Resummation of fermionic in-medium ladder diagrams to all orders

A system of fermions with a short-range interaction proportional to the scattering length a is studied at finite density. At any order an, we evaluate the complete contributions to the energy per particle 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 a4 are accurately reproduced. The emerging series in akf can be summed to all orders in the form of a double-integral over an arctangent function. In that representation the unitary limit a→∞ can be taken and one obtains the value ξ=0.5067 for the universal Bertsch parameter. We discuss also applications to the equation of state of neutron matter at low densities and mention further extensions of the resummation method.