Data Release for "First joint oscillation analysis of Super-Kamiokande atmospheric and T2K accelerator neutrino data"

This archive contains the electronic version in ROOT and pdf formats of the measurements of oscillation parameters obtained with the analyses from the paper “First joint oscillation analysis of Super-Kamiokande atmospheric and T2K accelerator neutrino data”.

It is published in Physical Review Letters and is available on the arXiv:2405.12488 [hep-ex]. 

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***** Results included in this release
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This release includes the results of the measurements of the different oscillation parameters obtained with the four analyses appearing in the paper. This corresponds to various 1D and 2D DeltaChi^2 and posterior probability maps as well as 2D confidence/credible regions for the parameters sin^2(theta13), sin^2(theta23), dm^2_32/|dm^2_31|, delta_cp, J_cp.

The results are separated into different files for the four analyses. Additional details on these analyses can be found below, but two of them use a Bayesian approach (producing posterior probabilities and credible intervals/regions) and two of them follow a frequentist approach (producing  DeltaChi^2 maps and confidence intervals/regions). A tag in the TGraph and histogram names also allow to differentiate the different types of intervals/regions: "cred" for credible interval from the Bayesian analysis, "conf" for confidence interval from the frequentist analysis. The tag “posterior” indicates that the object corresponds to a posterior probability distribution, while “chi2” indicates a DeltaChi^2 one.

Results for each mass ordering hypothesis are provided, denoted "NO" for normal ordering and "IO" for inverted ordering. The Bayesian files also include results marginalised over the mass ordering, denoted by the tag "both" in the object names. The frequentist files include results profiled over the mass ordering, indicated by a tag “profMO” in the object names.
The Bayesian and frequentist results use different conventions for the mass splitting in the inverted ordering: the Bayesian results are in term of #Deltam^{2}_{32} for both NO and IO, whereas the frequentist results are plotted versus #Deltam^{2}_{32} for the NO, and |#Deltam^{2}_{31}| for the IO. 

A constraint on theta13 from reactor experiment measurements is used for all results in this release. It corresponds to the value in the PDG 2019 review: sin^2(2theta_13)=(8.53+-0.27) x 10^{-2}. This is commonly referred to as "the reactor constraint", and a tag “wRC” is included in the name of the different objects as a reminder that it is used for these results.

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***** Brief descriptions of the four analyses
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Results are provided for the four analyses mentioned in the “Oscillation analysis” part of the paper. They were given names (Bayesian1, Bayesian2, Frequentist1, Frequentist2) based on the statistical approach they follow.

The Bayesian analyses are based on the two T2K analyses described in Eur. Phys. J. C 83, 782 (2023), extended to include the Super-Kamiokande atmospheric data, and with modifications to use the model described in the paper to which the present release is attached to. These analyses use Markov Chain Monte Carlo methods to compute marginal likelihoods for the parameter of interests.  

For the frequentist analyses, Frequentist1 is a modified version of Bayesian1, optimized for speed to be able to address the computational challenges of producing frequentist results from an ensemble of pseudo-experiments. Frequentist2 is based on the Super-Kamiokande atmospheric analysis described in PTEP 2019, 053F01 (2019), extended to include the T2K data, and also with modifications to follow the model described in the paper. These two analyses compute profile likelihood on a grid of oscillation parameters of interest to produce measurements of these parameters.

In terms of the differences between analyses mentioned in the paper, Bayesian2 is the analysis that does a simultaneous fit of the T2K near detector data with the events observed at SK, and the one for which the momentum scale uncertainty is not correlated between the atmospheric and T2K events observed at SK. The three other analyses use a covariance matrix to propagate the constraint on systematic uncertainties from T2K near detector data to the analysis of the events observed at SK, and treat the momentum scale uncertainty as correlated between atmospheric and T2K far detector events.

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***** Example codes
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Example codes are provided for each of the four analyses, showing how to produce the pdf file from this analysis from the corresponding ROOT file. How to run these example codes is indicated in the comments at the start of each of the example files.

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***** Objects inside the ROOT files
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The ROOT objects contained inside the files are named first with an identifier of which parameter(s) are being shown, followed by the reactor constraint tag, followed by the mass ordering tag.

For the Bayesian results, there is an additional tag to indicate if the results was obtained with a prior probability uniform in deltaCP (“flatdcp”) or uniform in sin(deltaCP) (“flatsindcp”)

A glossary is provided at the end of this readme.

