PINGU and the neutrino mass hierarchy: Statistical and systematic aspects

The proposed PINGU project (Precision IceCube Next Generation Upgrade) is expected to collect $O({10}^5)$ atmospheric muon and electron neutrino events in a few years of exposure, and to probe the neutrino mass hierarchy through its imprint on the event spectra in energy and direction. In the presence of non-negligible and partly unknown shape systematics, the analysis of high-statistics spectral variations will face subtle challenges that are largely unprecedented in neutrino physics. We discuss these issues both on general grounds and in the currently envisaged PINGU configuration, where we find that possible shape uncertainties at the (few) percent level can noticeably affect the sensitivity to the hierarchy. We also discuss the interplay between the mixing angle ${\theta }_{23}$ and the PINGU sensitivity to the hierarchy. Our results suggest that more refined estimates of spectral uncertainties are needed in next-generation, large-volume atmospheric neutrino experiments.

Figure 1

Widths (at $\pm 1\sigma$) of the resolution functions in energy and angle for ${\nu }_{\mu }$-like events in PINGU, in terms of logarithmic energy versus the zenith angle (left plot) or its cosine (right plot). See the text for details.

Figure 2

Breakdown of the main ingredients of typical muon (upper panels) and electron (lower panels) event spectra in energy and angle in PINGU. From left to right: the oscillation-independent factor $V_{\mathrm{eff}}^{\alpha }{\mathrm{\Phi }}^{\alpha }{\sigma }_{\mathrm{CC}}^{\alpha }$, the oscillation-dependent factor $P^{\alpha }$, their product (unsmeared event spectrum), and final results (smeared event spectrum). See the text for details.

Figure 3

Case of true normal hierarchy and ${\sin }^2{\theta }_{23}=0.6$. Left panels: Absolute event spectra (top: $\mu$ events; bottom: $e$ events). Middle panels: Statistical deviations with respects to the best-fit spectrum in the wrong (inverted) hierarchy, marginalized over oscillation and normalization systematics only. Right panels: The same deviations in percent values.

Figure 4

As in Fig. 3, but for true inverted hierarchy and ${\sin }^2{\theta }_{23}=0.4$.

Figure 5

PINGU sensitivity to the hierarchy ($N_{\sigma }$), for either true NH (top panels) or true IH (bottom panels), as a function of the live time $T$ in years. The abscissa is scaled as $\sqrt{T}$, so that the sensitivity bands (which span ${\sin }^2{\theta }_{23}{|}_{\text{true}}\in [0.4,0.6]$) would grow linearly for purely statistical errors. From left to right, the fit includes the following systematic errors: oscillation and normalization uncertainties, energy-scale and resolution-width errors, polynomial shape systematics (with up to quartic terms) at the 1.5% level, and uncorrelated systematics at the 1.5% level, as defined in Sec. 3.

Figure 6

As in Fig. 5, but for polynomial and uncorrelated systematic errors doubled (left) or halved (right), while statistical errors and systematics related to oscillation, normalization, and resolution uncertainties are assumed to be the same.

Figure 7

As in Fig. 5, but for ${\sin }^2{\theta }_{23}{|}_{\text{true}}\in [0.46,0.54]$.

Figure 8

Fitted values of ${\sin }^2{\theta }_{23}^{\mathrm{fit}}$ (at 1, 2, and $3\sigma$) versus the true value of ${\sin }^2{\theta }_{23}^{\text{true}}$, for the four possible cases where the test hierarchy (i.e., the one assumed in the fit) is either the true or the wrong one: (a) $\mathrm{NH}=\text{true}$, $\mathrm{NH}=\text{test}$; (b) $\mathrm{NH}=\text{true}$, $\mathrm{IH}=\text{test}$; (c) $\mathrm{IH}=\text{true}$, $\mathrm{IH}=\text{test}$; (d) $\mathrm{IH}=\text{true}$, $\mathrm{NH}=\text{test}$.