Leptonic CP Observables and Combined Inference

Leptonic CP and mass-ordering inference is a joint likelihood problem, not a reading of one oscillation probability. A valid combination fixes the PMNS and ordering conventions, predicts reconstructed event distributions from flux, cross section, propagation, detector response, and backgrounds, retains shared nuisance parameters and covariance, profiles or marginalizes every continuous and discrete degeneracy, and calibrates the test statistic when asymptotic assumptions fail. The procedure below is durable; it contains no current interval, ordering preference, experimental limit, global combination, or status claim.

Required background. Neutrino Oscillations, Coherence, and Matter Effects supplies the vacuum/matter probabilities and antineutrino sign rules. Validation and Theory Uncertainties supplies calibration, nuisance, covariance, and held-out-validation semantics.

Helpful background. Absolute Neutrino Mass and Majorana-Sensitive Probes clarifies which ordering and mass inputs can be shared without treating distinct probes as the same likelihood.

From oscillation parameters to event observables

For dataset $d$, reconstructed bin $b$, and channel $c$, a generic predicted count is

$$\begin{aligned} \lambda_{dbc}(\vartheta,\eta)= \int dE\,dE_{\rm rec}\;& \Phi_{dc}(E;\eta_\Phi) \sigma_c(E;\eta_\sigma) \epsilon_{dc}(E;\eta_\epsilon)\\ &\times R_{db}(E_{\rm rec}|E;\eta_R) P_c(E,L_d;\vartheta,\eta_\rho) +B_{dbc}(\eta_B). \end{aligned}$$

The physics vector $\vartheta$ includes mixing angles, mass splittings, the periodic phase $\delta$, and a discrete ordering label. The nuisance vector $\eta$ can include flux, interaction model, energy scale and resolution, efficiency, background, and matter-density parameters. Appearance channels carry direct interference sensitivity to $\delta$; disappearance and near-detector information constrain other combinations that break or rotate degeneracies. The experimentally observed object is the reconstructed distribution $n_{dbc}$, not $P_c$ alone.

A Poisson likelihood with a Gaussian auxiliary constraint has deviance

$$\begin{aligned} -2\ln\mathcal L={}&2\sum_{d,b,c} \left[ \lambda_{dbc}-n_{dbc} +n_{dbc}\ln\!\left(\frac{n_{dbc}}{\lambda_{dbc}}\right) \right]\\ &+(\eta-\eta_0)^{\mathsf T} C_\eta^{-1}(\eta-\eta_0), \end{aligned}$$

with the $n\ln(n/\lambda)$ term defined as zero when $n=0$. A covariance-form Gaussian likelihood may be suitable after verifying the count and systematic regime; it is not a universal replacement for the generative model.

Before combination, every input needs an observable contract:

Field	Required identity
Source and channel	Beam, reactor, solar, atmospheric, or other source; neutrino/antineutrino mode; appearance/disappearance flavor tags
Geometry and propagation	Baseline distribution, energy range and units, density profile and uncertainty, coherence/averaging treatment
Detector observable	Reconstructed variable, bin edges, response, efficiency, background, event selection, and exposure normalization
Theory convention	PMNS order and phase, $\Delta m_{ij}^2$ definition, ordering label, cross-section and nuclear model versions
Statistical object	Counts or summary statistic, likelihood form, covariance, nuisance priors/constraints, physical boundaries, blinded choices
Provenance	Dataset release, likelihood/software version, corrections, supersession, and overlap with every other input

Degeneracies are branches, not error bars

Several parameter transformations can give similar spectra over a bounded experiment:

• Intrinsic CP–amplitude degeneracy: changes in $\delta$ can be offset by mixing amplitudes or normalization over limited $L/E$.
• CP–ordering–matter degeneracy: changing the ordering alters matter enhancement, while $\delta$ and the density nuisance can compensate part of the spectral change.
• Octant degeneracy: disappearance can constrain a function close to $\sin^2(2\theta_{23})$, leaving $\theta_{23}$ and $\pi/2-\theta_{23}$ branches that appearance treats differently.
• Energy-scale–splitting degeneracy: shifting reconstructed energy can mimic a shifted $\Delta m^2$ in the oscillation phase.
• Flux–cross-section–efficiency degeneracy: event normalization constrains their product unless near-detector or external data resolve the factors.

The familiar intrinsic, ordering, and octant ambiguities arise already at the probability level and can produce disconnected solutions Burguet-Castell et al. 2001, §§ 2–4. A one-dimensional error on $\delta$ obtained after freezing the other branch is therefore conditional, not the combined result.

For a frequentist profile, minimize over nuisances and all alternate branches. For Bayesian marginalization, integrate over them with explicit priors and include the prior mass assigned to each ordering or octant. The two answer different questions; agreement is a useful robustness check, not an identity.

Exact synthetic correlated likelihood

A reproducible calculation supplies a bounded two-sector fixture. Let $o\in\{+1,-1\}$ be an ordering label, $\eta_N$ a shared normalization/cross-section nuisance, and $\eta_m$ a matter-density nuisance. Define

$$\boldsymbol r= \begin{pmatrix} \cos\delta+\eta_N\\ \sin\delta+o/2+\eta_N+\eta_m-3/2 \end{pmatrix}, \qquad C=\begin{pmatrix}1&1/2\\1/2&1\end{pmatrix},$$

and

$$\chi^2(\delta,o,\eta_N,\eta_m) =\boldsymbol r^{\mathsf T}C^{-1}\boldsymbol r +\eta_N^2+\eta_m^2.$$

