Quantifying the sensitivity of oscillation experiments to the neutrino mass ordering

TL;DR

This paper clarifies the statistical interpretation of sensitivity to neutrino mass ordering by situating it in a frequentist hypothesis-testing framework and introducing the Gaussian limit for the test statistic $T$. It demonstrates that the median sensitivity under Gaussian assumptions closely matches the conventional $\sqrt{\Delta\chi^2}$-based sensitivities and provides analytic expressions to compute power and crossing sensitivities, including for composite hypotheses. Through Monte Carlo simulations of JUNO, INO, PINGU, NO$\nu$A, and LBNE, the authors validate the Gaussian approximation for most setups and quantify deviations (notably in NO$\nu$A) while showing strong agreement for others. The study offers a practical, statistically rigorous toolkit to compare current and future experiments’ capabilities to determine the neutrino mass ordering, with implications for experimental design and the interpretation of sensitivities.

Abstract

Determining the type of the neutrino mass ordering (normal versus inverted) is one of the most important open questions in neutrino physics. In this paper we clarify the statistical interpretation of sensitivity calculations for this measurement. We employ standard frequentist methods of hypothesis testing in order to precisely define terms like the median sensitivity of an experiment. We consider a test statistic $T$ which in a certain limit will be normal distributed. We show that the median sensitivity in this limit is very close to standard sensitivities based on $Δχ^2$ values from a data set without statistical fluctuations, such as widely used in the literature. Furthermore, we perform an explicit Monte Carlo simulation of the INO, JUNO, LBNE, NOvA, and PINGU experiments in order to verify the validity of the Gaussian limit, and provide a comparison of the expected sensitivities for those experiments.

Figures (13)

• Figure 1: Left: Distribution of the test statistic $T$ for our default configuration of the JUNO reactor experiment discussed in Sec. \ref{['sec:juno']}. Histograms show the results of the MC simulation based on $10^5$ simulated experiments and black curves correspond to the Gaussian approximation discussed in Sec. \ref{['sec:gauss']}. Right: The value of $\alpha$ as a function of the critical value $T_{c}^\alpha$ required for rejecting inverted (blue) and normal (red) ordering for the JUNO reactor experiment. In the purple region both mass orderings are rejected at the CL ($1-\alpha$), in the white region both orderings are consistent with data at the CL ($1-\alpha$). The dashed lines in both panels indicate $T_c^\alpha$ for $\alpha = 0.01$ for both orderings. The dotted lines indicate the crossing point $T_c^{\rm NO} = T_c^{\rm IO}$. The dot-dashed line in the right panel shows an example (for $\alpha=0.1$) in which $T_{c,\textrm{IO}}^\alpha < T_{c,\textrm{NO}}^\alpha$.
• Figure 2: Gaussian approximation for the test statistics $T$. Left upper panel: critical values for rejecting normal ordering as a function of $T_0$, see Eq. \ref{['eq:Tca-gauss']}, for different values of $\alpha$ as labeled in the plot. Left lower panel: power of the test as a function of $T_0$ for different values of $\alpha$, see Eq. \ref{['eq:beta-gauss']}. Right panel: power of the test (left vertical axis) and the rate for an error of the second kind (right vertical axis) versus the CL ($1-\alpha$) for rejecting a given mass ordering for different values of $T_0$ as labeled in the plot. The vertical lines indicate the number of standard deviations, where we have used our standard convention Eq. \ref{['eq:sigma-alpha']} based on a 2-sided Gaussian for the solid lines and Eq. \ref{['eq:sigma-alpha-1sided']} based on a 1-sided Gaussian limit for the dashed lines. The dash-dotted red curve indicates $\alpha = \beta$, which follows in the Gaussian case from the condition $T_c^{\rm NO} = T_c^{\rm IO}$.
• Figure 3: Median sensitivity ($\beta=0.5$) as a function of $T_0$, see Eq. \ref{['eq:median-sens-gauss']}. The curves labeled "crossing" show the sensitivity corresponding to the condition $T_c^{\rm NO} = T_c^{\rm IO}$ according to Eq. \ref{['eq:alpha-crossing']}. The solid curves use the 2-sided Gaussian to convert $\alpha$ into $n\sigma$, Eq. \ref{['eq:sigma-alpha']}, whereas the dashed curves are based on the 1-sided test, Eq. \ref{['eq:sigma-alpha-1sided']}. The latter correspond to the "standard sensitivity" of $n = \sqrt{T_0}$ and $n = \sqrt{T_0}/2$ for the crossing sensitivity. The edges of the green and yellow bands are obtained from the conditions on the rate for an error of the second kind $\beta = 1/2 \pm 0.6827/2$ and $\beta = 1/2 \pm 0.9545/2$, respectively.
• Figure 4: Simulated distributions of the test statistic $T$ in the INO experiment. We use our default setup as defined in Tab. \ref{['tab:atm']} and assume $\theta_{23} = 45^\circ$. Solid curves show the Gaussian approximation from Eq. \ref{['eq:Tgauss1']}.
• Figure 5: Simulated distributions of the test statistic $T$ in the PINGU experiment with $\theta_{23} = 40^\circ, \, 45^\circ, \, 50^\circ$ for the left, middle, right panel, respectively. We use our default setup as defined in Tab. \ref{['tab:atm']}. Solid curves show the Gaussian approximation from Eq. \ref{['eq:Tgauss1']}.