  In quantum gases with contact repulsion, the distribution of momenta of the
atoms typically decays as $\sim 1/|p|^4$ at large momentum $p$. Tan's relation
connects the amplitude of that $1/|p|^4$ tail to the adiabatic derivative of
the energy with respect to the gas' coupling constant or scattering length.
Here it is shown that the relation breaks down in the one-dimensional Bose gas
with contact repulsion, for a peculiar class of stationary states. These states
exist thanks to the infinite number of conserved quantities in the system, and
they are characterized by a rapidity distribution which itself decreases as
$1/|p|^4$. In the momentum distribution, that rapidity tail adds to the usual
Tan contact term. Remarkably, atom losses, which are ubiquitous in experiments,
do produce such peculiar states. The development of the tail of the rapidity
distribution originates from the ghost singularity of the wavefunction
immediately after each loss event. This phenomenon is discussed for arbitrary
interaction strengths, and it is supported by exact calculations in the two
asymptotic regimes of infinite and weak repulsion.
