Joint neutrino oscillation analysis from the T2K and NOvA experiments

# Reproduction of NOvA & T2K Collaborations, Nature 646, 818 (2025) [arXiv:2510.19888]: joint NOvA+T2K
# fit with the Newtrinos.jl modules `nova` (2024 data) and `t2k` (2025 analysis) as they are.
# Run from this directory with the Newtrinos.jl environment at the pinned commit:
#   julia -t 8 --project=<path to Newtrinos.jl> reproduce.jl
using Newtrinos, Distributions, DataStructures, Accessors, FileIO, CairoMakie, HDF5, CSV, DataFrames, Printf

cd(@__DIR__); mkpath("ours"); mkpath("results"); mkpath("cache")

# CSV export of a profile scan: one row per grid point, ASCII column names, oscillation parameters in physical
# units (sin²θ, Δm²₃₂ = Δm²₃₁ − Δm²₂₁ in eV², δCP in rad), dchi2 = -2Δlog L w.r.t. `ref` (default: the scan's best fit)
function save_csv(path, r; ref = maximum(r.values.log_posterior))
    ascii(k) = replace(string(k), "θ" => "theta", "Δm²" => "dm2_", "δ" => "d", "₁" => "1", "₂" => "2", "₃" => "3", "CP" => "cp")
    cols = OrderedDict{String,Any}()
    for (k, v) in pairs(r.values)
        x = vec(v)
        if startswith(string(k), "θ")
            cols["sin2_" * ascii(k)] = sin.(x) .^ 2
        elseif k == :Δm²₃₁
            cols["dm2_32"] = x .- vec(r.values.Δm²₂₁)
        end
        cols[ascii(k)] = x
    end
    cols["dchi2"] = 2 .* (ref .- vec(r.values.log_posterior))
    haskey(cols, "llh") && (cols["log_likelihood"] = pop!(cols, "llh"))
    first_cols = filter(in(keys(cols)), ["sin2_theta23", "dm2_32", "dcp", "dchi2", "log_likelihood", "log_posterior"])
    CSV.write(path, DataFrame(cols)[:, vcat(first_cols, setdiff(collect(keys(cols)), first_cols))])
end

# draw each confidence level with its own line style (1σ solid, 2σ dashed, 3σ dotted)
function contours!(ax, res, levels; color, label)
    for (lv, ls) in zip(levels, (:solid, :dash, :dot))
        plot!(ax, res, levels = [lv], color = color, linestyle = ls, label = lv == first(levels) ? label : nothing)
    end
end

experiments = (nova = Newtrinos.nova.configure(), t2k = Newtrinos.t2k.configure())
likelihood = Newtrinos.generate_likelihood(experiments)

# as in the paper: sin²θ₁₂ = 0.307 and Δm²₂₁ = 7.53e-5 eV² fixed, sin²θ₁₃ = 0.0218 ± 0.0007, flat δCP
θ₁₃ = asin(sqrt(0.0218))
θ₂₃_range = (asin(sqrt(0.35)), asin(sqrt(0.68)))
p = merge(Newtrinos.get_params(experiments), (θ₁₂ = asin(sqrt(0.307)), Δm²₂₁ = 7.53e-5, θ₁₃ = θ₁₃))
priors = merge(Newtrinos.get_priors(experiments),
               (θ₁₂ = p.θ₁₂, Δm²₂₁ = p.Δm²₂₁, θ₁₃ = Normal(θ₁₃, 0.0007 / (2 * sin(θ₁₃) * cos(θ₁₃))), δCP = Uniform(0, 2π), θ₂₃ = Uniform(θ₂₃_range...)))

orderings = (NO = (Δm²₃₂ = (2.25e-3, 2.65e-3), start = (Δm²₃₂ = 2.43e-3, δCP = mod(-0.87π, 2π)), color = :steelblue),
             IO = (Δm²₃₂ = (-2.65e-3, -2.25e-3), start = (Δm²₃₂ = -2.48e-3, δCP = mod(-0.47π, 2π)), color = :darkorange))

function profile_octants(pr, p0, vars, cache_dir)
    oct = map((lower = (θ₂₃_range[1], π / 4), upper = (π / 4, θ₂₃_range[2]))) do r
        Newtrinos.profile(likelihood, @set(pr.θ₂₃ = Uniform(r...)), vars, @set(p0.θ₂₃ = sum(r) / 2), cache_dir = cache_dir)
    end
    # the two octant scans use θ₂₃ priors of different widths, which shifts each log_posterior by its own
    # constant −log(width): remove it before comparing the octants (nuisance-parameter priors are kept)
    widths = (lower = π / 4 - θ₂₃_range[1], upper = θ₂₃_range[2] - π / 4)
    oct = map((r, w) -> Newtrinos.NewtrinosResult(axes = r.axes, meta = r.meta,
                                                   values = merge(r.values, (log_posterior = r.values.log_posterior .+ log(w),))), oct, widths)
    better = oct.lower.values.log_posterior .>= oct.upper.values.log_posterior
    Newtrinos.NewtrinosResult(axes = oct.lower.axes, values = map((l, u) -> ifelse.(better, l, u), oct.lower.values, oct.upper.values), meta = oct.lower.meta)
end

