Most searches for ultra-high-energy astrophysical neutrinos look for radio emission from the electromagnetic and hadronic showers produced in their interactions. The radio frequency spectrum and angular distribution depend on the shower development, and so are sensitive to the interaction cross sections. At energies above about ${10}^{16}\mathrm{eV}$ (in ice), the Landau-Pomeranchuk-Migdal effect significantly reduces the cross sections for the two dominant electromagnetic interactions: bremsstrahlung and pair production. At higher energies, above about ${10}^{20}\mathrm{eV}$, the photonuclear cross section becomes larger than that for pair production, and direct pair production and electronuclear interactions become dominant over bremsstrahlung. The electron interaction length reaches a maximum around ${10}^{21}\mathrm{eV}$, and then decreases slowly as the electron energy increases further. In this regime, the growth in the photon cross section and electron energy loss moderates the rise in ${\nu }_e$ shower length, which rises from $\sim 10\mathrm{m}$ at ${10}^{16}\mathrm{eV}$ to $\sim 50\mathrm{m}$ at ${10}^{19}\mathrm{eV}$ and $\sim 100\mathrm{m}$ at ${10}^{20}\mathrm{eV}$, but only to $\sim 1\mathrm{km}$ at ${10}^{24}\mathrm{eV}$. In contrast, without photonuclear and electronuclear interactions, the shower length would be over 10 km at ${10}^{24}\mathrm{eV}$.

• Received 2 July 2010

DOI:https://doi.org/10.1103/PhysRevD.82.074017