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We review on a chiral power counting for in-medium chiral perturbation theorywith nucleons and pions as explicit degrees of freedom coupled to externalsources. It allows for a systematic expansion including both local andpion-mediated inter-nucleon interactions. One can identify from this powercounting classes of non-perturbative diagrams that require resummation. Anon-perturbative method based on Unitary Chiral Perturbation Theory was alsodeveloped for performing those resummations. This power counting andnon-perturbative techniques were firstly applied to calculate the pionself-energy, the pion-decay constants and the quark condensate in nuclearmatter up-to-and-including next-to-leading order (NLO) contributions. Thecancellation of the contributions at NLO to the pion self-energy and decayconstants from in-medium nucleon-nucleon (NN) interactions was derived. SomeNLO NN contributions survive for the quark condensate due to the quark-massdependence of the pion mass. Next, we discuss the calculation of the energydensity in the nuclear medium by employing the in-medium NN scatteringamplitudes. For symmetric and neutron matter it reproduces in good agreement,and without fine tuning, calculations from realistic NN potentials with a modelfor the three-nucleon interaction. These results are applied to derive theequation of state (EOS) for neutron stars and obtain an upper limit for aneutron mass slightly above 2 solar masses. Our results also fulfill otherconstraints from the detection of the gravitational waves in the eventGW170817, like the upper bound on the maximal mass of a neutron star and theallowed interval for the radius of a 1.4-solar-mass neutron star. The knowledgeof the neutron-matter EOS is also employed to give an upper bound of thegravitational constant within the strong gravitational field of a 2 solar-massneutron star.

An in-medium chiral power counting for nuclear matter and some applications