Resummation of fermionic in-medium ladder diagrams to all orders

Abstract

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 particleĒ(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 mediuminsertion 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. The emerging series in ak f 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.

Resummation of in-medium ladder diagrams: s-wave effective range and p-wave interaction

“…The emerging series in ak f could even be summed to all orders in the form of a double-integral over an arctangent-function. In that explicit representation the unitary limit, a → ∞, could be taken straightforwardly and the value ξ = 0.507 has been found for the Bertsch parameter [8]. This number is to be compared with the value ξ (pp) ≃ 0.237 resulting from a resummation of particle-particle ladder diagrams only [11].…”

“…In a recent work [8] the complete resummation of the (combined) particle-particle and holehole ladder diagrams generated by a contact-interaction proportional to the scattering length a has been achieved. A key to the solution of this (restricted) problem has been a different organization of the many-body calculation from the start.…”

“…This number is to be compared with the value ξ (pp) ≃ 0.237 resulting from a resummation of particle-particle ladder diagrams only [11]. Clearly, since only the (infinite) subset of particle-hole ladder diagrams [8] is computed in the normal phase one cannot expect accurate results for the unitary Fermi gas, whose true ground state is a superfluid. But interestingly, extrapolations of the equation of state of the unitary Fermi gas from finite to zero temperature, which smoothly pass over the pairing transition at T c give indications for a Bertsch parameter ξ n ≃ 0.5 in the normal phase [1,4,12].…”

Dense baryonic matter: Constraints from recent neutron star observations

“…Intermediate and long-range pion exchange dynamics (Fock terms from one-pion exchange and all explicit two-pion exchange processes in the presence of the medium) are computed up to three-loop order in the energy density. Contact terms (subject to resummations as described in [51]) are adjusted to properties of symmetric nuclear matter (the empirical binding energy per nucleon and the equilibrium density) and to the symmetry energy at k 0 F = 1.36 fm −1 . The "small" parameters, in addition to pion mass and momentum, now include the Fermi momenta, k p,n F /4πf π 1.…”

How neutron stars constrain the nuclear equation of state

“…Contact terms subsuming all unresolved short-distance nucleon-nucleon dynamics and three-nucleon interactions are properly considered. A resummation of in-medium ladder diagrams involving nucleon-nucleon contact interactions to all orders in the scattering length a is performed [15]. This framework is used to determine the energies per particle for both symmetric nuclear matter (SNM) and pure neutron matter (PNM).…”