Beyond modified Urca: The nucleon width approximation for flavor-changing processes in dense matter

Flavor-changing charged current (``Urca'') processes are of central importance in the astrophysics of neutron stars. The standard approach is to approximate the Urca rate as the sum of two contributions, direct Urca and modified Urca. Attempts to make Urca calculations more accurate have been blocked by an unphysical divergence in the mUrca rate at the dUrca threshold density. We argue that to make progress the traditional separation into ``direct'' and ``modified'' contributions should be replaced by a single calculation of the Urca rate that consistently includes collisional broadening of the nucleons in the medium. In this Letter we describe how to perform such a calculation in a systematically improvable way. In the simplest approximation this means including an imaginary part of the nucleon mass (nucleon width). The total Urca rate is then obtained via a straightforward generalization of the dUrca calculation, yielding results that agree with both direct and modified Urca at the densities where those approximations are valid. At low densities, we observe an enhancement of the rate by more than an order of magnitude, with important ramifications for neutron star cooling and other transport properties.