  The one dimensional $\delta$-function interacting Bose gas (the Lieb-Liniger
model) is an integrable system, which can model experiments with ultra cold
atoms in one dimensional traps. Even though the model is integrable,
integrability breaking effects are always present in the real world
experiments. In this work we consider the integrability breaking due to atomic
loss, which is the most relevant effect in the experiments. We set up a
framework for the exact computation of the losses of the canonical charges of
the model, and compute an exact result for the energy loss due to the local
$K$-body processes, valid for arbitrary $K$. Our result takes the form of
multiple integrals, which are explicitly factorized in the experimentally
relevant cases of $K=1,2,3$.
