Universal equation for Efimov states

Efimov states are a sequence of shallow three-body bound states that arise when the two-body scattering length is large. Efimov showed that the binding energies of these states can be calculated in terms of the scattering length and a three-body parameter by solving a transcendental equation involving a universal function of one variable. We calculate this universal function using effective field theory and use it to describe the three-body system of ${}^4\mathrm{He}$ atoms. We also extend Efimov’s theory to include the effects of deep two-body bound states, which give widths to the Efimov states.