A General Framework for Leggett--Garg Violation in Neutrino Oscillations and Matter Effects

Abstract

The Leggett--Garg inequality (LGI) constrains temporal correlations under the assumptions of macrorealism and noninvasive measurability. We study the four-time quantity \(K_4\) for two-flavor neutrino oscillations using a pseudo-density matrix (PDM) under static conditions, instead of non-invasive measurement. With this approach, it becomes possible to examine a Horodecki-type criterion for the temporal Pauli-correlation tensor. The Horodecki criterion had been proven for CHSH inequality (R. Horodecki et al. Phys. Lett. \textbf{A200}, 340), For any unitary qubit evolution, the corresponding scalar is \(M_R=2\). Although, this result characterizes the correlation tensor and the optimized temporal construction, it does not imply that a fixed flavor observable violates the LGI at every neutrino energy or baseline. We therefore evaluate \(K_4\) explicitly for flavor measurements in matter of constant density. The numerical examples illustrate how matter effects modify the temporal correlations, with a pronounced matter--vacuum separation near the MSW resonance for the chosen parameters and generally smaller differences in the selected off-resonance cases. This finding motivates further investigation of matter-induced temporal correlations in atmospheric neutrinos and their potential relevance to future studies with long-baseline and atmospheric-neutrino experiments, including DUNE and Hyper-Kamiokande.

The paper examines how constant-density matter affects the four-time Leggett–Garg quantity $K_4=C_{12}+C_{23}+C_{34}-C_{14}$ in two-flavor neutrino oscillations, whose macrorealistic bound is $|K_4|\leq2$. It uses pseudo-density matrices (PDMs) to represent sequential quantum correlations and assumes stationarity to reconstruct flavor correlators from oscillation probabilities at different propagation baselines. For arbitrary initial qubit states and unitary evolution, it derives that the two-time temporal Pauli-correlation tensor is orthogonal. Consequently, the sum of the two largest eigenvalues of $T_R^{\mathsf T}T_R$ is $M_R=2$, giving $2\sqrt{2}$ under the specified Horodecki-type optimization over measurement directions. This optimized construction is distinguished from measurements of the fixed flavor observable $Q=\sigma_z$, which need not violate the inequality. For flavor measurements, numerical calculations substitute the standard matter-modified mixing angle and mass splitting into $K_4$, using propagation intervals $L,3L,5L,9L$. At density $5.51\,\mathrm{g\,cm^{-3}}$ and electron fraction $Y_e=0.5$, the illustrative resonance energies are approximately $5.72$ GeV for atmospheric-scale parameters and $0.07$ GeV for solar-splitting parameters. Selected near-resonance cases show pronounced matter–vacuum differences; violation remains phase- and baseline-dependent. The calculation is idealized: it excludes three-flavor dynamics, realistic Earth density profiles, detector response, backgrounds, and statistical sensitivity. The $0.07$ GeV example is not accessible to solar-origin neutrinos. Stationarity does not establish noninvasiveness, and the predictions do not constitute an experimental rejection of macrorealism.