Comprehensive approach to tau-lepton production by high-energy tau neutrinos propagating through the Earth

There has been a recent surge in interest in the detection of $\tau$-lepton-induced air showers from detectors at altitude. When a $\tau$ neutrino (${\nu }_{\tau }$) enters the Earth, it produces $\tau$ leptons as a result of nuclear charged-current interactions. In some cases, this process results in a $\tau$ lepton exiting the surface of the Earth, which can subsequently decay in the atmosphere and produce an extensive air shower. These upward-going air showers can be detected via fluorescence, optical Cherenkov, or geomagnetic radio emission. Several experiments have been proposed to detect these signals. We present a comprehensive simulation of the production of $\tau$ leptons by ${\nu }_{\tau }$’s propagating through Earth to aid the design of future experiments. These simulations for ${\nu }_{\tau }$’s and leptons in the energy range from $10^{15}\mathrm{eV}$ to $10^{21}\mathrm{eV}$ treat the full range of incidence angles from Earth-skimming to diametrically traversing. Propagation of ${\nu }_{\tau }$’s and leptons includes the effects of rock and an ocean or ice layer of various thicknesses. The interaction models include ${\nu }_{\tau }$ regeneration and account for uncertainties in the Standard Model neutrino cross section and in the photonuclear contribution to the $\tau$ energy-loss rate.

Figure 1

Diagram of the ${\nu }_{\tau }$ propagation process. A ${\nu }_{\tau }$ can interact either via charged-current (CC) or neutral-current (NC) interaction. The CC interaction occurs $\sim 72\%$ of the time and results in the production of a $\tau$ lepton, carrying on average $\simeq 80\%$ of the energy of the ${\nu }_{\tau }$ at UHE and loses energy while propagates. The $\tau$ lepton can escape the Earth before it decays. If it decays in the Earth, it will produce a ${\nu }_{\tau }$, which will continue to propagate. NC interactions occur $\sim 28\%$ of the time and produce a ${\nu }_{\tau }$ which also continues to propagate. See text for more details.

Figure 2

Simulation geometry for a ${\nu }_{\tau }$ entering the Earth with incidence angle $\theta$. The model of the Earth assumes a sphere of radius $R_{\text{Earth}}$ and adjustable ocean or ice depth $D$. The ${\nu }_{\tau }$ is injected with a specified exit angle $\theta$, which determines the trajectory it will take across the various subsurface layers of varying density where the interactions will take place. Results are given in terms of the emergence angle $\overline{\theta }=90°-\theta$, which is the complement of the exit and entry angles.

Figure 3

Standard Model charged-current (CC) and neutral-current (NC) neutrino-nucleon cross sections ${\sigma }_{\nu N}$ as a function of energy from [50]. The upper and lower curves represent the upper and lower limits of the uncertainties due to the parton distribution function. In this work we consider the upper, middle, and lower curves in the calculation of the probability and energy distribution of exiting $\tau$ leptons.

Figure 4

Fractional $\tau$-lepton energy-loss rate $\frac{1}{E_{\tau }}⟨\frac{d{E}_{\tau }}{dx}⟩$, including ionization, pair production, bremsstrahlung, and photonuclear interactions as a function of energy. The two curves shown are for the ALLM [56] and ASW [57] photonuclear interaction models.

Figure 5

The probability $P_{\mathrm{exit}}$ that a $\tau$ lepton exits the Earth’s surface for emergence angles between 0.1° (Earth skimming) and 50° given a 4 km thick layer of ice with the standard cross section and energy-loss models. The feature at the emergence angle of 2° corresponds to the trajectory tangential to the rock layer beneath the 4 km thick layer of ice.

Figure 6

The exiting $\tau$ lepton energies corresponding to some of the energies shown in Fig. 5. The red line shows the most probable exiting tau lepton energy. The dark (light) gray band shows the 68% (95%) densest probability interval. The features in the curves are caused by regions where various interaction processes dominate. See Fig. 7 and text for details.

