Self-consistent computation of pair production from non-relativistic effective field theories in the Keldysh-Schwinger formalism

Sommerfeld-enhanced annihilation cross sections in the presence of nearly zero-energy bound states can become so large that perturbative partial-wave unitarity appears to be violated. Previous literature incorporated the short-distance annihilation potential self-consistently into the computation of the Schr\"odinger wave function at the origin, leading to the unitarization of the Sommerfeld effect in vacuum. We employ non-relativistic effective field theory methods and the Keldysh-Schwinger formalism to additionally include pair-creation effects in the self-consistent computation of four-point correlation functions, which renders the unitarization temperature dependent. Up to small thermal corrections in the non-relativistic and dilute regime of the pairs, we confirm the previous results based on the Schr\"odinger equation approach for scattering states in vacuum. For the first time, we analyze bound-state contributions beyond their leading decay via annihilation. Interestingly, our self-consistent computation of the four-point correlation function shows that bound states remain on-shell in their out-of-equilibrium decay, even though their spectral functions take the form of Breit-Wigner distributions due to finite decay widths. While this may appear paradoxical, it aligns with expectations from earlier results based on exact analytic solutions of the Kadanoff-Baym equations for a decaying elementary particle in a thermal environment.