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*** 2D regions
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Objects of the form:
gr2D_varX_varY_wRC_<NO,IO,both>_<conf,cred><68,90,955,997>(_N)
are TGraphs corresponding to the 2D confidence ("conf") or credible ("cred") regions for the 2 variables (varX, varY). 
N is the iterator for different TGraphs corresponding to the same region; these occur when confidence regions are discontinuous (for example when deltaCP loops over from +pi to -pi).

68, 90, 955, 997 are the percentage credible/confidence levels.

Most of the 2D frequentist regions were computed using the standard DeltaChi^2 values (from the Gaussian case), and therefore have only approximate coverage. However, the {sin^2(theta_23), deltaCP} confidence regions of analysis Frequentist1 were built using critical DeltaChi^2 values computed with the Feldman-Cousins method. To distinguish them from other confidence regions, a tag "FC" is included in the name of the corresponding TGraph.

The best fit markers are also provided for the 2D results:
gr2D_varX_varY_wRC_<NO,IO,both>_bestfit

The best fit markers and contour lines are generally for each MO *separately*, i.e. assuming DeltaChi^2 is 0 at the minimum or that the total posterior probability integrates to 1 in the mass ordering considered. There are some exceptions, in particular some 2D regions for (sin^2(theta_23), dcp) are also provided using a best fit over both MO to allow for comparisons with other experiments using this convention. This special set of contours has an extra tag "globalMO" in its name to distinguish it from the others.

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*** 1D and 2D histograms
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Objects of the form
h1D_var_<chi2,posterior>_wRC,_<NO,IO, both, profMO>
h2D_var1_var2_<chi2,posterior>_wRC_<NO,IO, both>
are respectively TH1D of the DeltaChi^2 ("chi2") or posterior probability ("posterior") for oscillation parameter "var" or TH2D for the couple of parameters (var1, var2)

The Bayesian and frequentist results use different conventions with respect to the mass ordering:
- DeltaChi^2 plots use a global minimum over both hierarchies
- Posterior probability plots integrate to unity *individually*

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***** Critical values for frequentist results
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For the 1D plots, critical delta chi2 values obtained with the Feldman-Cousins method are provided for theta23 and deltaCP 
grCritical_{variable}_chi2_wRC_{NO,IO,profMO}_conf{68, 90, 955}
variable: th23, dCP

The FC-corrected confidence intervals for these 2 variables can be obtained as the region for which the corresponding 1D DeltaChi^2 histogram is below the grCritical graph of a given level.

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***** Additional notes for Bayesian results
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For plots involving the mass splitting, the mass ordering is given by the sign:
  dm32>0 is normal hierarchy (Delta m^2_{32} > 0)
  dm32<0 is inverted hierarchy (Delta m^2_{32} < 0)

Note that the posteriors have not been smoothed, and may contain small discontinuities due to MCMC statistical uncertainties.

Plots with "_bestfit" appended indicate the point in the 2D parameter space (marginalized over the other parameters) with the highest posterior density, and is not necessarily the global minimum of the likelihood.

For the 1D posterior distributions, the user can freely calculate credible intervals from the distributions. It is recommended to start at the point of the highest posterior density, and moving down in posterior density to produce highest posterior credible intervals, which is the kind of credible intervals reported in the paper. 

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***** Glossary of tags used in objects names
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"wRC"   - Uses “reactor constraint” on theta13, sin^2(2theta_13)=(8.53+-0.27) x 10^{-2}
"FC"    - Feldman-Cousins
"NO"    - Normal mass Ordering
"IO"    - Inverted mass Ordering
"both"  - Marginalised over normal and inverted mass orderings
"profMO"  - Profiled over normal and inverted mass orderings
"cred"  - Credible interval
"conf"  - Confidence interval
"68"  - 68.3% (1 sigma)
"90"  - 90%
"955" - 95.5% (2 sigma)
"997" - 99.7% (3 sigma)
"chi2"  - DeltaChi^2 (-2lnL) for parameter
"Critical" - Critical DeltaChi^2 computed using Feldman-Cousins method
"th13"  - sin^2(theta_13)
"th23"  - sin^2(theta_23)
"dCP"   - delta CP
"dm2"   - Delta m^2_{23} (NO), |Delta m^2_{13} (IO)| for confidence intervals; used for frequentist analyses results.
"dm32"  - Delta m^{2_{23} regardless of mass ordering; in the Bayesian analyses, Delta m^2_{23} is always the variable that is plotted.
"jarlskog" - Jarlskog invariant
"flatdcp" - Using prior probability uniform in deltaCP
"flatsindcp" - Using prior probability uniform in sin(deltaCP)