Tau energy loss and ultrahigh energy skimming tau neutrinos

We consider propagation of high-energy earth-skimming taus produced in interactions of astrophysical tau neutrinos. For astrophysical tau neutrinos, we take generic power-law flux, $E^{-2}$ and the cosmogenic flux initiated by the protons. We calculate tau energy loss in several approaches, such as dipole models and the phenomenological approach in which parametrization of the $F_2$ is used. We evaluate the tau neutrino charged-current cross section using the same approaches for consistency. We find that uncertainty in the neutrino cross section and in the tau energy loss partially compensate giving very small theoretical uncertainty in the emerging tau flux for distances ranging from 2 to 100 km and for the energy range between $10^6$ and $10^{11}\mathrm{GeV}$, focusing on energies above $10^8\mathrm{GeV}$. When we consider uncertainties in the neutrino cross section, inelasticity in neutrino interactions and the tau energy loss, which are not correlated, i.e. they are not all calculated in the same approach, theoretical uncertainty ranges from about 30% and 60% at $10^8\mathrm{GeV}$ to about factors of 3.3 and 3.8 at $10^{11}\mathrm{GeV}$ for the $E^{-2}$ flux and the cosmogenic flux, respectively, for the distance of 10 km rock. The spread in predictions significantly increases for much larger distances, e.g., $\sim 1,000\mathrm{km}$. Most of the uncertainty comes from the treatment of photonuclear interactions of the tau in transit through large distances. We also consider Monte Carlo calculation of the tau propagation and we find that the result for the emerging tau flux is in agreement with the result obtained using analytic approach. Our results are relevant to several experiments that are looking for skimming astrophysical taus, such as the Pierre Auger Observatory, HAWC and Ashra. We evaluate the aperture for the Auger and discuss briefly application to the other two experiments.

Figure 1

The electromagnetic structure function $F_2(x,Q^2)$ for $Q^2=0.25{\mathrm{GeV}}^2$ (upper) and $1.5{\mathrm{GeV}}^2$ (lower) evaluated using the ALLM [49], and BDHM [53] parametrizations, and two dipole models (Soyez [58] and AAMQS [28]). The data are combined HERA data from H1 and ZEUS [62].

Figure 2

For muons, ${\beta }_{\mu }$ vs the energy of the muon $E$ for rock. The ${\beta }_{\mu }^{\mathrm{nuc}}$ for the dipole model (Soyez) is shown using dashed lines, and parametrizations of $F_2$ by ALLM (dotted), and BDHM (solid). The dot-dashed line comes from Bugaev and Shlepin [52]. The pair production ${\beta }_{\mu }^{\mathrm{pair}}$ and bremsstrahlung ${\beta }_{\mu }^{\mathrm{brem}}$ are also shown with the upper and lower dotted lines, respectively. For bremsstrahlung, ${\beta }_{\mu }^{\mathrm{brem}}=1.67\times {10}^{-6}{\mathrm{cm}}^2/\mathrm{g}$ and for pair production, ${\beta }_{\mu }^{\mathrm{pair}}=2.35\times {10}^{-6}{\mathrm{cm}}^2/\mathrm{g}$, essentially independent of energy for this energy range.

Figure 3

As in Fig. 2, for taus in rock. The bremsstrahlung contribution to ${\beta }_{\tau }$ is lower than the scales shown in this plot.

Figure 4

The average tau range in rock in km evaluated using the transport Monte Carlo with ALLM, BDHM and Soyez models for electromagnetic energy loss. Here, $E_{\tau }$ denotes the initial tau energy.

Figure 5

The number of taus per unit energy, assuming an initial flux scaling like $(E_{\tau }^i{)}^{-2}$, propagated with the Monte Carlo transport program through 1, 2, 10 and 20 km of rock in descending order in the figure (dashed), compared with the analytic results from Eq. (21) with ${\beta }_0=0.591\times {10}^{-6}{\mathrm{cm}}^2/\mathrm{g}$, ${\beta }_1=0.0404\times {10}^{-6}{\mathrm{cm}}^2/\mathrm{g}$ from the BDHM parametrization of the energy loss (dotted). The colored dots show the maximum $E_{\tau }$ for the analytic approach with $E_{\tau }^i<10^{12}\mathrm{GeV}$. The solid line shows the input tau flux to the Monte Carlo program.

