[Paper Review] Parametrization of the Cosmic Muon Flux for the Generator CMSCGEN

P. Biallass, T. Hebbeker|ArXiv.org|Jul 31, 2009

Astrophysics and Cosmic Phenomena7 references18 citations

TL;DR

This paper presents a new parametrization of the cosmic muon flux for the CMSCGEN generator, using CORSIKA simulations with EPOS and GHEISHA interaction models to model momentum and angular distributions via polynomial fits over 3–3000 GeV. The key result is a normalized flux parametrization with ±7% uncertainty in the central range, validated against experimental data and adjusted for charge ratio consistency.

ABSTRACT

The cosmic muon generator CMSCGEN is based on a parametrization of the differential muon flux at ground level, as obtained from the air shower simulation program CORSIKA. We present the underlying ansatz for this parameterization and provide an approximation of the momentum and angular distributions in terms of simple polynomials, in the momentum range 3 to 3000 GeV.

Motivation & Objective

• To improve the accuracy and uncertainty quantification of the cosmic muon flux parametrization in the CMSCGEN generator for use in collider experiments.
• To address limitations in prior parametrizations, particularly regarding momentum and angular distributions at high and low energies.
• To incorporate updated interaction models (EPOS, GHEISHA) and recent experimental data for better agreement with measurements.
• To ensure proper treatment of the positive-to-negative muon flux ratio, which is critical for detector simulation.

Proposed method

• Parametrizing the differential muon flux as $ \frac{d\Phi}{dp\,dc\,d\phi} = C_{\text{norm}} \cdot \frac{1}{p^3} \cdot s(L) \cdot z(c,L) \cdot \frac{1}{2\pi} $, where $ L = \log_{10}(p/\text{GeV}) $.
• Fitting the momentum spectrum function $ s(L) $ using a sixth-order polynomial to match CORSIKA simulation results across 3–3000 GeV.
• Modeling the zenith angle dependence via $ z(c,L) = b_0(L) + b_1(L)c + b_2(L)c^2 $, with $ c = \cos\theta $, normalized to unity over $ c \in [-1, -0.1] $.
• Calibrating $ C_{\text{norm}} $ using vertical muon flux measurements at 100 GeV, combining L3 and other experimental data to derive $ C_{\text{norm}} = 1.27 \times 10^3 \, \text{GeV}^2 \text{m}^{-2} \text{s}^{-1} \text{sr}^{-1} $.
• Applying a 'fudge factor' of 1.05 to improve agreement with L3 data at 46 and 100 GeV.
• Adjusting the charge ratio to $ R = 1.280 \pm 0.016 $, assigning 66.1% to positive and 43.9% to negative muons to match experimental fluxes.

Experimental results

Research questions

• RQ1 How accurately can a polynomial-based parametrization describe the cosmic muon momentum and angular distributions across 3–3000 GeV?
• RQ2 What is the impact of using modern interaction models (EPOS, GHEISHA) on the simulated muon flux compared to older models?
• RQ3 How well does the new parametrization reproduce experimental muon flux measurements, especially at vertical incidence?
• RQ4 What is the appropriate charge ratio of positive to negative muons, and how should it be implemented in the generator to match data?
• RQ5 What are the dominant sources of uncertainty in the parametrization, particularly at low and high momenta?

Key findings

• The momentum spectrum function $ s(L) $ is well approximated by a sixth-order polynomial: $ s(L) = -1 + 6.2218L - 13.940L^2 + 18.164L^3 - 9.2278L^4 + 1.9923L^5 - 0.15643L^6 $, with a fit accuracy of ~5% across 3–3000 GeV.
• The normalized flux at 100 GeV for vertical muons is $ (2.63 \pm 0.06) \times 10^{-3} \, \text{m}^{-2} \text{s}^{-1} \text{GeV}^{-1} \text{sr}^{-1} $, used to calibrate $ C_{\text{norm}} $.
• The normalization constant $ C_{\text{norm}} $ is determined to be $ 1.27 \times 10^3 \, \text{GeV}^2 \text{m}^{-2} \text{s}^{-1} \text{sr}^{-1} $, with a 1.05 fudge factor applied to align with L3 data.
• The charge ratio of positive to negative muons is set to $ R = 1.280 \pm 0.016 $, with 66.1% of muons assigned as positive and 43.9% as negative to match measurements.
• Uncertainty in the flux is ±7% in the central momentum range (10–500 GeV), increasing to ±25% at 3 GeV and ±50% at 3000 GeV due to lack of experimental data and model instability.
• The parametrization is valid for $ c \in [-1, -0.1] $, corresponding to $ \theta \in [0^\circ, 85.2^\circ] $, and should not be used below 3 GeV or above 3000 GeV.