Complex Analysis of Askaryan Radiation: UHE-$ν$ Identification and Reconstruction using the Hilbert Envelope of Observed Signals

The detection of ultra-high energy neutrinos (UHE-$ν$), with enegies above 10 PeV, has been a long-time goal in astroparticle physics. Autonomous, radio-frequency (RF) UHE-$ν$ detetectors have been deployed in polar regions that rely on the Askaryan effect in ice for the neutrino signal. The Askaryan effect occurs when the excess negative charge within a UHE-$ν$ cascade radiates in a dense medium. UHE-$ν$ can induce cascades that radiate in the RF bandwidth above thermal backgrounds. To identify UHE-$ν$ signals in data from Askaryan-class detectors, analytic models of the Askaryan electromagnetic field have been created and matched to simulations and laboratory measurements. These models describe the Askaryan electromagnetic field, but leave the effects of signal propagation through polar ice and RF channel response to simulations. In this work, a fully analytic Askaryan model that accounts for these effects is presented. First, formulas for the observed voltage trace and its Hilbert envelope are calculated. Second, the analytic model is compared to UHE-$ν$ signals at 100 PeV from NuRadioMC, a key Monte Carlo toolset in the field. Correlation coefficients between the analytic signal envelope and MC data in excess of $0.94$ are found, and 99.99% of UHE-$ν$ signals pass a correlation threshold of $ρ\geq 0.4$. Analysis of RF thermal noise reveals that just 0.2 background events have $ρ\geq 0.4$ in 5 years at a 1 Hz thermal trigger rate. Finally, we describe future work related to the measurement of the logarithm of the UHE-$ν$ cascade energy.