Zero-Range Effective Field Theory for Resonant Wino Dark Matter*

The most dramatic ``Sommerfeld enhancements'' of neutral-wino-pair annihilation occur when the wino mass is tuned to near critical values where there is a zero-energy S-wave resonance at the neutral-wino-pair threshold. If the wino mass is larger than the critical value, the resonance is a wino-pair bound state. If the wino mass is near a critical value, low-energy winos can be described by a zero-range effective field theory in which the winos interact nonperturbatively through a contact interaction. The parameters of the zero-range effective field theory can be determined by matching wino scattering amplitudes calculated by solving the Schr\"odinger equation for a nonrelativistic effective field theory in which the winos interact nonperturbatively through a potential due to the exchange of weak gauge bosons. The power of the zero-range effective field theory is illustrated by calculating the rate for formation of the bound state in the collision of two neutral winos through the emission of two soft photons.