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 E-(kf) 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.