Resummation of fermionic in-medium ladder diagrams to all orders

Understand

A system of fermions with a short-range interaction proportional to the scattering length $a$ is studied at finite density.

• At any order $a^n$, we evaluate the complete contributions to the energy per particle $\bar E(k_f)$ arising from combined (multiple) particle-particle and hole-hole rescatterings in the medium.
• This novel result is achieved by simply decomposing the particle-hole propagator into the vacuum propagator plus a medium-insertion and correcting for certain symmetry factors in the $(n-1)$-th power of the in-medium loop.
• Known results for the low-density expansion up to and including order $a^4$ are accurately reproduced.