Tau depolarization at very high energies for neutrino telescopes

The neutrino interaction length scales with energy, and becomes comparable to Earth’s diameter above 10’s of TeV energies. Over terrestrial distances, the tau’s short lifetime leads to an energetic regenerated tau neutrino flux, ${\nu }_{\tau }\to \tau \to {\nu }_{\tau }$, within the Earth. The next generation of neutrino experiments aim to detect ultrahigh energy neutrinos. Many of them rely on detecting either the regenerated tau neutrino, or a tau decay shower. Both of these signatures can be affected by the polarization of the tau through the energy distribution of the secondary particles produced from the tau’s decay. While taus produced in weak interactions are nearly 100% polarized, it is expected that taus experience some depolarization due to electromagnetic interactions in the Earth. In this paper, for the first time we quantify the depolarization of taus in electromagnetic energy loss. We find that tau depolarization has only small effects on the final energy of tau neutrinos or taus produced by high energy tau neutrinos incident on the Earth. Tau depolarization can be directly implemented in Monte Carlo simulations such as nupyprop and taurunner.

Figure 1

Initial polarization of taus produced from charged-current scattering. The average value of $⟨{\mathcal{P}}_{\nu ,z}^{\mathrm{CC}}(y)⟩$ as a function of tau energy fraction for ${\nu }_{\tau }$ CC scattering for different incident neutrino energies. The shaded region shows where 10% or less of the taus emerge with this energy fraction. In the massless lepton limit, $-⟨{\mathcal{P}}_{\nu ,z}^{\mathrm{CC}}(y)⟩=1$ for all $y$, corresponding to left-handed outgoing taus.

Figure 2

Average polarization of outgoing tau after a single scattering. The average polarization about $z$-axis $⟨{\mathcal{P}}_{\tau ,z}⟩$ (left) and the average total polarization $⟨P_{\mathrm{EM}}⟩$ (right) as a function of outgoing tau energy fraction for single photonuclear electromagnetic scattering of $\tau$’s with rock for different incident tau energies. Here, $⟨P^{\mathrm{EM}}⟩=(⟨{\mathcal{P}}_{\tau ,z}^{\mathrm{EM}}{⟩}^2+⟨{\mathcal{P}}_{\tau ,x}^{\mathrm{EM}}{⟩}^2{)}^{\frac{1}{2}}$.

Figure 3

Schematic of taus entering a slab of rock and propagating until they decay, and the distribution of final polarization before tau decay. Left: schematic of simulation of taus with $E_{\tau }^{\mathrm{in}}$ incident on a thick slab of rock, in which they propagate varying distances, then decay. Right: the differential number of taus as a function of final polarization from multiple EM interactions for ${10}^7$ $\tau$’s propagated through a slab of rock for each initial tau energy.

Figure 4

Effect of tau depolarization on outgoing tau neutrinos for multiple energies. Upper left: differential number of tau neutrinos as a function of ${\nu }_{\tau }$ energy fraction for simulated depolarization taus, shown for three different energies. Upper right: the orange (blue) markers show the ratio of the tau neutrino energy distribution with simulated (${\mathcal{P}}_z=0$) polarization to the tau neutrino energy distribution for left-handed tau decays as a function of $E_{\nu }/E_{\tau }^{\mathrm{in}}$ for taus of energy $E_{\tau }^{\mathrm{in}}={10}^9\mathrm{GeV}$ in rock. The dashed blue curve shows the approximate analytic evaluation of the ratio of decay neutrino distributions with ${\mathcal{P}}_z=0$ to ${\mathcal{P}}_z=-1$. The lower plots show these ratios for $E_{\tau }^{\mathrm{in}}={10}^{10}\mathrm{GeV}$ (left) and $E_{\tau }^{\mathrm{in}}={10}^{11}\mathrm{GeV}$ (right).

Figure 5

Earth-skimming tau neutrino with emerging tau, and impact of tau depolarization on the tau exit probability. Left: schematic diagram of a ${\nu }_{\tau }$ incident on the Earth that results in an emerging tau after series of CC interactions and tau decays (regeneration). The Earth emergence angle is $\beta$. Right: exit probability of taus as a function of different Earth emergence angles, for three different initial tau-neutrino energies. It shows a comparison when we consider LH polarization and simulated depolarization for EM interactions of the taus.

Figure 6

Final energy of exiting taus and their polarization from a neutrino beam. Differential number of exiting taus, with a normalization of $3/{\mathrm{N}}_{\tau }$, and average polarization ($-⟨{\mathcal{P}}_z⟩$) of exiting taus as a function of exiting tau’s energy fraction for initial neutrino energies $E_{\nu }={10}^9\mathrm{GeV}$ (left) and ${10}^{11}\mathrm{GeV}$ (right) and for $\beta =1°,5°$. The energy distribution normalization was chosen so the energy distributions and polarizations can appear in the same figures. Note that the $x$-axes have different ranges in the two panels.

Figure 7

Tau decay distribution as a function of $z_{\nu }=E_{\nu }/E_{\tau }$. The tau decay distribution as a function of $z_{\nu }$ for ${\mathcal{P}}_z=-1$, 0, 1 summed over all decay channels [solid line histograms, Eq. (4)] [49] and for the purely leptonic decay channels with unit branching fraction [dashed, Eq. (b2)].