE status and prospects Comprehensive Model Configurations Dedicated web page List of available configurations 1/49

13 GENIE status and prospects Comprehensive Model Configurations Details and configuration 11/49

14 Models Comprehensive Model Configurations Configurations of interest for this talk G_a - Default No MEC CCQE process is Llewellyn Smith Model Dipole Axial Form Factor - Depending on M A =.99 GeV Nuclear model: Fermi Gas Model - Bodek, Ritchie G16_1b - Default + MEC with Empirical MEC CCQE process is Llewellyn Smith Model Dipole Axial Form Factor - Depending on M A =.99 GeV Nuclear model: Fermi Gas Model - Bodek, Ritchie G16_a - Nieves, Simo, Vacas Model Theory motivated MEC CCQE process is Nieves Dipole Axial Form Factor - Depending on M A =.99 GeV Nuclear model: Local Fermi Gas Model Small variations changing FSI models Variation including Spectral Functions 1/49

15 Comparisons The Comparisons The GENIE suite contains a package devoted to comparing GENIE predictions against publicly released datasets. Provides the opportunity to improve and develop GENIE models Crucial database for new GENIE global fit to neutrino scattering data All sorts of possible formats and dimensions Can store correlations, even between different datasets 13/49

16 Comparisons The database Modern Neutrino Cross Section measurement nuclear targets typically flux-integrated differential cross-sections MiniBooNE, TK, MINERvA Historical Neutrino Cross Section Measurement Bubble chamber experiment Measurements of neutrino-induced hadronic system characteristics Forward/backward hadronic multiplicity distributions Multiplicity correlations... Measurements of hadron-nucleon and hadron-nucleus event characteristics (for FSI tuning) For pion, Kaons, nucleons and several nuclear targets Spanning hadron kinetic energies from few tens MeV to few GeV Semi-inclusive electron scattering data electron-nucleus QE data electron-proton resonance data 14/49

17 CC π datasets MiniBooNE CCQE Both ν and ν Phys. Rev. D81, 95 (1) Phys. Rev. D88, 31 (13) [GeV] T µ , GENIE - Double differential cross section flux integrated No correlations Preferred model is Nieves Model (G16_a) excellent agreement for ν χ = 11/137 DoF worse for ν χ = 176/78 DoF σ(ν CC π)/ Cosθ / µ µ -38 T µ [1 Data: miniboone_nuccqe_1 Cosθ µ cm /GeV/n] 1 15/49

18 µ CC π datasets MiniBooNE CCQE T µ [.4;.5 ] GeV Both ν and ν Phys. Rev. D81, 95 (1) Phys. Rev. D88, 31 (13) Double differential cross section flux integrated No correlations Preferred model is Nieves Model (G16_a) excellent agreement for ν χ = 11/137 DoF worse for ν χ = 176/78 DoF cm /GeV/n] -38 [1 T µ / µ Cosθ CC π)/ σ(ν Cosθ µ miniboone_nuccqe_1 G_a G16_1b χ = 4.6/17 DoF χ = 64.8/17 DoF G16_a χ = 14.3/17 DoF 15/49

19 CC π datasets TK ND8 π 3-17, GENIE - Double differential cross section flux integrated [GeV] P µ 6 Phys. Rev. D (16) 1 Fully correlated 4 Tensions between datasets Preferred model is G16_1b χ = 135/67 DoF.5 all models look reasonable "By eye" estimation correlation is complicated We can t ignore it! Cosθ µ -38 σ/ Cosθ µ / P µ [1 cm /GeV/n] Data: tk_nd8_numuccpi_15 16/49

20 CC π datasets TK ND8 π Cosθ µ [.9;.94 ] Double differential cross section flux integrated Fully correlated Tensions between datasets Preferred model is G16_1b χ = 135/67 DoF all models look reasonable "By eye" estimation correlation is complicated We can t ignore it! cm /GeV/n] -38 [1 P µ Cosθ µ / σ/ [GeV] P µ tk_nd8_numuccpi_15 G_a χ = 17./8 DoF G16_1b G16_a χ = 7.75/8 DoF χ = 5.8/8 DoF 16/49

21 Tuning Why tuning? Constraint parameters Provide specific tuning for experiments Liquid Argon tuning Expected Output: Best parameters Parameter covariance matrix To be used for prior constructions 17/49

