Parametric enhancement of flavor oscillation in a three-neutrino framework

When neutrinos travel through matter with a periodic density profile, the neutrino oscillation probability can be enhanced if certain conditions are satisfied. In a two-neutrino framework, the condition for parametric resonance is known. Herein, we consider the analogous parametric resonance condition within the context of a full three-neutrino framework with two oscillation scales. For energies in the range of hundreds of MeV to a few GeV, we find that neutrino oscillation can be parametrically enhanced if two approximate relations are satisfied. The first is similar to the two-neutrino parametric resonance condition while the second involves the other oscillation scale. Treating the Earth's density as piecewise constant, we show that oscillations in this energy range can be enhanced between two- and threefold.

Figure 1

The modulus of $u_3$ for $E=10$ MeV and various baselines $L_{a,b}$ with densities ${\rho }_a=4.5\text{g}/{\text{cm}}^3$ and ${\rho }_b=11.5\text{g}/{\text{cm}}^3$ in the two-neutrino limit $\theta =0.59,{\Delta }_{21}=7.54\times {10}^{-5}{\mathrm{eV}}^2$, and ${\Delta }_{31}=2.47\times {10}^{-3}{\mathrm{eV}}^2$. We mark the half-wavelength solutions with $\times$.

Figure 2

The modulus of $u_3$ for $E=500$ MeV and various baselines $L_{a,b}$ using the same data as Fig. 1. We mark the half-wavelength solutions with $\times$.

Figure 3

The modulus of $u_3$ for $E=100$ MeV and various baselines $L_{a,b}$ using the same data as Fig. 1. We mark the half-wavelength solutions with $\times$.

Figure 4

The modulus of $u_3$ for $E=200$ MeV and various baselines $L_{a,b}$ using the same data as Fig. 1. We mark the half-wavelength solutions with $\times$.

Figure 5

The oscillation probability ${\mathcal{P}}_{e\mu }$ for 200 MeV neutrinos through a castle-wall profile with (a) $L_a=3100$ km and $L_b=1548$ km and (b) $L_a=1550$ km and $L_b=2477$ km. The shaded regions in the plots indicate the region of density ${\rho }_b=11.5\text{g}/{\text{cm}}^3$.

Figure 6

A plot of $\overline{u}$ for $E=200$ MeV and various baselines $L_{a,b}$ with densities ${\rho }_a=4.5\text{g}/{\text{cm}}^3$ and ${\rho }_b=11.5\text{g}/{\text{cm}}^3$ with $\theta =0.59,\varphi =0.15,{\Delta }_{21}=7.54\times {10}^{-5}{\mathrm{eV}}^2$, and ${\Delta }_{31}=2.47\times {10}^{-3}{\mathrm{eV}}^2$.

Figure 7

A plot of $\overline{u}$ for $E=500$ MeV and various baselines $L_{a,b}$ using the same data as Fig. 6. We mark the approximate minima of $\overline{u}$, solutions to the equations Eqs. (45) and (46), with the symbol $\times$.

Figure 8

A plot of $\overline{u}$ for $E=1$ GeV and various baselines $L_{a,b}$ using the same data as Fig. 6. We mark the approximate minima of $\overline{u}$, solutions to the equations Eqs. (47) and (48), with the symbol $\times$.

Figure 9

The oscillation probability ${\mathcal{P}}_{e\mu }$ for 1 GeV neutrinos through a castle-wall profile with $L_a=3220$ km and $L_b=1819$ km. The shaded areas indicate regions of density ${\rho }_b=11.5\text{g}/{\text{cm}}^3$.

Figure 10

The oscillation probability ${\mathcal{P}}_{e\mu }$ for 1 GeV neutrinos through a castle-wall profile with $L_a=1800$ km and $L_b=2296$ km. The shaded areas indicate regions of density ${\rho }_b=11.5\text{g}/{\text{cm}}^3$. We set $\theta =0.59,\varphi =0$, and $\psi =0.72$.

Figure 11

A plot of $\overline{u}$ for $E=1$ GeV and various baselines $L_{a,b}$ with $\theta =0.59,\varphi =0.15$, and $\psi =0.72$ in the castle-wall profile.

Figure 12

The oscillation probability ${\mathcal{P}}_{e\mu }$ for 1 GeV neutrinos through a castle-wall profile with $L_a=1625$ km and $L_b=2410$ km. We set $\theta =0.59,\varphi =0.15$, and $\psi =0.72$. We omit the shaded regions which typically indicate the higher density region.

Figure 13

(a) We plot $\overline{u}$ for core-crossing trajectories. The dotted (black) curve is for neutrino energy 600 MeV; the dashed (red) curve is for neutrino energy 800 MeV; the solid (blue) curve is for neutrino energy 1 GeV. (b) The oscillation probability ${\mathcal{P}}_{e\mu }$ at a detector after traveling a distance of $2L_a+L_b$ along a core-crossing trajectory. The curves are averaged over a flat energy spectrum with a width of 200 MeV. The dotted (black) curve's spectrum is centered on 600 MeV; the dashed (red) curve's spectrum is centered on 800 MeV; the solid (blue) curve's spectrum is centered on 1 GeV.