results = map(keys(orderings), values(orderings)) do mo, o
    pr = @set priors.Δm²₃₁ = Uniform((o.Δm²₃₂ .+ p.Δm²₂₁)...)
    p0 = merge(p, (Δm²₃₁ = o.start.Δm²₃₂ + p.Δm²₂₁, δCP = o.start.δCP, θ₂₃ = asin(sqrt(0.56))))
    dcp = profile_octants(pr, p0, OrderedDict(:δCP => 41), "cache/$(mo)_dcp")
    dcpth23 = Newtrinos.profile(likelihood, pr, OrderedDict(:δCP => 25, :θ₂₃ => 21), p0, cache_dir = "cache/$(mo)_dcpth23")
    FileIO.save("results/nova_t2k_$(mo).jld2", Dict("dcp" => dcp, "dcpth23" => dcpth23))
    mo => (; dcp, dcpth23)
end |> NamedTuple
# Δχ² references: each scan type carries the log density of its own scanned-parameter priors, so a scan is
# referenced to the best fit over both orderings of the same scan type
scan_max(k) = maximum(r -> maximum(r[k].values.log_posterior), results)
global_max = maximum(r -> maximum(r.dcp.values.log_posterior), results)
for mo in keys(results)   # dchi2 relative to the global best fit over both orderings
    save_csv("results/nova_t2k_$(mo)_dcp.csv", results[mo].dcp; ref = global_max)
    save_csv("results/nova_t2k_$(mo)_dcp_ssth23.csv", results[mo].dcpth23; ref = scan_max(:dcpth23))
end

# Fig. 3: δCP – sin²θ₂₃, 1σ / 2σ / 3σ, per ordering, with the official Bayesian regions (data/, extracted from the vector figure)
wrap(x) = mod(x + π, 2π) - π
regions = h5open("data/joint_fig3_regions.h5") do f
    Dict(mo => Dict(l => [(read(f["$mo/$l/$i/dcp"]), read(f["$mo/$l/$i/ssth23"])) for i in keys(f["$mo/$l"])] for l in keys(f[mo])) for mo in keys(f))
end
fig = Figure(size = (1200, 480))
for (i, mo) in enumerate(keys(orderings))
    ax = Axis(fig[1, i], title = "NOvA+T2K, $(mo): 1σ, 2σ, 3σ", xlabel = "δCP", ylabel = "sin²θ₂₃")
    for (lv, a) in (("3sigma", 0.15), ("2sigma", 0.3), ("1sigma", 0.5)), poly in get(regions[string(mo)], lv, [])
        poly!(ax, Point2f.(poly...), color = (:gray, a), strokewidth = 0)
    end
    poly!(ax, Point2f[(10, 10), (10.1, 10), (10, 10.1)], color = (:gray, 0.4), label = "NOvA+T2K (Bayesian)")
    r = results[mo].dcpth23; d = wrap.(r.axes.δCP); k = sortperm(d)
    contours!(ax, Newtrinos.NewtrinosResult(axes = (dcp = d[k], ss23 = sin.(r.axes.θ₂₃) .^ 2), values = map(v -> v[k, :], r.values)), [0.6827, 0.9545, 0.9973]; color = orderings[mo].color, label = "Newtrinos (profile)")
    xlims!(ax, -π, π); ylims!(ax, 0.38, 0.65)
    axislegend(ax, position = :lb)
end
save("ours/fig3_dcp_ssth23.png", fig)

# best fits vs. Extended Data Table III (Bayesian highest-posterior values)
official = (NO = (dm2 = 2.43, ss23 = 0.561, dcp = -0.87), IO = (dm2 = 2.48, ss23 = 0.563, dcp = -0.47))
open("results/bestfit_vs_official.toml", "w") do io
    for mo in keys(orderings)
        r = results[mo].dcp; i = argmax(r.values.log_posterior)
        @printf(io, "[%s]\nabs_dm2_32 = %.4f  # official %.2f (1e-3 eV²)\nsin2_theta23 = %.4f  # official %.3f\ndcp_over_pi = %.3f  # official %.2f\nminus2_dlogL_vs_global = %.3f\n\n",
                mo, abs(r.values.Δm²₃₁[i] - p.Δm²₂₁) * 1e3, official[mo].dm2, sin(r.values.θ₂₃[i])^2, official[mo].ss23,
                wrap(r.axes.δCP[i]) / π, official[mo].dcp, 2 * (global_max - r.values.log_posterior[i]))
    end
end