Atmospheric Neutrino Oscillation Parameters

• Atmospheric neutrino oscillation parameters are fundamental metrics that quantify the probability of neutrino flavor changes over long baselines using energy-dependent oscillation formulas.
• Their determination involves high-statistics atmospheric samples, detailed flux modeling, cross-section evaluations, and advanced statistical inference to control systematic uncertainties.
• Enhanced detector techniques, including CNN-based event reconstruction and magnetized charge identification, complement accelerator experiments to refine mass splitting and mixing angle measurements.

Atmospheric neutrino oscillation parameters quantify the flavor transformation probabilities of neutrinos traversing the Earth as a function of their energy and propagation length. These parameters, most notably the atmospheric mass-squared splitting ($\Delta m^2_{32}$ or $\Delta m^2_{31}$) and the atmospheric mixing angle ($\theta_{23}$), are fundamental inputs to the three-flavor neutrino oscillation framework. High-statistics atmospheric samples—especially muon neutrino disappearance at energies from a few to several tens of GeV—enable precise, complementary measurements to accelerator-based long-baseline oscillation programs. The determination of these parameters requires detailed modeling of neutrino fluxes, interaction cross sections, detector response, and a robust statistical inference methodology, all while incorporating profound sources of systematic uncertainties.

1. Theoretical Formulation and Oscillation Probability

The survival probability for muon neutrinos ($\nu_\mu$) in the context of atmospheric oscillations, in the two-flavor approximation, is customarily expressed as

$$P(\nu_\mu \to \nu_\mu) = 1 - \sin^2 2\theta_{23}\;\sin^2\left(\frac{\Delta m^2_{32} L}{4E} \right)$$

where $L$ is the baseline (path through the Earth, a function of zenith angle), $E$ is the neutrino energy, $\theta_{23}$ is the atmospheric mixing angle, and $\Delta m^2_{32}$ is the atmospheric mass-squared splitting. In contemporary analyses, the full three-flavor framework is employed, incorporating subdominant effects from $\theta_{13}$, $\Delta m^2_{31}$0, matter-induced oscillation modifications, and subleading mass differences (e.g., $\Delta m^2_{31}$1 effects): $\Delta m^2_{31}$2 with $\Delta m^2_{31}$3 from the PMNS matrix. At atmospheric energies (several to tens of GeV), the oscillation is dominated by $\Delta m^2_{31}$4 and $\Delta m^2_{31}$5, though precise data require inclusion of all standard oscillation parameters.

2. Experimental Access and Data Binning Strategies

Atmospheric neutrinos are detected via their charged-current interactions in large-volume detectors such as IceCube-DeepCore, Super-Kamiokande, ANTARES, NOvA, MINOS, and ICAL/INO. The basic experimental signature is the observation of $\Delta m^2_{31}$6-induced muon tracks or cascades. Detectors exploit the wide range of neutrino zenith angles (hence baselines up to Earth's diameter, $\Delta m^2_{31}$7 km) and energies ($\Delta m^2_{31}$8–$\Delta m^2_{31}$9 GeV) typical in atmospheric samples. Data are binned in reconstructed energy and zenith angle—often with $\theta_{23}$05 GeV energy bins and a few zenith-angle bins, with more granularity in the oscillation-sensitive regions. For example, IceCube DeepCore studies utilize energy bins of $\theta_{23}$1 GeV and prioritize upward-going neutrinos, which have traversed longer paths and thus are more likely to exhibit oscillation-induced depletion of $\theta_{23}$2 flux (Fernandez-Martinez et al., 2010).

Such binned distributions (energy and zenith angle) are critical because the position and depth of the oscillation “dip” in event spectra directly constrain $\theta_{23}$3 and $\theta_{23}$4 (Thakore et al., 2013, Whitehead, 2016, Collaboration et al., 4 Sep 2025). Event categories (track/cascade, contained/entering/leaving) and advanced particle identification based on Cherenkov topology and machine-learning classifiers further enhance sensitivity by improving flavor tagging and removing contaminant backgrounds.

3. Statistical Inference, Systematics, and Nuisance Parameters

Extracting oscillation parameters from atmospheric neutrino data relies on likelihood or $\theta_{23}$5-minimization methods comparing observed binned distributions to predictions from oscillation models, flux calculations, cross sections, and full detector simulation. Systematic uncertainties are incorporated as additional error terms in quadrature (Gaussian penalty terms) or via “nuisance parameters” profiled or marginalized in the fit.

