The Neutrino Day-Night Effect

Abstract

The day-night effect describes the difference in state of solar neutrinos observed during the daytime compared to nighttime. The asymmetry arises because neutrinos observed at night must propagate through the entire Earth to reach the detector, interacting with the planet's electron-dense matter and undergoing flavor oscillations. We derive a computational model that simulates the day-night effect, which works with 2, 3, or 4 neutrino flavors. The model takes the Earth's electron density to be either uniform or split into two uniform subsections, the core and the mantle. We show that including the more electron-dense core leads to measurably different flavor change, implying the day-night effect can be used to scan the Earth's internal composition. We also demonstrate that the day-night effect is sensitive to the inclusion of a sterile 4th neutrino and can be used to constrain the sterile mass-squared difference.

Introduction

Neutrinos are fundamental particles in the Standard Model (SM) that lie at the forefront of modern theoretical physics research. They come in 3 ’flavors’ — electron, muon, and tau — and are particularly unique in that they do not couple to the electromagnetic interaction, and can therefore propagate through matter endlessly without slowing down or being absorbed. Neutrinos do participate in the weak nuclear force, being absorbed or emitted in any weak nuclear reaction. We can thus observe a neutrino flux coming from modern nuclear reactors, as well as from nuclear fusion in stellar cores. Neutrinos are also critical to many high-energy astrophysical phenomena, such as core-collapse supernovae and neutron star mergers, where they are produced through neutronization and other nuclear processes and carry off tremendous amounts of energy.

But possibly the most captivating aspect of neutrinos, and the reason they are such a highly active area of research, is that their theoretically predicted behavior in the Standard Model disagrees with experimental observations: they are one of the only cases where the SM fails to accurately model reality on the small scale. Neutrinos are theoretically predicted to be massless, yet data from solar neutrinos suggests that they oscillate in flavor while they propagate, which is impossible if they have no mass. This discrepancy between observation and theory implies neutrinos may be a doorway to new physics beyond the standard model.

In this context, the Day-Night Effect is an observed neutrino phenomenon that can help us constrain unknown neutrino parameters as well as probe the composition of the Earth’s core and mantle. The effect describes the difference between the solar neutrinos a detector observes in the day and those observed at night. Neutrinos emitted in the sun and observed at nighttime have travelled through the Earth to get to the detector, changing flavor as they travel through the planet. As a consequence, the flavors of observed solar neutrinos vary periodically with the time of day. This fluctuation depends on neutrino parameters as well as the electron density of the Earth.

In this paper we derive a computational model that simulates the flavor oscillation of a neutrino travelling through the Earth. The model allows us to calculate the expected solar neutrino flux at a detector as a function of time, the latitude and longitude of the detector, the declination of the Earth, the neutrino masses and mixing angles, and the composition of the Earth’s mantle and core. We propose how our theoretical model could be combined with observed neutrino flux data to place constraints on the neutrino mass-squared differences, to probe new neutrino physics such as the possibility of a sterile 4th neutrino flavor, or to probe the composition of the earth, including its electron density and the radius of its core.

The Hyper-Kamiokande neutrino detector, which comes online in 2028, will be able to measure the day-night effect by detecting the Sun’s neutrino flux. Hyper-Kamiokande is a water Chernekov detector -- essentially a giant water tank lined with photomultiplier tubes. When a neutrino interacts with an electron in the water, the electron is imparted with sufficient energy to travel faster than the speed of light in water, releasing high-energy electromagnetic radiation in a 'photonic boom' which is detected by the photomultipliers. We can expect about 130 detections per day, meaning it will take years of observation to reliably measure the strength of the Day-Night effect.

Background

The three neutrino flavor states are non-diagonal in the neutrino mass eigenbasis, leading to flavor oscillations in vacuum. The unitary Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix U_mns allows for a change of basis between neutrino mass and flavor states: \begin{equation} \ket{\nu_\alpha} = [U_{mns}]_{\alpha i}\ket{\nu_i} \end{equation} where the α (i) subscript runs over flavor (mass) eigenstates. Taking an ultrarelativistic approximation, the flavor-evolution of a neutrino is governed by a Schrodinger-like equation, written in the mass basis as \begin{equation} i\frac{\mathrm{d}}{\mathrm{d}L}\ket{\nu_i} = \frac{m_i^2}{2E}\ket{\nu_i}. \end{equation} Switching to the flavor basis through the PMNS matrix, we have \begin{equation} i\frac{\mathrm{d}}{\mathrm{d}L}\ket{\nu_\beta} = [H_{vac}]_{\beta \alpha} \ket{\nu_\alpha} \end{equation} where \begin{equation} [H_{vac}]_{\beta \alpha} = [U_{mns}]_{\beta i}\frac{m_i^2}{2E} [U_{mns}^\dagger]_{i \alpha} \end{equation} is the Hamiltonian operator encoding neutrino flavor oscillations in vacuum.