This is a dimensionless synthetic likelihood, not experimental evidence. Writing $\boldsymbol r=\boldsymbol d+A\boldsymbol\eta$ gives the analytic profile condition

$$\left(A^{\mathsf T}C^{-1}A+I\right) \widehat{\boldsymbol\eta} =-A^{\mathsf T}C^{-1}\boldsymbol d, \qquad A=\begin{pmatrix}1&0\\1&1\end{pmatrix}.$$

The exact checkpoints are

$(\delta,o)$	$(\widehat\eta_N,\widehat\eta_m)$	$\chi^2_{\rm prof}$
$(\pi/2,+1)$	$(0,0)$	$0$
$(0,+1)$	$(-4/15,14/15)$	$32/15$
$(0,-1)$	$(-2/15,22/15)$	$68/15$
$(\pi,+1)$	$(8/15,2/15)$	$8/15$
$(\pi,-1)$	$(2/3,2/3)$	$4/3$
$(\pi/2,-1)$	$(2/15,8/15)$	$8/15$

The wrong-ordering point can fit partly by moving correlated nuisances, and the two CP-conserving phases $0$ and $\pi$ have different best branches. Dropping the off-diagonal covariance, suppressing $o=-1$, or minimizing the two sectors with independent copies of $\eta_N$ changes the problem rather than approximating it.

Hypothesis tests and coverage

A CP-conservation statistic must minimize over both CP-conserving phases and every allowed branch:

$$q_{\rm CP}= \min_{\substack{\delta\in\{0,\pi\}\\o,\eta}} [-2\ln\mathcal L] -\min_{\delta,o,\eta}[-2\ln\mathcal L].$$

An ordering statistic compares two discrete, nonnested hypotheses after separately profiling or marginalizing their nuisances. Neither statistic is automatically distributed as a one-degree-of-freedom $\chi^2$. Periodicity, physical boundaries, discrete orderings, disconnected degeneracies, small samples, and nuisance parameters weakly identified under a null can all violate regular Wilks conditions. Calibrate with parameter-dependent ensembles or an exact confidence construction when these effects matter; Qian et al. analyze the special nonnested ordering problem in Qian et al. 2012, §§ II–V.

Asymptotic likelihood formulas remain useful when their regularity and large-sample hypotheses have been tested Cowan et al. 2011, §§ 2–3. A confidence level is long-run coverage under a specified construction, not the posterior probability of a hypothesis. A Bayesian ordering probability additionally depends on model priors and prior volume. If analysis regions, channels, orderings, or hypotheses were selected after inspecting data, the trials or selection procedure must enter the calibration.

Combining experiments without double use

Likelihoods multiply only conditionally on a complete set of shared parameters. For two Gaussian summaries,

$$C_{AB}^{\rm joint}= \begin{pmatrix} C_A&C_{AB}\\ C_{AB}^{\mathsf T}&C_B \end{pmatrix}$$

must be positive semidefinite and preserve cross-experiment correlations. Common flux calculations, interaction data, detector calibrations, external oscillation constraints, matter models, and simulated event samples can make $C_{AB}\ne0$. If a published likelihood already contains an external constraint, adding the same source again double counts it.

A robust combination workflow is:

1. reconstruct a dependency graph from raw or released inputs to each likelihood term;
2. replace duplicated external constraints by one shared nuisance model;
3. align units, PMNS phase, mass-splitting, ordering, and neutrino/antineutrino conventions;
4. validate each component on collaboration-supplied benchmarks before combining;
5. test nuisance pulls and goodness of fit by dataset and channel, not only globally;
6. run ensembles at degeneracy, boundary, and low-count points; and
7. publish the full likelihood or sufficient reproducible surrogate with version and supersession information.

Evidence boundary

The likelihood construction, degeneracy taxonomy, and synthetic fixture above are stable methods. Experimental exposures, calibrations, event samples, systematics, likelihood releases, confidence regions, preferred phases or orderings, and global combinations are mutable. Any statement about them requires a dated evidence record naming the dataset and likelihood versions, covariance, corrections, data cutoff, analysis convention, validation, and supersession. No current interval, preference, limit, combination, or status is asserted here because that governed record is absent.

For a runnable convention and profiling check, reproduce the synthetic likelihood with both ordering branches. For versioned neutrino and Standard Model status, continue to Effective Field Theory and Tests of the Standard Model. Return to the propagation sibling when the issue is a matter sign, coherence assumption, or probability rather than a likelihood.

Failure modes

Freezing a discrete branch. Reporting $\Delta\chi^2(\delta)$ for one ordering or octant can hide a competing minimum. Display or marginalize every declared branch and state the conditional result if one is fixed externally.

Duplicating a shared constraint. The same cross-section or oscillation dataset can enter multiple released likelihoods. Trace inputs and represent it once as a shared nuisance or covariance block.

Invoking Wilks by parameter count. Counting one scanned phase does not establish a $\chi_1^2$ law when the phase is periodic, the null is a two-point set, and ordering is discrete. Verify the sampling distribution under the actual nulls.

Turning a synthetic check into evidence. Exact profile values validate algebra and code only. They say nothing about present leptonic CP violation or mass ordering.

References

• Burguet-Castell, J., M. B. Gavela, J. J. Gómez-Cadenas, P. Hernández, and O. Mena. “On the Measurement of Leptonic CP Violation.” Nuclear Physics B 608 (2001): 301–318. DOI.
• Cowan, Glen, Kyle Cranmer, Eilam Gross, and Ofer Vitells. “Asymptotic Formulae for Likelihood-Based Tests of New Physics.” European Physical Journal C 71 (2011): 1554. DOI.
• Qian, X., A. Tan, W. Wang, J. J. Ling, R. D. McKeown, and C. Zhang. “Statistical Evaluation of Experimental Determinations of Neutrino Mass Hierarchy.” Physical Review D 86 (2012): 113011. DOI.