Title: Thermal fading of the $1/k^4$-tail of the momentum distribution induced by the hole anomaly

Abstract: We provide the ab-initio Path Integral Monte Carlo calculation of the momentum distribution in a one-dimensional repulsive Bose gas at finite temperatures. We explore all interaction and thermal regimes. An important reference temperature is that of the hole anomaly, observed as a peak in the specific heat and a maximum in the chemical potential. We find that at large momentum $k$ and temperature above the anomaly threshold, the universal tail $\mathcal{C}/k^4$ of the distribution (proportional to the Tan's contact $\mathcal{C}$) is screened by the $1/|k|^3$-term due to a dramatic thermal increase of the internal energy. The same fading is consistently revealed in the short-distance behavior of the one-body density matrix (OBDM) where the $|x|^3$-dependence disappears for temperatures above the anomaly. At very high temperatures, the OBDM and the momentum distribution approach the Gaussian of classical gases. We obtain a new and general analytic tail for the momentum distribution and a minimum $k$ fixing its range of validity, both calculated with Bethe-Ansatz and valid for any interaction strength and temperature.