Erratum: A Comprehensive Approach to Tau-Lepton Production by High-Energy Tau Neutrinos Propagating Through Earth [Phys. Rev. D 97, 023021 (2018), arXiv:1707.00334]

TL;DR

This paper reports a correction to a computational code used for simulating tau-lepton production by high-energy tau neutrinos passing through Earth, addressing a density tracking error that affected previous results.

Contribution

The authors identify and fix a coding error in the NuTauSim simulation tool that impacted the accuracy of tau neutrino propagation modeling.

Findings

The corrected code provides more accurate density tracking during particle propagation.

Previous results underestimated densities in Earth's inner layers due to the error.

Updated simulations improve the reliability of tau neutrino interaction predictions.

Abstract

We report an error found during independent review of the NuTauSim publicly available code \url{https://github.com/harmscho/NuTauSim} that forms the basis of this publication. The error in the code was in tracking the density of the medium during particle propagation. After the first interaction, the code was referencing the depth of penetration back to the surface of the Earth rather than the location of the last interaction. The results were obtained using densities that were systematically underestimated when the particle was traversing the inner layers of the Earth by assigning the density of either ice or bedrock, depending on the particle energy or ice thickness of the simulation. This error was fixed and the repository updated on September 29, 2018.

Full text

Erratum: A Comprehensive Approach to Tau-Lepton Production by High-Energy Tau Neutrinos

Propagating Through Earth

Phys. Rev. D 97, 023021 (2018)

Jaime Alvarez-Muñiz1, Washington R. Carvalho Jr.2,1, Austin L. Cummings3, Kévin Payet4, Andrés Romero-Wolf5, Harm Schoorlemmer6, Enrique Zas1

1Departamento de Física de Partículas & Instituto Galego de Física de Altas Enerxías, Univ. de Santiago de Compostela, 15782 Santiago de Compostela, Spain

2Departamento de Física, Universidade de São Paulo, São Paulo, Brazil

3 Gran Sasso Science Institute, School of Advanced Studies, L’Aquila, Italy

4 Université Joseph Fourier (Grenoble I); Currently at La Javaness, 75010, Paris, France

5 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA

6 Max-Planck-Institut für Kernphysik, 69117, Heidelberg, Germany

We report an error found during independent review of the publicly available code111https://github.com/harmscho/NuTauSim that forms the basis of this publication. The error in the code was in tracking the density of the medium during particle propagation. After the first interaction, the code was referencing the depth of penetration back to the surface of the Earth rather than the location of the last interaction. The results were obtained using densities that were systematically underestimated when the particle was traversing the inner layers of the Earth by assigning the density of either ice or bedrock, depending on the particle energy or ice thickness of the simulation. This error was fixed and the repository updated on September 29, 2018.

The main impact of the coding error is that the $\tau$ lepton exit probability $P_{\mathrm{exit}}$ was being overestimated for emergence angles greater than the value corresponding to the interface between the outermost layer of the Earth model and the layer below. In the case of a layer of ice with 4 km thickness, the impact is on emergence angles $\bar{\theta}>2^{\circ}$. All of the figures that used simulation results in the original paper are reproduced here with the same Figure number to facilitate comparison. In the updated Figure 5, $P_{\mathrm{exit}}$ is unaffected for $\bar{\theta}\leq 2^{\circ}$ but suppressed compared to the original result for $\bar{\theta}>2^{\circ}$. The suppression increases with emergence angle and depends on the tau neutrino energy $E_{\nu_{\tau}}$, reaching up to a factor of $\sim$ 5 at $\bar{\theta}=30^{\circ}$. The shape of the distribution of exiting $\tau$-lepton energies (gray bands in Figure 6) did not significantly change after the update. The reason for this is that exiting $\tau$ leptons are produced near the surface and mostly propagate within the ice because their their range is limited to $\sim$50 km at $10^{21}$ eV and $\sim$5 km at $10^{17}$ eV. The error in the code was mis-assigning the density only at layers below the first surface layer. While this could modify the shape of the exiting $\tau$ lepton energy distribution for high emergence angles, it is not very relevant because $P_{\mathrm{exit}}$ is already highly suppressed. The mean number of CC and NC interactions and tau decays, shown in Figure 7, is unaffected for $\bar{\theta}\leq 2^{\circ}$ but shows a slight increase for $\bar{\theta}>2^{\circ}$, as expected from an increase in density after the coding error fix.

The general conclusions about the effect of $\nu_{\tau}$ regeneration are not modified. In the updated Figure 9, the effect of not including the effects of regeneration is still to significantly suppress $P_{\mathrm{exit}}$ for $\bar{\theta}>2^{\circ}$, increasingly so with higher emergence angle. The updated figure includes additional suppression due to the coding error discussed above. The distributions of $\tau$ lepton energies shown in Figure 9 are not significantly modified after the coding error fix for the same reasons discussed the previous paragraph.

