We examine the sensitivity of a large scale two-phase liquid argon detector to the directionality of the dark matter signal. This study was performed under the assumption that, above 50 keV of recoil energy, one can determine (with some resolution) the direction of the recoil nucleus without head-tail discrimination, as suggested by past studies that proposed to exploit the dependence of columnar recombination on the angle between the recoil nucleus direction and the electric field. In this paper we study the differential interaction recoil rate as a function of the recoil direction angle with respect to the zenith for a detector located at the Laboratori Nazionali del Gran Sasso and we determine its diurnal and seasonal modulation. Using a likelihood-ratio based approach we show that, with the angular information alone, 100 events are enough to reject the isotropic hypothesis at three standard deviation level. For an exposure of 100 tonne years this would correspond to a spin independent WIMP-nucleon cross section of about 10^-46 cm^2 at 200 GeV WIMP mass. The results presented in this paper provide strong motivation for the experimental determination of directional recoil effects in two-phase liquid argon detectors.
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