Studies of resonance conditions on neutrino oscillations in matter

TL;DR

This work develops a discriminant-based framework to identify resonance conditions in matter for 2-, 3-, and 4-neutrino oscillations, including a sterile state, by analyzing the characteristic equation of the effective Hamiltonian. It derives explicit forms for the discriminants $D_{(2)}$, $D_{(3)}$, and $D_{(4)}$ and shows how resonances correspond to local minima of these discriminants, with the energy–density relation $E A \approx \tfrac{\Delta m^2}{2}\cos2\theta$ governing the resonance location. Numerical analyses in a $(3+1)_1$ mass scheme illustrate how small mixings sharpen resonances and how the relevant densities scale with $E$; the study connects the discriminant formalism to observable survival probabilities, e.g., $P_{ee}$. The results clarify resonance structure beyond the standard two-neutrino MSW picture and assess the practicality of observing 4-neutrino resonances under Earth- and Sun-like densities, suggesting that such resonances may be difficult to realize in typical experimental setups.

Abstract

We analytically discuss the resonance conditions among several neutrinos in matter. The discriminant for the characteristic equation of the Hamiltonian is expressed by the coefficients of the equation. The result of the computation for the discriminants tells us that the neutrino energy and the matter density are in inverse proportion to each other at the resonance states in not only 2- but also 3- and 4-neutrino models.

Figures (4)

• Figure 1: Several mass patterns for 4-neutrino schemes.
• Figure 2: Normalized discriminants $D_n$$(n=2,3,4)$ as a function of the matter density $A$ for a $(3+1)_1$-scheme, where $\theta_{12} = \frac{\pi}{4}$, $\theta_{23} = \frac{\pi}{4}$, $\theta_{13} = \frac{5}{180}\pi$, $\theta_{14} = \frac{3}{180}\pi$, $\theta_{24} = 0$, $\theta_{34} = 0$, $\Delta_{13} = \Delta_{24} = \Delta_{34} = 0$, $\Delta m_{21}^2 \simeq 7 \times 10^{-5} \,\mathrm{eV^2}$, $\Delta m_{32}^2 \simeq 3 \times 10^{-3} \,\mathrm{eV^2}$, $\Delta m_{41}^2 \simeq 1 \,\mathrm{eV^2}$, $E=10$ GeV. The solid, broken and dotted lines are normalized discriminants $D_4$, $D_3$ and $D_2$, respectively.
• Figure 3: Normalized discriminants $D_n$$(n=2,3,4)$ as a function of the matter density $A$ for a $(3+1)_1$-scheme, where $\theta_{12} = \frac{\pi}{4}$, $\theta_{23} = \frac{\pi}{4}$, $\theta_{13} = \frac{5}{180}\pi$, $\theta_{14} = \frac{3}{180}\pi$, $\theta_{24} = 0$, $\theta_{34} = 0$, $\Delta_{13} = \Delta_{24} = \Delta_{34} = 0$, $\Delta m_{21}^2 \simeq 7 \times 10^{-5} \,\mathrm{eV^2}$, $\Delta m_{32}^2 \simeq 3 \times 10^{-3} \,\mathrm{eV^2}$, $\Delta m_{41}^2 \simeq 1 \,\mathrm{eV^2}$, $E=10$ MeV. The solid, broken and dotted lines are normalized discriminants $D_4$, $D_3$ and $D_2$, respectively.
• Figure 4: The surviving probability $P_{ee}$ of the electron neutrino transition as a function of the matter density $A$ for the $(3+1)_1$-scheme, where $\theta_{12} = \frac{\pi}{4}$, $\theta_{23} = \frac{\pi}{4}$, $\theta_{13} = \frac{5}{180}\pi$, $\theta_{14} = \epsilon$, $\theta_{24} = 0$, $\theta_{34} = 0$, $\Delta_{13} = \Delta_{24} = \Delta_{34} = 0$, $\Delta m_{21}^2 \simeq 7 \times 10^{-5} \,\mathrm{eV^2}$, $\Delta m_{32}^2 \simeq 3 \times 10^{-3} \,\mathrm{eV^2}$, $\Delta m_{41}^2 \simeq 1 \,\mathrm{eV^2}$, $E=10\, \mathrm{GeV}$ and $L=10000 \, \mathrm{km}$. The solid and broken lines show the transition probabilities for $\epsilon = \frac{3}{180}\pi, 0$, respectively, which correspond to the 4-neutrino and the 3-neutrino cases.