The biggest impact on the code fix is on the behavior of $P_{\mathrm{exit}}$ with ice thickness, as shown in Figure 10. This effect was originally studied in Palomares-Ruiz_2006 . Prior to the code fix, we had concluded that a layer of ice increased $P_{\mathrm{exit}}$ at emergence angles $\bar{\theta}\gtrsim 0.3^{\circ}-1.0^{\circ}$ (depending on energy) for neutrino energies $E_{\nu_{\tau}}>3\times 10^{18}$ eV. With the updated simulation, this range of emergence angles where $P_{\mathrm{exit}}$ benefits from a layer of ice is restricted to $0.3^{\circ}\lesssim\bar{\theta}\lesssim 3^{\circ}$. Note that this range of emergence angles corresponds to the majority of the area observed by a detector at altitude Romero-Wolf_2018 . The reason for this change is that the densities at higher emergence angles were systematically being underestimated resulting in suppression of the neutrino interaction probability in the subsurface rock layers. At energies $E_{\nu_{\tau}}<3\times 10^{18}$ eV, the code fix results in $P_{\mathrm{exit}}$ being higher for bare rock than ice at all emergence angles. For $E_{\nu_{\tau}}=10^{17}$ eV, we find that bare rock has approximately twice the $P_{\mathrm{exit}}$ than for a layer of ice. This behavior is explained by the probability of neutrino interaction being higher in rock by a factor of $\sim 2.8$ while the tau range being only $\sim 30\%$ longer in ice.

To better understand the behavior at lower energies, we ran a set of simulations with $E_{\nu_{\tau}}=10^{17}$ eV and constant $\bar{\theta}=10^{\circ}$ for various ice thicknesses (Figure 15). For ice thickness $D\geq 1$ km, $P_{\mathrm{exit}}$ is constant. This is because leptons produced in rock are mostly absorbed in the ice and only leptons produced in the ice $\sim$3 km away from the surface (the $\tau$-lepton range in this energy scale) are able to exit. Note that for this geometry ($\bar{\theta}=10^{\circ}$), the track length $L$ in ice, after traversing rock, is $L\geq 6$ km for $D\geq 1$ km, and increases with increasing ice thickness. In this case, $\tau$ leptons produced by neutrino interactions in the ice dominate the contribution to $P_{\mathrm{exit}}$. For ice thickness $\lesssim 1$ km, the range of $\tau$ leptons produced by interactions in rock (where the $\nu_{\tau}$ interaction probability is higher) is sufficiently large that they have a high probability of escaping the ice layer into the atmosphere and therefore contribute significantly to $P_{\mathrm{exit}}$ compared to neutrino interactions in the ice layer. As the ice thickness is reduced to $\sim 0.01$ km, the fraction of neutrinos interacting in ice compared to rock is negligible thereby making $P_{\mathrm{exit}}$ approximately constant for ice thickness $\lesssim 0.01$ km.

The relative differences between cross-section and $\tau$-lepton energy loss models (Figure 11) does not result in any significant changes after the code fix other than the behavior of $P_{\mathrm{exit}}$ vs emergence angle already discussed in Figure 5. The distribution of exiting $\tau$ leptons for the various models, shown in Figure 12, also does not show significant differences after the code fix.

The exiting $\tau$ lepton flux resulting from an incoming $\nu_{\tau}$, shown in Figure 13, behaves as expected based on the discussion above: for emergence angle $\bar{\theta}=1^{\circ}$ the results are unchanged while for $5^{\circ}$ and $10^{\circ}$, the spectra retain the same shape but with a lower integrated flux by a factor of $\sim 3$, consistent with the change in $P_{\mathrm{exit}}$ in the old and new Figure 5. The fluxes with interaction histories shown in Figure 14 are unchanged after the code fix for $\bar{\theta}=0.2^{\circ}$, $1.0^{\circ}$, and $2.0^{\circ}$, as expected. For $\bar{\theta}=5.0^{\circ}$, the exiting $\tau$ lepton fluxes for bare rock (dashed lines) are not modified since the subsurface layers do not significantly change in density for these trajectories. For the 4 km thick ice layer, however, the flux curves are suppressed by a factor of 2, which is consistent with the change in $P_{\mathrm{exit}}$ in Figure 5 after the code fix. The contribution from trajectories that underwent exactly one CC interaction (green), corresponding to events with no $\nu_{\tau}$ regeneration, was not changed since these tend to occur near the surface. The contribution from trajectories that had at least one NC interaction (red) or at least one $\tau$ lepton decay (black) are suppressed by a factor of $\sim 3$, which is consistent with the coding error that was incorrectly assigning the surface ice density rather than the subsurface bedrock density.

Making the code publicly available has succeeded at motivating independent review and improving the quality of the simulations. The code fix has left most conclusions of the original paper qualitatively unchanged with one slight modification: the benefit of having a layer of ice for $E_{\nu_{\tau}}>3\times 10^{18}$ eV is limited to emergence angles from $\sim 0.3^{\circ}$ to $\sim 3.0^{\circ}$ and not to higher emergence angles as originally stated.

Acknowledgements Part of this work was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. J. A-M and E.Z. thank Ministerio de Economía (FPA 2015-70420-C2-1-R and FPA2017-85114-P), Consolider-Ingenio 2010 CPAN Programme (CSD2007-00042), Xunta de Galicia (GRC2013-024 and ED431C 2017/07), Feder Funds, $7^{\rm th}$ Framework Program (PIRSES-2009-GA-246806) and RENATA Red Nacional Temática de Astropartículas (FPA2015-68783-REDT). W.C. thanks grant #2015/15735-1, São Paulo Research Foundation (FAPESP). We thank N. Armesto and G. Parente for fruitful discussions on the neutrino cross-section and $\tau$ lepton energy-loss models.

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