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Atmospheric MUons from PArametric formulas: a fast GEnerator for neutrino telescopes (MUPAGE)

G. Carminati, A. Margiotta, M. Spurio

arXiv:0802.0562v2physics.ins-detastro-phhep-exhep-ph

TL;DR

Atmospheric muons are both essential calibration events and a major neutrino-telescope background, motivating a fast bundle generator. MUPAGE uses shower-derived parametric formulas to generate muon-bundle kinematics on a detector-surrounding cylinder, and demonstrates year-scale high-energy atmospheric-muon production at 27 Hz. Its bundle model assumes parallel muons, and one sampling approximation is limited when the can height greatly exceeds its diameter.

  • Problem

    Atmospheric muons are the most abundant calibration signal in neutrino telescopes but also a major background, while existing parameterizations lacked multi-muon bundle simulation.

  • Method

    MUPAGE uses parametric formulas derived from full cosmic-ray shower simulations to generate bundle multiplicity, geometry, energies, and kinematics on a cylindrical can.

  • Results

    27 Hz was achieved for atmospheric-muon bundles above 3 TeV on a 1.4 km^2 cylinder at 2475 m depth, with 8.5×10^8 events corresponding to one year produced.

  • Takeaways & Limitations

    MUPAGE provides a fast tool for producing large atmospheric-muon samples for underwater or ice neutrino-telescope simulation and detector testing.

  • Takeaways & Limitations

    The hit-or-miss approximation for Nmax is invalid when the can height is much larger than the extended-can diameter, although this is not the case for present neutrino-telescope configurations.

Abstract

from arXiv · show

Neutrino telescopes will open, in the next years, new opportunities in observational high energy astrophysics. For these experiments, atmospheric muons from primary cosmic ray interactions in the atmosphere play an important role, because they provide the most abundant source of events for calibration and test. On the other side, they represent the major background source. In this paper a fast Monte Carlo generator (called MUPAGE) of bundles of atmospheric muons for underwater/ice neutrino telescopes is presented. MUPAGE is based on parametric formulas [APP25(2006)1] obtained from a full Monte Carlo simulation of cosmic ray showers generating muons in bundle, which are propagated down to 5 km w.e. It produces the event kinematics on the surface of a user-defined virtual cylinder, surrounding the detector. The multiplicity of the muons in the bundle, the muon spatial distribution and energy spectrum are simulated according to a specific model of primary cosmic ray flux, with constraints from measurements of the muon flux with underground experiments. As an example of the application, the result of the generation of events on a cylindrical surface of 3 km^2 at a depth of 2450 m of water is presented.

PROGRAM SUMMARY

MUPAGE is a fast C++ generator for atmospheric muon bundles in underwater or ice neutrino telescopes. It addresses both the calibration value and major background contribution of atmospheric muons by producing detector-ready event kinematics.

  • PROGRAM SUMMARY: MUPAGE generates atmospheric muon events for underwater and ice neutrino telescopes using parametric formulas for flux, multiplicity, radial distribution, and energy spectrum.The program is implemented in C++ and is intended for fast simulation.
  • PROGRAM SUMMARY: Atmospheric muons are the most abundant neutrino-telescope signal for calibration and detector-response checks, but they also form a major background.Downward-going muons can be reconstructed as upward-going particles, mimicking high-energy neutrino interactions.
  • PROGRAM SUMMARY: MUPAGE outputs event kinematics on the surface of a user-defined cylindrical can surrounding the detector.The output can feed detector-dependent simulations of light production and signal detection.
  • PROGRAM SUMMARY: The program structure includes event generation on the can surface, separate treatment of single and multiple muons, livetime computation, and an ANTARES application.Events are generated with the same relative weight, and each run's livetime is computed.

2 Simulation of bundles of atmospheric muons

The simulation models underwater muon bundles with parametric formulas derived from full cosmic-ray shower simulations. These formulas describe flux, angular behavior, multiplicity, radial structure, and energy distributions across the generator’s depth and angle range.

