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

arXiv:2606.09116 · astro-ph.IM, astro-ph.HE · Submitted 2026-06-08 · Read on arXiv

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Introduction to the show: ident: Astrophysics Radio. Generated commentary on the latest astrophysics papers.

Vera: I'm Vera, and with me are Jocelyn and Subrahmanyan, guest researcher.

Jocelyn: Today's paper: "On the Nonlinear Dependence of Underground Muon Rate on Atmospheric Temperature Observed at Daya Bay".

Vera: The study investigates the atmospheric temperature dependence of underground cosmic-ray muon rates observed at Daya Bay, revealing that existing theories predict a nonlinear relationship,

Jocelyn: First, who's behind it and why it matters.

Title and authors: Vera: So we're looking at this paper today, "On the Nonlinear Dependence of Underground Muon Rate on Atmospheric Temperature Observed at Daya Bay," and it seems like they are tackling a really specific puzzle regarding how the muon rate changes with temperature. It’s about taking something that used to look nonlinear in their data and trying to figure out why.

Jocelyn: I was reading the title, and it immediately tells me this work is focusing on the underground muon rate at Daya Bay, which is a bit different from many of the other cosmic ray studies we've seen. It suggests that temperature isn't just a simple multiplier; there’s a deeper complexity involved in how that relationship behaves.

Subrahmanyan: From a theoretical viewpoint, when we talk about atmospheric muons, temperature certainly plays a role because it affects the density profile and the interaction cross-sections of the air they pass through. This paper seems to be pushing beyond the standard models by looking at how that temperature profile influences things across different depths.

Vera: Exactly, Subrahmanyan; it’s about moving past just looking at where muons are made and focusing on how every layer of atmosphere contributes to the final count, which is what this paper suggests we need to consider.

Jocelyn: And the authors themselves are Lei Liao, Taichong Ge, and Zhe Wang from Tsinghua University, so we're getting input from experts in high-energy physics and engineering physics on how they tackle this specific problem.

Subrahmanyan: I think the real significance here is their attempt to create a more general solution for the cascade equations because existing theories only look at local effects during production, which seems too simple for what we're seeing at Daya Bay.

Vera: That makes sense; they’re suggesting that focusing only on the point where mesons are born isn't enough to capture the whole story of how those particles evolve through the entire atmosphere.

Jocelyn: It sounds like they want to show that temperature dependence isn't inherently nonlinear in a fundamental way, but rather appears that way because of the limitations in how we’ve modeled it so far.

Subrahmanyan: Precisely; their goal is to provide a framework where the entire temperature profile has its influence captured, which should allow us to see if the relationship simplifies nicely under the right mathematical treatment.

The paper's summary: Vera: Now that we know what they’re aiming for, let’s talk about what they actually found in this paper. Essentially, this work by Lei Liao and colleagues provides a new, more general solution to the cascade evolution equations that takes the entire temperature profile into account when modeling how muons are produced and how they propagate down to the ground.

Jocelyn: What I picked up is that they introduced new ways to define things like the effective temperature weight and the temperature coefficient, which are derived from a functional derivative approach based on the actual atmospheric mean profile, T0(Y).

Subrahmanyan: That’s where it gets interesting for me; this functional derivative approach allows them to decompose the total influence into two distinct parts: a local term at the observation depth and a propagation term that accumulates along the meson's history.

Vera: It sounds like they’re saying that even if there isn't a significant production or decay happening right at your observation point, mesons created higher up still have an effect on what you measure down there, which is a key insight for me as someone who studies how things travel through the atmosphere.

Jocelyn: And by using this new decomposition, they managed to show that when they apply these definitions derived from the functional derivative to real atmospheric data, the underground muon rate recovers a linear dependence on atmospheric temperature.

Subrahmanyan: So, they’re taking something that was empirically observed as nonlinear modulation and showing mathematically how it becomes linear when you use this more complete treatment of the physics involved in the cascade equations.

Vera: That's quite a result, especially since they used MCEq simulations with real atmospheric data from ERA5 to test their new framework, and it reproduced the nonlinear modulation when using old methods.

Jocelyn: But when they switch to these new definitions—the Teff, ∆Teff, and α derived from that functional derivative—they find that the underground muon rate recovers a linear dependence on atmospheric temperature.

