On the Nonlinear Dependence of Underground Muon Rate on Atmospheric Temperature Observed at Daya Bay

Technical Summary: On the Nonlinear Dependence of Underground Muon Rate on Atmospheric Temperature Observed at Daya Bay

Problem Statement
The modulation of underground cosmic-ray muon flux by atmospheric temperature is a well-established phenomenon, traditionally explained by theories from Barrett, Gaisser, and others. These theories posit a positive correlation between muon rate and atmospheric temperature, driven by the competition between meson (pion and kaon) decay and interaction. However, data from the Daya Bay Neutrino Experiment, particularly in the shallower Experimental Halls 1 and 2, exhibit a nonlinear modulation effect: the correlation appears stronger during warmer months than cooler months. Existing theoretical frameworks, which often rely on isothermal atmosphere assumptions or asymptotic projections of temperature perturbations onto one-dimensional depth weights, fail to account for this nonlinearity when applied to realistic, non-isothermal temperature profiles.

Methodology
The authors propose a revised theoretical framework and a numerical verification strategy:

1. Improved Analytical Solution: The paper derives an improved solution to the atmospheric cascade evolution equations for meson and muon energy spectra. Unlike previous approximations that link high- and low-energy limits or assume isothermal conditions, this solution retains the full non-local dependence on the atmospheric temperature profile. Specifically, it treats the decay length as an explicit functional of the density-temperature relation, preserving the integral of temperature history along the meson trajectory from production to decay.
2. New Definitions: Based on this improved solution, the authors redefine the effective temperature ($T_{eff}$), its variation ($\Delta T_{eff}$), and the temperature coefficient ($\alpha_T$). Instead of using isothermal projected quantities, these are defined via the functional derivative of the muon rate with respect to the actual, time-varying temperature profile $T(Y)$. This approach decomposes the response into a local term (decay probability at the production layer) and a propagation term (meson survival history).
3. Numerical Simulation (MCEq): To validate the theory, the authors utilize the Matrix Cascade Equations (MCEq) toolkit. They input real atmospheric data from the ERA5 database (spanning 2011–2021) and specific terrain data for the Daya Bay experimental halls (EH1, EH2, EH3).
   • The simulation accounts for primary cosmic ray flux (Hillas Gaisser 2012 model), hadronic interactions (SIBYLL2.3C), and rock overburden.
   • The study reproduces the muon rate variations over 11 years.
   • Two calculation paths are compared: one using conventional Grashorn-style weighting (which projects perturbations onto a 1D depth weight) and one using the new functional derivative method (numerically calculated via finite differences in MCEq).

Key Results

• Reproduction of Nonlinearity: Using conventional cascade solutions and the standard Grashorn weighting method, the authors successfully reproduce the nonlinear correlation between the relative change in underground muon rate and the relative change in effective temperature observed at Daya Bay. The temperature coefficient $\alpha_T$ is shown to be larger at higher temperatures.
• Restoration of Linearity: When the muon rate variation is calculated using the new functional derivative-based weights (which account for the full non-local temperature history), the relationship between the muon rate and the effective temperature becomes linear.
• Validation: The numerical calculation of the functional derivative using MCEq confirms the theoretical predictions derived in Section 3. The linear dependence is recovered across different interaction models (SIBYLL2.3C, 2.1, 2.3E) and energy thresholds.

Significance and Claims
The paper claims that the observed nonlinearity at Daya Bay is not a fundamental breakdown of the temperature-muon correlation physics, but rather an artifact of the approximations used in conventional effective-temperature definitions. By treating the temperature profile as a functional input rather than projecting it onto a simplified weight, the authors demonstrate that the underlying physical relationship is linear.

The work provides:

1. A more general solution to cascade equations that fully depicts how the entire temperature profile influences the final muon rate.
2. Refined definitions for effective temperature and the temperature coefficient that are valid for non-isothermal atmospheres.
3. A calculation framework that can explain the nonlinear effects found at Daya Bay and offers a more precise method for calculating temperature coefficients for other underground experiments.

The authors conclude that their method recovers the expected linear dependence, suggesting that previous interpretations of nonlinear modulation were limited by the isothermal projection assumptions inherent in standard analytic treatments.