⚛️ phenomenology

Earth-density effects in long-baseline neutrino experiments in a four-flavor (3+1) sterile-neutrino framework

This paper analytically and numerically investigates how Earth's realistic density profile and a four-flavor (3+1) sterile neutrino framework with eV-scale mass splitting and non-standard neutral current interactions influence the systematic bias in reconstructing the CP-violating phase $\delta_{13}$ in long-baseline experiments, finding that while the bias remains negligible for baselines up to 5000 km, it grows for trajectories sampling the lower mantle and core due to a specific interference mechanism.

Original authors: Bipin Singh Koranga, Aditya Pant, Pranav Kumar, Vivek Kumar Nautiyal

Published 2026-09-24

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CC0 1.0

Original authors: Bipin Singh Koranga, Aditya Pant, Pranav Kumar, Vivek Kumar Nautiyal

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep beneath the Earth's crust, invisible particles known as neutrinos are constantly streaming through the planet, passing through rock and metal as if they were made of air. These ghostly particles come in three known types, or flavors, and as they travel long distances, they have a peculiar habit of changing from one flavor into another. This phenomenon, called oscillation, is not random; it is governed by the fundamental properties of the particles and the environment they move through. Scientists have long used massive detectors to catch these changing particles, hoping to measure a specific property called a "CP-violating phase." This measurement is crucial because it could explain why the universe is made of matter rather than antimatter. However, to get an accurate reading, researchers must account for the Earth itself. As neutrinos travel through the planet, the varying density of the rock and the core acts like a lens, subtly altering their behavior. If scientists treat the Earth's density as a simple, uniform average, they risk introducing a significant error into their calculations, especially for neutrinos traveling thousands of kilometers.

A team of physicists from India has now expanded this line of inquiry to include a hypothetical fourth type of neutrino, one that is "sterile" and does not interact with normal matter in the same way the other three do. While the existence of these sterile neutrinos remains unproven, they are a compelling idea that could explain several puzzling anomalies observed in past experiments. The researchers wanted to know: if these extra particles exist, how would they change the way Earth's density affects our measurements of the known neutrinos? They built a sophisticated computer model to simulate neutrinos traveling through a realistic, continuous model of the Earth's interior, ranging from the thin crust to the dense core. Instead of using a simplified, blocky approximation of the planet's layers, they used a detailed mathematical description that captures the smooth changes in density as one moves deeper underground. They then compared the results of this realistic simulation against a simpler model that assumes the Earth has a constant average density, testing various scenarios where the sterile neutrino might be present.

The study reveals that the presence of a sterile neutrino does not simply make the Earth's density effect stronger or weaker in a uniform way. Instead, it creates a complex pattern of interference that shifts the error up and down depending on exactly how far the neutrino travels. For shorter journeys, up to about 5,000 kilometers, the error remains small and manageable, similar to what scientists see in the standard three-neutrino model. However, once the neutrino's path takes it deep into the lower mantle and the core, the situation changes. The researchers found that the sterile neutrino causes the measurement error to fluctuate unpredictably. In some cases, the error is smaller than expected; in others, it becomes significantly larger. This happens because the sterile neutrino interacts with the Earth's matter in a unique way, creating a subtle tug-of-war between different physical forces that depends on the precise shape of the Earth's density profile along the particle's path.

The team discovered that this fluctuation is driven by a specific mechanism where the sterile neutrino's interaction with the Earth's matter creates a new kind of influence that competes with the standard effects. This competition changes its nature as the neutrino travels deeper, flipping from one type of influence to another at specific points, particularly where the density of the Earth jumps sharply, such as at the boundary between the mantle and the core. The researchers tested four different possible values for a hidden property of the sterile neutrino and found that while the exact point where the error flips changes slightly with each value, the overall pattern of fluctuation remains the same. This suggests that the effect is a fundamental feature of a universe with four neutrino types, rather than a coincidence of a single specific setup.