22 Tuning Why tuning? Constraint parameters Provide specific tuning for experiments Liquid Argon tuning Expected Output: Best parameters Parameter covariance matrix To be used for prior constructions Requirements granted by the comparisons: Data Metric Minimizer? Old problem in High Energy Physics CPU demanding Solution found in the Professor suite Numerical assistant Developed for ATLAS experiment 17/49

23 Professor Professor Parametrization instead of a full MC 18/49

24 Professor Professor Parametrization instead of a full MC 1 Select points of param space p 18/49

25 Professor Professor Parametrization instead of a full MC 1 Select points of param space Evaluate bin s behaviour with brute force p 18/49

26 Professor Professor Parametrization instead of a full MC 1 Select points of param space Evaluate bin s behaviour with brute force 3 Parametrization I(p) interpolated value p 18/49

27 Professor Professor Parametrization instead of a full MC 1 Select points of param space Evaluate bin s behaviour with brute force 3 Parametrization I(p) Repeat for each bin a parameterization I j (p) for each bin N dimension polynomial Including all the correlation terms up to the order of the polynomial interpolated value Minimize according to I(p) p 15 parameters Special thanks to H. Schulz 18/49

28 Professor Advantages Highly parallelizable independent from the minimization All kind of parameters can be tuned Not only reweight-able 19/49

29 Professor Advantages Highly parallelizable independent from the minimization All kind of parameters can be tuned Not only reweight-able Advanced system Take into account correlations weights specific for each bin and/or dataset Proper treatment while handling multiple datasets Restrict the fit to particular subsets Priors can be included Avoid unphysical result Nuisance rescaling parameters can be inserted proper treatment for datasets without correlations (MiniBooNE) Reliable minimization algorithm based on Minuit 19/49

30 The first tuning Tuning against CC π datasets /49

31 Inputs Datasets data points MiniBooNE ν µ CCQE D histogram 137 points No correlation matrix MiniBooNE ν µ CCQE D histogram 78 points No correlation matrix TK ND8 π (16) V D histogram 8 points full covariance matrix MINERvA ν µ CCQE 1D histogram 8 points full covariance matrix MINERvA ν µ CCQE 1D histogram 8 points full covariance matrix Data Missing Covariance between Neutrino and antineutrino data Minerva released this information! 1/49

32 Inputs Models and parameters Default + Empirical MEC G16_1b in naming scheme Full Nieves Model G16_a in naming scheme /49

33 Inputs Models and parameters Default + Empirical MEC G16_1b in naming scheme Full Nieves Model G16_a in naming scheme Parameter Range Default value QEL-M A (GeV/c ) [.7 ; 1.8].99 QEL-CC-XSecScale [.8 ; 1.] 1 RES-CC-XSecScale [.5 ; 1.5] 1 FSI-PionMFP-Scale [.6 ; 1.4] 1 FSI-PionAbs-Scale [.4 ; 1.6] 1 MEC-FracCCQE - G16_1b only [ ; 1].45 MEC-CC-XSecScale - G16_a only [.7; 1.3] 1 /49

34 Inputs Models and parameters Default + Empirical MEC G16_1b in naming scheme Full Nieves Model G16_a in naming scheme Parameter Range Default value QEL-M A (GeV/c ) [.7 ; 1.8].99 QEL-CC-XSecScale [.8 ; 1.] 1 RES-CC-XSecScale [.5 ; 1.5] 1 FSI-PionMFP-Scale [.6 ; 1.4] 1 FSI-PionAbs-Scale [.4 ; 1.6] 1 MEC-FracCCQE - G16_1b only [ ; 1].45 MEC-CC-XSecScale - G16_a only [.7; 1.3] 1 Other inputs: Nuisance scaling parameters 3 % for MiniBooNE Dataset Priors on QEL-CC-XSecScale and RES-CC-XSecScale Gaussian with sigma.1 /49

