Constraints on the diffuse flux of ultrahigh energy neutrinos from four years of Askaryan Radio Array data in two stations

The Askaryan Radio Array (ARA) is an ultrahigh energy (UHE, >10¹⁷ eV) neutrino detector designed to observe neutrinos by searching for the radio waves emitted by the relativistic products of neutrino-nucleon interactions in Antarctic ice. In this paper, we present constraints on the diffuse flux of ultrahigh energy neutrinos between 10¹⁶ and 10²¹ eV resulting from a search for neutrinos in two complementary analyses, both analyzing four years of data (2013--2016) from the two deep stations (A2, A3) operating at that time. We place a 90% CL upper limit on the diffuse all flavor neutrino flux at 10¹⁸ eV of EF(E)=5.6×10^-16 cm^-2 s^-1 sr^-1. This analysis includes four times the exposure of the previous ARA result and represents approximately $1/5$th the exposure expected from operating ARA until the end of 2022.