The findings have important implications for future experiments designed to measure the properties of neutrinos with extreme precision. Current and planned detectors, which aim to send neutrino beams through the Earth over distances of up to 12,000 kilometers, must account for this complex behavior. If scientists continue to use a simplified model of the Earth's density, they risk misinterpreting their data, potentially leading to incorrect conclusions about the nature of matter in the universe. The study confirms that for long-distance neutrino experiments, the detailed, continuous structure of the Earth matters more than ever, and the potential existence of sterile neutrinos adds a layer of complexity that cannot be ignored. By mapping out exactly how these errors behave across different distances, the researchers have provided a clearer roadmap for future experiments, ensuring that when they finally pin down the secrets of the neutrino, they are looking through the clearest possible lens.

Technical Summary: Earth-Density Effects in Long-Baseline Neutrino Experiments within a Four-Flavor (3+1) Sterile-Neutrino Framework

Problem Statement
Long-baseline (LBL) neutrino experiments (e.g., T2K, NOνA, DUNE) rely on the Mikheyev–Smirnov–Wolfenstein (MSW) matter effect to determine the leptonic CP-violating phase $\delta_{13}$ and the neutrino mass ordering. A known systematic limitation in these analyses is the treatment of Earth's matter density. Previous work (Ref. [1]) demonstrated that approximating Earth's density as a constant path-length average is a valid approximation for baselines $L \lesssim 5000$ km but introduces a significant systematic bias in $\delta_{13}$ (tens of degrees) when trajectories sample the lower mantle and core ( $L \gtrsim 9000$ km).

Simultaneously, anomalies in short-baseline experiments (LSND, MiniBooNE, reactor, and gallium anomalies) motivate the existence of light sterile neutrinos ( $\Delta m^2 \sim \mathcal{O}(1)$ eV $^2$ ). While sterile neutrinos do not participate in neutral-current (NC) weak interactions, their mixing with active flavors introduces an additional matter potential term. The central problem addressed in this paper is whether the inclusion of a light sterile neutrino in a (3+1) framework modifies, enhances, or suppresses the Earth-density systematic bias identified in the standard three-flavor paradigm.

Methodology
The authors extend the analysis of Ref. [1] from a three-flavor to a four-flavor (3+1) framework using a full $4 \times 4$ matrix-exponentiation approach. Key methodological updates include:

Formalism: The study utilizes a unitary $4 \times 4$ mixing matrix $U$ parametrized with six rotations, introducing new mixing angles ( $\theta_{14}, \theta_{24}, \theta_{34}$ ) and CP phases ( $\delta_{14}$ ). The Hamiltonian includes both Charged-Current (CC) and Neutral-Current (NC) matter potentials. Crucially, the NC potential $A_{NC}$ acts on active flavors but not on the sterile state, creating a potential difference proportional to the neutron density.
Analytical Derivation: The authors derive a leading-order analytical expression for the reconstructed $\delta_{13}$ bias. By adiabatically integrating out the fast oscillation associated with $\Delta m^2_{41}$ , they demonstrate that the NC potential generates an effective non-standard-interaction (NSI)-like term in the $3 \times 3$ active Hamiltonian. This term interferes with the standard $s_{13}$ -driven CP term.
Earth Density Modeling: The paper replaces the discrete four-shell Earth model used in previous work with the continuous, polynomial PREM (Preliminary Reference Earth Model) profile by Dziewonski & Anderson (1981). This allows for a more precise calculation of the path-integrated density and the removal of artificial discontinuities at shell boundaries.
Parameter Scan: The analysis scans the sterile CP phase $\delta_{14} \in \{0, \pi/2, \pi, 3\pi/2\}$ and the mixing angle $\theta_{34} \in \{0^\circ, 5^\circ\}$ , in addition to the standard benchmark parameters ( $\Delta m^2_{41} = 0.30$ eV $^2$ , $\theta_{14} = 8.13^\circ$ , $\theta_{24} = 5.40^\circ$ ).
Numerical Implementation: Results are computed on a high-resolution $400 \times 400$ energy-baseline grid ( $L = 1000–12000$ km) to ensure numerical precision and minimize scan noise.