  • 2 Simulation of bundles of atmospheric muons: Existing underwater-flux parameterizations did not simulate bundles containing two or more muons, whereas full shower simulations could do so at very high CPU cost.MUPAGE was developed to overcome this limitation with a faster parametric approach.
  • 2 Simulation of bundles of atmospheric muons: MUPAGE parameterizes bundle flux, angular distribution, and energy spectra for vertical depths from 1.5 to 5 km w.e., using formulas derived from HEMAS shower simulations.The muon energy depends on depth and zenith angle, and for bundles also on multiplicity and radial distance from the shower axis.
  • 2 Simulation of bundles of atmospheric muons: A generated event first specifies bundle multiplicity, shower-axis direction, and can impact point; individual muon distances, energies, and impact coordinates are then calculated.Muons in a bundle are modeled as parallel to the shower axis and reaching a plane perpendicular to it simultaneously.
  • 2.1 Parametric formulas for the flux, energy spectrum and radial distance of muons: The bundle flux is parameterized as Φ(h, ϑ, m) = K(h, ϑ) m^ν(h,ϑ), with depth, zenith angle, and multiplicity as the central variables.The formula is evaluated for different multiplicities and zenith angles, including at h = 2.5 km w.e.
  • 2.1 Parametric formulas for the flux, energy spectrum and radial distance of muons: The parameterization is compared with AMANDA-II measurements for Eµ > 20 GeV at h = 1.64 km w.e. and with calculations from at several depths.The comparison uses flux versus zenith-angle cosine at depths h = 1.64, 2.0, and 3.0 km w.e.
  • 2.1 Parametric formulas for the flux, energy spectrum and radial distance of muons: For multimuon events, energy sampling depends on depth, zenith angle, multiplicity, and each muon’s radial distance from the shower axis.The radial-distance dependence reflects the modeled energy distribution within a bundle.

3 Program structure

MUPAGE is organized as a C++ program with geometry, parameter, input-decoding, and muon-generation components. Users configure detector and simulation ranges through an input file and obtain event and livetime outputs.

  • 3 Program structure: MUPAGE is written in C++ and uses ROOT libraries, with source components for geometry, parameters, input decoding, muon generation, and the main program.The packaged directory includes headers, source files, a Makefile, README, and launch scripts.
  • 3 Program structure: The program is launched through C-shell or Bash scripts that accept a random seed, input file, output files, and event count.The scripts also allow users to set the run number and output directories.
  • 3.1 Description of the input file: The input file contains detector geometry, medium, multiplicity, zenith-angle, and muon-energy simulation parameters.Default values refer to the ANTARES experiment.
  • 3.1 Description of the input file: MULTmax defaults to 1000 and is the only parameter identified as significantly affecting CPU time.Splitting event files across multiplicity ranges requires livetime normalization when histograms are filled.
  • 3.2 Description of the output files: The event output is a formatted ASCII table, while a second output file records simulated livetime and its statistical error.All events have weight 1, and the geometry uses an extended can surrounding the detector volume.

event

Each output event records the bundle and the kinematics of every muon reaching the can. The event data include identifiers, impact coordinates, direction, energy, timing, and particle identification.

  • event: The event identifier and bundle multiplicity specify the event and the number of muons at the shower-axis intersection with the can.The multiplicity is defined at the depth where the shower axis hits the can.
  • event: For each muon, the output records an identifier and impact coordinates (xi, yi, zi) on the can surface.Only muons intercepting the can are included in the per-muon records.
  • event: The output gives direction cosines (vx, vy, vz) coincident with the bundle axis and the muon energy Ei in GeV.The shared direction representation follows the parallel-muon bundle model.
  • event: Each muon record includes a time delay ti relative to the first listed muon, which may be positive or negative, plus a particle identification number.The particle identifier supports detector-dependent simulations following the MUPAGE step.

Parameter

Table 2 documents the detector, medium, simulation-range, geometry, and output settings used by MUPAGE.

  • Parameter: The configuration includes the can’s vertical coordinates, radius increase, and mean medium density.
  • Parameter: Users can set minimum and maximum bundle zenith angles, lateral spreads, muon energies, and multiplicities.
  • Parameter: The input also defines muon identifiers and a multiplicative factor for special geometries.The default muon identifier is the GEANT3 code, while other Monte Carlo systems can use their appropriate identifier.
  • Parameter: The parameter file specifies detector configuration, medium properties, and simulation ranges for multiplicity, zenith angle, and muon energy.Parameters are described with names, default values, and units.
  • Parameter: MUPAGE writes a livetime file with statistical error, and all generated events have weight 1.

4 Generation of muon bundles on the can surface

MUPAGE samples bundle direction, multiplicity, and impact point on an enlarged cylindrical can, then uses Hit-or-Miss sampling to reproduce the atmospheric-muon flux.

  • 4 Generation of muon bundles on the can surface: The generation cylinder is enlarged by Recr to include peripheral muons in large bundles, with Rext = Rcan + Recr.For a can matching the detector cylinder, Recr ≃ 10λabs is recommended.
  • 4.1 Sampling the bundle direction and impact point on the can surface: The generator first samples bundle multiplicity, zenith, and azimuth from uniform ranges in the detector frame.Only downward-going particles are generated.
  • 4.1 Sampling the bundle direction and impact point on the can surface: The shower-axis interception point is sampled uniformly on the cylinder projection perpendicular to the bundle direction.The projected-area geometry determines whether the point belongs to the upper disk or lateral surface.
  • 4.2 Hit-or-Miss method to sample the impact point: Hit-or-Miss acceptance corrects uniformly sampled depth, zenith, and multiplicity according to Φ(h, ϑ, m)·S(ϑ)·∆Ω(ϑ).The acceptance uses an upper bound Nmax for this function.
  • 4.2 Hit-or-Miss method to sample the impact point: The Nmax approximation is invalid when Hcan ≫ Rext, although this is not the case for present neutrino-telescope configurations.A multiplicative input factor is provided to correct the procedure for special geometries.