35 Outputs Sheer results G16_1b - Default + MEC G16_a - Full Nieves Model Parameter Best fit Nominal M A (GeV/c ) 1.17 ±.3.99 ±.1 QEL-CC-XSecScale.9 ±. 1 RES-CC-XSecScale 1. ±.7 1 MEC-FracCCQE.55 ±.6.45 FSI-PionMFP-Scale.86 ±.4 1. ±. FSI-PionAbs-Scale.76 ±.9 1. ±.3 Parameter Best fit Nominal M A (GeV/c ) 1. ±.3.99 ±.1 QEL-CC-XSecScale.91 ±. 1 RES-CC-XSecScale 1.1 ±.4 1 MEC-CC-XSecScale 1.18 ±. 1 FSI-PionMFP-Scale 1.17 ±.4 1. ±. FSI-PionAbs-Scale 1. ±.9 1. ±.3 M A is reasonably low Nieve s model is compatible with free nucleons fit Precision of M A reduced Our choice not to add a strong prior QEL reduced by 1% MEC increased by % FSI parameters strongly correlated They are better constrained than the GENIE prior 3/49

36 Agreement Agreement with respect to datasets G16_1b - Default + MEC G16_a - Full Nieves Model Dataset Best fit χ Nominal χ Miniboone ν µ CC π 177 / / 137 MiniBooNE ν µ CC π 66. / / 78 TK ND 8 CC π 94 / / 8 Total 337 / / 95 Dataset Best fit χ Nominal χ Miniboone ν µ CC π 89.3 / / 137 MiniBooNE ν µ CC π 48.1 / / 78 TK ND 8 CC π 1 / / 8 Total 39 / / 95 Improvement possible for both models The fit is working Fit driven by MiniBooNE datasets Lowest information No correlations Room for improvement These TK and Minerva datasets cannot be fit on their own They cover a small phase space region Parameters goes to the boundaries 4/49

37 Best fit plots Best fit - G16_1b - MiniBooNE ν µ CCQE Cosθ µ [.8;.9 ] Cosθ µ [.9; 1 ] -38 / µ [1 T µ Cosθ cm /GeV/n] CC π)/ µ -38 [1 / µ T µ cm /GeV/n] 1 Cosθ CC π)/ µ 1 σ(ν σ(ν miniboone_nuccqe_1 T µ [GeV] miniboone_nuccqe_1 T µ [GeV] trunk:g16_1b:miniboone_fhc χ = 37.3/16 DoF trunk:g16_1b:miniboone_fhc χ = 95.8/18 DoF trunk:tuned:miniboone_fhc χ = 7.46/16 DoF trunk:tuned:miniboone_fhc χ = 11.9/18 DoF Fit has a big impact 5/49

38 Best fit plots Best fit - G16_1b - MiniBooNE ν µ CCQE Cosθ µ [.6;.7 ] Cosθ µ [.7;.8 ] cm /GeV/n] -38 [1.4.3 cm /GeV/n] -38 [1.6.4 T µ T µ / µ Cosθ. / µ Cosθ CC π)/.1 CC π)/. σ(ν µ σ(ν µ miniboone_nubarccqe_13 T µ [GeV] miniboone_nubarccqe_13 T µ [GeV] trunk:g16_1b:miniboone_rhc χ = 6.8/8 DoF trunk:g16_1b:miniboone_rhc χ = 9.6/9 DoF trunk:tuned:miniboone_rhc χ = 6.1/8 DoF trunk:tuned:miniboone_rhc χ = 11.8/9 DoF Improvement not really necessary in this case 6/49

39 Best fit plots Best fit - G16_1b - MINERvA /neutron] cm /GeV -38 [1 QE dσ/dq Neutrinos 3-17, GENIE - PRL 111, 51 (13) MINERvAExDataCCQEQ QQE [GeV ] /proton] cm /GeV -38 [1 QE dσ/dq Antineutrinos 3-17, GENIE - PRL 111, 5 (13) MINERvAExDataCCQEQ QQE [GeV ] trunk:g16_1b:minerva_numu_13 χ = 17.5/8 DoF trunk:g16_1b:numubar_13 χ = 6.3/8 DoF trunk:tuned:minerva_numu_13 χ = 1.9/8 DoF trunk:tuned:numubar_13 χ = 4.7/8 DoF "Eye evaluation" wouldn t prefer a model over the other 7/49

