Atmospheric Muon Flux at Sea Level, Underground, and Underwater

TL;DR

This work develops a unified nuclear-cascade framework to predict atmospheric muon fluxes from sea level to 18 km w.e., incorporating a semiempirical NSU primary spectrum, energy-dependent hadronic cross sections, and detailed kaon/nucleon/ pion transport. It then analyzes charm-hadron–driven prompt muons using three models (RQPM, QGSM, VFGS) to quantify PM contributions and provides fitting parametrizations for both differential and integral PM fluxes. By solving the muon transport equation with a semi-analytical method, the authors compare predictions to an extensive dataset spanning ground-based, underground, and underwater detectors, finding that conventional $\pi$ and $K$ muons describe many measurements up to about 6–7 km w.e., while deeper data and high-energy sea-level measurements show tensions that cannot be reconciled by a single charm model. The study highlights substantial experimental and model uncertainties in the PM sector, emphasizing the role of underground data in normalizing atmospheric neutrino flux predictions for high-energy neutrino astronomy. Overall, it provides a comprehensive framework and practical parametrizations to interpret muon fluxes and informs future neutrino-background assessments for large-scale detectors.

Abstract

The vertical sea-level muon spectrum at energies above 1 GeV and the underground/underwater muon intensities at depths up to 18 km w.e. are calculated. The results are particularly collated with a great body of the ground-level, underground, and underwater muon data. In the hadron-cascade calculations, the growth with energy of inelastic cross sections and pion, kaon, and nucleon generation in pion-nucleus collisions are taken into account. For evaluating the prompt muon contribution to the muon flux, we apply two phenomenological approaches to the charm production problem: the recombination quark-parton model and the quark-gluon string model. To solve the muon transport equation at large depths of homogeneous medium, a semi-analytical method is used. The simple fitting formulas describing our numerical results are given. Our analysis shows that, at depths up to 6-7 km w. e., essentially all underground data on the muon intensity correlate with each other and with predicted depth-intensity relation for conventional muons to within 10%. However, the high-energy sea-level data as well as the data at large depths are contradictory and cannot be quantitatively decribed by a single nuclear-cascade model.

Figures (14)

• Figure 1: Vertical differential momentum spectra of conventional muons at sea level calculated by Volkova et al.Volkova79, Dar Dar83, Butkevich et al.Butkevich89, Lipari Lipari93, Agrawal et al.Agrawal96, and in present work.
• Figure 3: Fragmentation of quark chains into $D$ mesons in the QGSM: (a,b) favored fragmentation into $\overline{D}{}^0$; (c) unfavoured fragmentation into $D^-$ and $D^0$.
• Figure 4: Vertical differential momentum spectrum of muons at sea level. The direct data are taken from Refs. Baber68Bateman71Allkofer71Nandi72MARS75MARS77Rastin84MASS93L3Cosmic93 and indirect (underground) data are from Refs. MACRO95LVD98Collapse85BNO92Frejus94MSU94. The shaded areas are for the MACRO fit MACRO95. The solid curves represent the results of this work for the conventional ($\pi,K$) differential muon spectrum and for the $\pi,K$ muon spectrum plus the PM contribution calculated according to QGSM, RQPM, and VFGS.
• Figure 5: Vertical integral momentum spectrum of muons at sea level. The direct data are taken from Refs. MARS63Baber68Nandi72MARS75Rastin84MASS93EAS-TOP95 and indirect (underground) data are from Refs. BNO90KGF90aKGF64bBNO92. The solid curves represent the results of this work for the conventional ($\pi,K$) integral muon spectrum and for the $\pi,K$ muon spectrum plus the PM contribution calculated according to QGSM, RQPM, and VFGS.
• Figure 6: Muon intensity vs standard rock thickness. The data are from Refs. Wilson38Clay39Bollinger50Randall51Avan55Castagnoli65Stockel69Bergamasco71Crookes73Crouch87. The dashed curve represents our $\pi,K$-muon DIR, the solid curve represents the same plus the neutrino-induced muon background after Crouch Crouch87.