In the presence of an electron-dense matter background, the Hamiltonian has an additional matter term arising from charged-current interactions. The matter Hamiltonian is given in the 3-flavor case by \begin{equation} \quad H_{mat} = \mathrm{Diag}(\sqrt{2}G_FN_e,0,0) \end{equation} in the flavor basis, where G_F is the Fermi coupling constant and N_e is the electron number density of the matter background. The total Hamiltonian is then H = H_vac + H_mat. The matter contribution can lead to significant flavor change for solar neutrinos travelling through the earth. The strength of the effect depends on the mixing angles, mass-squared differences, and electron number density of the medium.

The total Hamiltonian gives rise to an evolution operator _U(L), given in the eigenbasis of H by \begin{equation} U(L) = \int -i H \mathrm{d}L = \mathrm{Diag}(e^{-i\lambda_1},e^{-i\lambda_2},e^{-i\lambda_3}) \end{equation} where λ_1,2,3 are the eigenvalues of H.

The quantum mechanical wave function oscillations generated by H only manifest physically as a modified probability of observing each flavor at the detector. The probability that a neutrino arriving from the sun in mass state i will be observed in flavor state α after propagating for a distance L through the Earth is given by \begin{equation} P_{i \alpha} = |\bra{\nu_\alpha}U_{mat}^\dagger U(L) U_{mat} U_{mns} \ket{\nu_i}|^2, \end{equation} where U_mat is the unitary change of basis operator satisfying \begin{equation} [U_{mat}]_{\alpha \lambda_i}\ket{\nu_\alpha} = \ket{\nu_{\lambda_i}}. \end{equation} In general, the solar neutrino flux arriving at the earth will be energy-dependent and in a superposition of mass eigenstates.

Further, water Chernekov detectors like Hyper-Kamiokande can't differentiate one flavor from another directly: there is a cross-section for the detection of each flavor, and all detections contribute to a total count of neutrinos N observed per unit time. To theoretically calculate N, we must integrate the expected flux per flavor over all observable energies, average over the time period that observations will be taken, factor in the cross-section that a neutrino will be observed, and sum over flavors: \begin{equation} N = \sum_{\alpha,i}\int_{E_i}^{E_f}\int_{t_i}^{t_f}\mathrm{d}E_{\nu}\mathrm{d}t \ \frac{1}{t_f-t_i}\frac{\mathrm{d}\Phi_i}{\mathrm{d}E_{\nu}} \ P_{i\alpha}(E_{\nu},t) \ \sigma_\alpha \label{eq:totalcount} \end{equation} where dΦ_i / dE_ν is the normalized neutrino flux energy density arriving from the sun in mass state i and σ_α is the cross-section for observation of a neutrino in flavor state α. The difference in cross-section is the only reason that flavor oscillations will actually manifest as a change in $N$; if the cross-section was the same for all flavors then oscillating between one flavor and another wouldn't affect observations at the detector.

The observed day-night effect will depend on the distance a neutrino has to travel through the earth to arrive at a detector, which in turn depends on the position of the detector on the Earths surface as well as the time of year and, of course, the time of day. Using simple trigonometry we can calculate that the distance travelled through the Earth by a neutrino propagating from the sun will be \begin{equation} \begin{aligned} L &= 2(x \cos(\delta) - y\sin(\delta)), \\ x &= R\cos(\theta)\cos(\phi), \\ y &= R \sin(\theta) \end{aligned} \end{equation} where R is the radius of the Earth (approximated as a sphere), θ is the latitude of the detector, phi is a shifted longitude angle defined such that φ = 0 is the longitude of points on the exact opposite side of the Earth from the sun (so φ = 0 at solar midnight), and δ is the declination of the Earth, which varies seasonally. As a function of time of day and year, we can write \begin{equation} \begin{aligned} \phi =& \frac{360^\circ}{24}t, \\ \delta = -23.45^\circ&\cos(\frac{360}{365}(d+10)), \end{aligned} \end{equation} where t is the time in hours and d is the day of the year, with d=1 on January 1st. The strength of matter-induced flavor oscillations depends on the electron number density N_e of background matter, which is not constant throughout the Earth: a good approximation, however, is that it varies as a step function, taking a constant value throughout the mantle and then jumping to a higher value in the denser core. To factor this in, we must know how far a neutrino travels through the core in addition to its distance travelled through the whole planet. Again, some trivial trigonometry shows that the distance propagated through the core L_c will be \begin{equation} \begin{aligned} L &= 2\sqrt{R_c^2 - h^2 - d^2}, \\ h &= R\sin(\theta+\delta)\cos(\phi), \\ d &= R \sin(\phi), \end{aligned} \end{equation} where R_c is the radius of the core.