Subrahmanyan: That recovery of linearity is what makes this paper important because it suggests the previous non-linear observations weren't due to some unknown physics in the cascade itself, but rather an artifact of using an incomplete model for temperature influence.

The paper's improvements: Vera: So, focusing on what they actually improved, their primary contribution is developing this "improved solution of the cascade evolution equations" which is designed to cover both high-energy and low-energy meson energy spectra naturally.

Jocelyn: They introduced a specific functional path-length functional, L(X′, X, T), which incorporates the temperature profile through the density term in the decay rate, specifically using that physical path-length functional defined as L(X′, X, T) ≡ Z X / X′ dX′′ ρ(X′, T).

Subrahmanyan: The paper highlights that this non-local nature of this functional means a perturbation at depth Y can affect mesons produced above Y and observed below it, which is a crucial physical aspect they are modeling correctly.

Vera: That idea of the non-local effect—that something happening high up affects things low down—is what makes their solution more general than before because it respects the complete temperature history along meson trajectories.

Jocelyn: Furthermore, they derived new definitions for W(Y), the effective-temperature weight, which is calculated by taking the functional derivative of the muon rate with respect to the actual mean atmospheric profile T0(Y).

Subrahmanyan: And that weight definition is key because it decomposes into two parts: a local term WM,loc(Y) and a propagation term WM,prop(Y), which helps clarify where the temperature influence is coming from.

Vera: It’s fascinating how they managed to use this functional derivative approach to define these terms, which moves away from just using an isothermal projection for things like Teff.

Jocelyn: So, in short, the improvements are providing a mathematically rigorous way to link the muon rate directly to temperature variations across the atmosphere rather than relying on simpler approximations.

Conclusion: Vera: We’ve covered a lot today regarding this paper on "On the Nonlinear Dependence of Underground Muon Rate on Atmospheric Temperature Observed at Daya Bay," and it seems their main point is that using a more general cascade solution resolves the previously seen nonlinearity by showing linearity when you use the correct functional definitions.

Jocelyn: It really boils down to how they defined those effective temperature weights using functional derivatives, which allowed them to see that the observed nonlinear modulation was just an artifact of an incomplete way of modeling temperature's effect.

Subrahmanyan: From a theoretical perspective, this implies that the underlying physics governing muon production and propagation is consistent with a linear relationship once you account for the full non-isothermal nature of the environment, which should give us more confidence in applying these models to other cosmic ray phenomena.

Vera: That’s what I’m taking away; it's about refining our tools so that we can extract more reliable physical parameters from the data we observe in the sky.

Jocelyn: It gives us a better way to interpret the data, especially when we think about how these muon rates relate to temperature variations across different air depths.

Subrahmanyan: This kind of work helps bridge the gap between complex theoretical modeling and real-world experimental observations, which is essential for understanding the bigger cosmic picture.

Vera: Well, this paper on "On the Nonlinear Dependence of Underground Muon Rate on Atmospheric Temperature Observed at Daya Bay" provides a solid framework for how we should approach these types of problems going forward.

Jocelyn: It’s definitely a piece that helps us refine our understanding of atmospheric effects in cosmic ray physics.

Subrahmanyan: We look forward to seeing how this new methodology is applied elsewhere in the future.

Lei Liao, Taichong Ge, Zhe Wang

Department of Engineering Physics, Tsinghua University

astro-ph.IM, astro-ph.HE

Submitted: 2026-06-08

Updated: 2026-10-05

Comments: 16 pages, 8 figures

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 80/100

The gist: The study investigates the atmospheric temperature dependence of underground cosmic-ray muon rates observed at Daya Bay, revealing that existing theories predict a nonlinear relationship, which this

Key concepts

Improved Cascade Solution
This is a more general mathematical solution for how cosmic rays and mesons evolve through the atmosphere. Unlike previous approximate solutions, this improved method accurately tracks the complete temperature history along every meson's trajectory, covering both high and low-energy regimes.
Functional Derivative Approach
This technique is used to define new measures of temperature dependence (like Teff). Instead of using a simple isothermal projection, this method calculates how the muon rate changes when the actual atmospheric temperature profile is slightly altered. This captures both local effects at a specific depth and how those effects propagate through the meson's history.
Effective-Temperature Weight W(Y)
This new definition quantifies how sensitive the underground muon rate is to temperature at a specific depth Y. It is calculated by taking the functional derivative of the muon rate with respect to that temperature profile, allowing researchers to decompose this sensitivity into a local production term and a propagation term.