40 Best fit plots Best fit - G16_1b - TK ND8 P µ [.8;.95 ] GeV cm /GeV/n].8 agreement with TK has worsened not surprising Tensions already highlighted χ : / 8 DoF -38 [1 P µ Cosθ µ / σ/ tk_nd8_numuccpi_15_rps Cosθ µ trunk:g16_1b:tk_nd8_numu_fhc χ = 6.9/8 DoF trunk:tuned:tk_nd8_numu_fhc χ = 9.3/8 DoF 8/49

41 Best fit plots Best fit - G16_a Cosθ µ [.9; 1 ] cm /GeV/n] [1 Nieves model already works well Agreement is preserved Notable improvement only w.r.t. MiniBooNE ν µ T µ / µ Cosθ CC π)/ σ(ν µ miniboone_nubarccqe_13 T µ [GeV] trunk:g16_a:miniboone_rhc χ = 53.8/15 DoF trunk:g16_a_tuned:miniboone_rhc χ = 13.5/15 DoF 9/49

42 Tuning program Next steps More tunings can be done hadronization retune Pythia 6 and 8 (implementation is ongoing) Tune of FSI Both hn and ha intranuke Free nucleon cross section model including dσ/dq data 3/49

43 Tuning program Next steps More tunings can be done hadronization retune Pythia 6 and 8 (implementation is ongoing) Tune of FSI Both hn and ha intranuke Free nucleon cross section model including dσ/dq data Data from Liquid argon experiments Part of GENIE collaboration is in SBND Plan for argon tunings Look forward to more data Release these results Paper is in preparation Implementation in GENIE v3 3/49

44 Conclusions Conclusion We are renewing GENIE New models Systematic validation against Cross section data Maintained and rich database We have a very powerful fitting machinery Validated A new branch of analyses Alternative tool to propagate systematic uncertainties What we can do depends on data quality Look forward a promising collaboration between generators, experiments and theorists 31/49

45 Backup Backup slides 3/49

46 Backup Parametrization residuals 35 /global Good Bad 33/49

47 Backup Data covariance Data Covariance bin bin 34/49

48 Backup Model dependencies Model dependencies in Oscillation Analyses A simple ratio between Near and Far spectra is not enough Detectors exposed to different flux functionally identical detectors do not exists Near flux has to be fitted at the near detector and then propagated Models required 35/49

49 Backup Model dependencies Model dependencies in Oscillation Analyses () CCQE is a -body reaction E ν depends is just a function of lepton momentum and angle p/h is not a -body reaction low energy tails in reconstructed energy distributions p/h also relevant for CP searches np-nh is different for ν/ ν E ν = m p (mn E b) m l + (mn E b)e l (m n E b E l + p l cosθ l ) p/h modelling is important to achieve required precision Martini et al. 36/49

50 Backup Model Comparisons Model comparison 37/49

51 Backup Model Comparisons Model comparison [M.Martini, FUNFACT J Lab workshop] 38/49

52 Backup Importance of Covariance Importance of the covariance - an example 3-17, GENIE - /neutron] cm /GeV -38 [1 QE dσ/dq QQE [GeV ] Real dataset 8 points Which is the best agreeing curve? Black Red Difference in terms of sigma? < 1 > 1 Black χ = 17.5/8 DoF Red χ = 1.9/8 DoF Almost σ 39/49

53 Backup Importance of Covariance Example of easy improvement Minerva experiment Cross sections of CC 1-proton on different targets C, Fe, Pb Wonderful dataset p/h and FSI tuning Covariance matrices for each target Best format among present data releases /nucleon) /GeV ( cm dσ C /dq /nucleon) /GeV ( cm dσ Pb /dq arxiv: v1 3.6e+ Data POT - ν µ C µ p Data GENIE with FSI GENIE No FSI NuWro with FSI NuWro No FSI..4.6 Qp (GeV.8 ) e+ Data POT - ν µ Pb µ p Data GENIE with FSI GENIE No FSI NuWro with FSI NuWro No FSI..4.6 Qp (GeV.8 ) /49