Modeling the Day-Night Effect

We developed a computational model that evolves the flavor state of a neutrino passing through matter of a uniform electron density. The model lets us predict the expected survival probabilities for each flavor at a detector as a function of the time of day, or as a function of the distance travelled through a electron-dense medium such as the Earth. Figure 1 shows a plot of the neutrino flavor probability amplitudes against distance travelled through matter, using the characteristic electron density of iron.

Figure 2 shows how the probabilities vary at a detector moving around the earth, graphed against time of day with t = 0:00 corresponding to solar midnight. The initial neutrino state in both these plots is taken to be the second mass state: an accurate 0th order approximation thanks to the Mikheyev-Smirnov-Wolfenstein (MSW) effect which causes solar neutrinos to project onto the second mass state while they leave the sun. Our model allows us to easily modify the initial condition to any other state as well.

Dawn is at roughly t = 6:00 AM, when observation probabilities become constant, so the neutrinos are not influenced by matter. The variation at nighttime is a reparametrization of the distribution in figure 1, since the distance travelled through the Earth varies with time.

The Hyper-Kamiokande experiment will be able to measure a total neutrino count throughout the day, with a nonzero detection cross-section for all 3 flavors. However, these cross-sections are not equal: electron neutrino detections are far more likely. We can use this fact to extrapolate neutrino survival probabilities from the total observed count and thus measure the strength of the day-night effect.

Including the Core

Our simulation can be modified to include the Earth's core, which leads to a different observed neutrino probability since the core has a higher electron density than the mantle. The observed neutrino flux is then sensitive to the radius and density of the Earth's core.

Figure 3 shows how neutrino flavor oscillations over time are impacted by the inclusion of a more electron-dense core. The oscillations are sensitive to the density and radius of of the core as well as the density of the mantle. Neutrino observation probabilities oscillate far more strongly at times when neutrinos must traverse the core to reach the detector. The key takeaway from this data is that we can use neutrinos to probe inner properties of the Earth. A more complete analysis would separate the core into inner and outer layers and account for the non-uniform electron density of the mantle.

Adding a Sterile 4th Neutrino Flavor

The Day-Night Effect could let us probe new physics, such as a fourth neutrino flavor. A sterile fourth neutrino has been suggested as a potential dark matter candidate. Several ongoing neutrino experiments are searching for evidence of this fourth flavor. The Day-Night Effect is sensitive to the presence of a sterile neutrino, and could be used to determine its existence.

To include a 4th flavor, we undergo the same process outlined in the Methods section but upgrade the MNS matrix to its 4-flavor analogue, including 3 additional mixing angles. The new matter component of the Hamiltonian is \begin{equation} H_m = \mathrm{Diag}(\sqrt{2}G_F N_e,0,0,\frac{1}{\sqrt{2}}G_F N_e). \end{equation} where there is an additional component arising from the fact that the sterile neutrino flavor does not experience neutral-current interactions with background nucleons and electrons (and all other flavors do). Figure 4 shows the result of including a 4th neutrino flavor in the time plot of probability oscillations. The three plots show different choices of neutrino parameters for the sterile flavor.

Conclusion

The Neutrino Day-Night effect is a powerful tool for probing the composition of the Earth, constraining unknown neutrino parameters, and testing new physics such as a sterile neutrino flavor. We derived a computational model that calculates the specific time dependency of this effect given an initial conditions and a choice of parameters. This model would provide a theoretical description of the effect against which real data from Hyper-Kamiokande could be compared. There is work left to be done on our project: we hope to develop a more precise and complete description of the sensitivity of the day-night effect to the 4th neutrino flavor.

About Me

Hi! I'm Henry Purcell, an undergraduate at University of California, Berkeley. I had the awesome opportunity to research the Day-Night effect with André de Gouvêa this Summer. Being a part of CIERA REU has been an incredible experience, and I hope to continue working with André over the coming months. Please feel free to reach out to me at henrypurcell [at] berkeley.edu with any questions!

This material is based upon work supported by the National Science Foundation under Grant No. AST-2149425. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.