Terminology

Summary

The study investigates the atmospheric temperature dependence of underground cosmic-ray muon rates observed at Daya Bay, revealing that existing theories predict a nonlinear relationship, which this work addresses by providing a more general solution to cascade equations that accounts for the entire temperature profile.

The Gist

An improved solution of the cascade evolution equations and new definitions of effective temperature weight and temperature coefficient are provided, which demonstrate that when applied to real atmospheric data, the underground muon rate recovers a linear dependence on atmospheric temperature.

Improved Cascade Solution for Meson Energy Spectra

The research develops an improved solution of the cascade evolution equations for the meson and muons energy spectra that naturally covers both high-energy and low-energy regimes. The temperature profile enters this solution through the density term in the decay rate, specifically through a physical path-length functional (X′, X, T), which is defined as L(X′, X, T) ≡ Z X / X′ dX′′ ρ(X′, T). This non-local nature means that a perturbation at depth Y can affect mesons produced above Y and observed below it.

Muon Production Source Term

The conversion from the meson spectrum to a local muon production source, denoted as Qµ,M(Eµ, X), is derived by folding the decay rate with the two-body decay distribution. The resulting expression for this source term incorporates the non-isothermal solution: Qµ,M(Eµ, X) = ZNMmMc2/λN cτMkMρ(X, T)(1 − rM) Z X / X′ exp-X'ΛN - X - X'ΛM Eµ/rM Eµ dEM/E2 M ϕN (EM). This formula combines the two-body decay kinematic interval in energy and the possible production depths of the parent meson.

New Definitions for Temperature Dependence

To capture the functional dependence on temperature, new definitions are introduced based on a first-order expansion around the actual mean atmospheric profile, T0(Y). The effective-temperature weight is redefined as W(Y) ≡ δRµ / δT0(Y), which is calculated by taking the functional derivative of the muon rate with respect to the temperature profile. This functional derivative, when evaluated at a specific point Y, decomposes into two terms: the first local term WM,loc(Y) and the second propagation term WM,prop(Y).

Numerical Verification and Results

The theoretical framework is tested using MCEq [17] with real atmospheric temperature input from ERA5. The simulation reproduces the nonlinear modulation observed at Daya Bay when using conventional cascade solutions. However, when employing the new definitions of Teff, ∆Teff, and α derived from the functional derivative (Eq. 26), the underground muon rate recovers a linear dependence on atmospheric temperature. This linearity is verified through numerical calculation where the weight W(Y) is calculated using MCEq output (Eq. 34), demonstrating that linearity is recovered. The work concludes that the nonlinearity observed previously can be explained by this more general treatment, and the temperature coefficient αT is larger at higher temperatures.

Summary of Key Contributions

  1. Providing an improved solution of the cascade evolution equations to cover both high-energy and low-energy regimes.

  2. Introducing a functional derivative approach to define Teff, ∆Teff, and α based on the actual temperature profile rather than an isothermal projection.

  3. Demonstrating that the nonlinear correlation between underground muon rate and effective temperature is restored to a linear dependence when the method of functional derivative is applied.

  4. Decomposing the weight W(Y) into a local term (WM,loc(Y)) and a propagation term (WM,prop(Y)).

How it works

The paper establishes that conventional solutions are approximate solutions obtained under simplified assumptions. The improved solution preserves the complete temperature history along meson trajectories. The key to the new definitions lies in calculating the functional derivative of the muon production term, which is shown to contain both a local contribution at the observation depth and a propagation contribution accumulated along the meson history. This decomposition clarifies that conventional effective-temperature weights are not purely local production-layer factors.

Numerical Calculation by MCEq

The study utilizes MCEq [17] for simulations, adopting the SIBYLL23C interaction model and H3a primary flux. The simulation scans air density ρ(h, θ) at zenith angle θ to obtain the surface muon spectrum Φµ(Eµ, θ), which is then integrated over energy from a threshold energy Eth(θ, ϕ). By calculating the functional derivative of the production spectrum Qµ for each depth X using MCEq output (Eq. 34), the weight W(Y) is calculated.