5 Single muons

For single-muon events, MUPAGE uses the shower axis as the muon direction and samples the muon energy with a depth- and angle-dependent distribution.

  • 5 Single muons: For m = 1, the muon direction is assumed to coincide with the shower axis, and its can impact point is the bundle impact point.The arrival time on the can surface is set to t = 0.
  • 5 Single muons: The muon energy distribution depends on the vertical depth h of the impact point and the muon zenith angle ϑ.
  • 5 Single muons: Energy is sampled in log10 E between the configured minimum and maximum and accepted with a Hit-or-Miss test.The acceptance compares a random ordinate with the maximum of the energy distribution at h and ϑ.

6 Multiple muons

For multimuon bundles, MUPAGE samples each muon’s radial position and energy, transforms bundle-frame coordinates into the laboratory frame, finds can intersections, and computes relative arrival times.

  • 6.1 The radial distance of muons with respect to the bundle axis: Each muon’s radial distance from the bundle axis is sampled from a distribution depending on depth, multiplicity, and zenith angle.The accepted radius and a random azimuth define coordinates in the Bundle Axis Frame.
  • 6.2 Coordinates of the multiple muons on the can surface: Each muon’s straight-line trajectory is intersected with the can surface to determine its impact point or whether it misses the can.For lateral-surface intersections, the quadratic discriminant determines whether solutions exist.
  • 6.2 Coordinates of the multiple muons on the can surface: A rotation matrix transforms each muon’s Bundle Axis Frame coordinates into laboratory-frame coordinates.The transformation uses Euler-angle rotations related to the bundle zenith and azimuth.
  • 6.2 Coordinates of the multiple muons on the can surface: The number of muons that intercept the can defines the observed bundle multiplicity mc ≤ m.
  • 6.3 Arrival time of the muons in the bundle: Relative delays are computed from signed distances between each muon’s generated point and can impact point, relative to the first muon.The light velocity used is c = 2.99×10^8 m/s, so delays may be positive or negative.
  • 6.4 Energy of the muons in a bundle: Each multimuon energy depends on depth, zenith angle, bundle multiplicity, and radial distance from the shower axis.The single-muon energy-sampling procedure is repeated for every muon.

7 Livetime of the simulation

MUPAGE computes detector livetime from generated events on the can surface by evaluating rates across 33 angular bins and averaging the resulting bin livetimes.

  • MUPAGE automatically derives detector livetime from the number of generated events on the can surface.
  • The simulation evaluates event counts in 33 angular bins for minimum multiplicity events whose shower axis intersects the can upper disk.
  • Each bin’s equivalent livetime is calculated using the expected event rate and the projected area of the can’s upper disk.
  • The reported livetime is a weighted average of the 33 bin values, with its statistical error, written to OUTFILE2.

8 An example of application: the case of ANTARES

The ANTARES application demonstrates that MUPAGE can generate large atmospheric-muon samples for detector testing and high-energy-neutrino background studies, with multiplicity and energy-threshold choices controlling rate and computing cost.

  • ANTARES operated a 12-line neutrino telescope at 2475 m depth, with an instrumented area of approximately 0.06 km2.
  • A one-month-equivalent sample used 358 files, each corresponding to 7260 ± 4 s and requiring about one hour of CPU time to produce.
  • The default generation rate was 1240 Hz above a 20 GeV surface threshold, nearly 1000 times the observed rate after realistic triggering.
  • For one-year high-energy-neutrino background studies, an energy threshold above 3 TeV reduced the default-cylinder event rate to 27 Hz.

9 Conclusions

The paper concludes that MUPAGE is a fast atmospheric-muon-bundle generator based on parametric flux and energy-spectrum formulas, producing large samples for underwater and ice neutrino telescopes.

  • MUPAGE generates atmospheric-muon-bundle kinematics using parametric formulas valid from 1.5 to 5.0 km water equivalent and zenith angles up to 85°.
  • The simulated energy spectrum accounts for both shower multiplicity and each muon’s distance from the shower axis.
  • For a 1.4 km2 cylinder at 2475 m depth, the generation rate above 3 TeV was 27 Hz.
  • MUPAGE produced 8.5×10^8 events corresponding to one year of data using 232 CPU hours on a 2.33 GHz processor.
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