Figure 6

The neutrino-nucleon charged current interaction ${\sigma }_{\nu N}$ as a function of neutrino energy $E_{\nu }$ from NLO perturbative QCD [23] (dot-dashed), using the Soyez dipole (dashed) and the BDHM (solid) and ALLM (dotted) parametrizations.

Figure 7

The mean inelasticity $⟨y⟩$ for neutrino scattering with isoscalar nucleons as a function $E_{\nu }$ using the Soyez dipole (dotted) and parametrizations of BDHM (dashed) and ALLM (solid).

Figure 8

The ratio of outgoing tau flux to incident tau neutrino flux with an incident flux that scales as $E_{\nu }^{-2}$ for a depth of 2, 5 and10 km rock, as a function of lepton energy, using the BDHM (upper), Soyez (middle) and ALLM (lower) approaches to both tau energy loss and the tau neutrino cross section. The solid lines use the analytic survival probability and the numerical differential neutrino cross section. The dashed lines use a Monte Carlo evaluation of the neutrino interaction and the tau energy loss, with a tau neutrino energy cutoff of $E_{\nu }=10^{12}\mathrm{GeV}$.

Figure 9

The ratio of outgoing tau flux to incident tau neutrino flux for a depth of 10 km rock, as a function of lepton energy, using the BDHM (upper) and ALLM (lower) tau energy loss parametrizations. The solid lines use neutrino-nucleon cross sections based on the same BDHM and ALLM parametrizations, while the dashed lines show the flux ratio if the perturbative NLO neutrino-nucleon cross section is used.

Figure 10

The ratio of outgoing tau flux to incident tau neutrino flux for depths of 2, 5 and 10 km rock, as a function of lepton energy, using the BDHM, Soyez and ALLM parametrizations and an incident neutrino flux scaling as $E_{\nu }^{-2}$.

Figure 11

The ratio of the emerging tau flux to an incident cosmogenic neutrino flux from the pure proton cosmic ray model with the AGN evolution scenario of Ref. [72].

Figure 12

Uncertainties of the transmission function for 10 km rock due to the cross section and the inelasticity for neutrino interaction and the tau energy loss.

Figure 13

The ratio of the emerging tau flux to an incident neutrino flux scaling as $E_{\nu }^{-2}$, using the BDHM energy loss and neutrino cross section (upper) and all three approaches (lower). In the upper figure, the dashed lines are from the Monte Carlo simulation of neutrino interactions and tau propagation with $E_{\nu }^{\max }=10^{12}\mathrm{GeV}$, while the dotted lines use the Monte Carlo program only for the neutrino interaction, for 30 (upper), 50 (middle) and 100 (lower curves) km. In the lower figure, we show the BDHM (dashed), Soyez (dot-dashed) and ALLM (solid) approaches for the same distances.

Figure 14

The transmission function calculated analytically for incident cosmogenic neutrino flux and with tau energy loss in the Soyez approach, for a depth of 30 and 100 km rock.

Figure 15

Uncertainties of the transmission function for 100 km rock due to the neutrino cross section and the neutrino inelasticity for the tau energy loss obtained in Soyez approach.

Figure 16

Chord length through the Earth $D$ for nadir angle $\theta$ and radius of the Earth $R_{\oplus }$.

Figure 17

The transmission function for an incident neutrino flux scaling as $E_{\nu }^{-2}$ for $E_{\nu }<10^{12}\mathrm{GeV}$ for $\alpha =\pi /2-\theta =1°$, 2°, 4° and 6° for the BDHM, Soyez and ALLM approaches.

Figure 18

Comparison of the transmission functions for incident neutrino flux scaling as $E_{\nu }^{-2.5}$ and $E_{\nu }^{-2}$ for 10 km and 100 km.

Figure 19

The geometry used to approximate the Auger aperture. Here, $h$ is the altitude a distance $x_0$ from the tau decay point, for taus emerging from the Earth with a nadir angle $\theta$.

Figure 20

The effective aperture for the Pierre Auger Observatory for $A=\mathrm{3,000}{\mathrm{km}}^2$ using the analytic approximation to the tau survival probability in rock and an approximate efficiency for detection, for Soyez, BDHM and ALLM models for both energy loss and neutrino cross section (solid line), and with the NLO perturbative neutrino cross section substituted (dashed lines).