54 Backup Importance of Covariance Example of easy improvement Minerva experiment Cross sections of CC 1-proton on different targets C, Fe, Pb Wonderful dataset p/h and FSI tuning Covariance matrices for each target Best format among present data releases Not a full covariance matrix Neglecting the same flux Same detector/reconstruction We can check agreement we can not fit these data without neglecting a correlation /nucleon) /GeV ( cm dσ C /dq /nucleon) /GeV ( cm dσ Pb /dq arxiv: v1 3.6e+ Data POT - ν µ C µ p Data GENIE with FSI GENIE No FSI NuWro with FSI NuWro No FSI..4.6 Qp (GeV.8 ) e+ Data POT - ν µ Pb µ p Data GENIE with FSI GENIE No FSI NuWro with FSI NuWro No FSI..4.6 Qp (GeV.8 ) /49

55 Backup π puzzle CC Quasi-Elastic - π on single nucleons Theoretically well understood One diagram μ N A, B and C are form factors They have to be measured B and C are known from e-n scattering A to be extracted from ν data Ʋ W+ N Axial Form factor Dipole standard g A = 1.6 from neutron β decay parameterization ( ) fitted based on σ/ Q data A(Q ) = g A 1 + Q M A 41/49

56 Backup π puzzle CC Quasi-Elastic - Data 3-17, GENIE - cm ] Hydrogen / Deuterium data from.1 GeV to 1 GeV For both Neutrinos and Anti-neutrinos Critical parameter: M A M A 1 GeV -38 CCQE [1 ν µ ANL_1FT,1 E ν [GeV] ANL_1FT,3 BEBC,1 BNL_7FT,3 FNAL_15FT,3 NOMAD, SERP_A1,1 Gargamelle, SERP_A1, SKAT,8 4/49

57 Backup π puzzle CC Quasi-Elastic - Data Hydrogen / Deuterium data from.1 GeV to 1 GeV For both Neutrinos and Anti-neutrinos (1 38 cm / GeV ) < dσ / dq > ν µ n µ p Allasia et al., CERN WA5 (BEBC) 199 Target: D (converted to free neutron) Event number: 55 5 < E ν < 15 GeV, < E ν > 54 GeV M A =.999 ±.11 GeV (global fit) M A =.89 ±.44 GeV (best fit) Critical parameter: M A M A 1 GeV Q (GeV ) 4/49

58 Backup π puzzle π on heavy nuclei MiniBooNE data AIP Conf. Proc. 1189: (9); Phys. Rev. D 81, 95 (1) On heavy nuclei things got complicated MiniBooNE first evidence Carbon target Possible explanation from enhanced M A incompatibility with "historical" datasets 43/49

59 Backup π puzzle π on heavy nuclei - Solution MoniBooNE is Cherenkov detector Not able to see nucleons p-h scheme miniboone dataset is a CCQE-like sample genuine CCQE Multinucleon Emission np-nh Leading contribution is p-h ( particles - holes) 44/49

60 Backup π puzzle Particles - Holes NN correlations MEC M. Martini NN correlation-mec interference 16 diagrams 49 diagrams 56 diagrams Not easy to have a complete model Different approaches include different diagrams 45/49

61 Backup Data from comparisons Hadronization example 46/49

62 Backup Data from comparisons Hadronization example 47/49

63 Backup Data from comparisons Hadronization example <n π > W (GeV /c ) FNAL 15 ν Ne+H BEBC ν Ne+H BEBC νh νp G_a νn G_a νp G16_a νn G16_a /49

64 Backup Systematic propagation Tuning Output Parameters best fit Parameters covariance Prediction covariance due to the propagation of parameter covariance 49/49

65 Backup Systematic propagation Tuning Output Parameters best fit Parameters covariance Prediction covariance due to the propagation of parameter covariance Data Constraints for Oscillation analyses Events Muon Angle for π events Default Cos θ 49/49

66 Backup Systematic propagation Tuning Output Parameters best fit Parameters covariance Prediction covariance due to the propagation of parameter covariance Data Constraints for Oscillation analyses Propagate the result to other observables Events Muon Angle for π events Events Muon Angle for π events Default Default Tuned Cos θ Cos θ 49/49

67 Backup Systematic propagation Tuning Output Parameters best fit Parameters covariance Prediction covariance due to the propagation of parameter covariance Data Constraints for Oscillation analyses Propagate the result to other observables Propagate parameters uncertainty through the parameterization bin Correlation bin Events Muon Angle for π events Default Tuned Cos θ 49/49