Key systematics in atmospheric analyses include:

• Absolute and relative normalization of the atmospheric neutrino flux (uncertainty $\theta_{23}$610–25%, with smaller uncertainty on the $\theta_{23}$7 ratio, $\theta_{23}$85%)
• Neutrino-matter cross section uncertainties (particularly at transition energies between quasi-elastic, resonance, and deep inelastic scattering)
• Detector response (e.g., efficiency, optical module calibration for Cherenkov detectors, energy/angular resolutions)
• Event classification and muon charge identification (critical for experiments such as MINOS and ICAL that separate $\theta_{23}$9 from $\nu_\mu$0 events)
• Treatment of atmospheric muon contamination and non-neutrino backgrounds

Self-calibration procedures exploit less-oscillated event samples (e.g., downward-going neutrinos) to constrain flux normalization (Fernandez-Martinez et al., 2010), and external priors (e.g., reactor $\nu_\mu$1 value) further break parameter degeneracies (Collaboration et al., 2023).

Table: Example Structure of Fit Nuisance Parameters in Atmospheric Oscillation Analyses

Nuisance Parameter Type	Typical Magnitude	Experimental Control Method
Neutrino Flux Normalization	10%–25%	Unoscillated event rate, MC tuning
$\nu_\mu$2 ratio	~5%	Magnetized detector separation
Detector Efficiency	few %	Calibration sources, simulation
Energy Scale	few %	Test beams, Michel electrons

Such thorough systematics treatment is essential for achieving the reported sub-5% error on $\nu_\mu$3 and $\nu_\mu$410% on $\nu_\mu$5 in recent atmospheric samples (Collaboration et al., 2023, Collaboration, 2024, Collaboration et al., 4 Sep 2025).

4. Precision Measurements, Complementarity, and Global Results

Results from modern atmospheric neutrino experiments demonstrate consistency and complementarity with long-baseline accelerator experiments. The current world-leading constraints, achieved by NOvA and IceCube-DeepCore using joint fits or machine-learning-enhanced reconstruction of atmospheric data, yield (for normal ordering): $\nu_\mu$6 (Collaboration et al., 4 Sep 2025)

$\nu_\mu$7

(Collaboration, 2024)

These values result from the precise localization of the oscillation dip in $\nu_\mu$8 distributions and the control of systematics. Maximal mixing ($\nu_\mu$9) is favored within current uncertainties, with best-fit values showing a mild preference for the higher octant but without strong statistical significance (Collaboration et al., 4 Sep 2025). Combined analyses (e.g., joint SK-T2K fits (Super-Kamiokande et al., 2024), inclusion of reactor constraints (Collaboration et al., 2023)) further refine the measurement and help address degeneracies in $$P(\nu_\mu \to \nu_\mu) = 1 - \sin^2 2\theta_{23}\;\sin^2\left(\frac{\Delta m^2_{32} L}{4E} \right)$$0 and the mass ordering.

Moreover, atmospheric datasets have unique sensitivity to Earth matter effects, critically informing the neutrino mass ordering through the observation of resonance-enhanced flavor transitions in upward-going, multi-GeV samples (Winter, 2015, Collaboration et al., 2023).

5. Detector-Specific Methodologies and Technological Advances

Atmospheric neutrino oscillation measurements leverage innovations in detector design, reconstruction algorithms, and statistical methodology:

• Dense instrumentation (e.g., IceCube DeepCore, PINGU) enables low-threshold energy detection (as low as $$P(\nu_\mu \to \nu_\mu) = 1 - \sin^2 2\theta_{23}\;\sin^2\left(\frac{\Delta m^2_{32} L}{4E} \right)$$15 GeV) crucial for accessing the oscillation dip.
• Use of convolutional neural networks (CNNs) for event reconstruction allows for high-statistics, efficient, and high-fidelity event classification—e.g., IceCube DeepCore employs dedicated CNNs for energy, zenith, vertex, PID, and atmospheric muon rejection, delivering a $$P(\nu_\mu \to \nu_\mu) = 1 - \sin^2 2\theta_{23}\;\sin^2\left(\frac{\Delta m^2_{32} L}{4E} \right)$$25,000-fold increase in throughput over previous likelihood-based methods, while maintaining or improving reconstruction precision (Collaboration, 2024).
• Magnetized detectors (MINOS, ICAL/INO) permit explicit separation of $$P(\nu_\mu \to \nu_\mu) = 1 - \sin^2 2\theta_{23}\;\sin^2\left(\frac{\Delta m^2_{32} L}{4E} \right)$$3 and $$P(\nu_\mu \to \nu_\mu) = 1 - \sin^2 2\theta_{23}\;\sin^2\left(\frac{\Delta m^2_{32} L}{4E} \right)$$4 samples, constraining both absolute rates and ratios—unique for systematics handling and for searches for CPT violation or non-standard interactions (Collaboration et al., 2012, Thakore et al., 2013, Mohan et al., 2016).
• Advanced neutron tagging (Super-Kamiokande IV/V) enhances $$P(\nu_\mu \to \nu_\mu) = 1 - \sin^2 2\theta_{23}\;\sin^2\left(\frac{\Delta m^2_{32} L}{4E} \right)$$5 statistical separation, critical for matter effect sensitivity and mass ordering in water Cherenkov detectors (Collaboration et al., 2023).
• Analysis techniques include maximum likelihood fits, incorporation of systematic penalty terms, use of “pull” methods for nuisance parameters, and exploitation of off-diagonal “self-calibration” in multidimensional (energy, zenith, topology, flavor) distributions.

