Neutrino Oscillations as a Probe of Macrorealism

TL;DR

This work reframes Leggett-Garg inequalities as a robust test of macrorealism for neutrino flavor oscillations by generalizing LG strings to an optimal family $K_n({\boldsymbol\sigma})$ and leveraging stationary correlators. It introduces two macrorealistic background models, $\mathcal{H}_0^a$ with $\mathcal{C}(\tau)=1$ and $\mathcal{H}_0^b$ with $\mathcal{C}(\tau)=e^{-\Gamma \tau}$, to generate pseudo-data that reflect realistic fluctuations without unphysical correlators. A new test statistic, the RMS $z$-score $z_{\rm RMS}$, quantifies LGI violations across phase-matched sequences ${\bf s}$, yielding post-trial significances of about $2.1\sigma$ (for $\mathcal{H}_0^a$) and $3.7\sigma$ (for $\mathcal{H}_0^b$) when applied to MINOS/MINOS+ muon-neutrino survival data. The results show more conservative evidence for LGI violations than earlier claims, highlighting the importance of background modelling and providing a general framework for reanalysis of related neutrino data sets with macrorealism tests.

Abstract

The correlations between successive measurements of a quantum system can violate a family of Leggett-Garg Inequalities (LGIs) that are analogous to the violation of Bell's inequalities of measurements performed on spatially separated quantum systems. These LGIs follow from a macrorealistic point of view, imposing that a classical system is at all times in a definite state and that a measurement can, at least in principle, leave this state undisturbed. Violations of LGIs can be probed by neutrino flavour oscillations if the correlators of consecutive flavour measurements are approximately stationary. We discuss here several improvements of the methodology used in previous analyses based on accelerator and reactor neutrino data. We argue that the strong claims of LGI violations made in previous studies are based on an unsuitable modelling of macrorealistic systems in statistical hypothesis tests. We illustrate our improved methodology via the example of the MINOS muon-neutrino survival data, where we find revised statistical evidence for violations of LGIs at the $(2-3)σ$ level, depending on macrorealistic background models.

Figures (3)

• Figure 1: Illustration of LG strings and their violation of LGIs in QM. We consider $n=4$ consecutive measurements (see top panels) with phase steps $\phi_a$, $\phi_b$, and $\phi_c$ and assume that the correlators follow $\mathcal{C}_{ij} = \cos(\phi_i-\phi_j)$ for maximal two-level oscillations in QM. The total phase $\phi_{\rm tot} = \phi_a+\phi_b+\phi_c$ is fixed to $\phi_{\rm tot}=3\pi/2$, comparable to the phase range covered by the MINOS/MINOS+ data. The ternary plots indicate the relative combinations of $\phi_a$, $\phi_b$, and $\phi_c$ where we expect that the corresponding LG strings violate (obey) the LGIs indicated by the red-shaded (blue-shaded) regions. The left panel shows $K_4$ as in Eq. (\ref{['eq:Kn']}) which does not violate the LGI $K_4\leq 2$ (nor $-2\leq K_4$) for this example. The center (right) panel shows $K_4^{\rm max}$ in Eq. (\ref{['eq:Knmax']}) for two different orders of measurements ("$1+3$" or "$2+2$") as indicated in the top panel.
• Figure 2: Flavour correlations observed in MINOS/MINOS+ data Sousa:2015bxa (black data) in comparison to the prediction of the best-fit oscillation models assuming normal ordering Esteban:2020cvm (in green) and two classical models: a constant model (in dotted blue) and the best-fit Markovian model (in dashed purple).
• Figure 3: Normalized distributions of the fraction of LGI violations (left panels) and corresponding RMS $z$-scores (right panels) based on $10^7$ pseudo-samples. Results are shown separately for LG strings in Eq. (\ref{['eq:Kmax']}) for five different combinations of $n_a$ and $n_b$ with $n_a+n_b = 3$, $4$ and $5$ listed in Table \ref{['tab1']}. The results from MINOS/MINOS+ data are shown as vertical lines with $\star$ symbols. The top panels show the distributions expected from neutrino oscillations assuming normal ordering (NO) Esteban:2020cvm. The bottom panels show those expected for the background (BGR) models $\mathcal{H}_0^a$ with $\mathcal{C}(\tau)=1$ (filled histograms) and $\mathcal{H}_0^b$ with $\mathcal{C}(\tau)=e^{-\Gamma \tau}$ (open histograms). The BGR distributions of RMS $z$-scores allow us to estimate the chance probabilities ($p$-values) of LGI violations under the BGR hypotheses at the level observed in MINOS/MINOS+ data. Treating the five $p$-values as independent trials, we estimate the post-trial significance at the level of $2.1\sigma$ ($3.7\sigma$) for the BGR model $\mathcal{H}_0^a$ ($\mathcal{H}_0^b$).