Constraints on the Diffuse Flux of Ultra-High Energy Neutrinos from Four Years of Askaryan Radio Array Data in Two Stations

The Askaryan Radio Array is an ultra-high energy (UHE, $>10^{17}$\,eV) neutrino detector designed to observe neutrinos by searching for the radio waves emitted by the relativistic byproducts of neutrino-nucleon interactions in Antarctic ice. In this paper, we present constraints on the diffuse flux of ultra-high energy neutrinos between $10^{16}-10^{21}$\,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 neutrino flux at $10^{18}$ eV of $EF(E)=3.9\times10^{-16}$~$\textrm{cm}^2$$\textrm{s}^{-1}$$\textrm{sr}^{-1}$. The analysis leverages more than quadruple the livetime of the previous ARA limit, and at this energy is the most sensitive reported by an in-ice radio neutrino detector by more than a factor of four. Looking forward, ARA has yet to analyze more than half of its archived data, and expects that dataset to double in size in the next three years as livetime accumulates in the full five-station array. With this additional livetime, we project ARA will have world-leading sensitivity to neutrinos above $10^{17}$ eV by 2022.