Improvements for AI systems

As a fastidious and diligent researcher, I have analyzed this paper, On the Nonlinear Dependence of Underground Muon Rate on Atmospheric Temperature Observed at Daya Bay, focusing on its theoretical advancements in atmospheric cascade modeling and statistical parameterization.

The core contribution is moving from simplified isothermal approximations to a solution that naturally incorporates the full non-isothermal temperature profile into the muon production spectrum. This enables a transition from an empirically observed nonlinear correlation to a theoretically predictable linear dependence when using new definitions of effective temperature weighting.

Here are the specific improvements for AI systems, categorized by capability:

) 1. Enhanced Atmospheric Cascade Modeling and Physics Simulation

The paper introduces a more general solution to the cascade evolution equations (Eqs. 5-9), which explicitly accounts for non-local effects in both production depth and decay history, incorporating the temperature profile through the density relation derived from hydrostatic equilibrium (Eqs. 12, 13).

) The improved AI system can perform:

  • Predicting underground muon rates under realistic, spatially varying atmospheric temperature profiles (using ERA5 or similar global reanalysis data).

  • Modeling the propagation history of mesons, allowing the AI to distinguish between local production effects and attenuation effects based on a continuous temperature field.

) 2. Development of Non-Isothermal Effective Temperature Weighting (W(Y))

The system introduces a rigorous definition for the effective temperature weight, derived as the functional derivative of the muon rate with respect to the actual temperature profile, decomposing it into local production and propagation terms (Eqs. 27-28).

) The improved AI system can perform:

  • Calculating a spatially dependent Effective Temperature that accurately reflects where in the atmosphere (at depth Y) temperature variations most influence the final muon count.

  • Quantifying the ratio between the relative change in muon rate and effective temperature change, providing a rigorous metric for atmospheric sensitivity.

) 3. Linearization of Nonlinear Effects via Functional Derivatives

The central achievement is demonstrating that the observed nonlinear modulation is mathematically equivalent to a linear dependence when using these new functional definitions (Eq. 20: ∆Rµ/Rµ0 = αTef f/Tef f).

) The improved AI system can perform:

  • Taking complex, non-linear experimental data (like Daya Bay's rate vs. temperature correlation) and applying the derived functional derivative operator to systematically linearize the relationship, effectively extracting a constant temperature coefficient without relying on potentially flawed isothermal projections.

) 4. High-Fidelity Numerical Simulation and Model Validation

The paper utilizes MCEq for numerical simulation, comparing conventional treatments with the new framework (Section 4.3 and 4.4). It successfully reproduces the nonlinear behavior in conventional treatments but recovers linearity with the new method (Fig. 7).

) The improved AI system can perform:

  • Acting as a high-fidelity surrogate model for underground muon flux calculation, capable of simulating atmospheric interactions using complex hadronic models (e.g., SIBYLL23C) while simultaneously calculating the temperature sensitivity using the novel functional derivatives.

  • Validating and benchmarking other atmospheric interaction models by comparing their resulting effective temperature coefficients against the theoretically derived results (Grashorn [13]).

In summary, this research moves AI from merely fitting existing data patterns to understanding the underlying physics governing those patterns. The improved AI system will be a sophisticated tool for cosmic ray physics that can accurately predict and explain atmospheric temperature effects with high precision, specifically by rigorously separating local production from propagation history within a non-isothermal environment.

Abstract

The underground cosmic-ray muon rate is modulated by atmospheric temperature. It can be explained by the theories of Barrett, Gaisser, and others. However, the Daya Bay Neutrino Experiment observes a nonlinear temperature dependence. We find that, when deriving the temperature dependence of the muon rate, existing theories consider only the impact of the local temperature on muon production at the layer where muons are produced. In this work, we provide a more general solution to the cascade equations that fully depicts how the entire temperature profile influences the final muon rate. The corresponding definitions of the effective temperature weight and temperature coefficient are also presented. We have examined the results using the numerical tool MCEq with real atmospheric temperature inputs. A linear modulation is recovered and verified. This work can help explain the nonlinear effect observed at Daya Bay and provide a more refined theoretical framework for calculating the temperature coefficient in other experiments.

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