# Zero-Range Effective Field Theory for Resonant Wino Dark Matter I. Framework

## TL;DR

This paper develops a zero-range effective field theory to describe resonant wino dark matter interactions near a critical mass, capturing nonperturbative effects and enabling calculations of bound state formation rates.

## Contribution

It introduces a zero-range EFT framework for resonant wino interactions, including a method to determine parameters via scattering amplitude matching and applies it to bound state formation.

## Key findings

- Effective field theory captures Sommerfeld enhancements near critical mass.
- Parameters determined by matching scattering amplitudes.
- Calculated formation rate of wino bound states.

## Abstract

The most dramatic "Sommerfeld enhancements" of neutral-wino-pair annihilation occur when the wino mass is near a critical value where there is a zero-energy S-wave resonance at the neutral-wino-pair threshold. Near such a critical mass, low-energy winos can be described by a zero-range effective field theory in which the winos interact nonperturbatively through a contact interaction. The effective field theory is controlled by a renormalization-group fixed point at which the neutral and charged winos are degenerate in mass and their scattering length is infinite. The parameters of the zero-range effective field theory can be determined by matching wino-wino scattering amplitudes calculated by solving the Schr\"odinger equation for winos interacting through a potential due to the exchange of weak gauge bosons. If the wino mass is larger than the critical value, the resonance is a wino-pair bound state. 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.