Key Contributions and Analytical Insights
The primary theoretical contribution is the identification of an interference mechanism that governs the sterile-induced modification of the $\delta_{13}$ bias.

Effective NSI Term: The integration of the fast sterile oscillation yields an effective Hamiltonian term proportional to $A_{NC}(x)$ . This term couples $\nu_e$ and $\nu_\mu$ via the product of mixing angles $s_{14}s_{24}$ and the phase $\delta_{14}$ .
Interference Phase: The bias is modulated by a relative phase $\Phi_{NC}(L) = \int_0^L A_{NC}(x) dx - \Delta_{31}(L)$ . Because $A_{NC}(x)$ tracks the non-linear density profile of the Earth (specifically the sharp jumps at the core-mantle and core boundaries), while $\Delta_{31}(L)$ grows linearly with $L$ , the interference term $\cos[\delta_{13} - \delta_{14} + \Phi_{NC}(L)]$ changes sign as the baseline increases.
Redistribution vs. Rescaling: The paper demonstrates that the sterile sector does not simply uniformly amplify or suppress the Earth-density bias. Instead, it redistributes the bias non-monotonically across the baseline. Depending on the baseline and the specific values of $\delta_{14}$ and $\theta_{34}$ , the bias can be suppressed (destructive interference) or enhanced (constructive interference) relative to the three-flavor case.

Results

Baseline Dependence: For $L \lesssim 5000$ km, the bias in the four-flavor framework remains comparable to or below the three-flavor benchmark. As trajectories sample the lower mantle and core ( $L > 6000$ km), the bias grows, but its magnitude and sign vary significantly based on the sterile parameters.
Sign Flips: The analysis confirms that the sign of the sterile-induced modification flips at specific baselines (e.g., between 9000 km and 11000 km) where the trajectory crosses the outer core. This flip is a direct consequence of the density profile shape, not an artifact of the numerical scan.
Parameter Robustness: Scanning $\delta_{14}$ and $\theta_{34}$ confirms that the redistribution effect is a generic feature of the (3+1) framework. While varying these parameters shifts the baseline at which the sign flip occurs, it does not eliminate the qualitative pattern of non-uniform redistribution.
Quantitative Bias: Table II presents specific bias values. For instance, at $L=7000$ km, the four-flavor bias is significantly reduced (8.0 $^\circ$ ) compared to the three-flavor case (118.0 $^\circ$ , noting a secondary $\chi^2$ minimum in the latter), whereas at $L=11000$ km, the bias is enhanced (32.0 $^\circ$ vs 40.0 $^\circ$ ).

Significance and Claims
The paper claims that the presence of an eV-scale sterile neutrino fundamentally alters the nature of the Earth-density systematic in LBL experiments. Rather than acting as a simple scaling factor, the sterile sector introduces a baseline-dependent interference mechanism that redistributes the systematic bias.

Structural Generality: The authors assert that this redistribution is a robust, structurally generic feature of the (3+1) framework, driven by the interplay between the NC potential and the Earth's density profile, rather than a numerical coincidence tied to specific parameter choices.
Analytical Grounding: The work provides an analytical explanation (via Eq. 6 and 7) for why the bias behaves non-monotonically, linking it to the phase difference between the NC-driven coupling and the standard CC-driven oscillation.
Experimental Implications: The results suggest that assuming a constant density or a simple three-flavor model in the presence of sterile neutrinos could lead to complex, non-intuitive errors in the reconstruction of $\delta_{13}$ , particularly for experiments with baselines traversing the Earth's core.

The paper concludes by noting that while the specific numerical pattern depends on the assumed sterile parameters, the underlying mechanism of redistribution is intrinsic to the (3+1) model. Future work is suggested to include joint fits of $\delta_{13}$ and sterile parameters, and to extend the analysis to atmospheric neutrino and IceCube/KM3NeT datasets.

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