Recent detector upgrades, expansion of fiducial volumes, improved calibration of optical module responses (including in-situ SPE charge distribution and depth-dependent ice properties), and removal of delayed/scattered photon hits have collectively advanced the achievable precision (Collaboration et al., 2023).

6. Impact, Limitations, and Prospects

Atmospheric neutrino oscillation parameter measurements have yielded:

• Precise, independent determination of $$P(\nu_\mu \to \nu_\mu) = 1 - \sin^2 2\theta_{23}\;\sin^2\left(\frac{\Delta m^2_{32} L}{4E} \right)$$6 ($$P(\nu_\mu \to \nu_\mu) = 1 - \sin^2 2\theta_{23}\;\sin^2\left(\frac{\Delta m^2_{32} L}{4E} \right)$$71.5% error) and $$P(\nu_\mu \to \nu_\mu) = 1 - \sin^2 2\theta_{23}\;\sin^2\left(\frac{\Delta m^2_{32} L}{4E} \right)$$8 (error floor set by systematics, $$P(\nu_\mu \to \nu_\mu) = 1 - \sin^2 2\theta_{23}\;\sin^2\left(\frac{\Delta m^2_{32} L}{4E} \right)$$90.02–0.06) in agreement with accelerator experiment results (Collaboration et al., 4 Sep 2025, Collaboration, 2024).
• No statistically significant deviation from maximal mixing nor clear resolution of the $L$0 octant, though mild preferences are observed in some fits.
• Mild to moderate preference for normal mass ordering as statistical power grows and constraints from reactor mixing angle measurements (e.g., Daya Bay) are incorporated (Collaboration et al., 4 Sep 2025, Collaboration et al., 2023).
• Advanced exclusion of scenarios with large $L$1 appearance asymmetries, thus constraining the allowed parameter region in $L$2 and mass ordering space (Catano-Mur, 2022).
• Rigorous tests for new physics scenarios, e.g., CPT violation (no evidence found (Collaboration et al., 2012)), sterile neutrinos (limits from multi-year ANTARES data (Collaboration et al., 2018)), and measurements of Earth density profiles via matter-induced oscillation effects (Winter, 2015).

Limitations arise from residual uncertainties in flux, cross sections at higher energies, finite detector resolutions, and, for $L$3 and octant/ordering, the interplay among three-flavor effects and matter interactions. Further reduction of uncertainties and sensitivity to new physics will require continued development of advanced reconstruction, systematic modeling, and joint analyses combining atmospheric and accelerator data (Super-Kamiokande et al., 2024).

7. Summary Table of Recent Atmospheric Oscillation Parameter Measurements

Experiment/Analysis	Data Period	$L$4 eV$L$5]	$L$6	Mass Ordering Pref.	Reference
NOvA (10 y, final)	up to 2025	$L$7 (N)	$L$8	Normal, Bayes 6.6	(Collaboration et al., 4 Sep 2025)
IceCube DeepCore (CNN, 9.3 y)	2012–2021	$L$9	$E$0	Normal	(Collaboration, 2024)
IceCube DeepCore (calibrated, 8 y)	2011–2019	$E$1	$E$2	Normal	(Collaboration et al., 2023)
Super-Kamiokande I–V (full)	1996–2020	(see Table in paper)	(see Table in paper)	Normal, 70–80% CL	(Collaboration et al., 2023)

These results define the current landscape of atmospheric neutrino oscillation precision and demonstrate the maturity of both experimental detection and multivariate statistical interpretation in this sector.