Towards a theory of dissipative Dark Matter I: the Born limit

We derive the energy-differential cross section and energy loss rate for dissipative self-interacting dark matter (dSIDM) models within the Born regime using perturbative quantum field theory. Six dissipative scenarios are considered, incorporating the emission of particles that may be either massless or possess a kinematically allowed light mass. Both short-range and long-range force-mediated dSIDM interactions are examined. In the non-relativistic regime we obtain closed-form expressions of the energy-differential cross sections by a controlled expansion in the initial relative dark matter velocity. Up to trivial factors, the leading-order squared emission amplitude is model-independent for massless emissions. Model dependence arises for massive particle emission and at the next-to-leading order. The latter reduces to three distinct cases. The derived analytical expressions for dipole and quadrupole emission exhibit excellent agreement with numerical computations, providing simple, ready-to-use formulas. These expressions are benchmarked against full numerical results to confirm their accuracy. Furthermore, we analyze the behavior of these processes in the soft emission limit. Our results show that additional corrections are necessary when applying factorization to ensure consistency between the soft energy-differential cross section and the full counterparts across a broad energy range. Finally, we investigate the regime of perturbative validity in terms of the model parameters, identifying the conditions under which our results are applicable.