  We study the thermal behavior of correlations in a one-dimensional Bose gas
with tunable interaction strength, crossing from weakly-repulsive to
Tonks-Girardeau regime. A reference temperature in this system 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 momenta $k$ and temperature above the
anomaly threshold, the tail $\mathcal{C}/k^4$ of the momentum distribution
(proportional to the Tan contact $\mathcal{C}$) is screened by the
$1/|k|^3$-term due to a dramatic thermal increase of the internal energy
emerging from the thermal occupation of spectral excitation states. The same
fading is consistently revealed in the behavior at short distances $x$ of the
one-body density matrix (OBDM) where the $|x|^3$-dependence disappears for
temperatures above the anomaly. We obtain a new general analytic tail for the
momentum distribution and a minimum $k$ fixing its validity range, both
calculated with exact Bethe-Ansatz method and valid in all interaction and
thermal regimes, crossing from the quantum to the classical gas limit. Our
predictions are confirmed by comparison with ab-initio Path Integral Monte
Carlo calculations for the momentum distribution and the OBDM exploring a wide
range of interaction strength and temperature. Our results unveil a novel
connection between excitations and correlations. We expect them to be of
interest to any cold atomic, nuclear, solid-state, electronic and spin system
exhibiting an anomaly or a thermal second-order phase transition.
