  Recently we have developed a novel chiral power counting scheme for an
effective field theory of nuclear matter with nucleons and pions as degrees of
freedom [1]. It allows for a systematic expansion taking into account both
local as well as pion-mediated multi-nucleon interactions. We apply this power
counting in the present study to the evaluation of the pion self-energy and the
energy density in nuclear and neutron matter at next-to-leading order. To
implement this power counting in actual calculations we develop here a
non-perturbative method based on Unitary Chiral Perturbation Theory for
performing the required resummations. We show explicitly that the contributions
to the pion self-energy with in-medium nucleon-nucleon interactions to this
order cancel. The main trends for the energy density of symmetric nuclear and
neutron matter are already reproduced at next-to-leading order. In addition, an
accurate description of the neutron matter equation of state, as compared with
sophisticated many-body calculations, is obtained by varying only slightly a
subtraction constant around its expected value. The case of symmetric nuclear
matter requires the introduction of an additional fine-tuned subtraction
constant, parameterizing the effects from higher order contributions. With
that, the empirical saturation point and the nuclear matter incompressiblity
are well reproduced while the energy per nucleon as a function of density
closely agrees with sophisticated calculations in the literature.