Figure 7

The mean number of CC, NC interactions, and tau lepton decays as a function of emergence angle for various incident neutrino energies corresponding to Figs. 5 and 6. Top panel: The mean number of CC interactions must be at least 1 since we are selecting particles resulting in a $\tau$ lepton exiting the Earth’s surface. Middle panel: The mean number of neutral-current interactions. The sharp transition at emergence angle $\overline{\theta }=2°$ corresponds to the direction tangential to the subsurface rock beneath a 4 km thick layer of ice. Bottom panel: The mean number of $\tau$ lepton decays also show a feature at $\overline{\theta }=2°$. Note that for $\overline{\theta }<2°$ the particle traverses ice only, while for $\overline{\theta }>2°$ the particle traverses a combination of rock and ice, which affects the behavior of the $\tau$-lepton and ${\nu }_{\tau }$ interactions.

Figure 8

The probability that a $\tau$ lepton exits Earth’s surface including and excluding the effect of ${\nu }_{\tau }$ regeneration given a 4 km thick layer of ice and standard neutrino cross section and tau-lepton energy-loss models. Excluding regeneration significantly underestimates the probability of exiting $\tau$ leptons for $\overline{\theta }>2°$, where the trajectories propagate through rock rather than pure ice.

Figure 9

The exiting $\tau$-lepton energies corresponding to $E_{\nu }={10}^{20}\mathrm{eV}$ in Fig. 8 with and without regeneration. Excluding regeneration suppresses exiting $\tau$ leptons with energy $E_{\tau }<{10}^{17}\mathrm{eV}$.

Figure 10

The probability that a $\tau$ lepton exits the Earth’s surface for various energies and ice thicknesses, including bare rock, assuming standard cross section and energy-loss models. From top to bottom, the input neutrino energies are ${10}^{20}$, ${10}^{19}$, ${10}^{18}$, and ${10}^{17}\mathrm{eV}$. A layer of ice is favorable to exiting $\tau$ leptons for neutrino energies $>{10}^{18}\mathrm{eV}$, while bare rock is favorable for neutrino energies $<{10}^{18}\mathrm{eV}$. See text for details.

Figure 11

The probability that a $\tau$ lepton exits the Earth’s surface for various combinations of neutrino cross section and $\tau$-lepton energy-loss models given a 4 km thick ice layer for a $10^{20}\mathrm{eV}$ injected ${\nu }_{\tau }$. Lowering the cross section has the general effect of reducing the $\tau$ lepton exit probability for emergence angles below where the trajectory is tangential to the subsurface rock layer while increasing the probability for larger emergence angles. The ASW energy-loss-rate model, which is suppressed compared to the more standard ALLM model, results in an overall increase in $\tau$ lepton exit probability.

Figure 12

The exiting $\tau$ lepton energies for various models corresponding to Fig. 11. On each panel, the cross-section model and energy-loss-rate models are labeled in the top right corner. The variance in exiting $\tau$-lepton energies tends to increase as the cross section increases for trajectories that traverse mostly rock. The energy-loss model changes the range of emergence angles where the most probable energies plateau.

Figure 13

The range of cosmogenic neutrino fluxes from Kotera 2010 [7] and the resulting flux of $\tau$ leptons for emergence angles $\overline{\theta }=1°$, 5°, and 10° (see Fig. 2). The results use the middle neutrino-nucleon cross-section curve (Fig. 3), ALLM energy-loss rate (Fig. 4), and $D=4\mathrm{km}$ thick ice.

Figure 14

The resulting flux of $\tau$ leptons for a cosmogenic neutrino flux in the middle of the flux ranges from Kotera 2010 [7] (gray band in Fig. 13). The different line colors indicate the interaction history that led to the exiting $\tau$ leptons (see text for more details). We show the effect of a 4 km thick ice layer (dashed lines) versus bare rock (solid lines) for four different emergence angles as indicated on the panels. These results are obtained using the middle neutrino-nucleon cross-section curve and ALLM energy-loss rate.