Conserving Approximations to Dilute Equilibrium Systems. Pair Interaction Potential

Within the conserving Φ-derivable and virial approaches to equilibrium dilute strongly-interacting systems we study a system of non-relativistic fermions of one kind (e.g., neutron matter) interacting via a pair potential in the limit \(n{{\lambda }^{3}} \ll 1\), where n is the fermion number density and λ is the thermal wavelength. The variety of potentials of the quantum and classical origin is used for the quantitative comparison of results obtained within both approaches. In all cases relevant to the nucleon-nucleon interaction, the attraction volume of the interaction is larger than the repulsion one. General properties of the Φ derivable approach are studied. Expressions for the second and third virial coefficients are derived and analyzed for the Φ functional including the tadpole and sandwich diagrams. The classical high-temperature limit of the Φ functional is reproduced by the tadpole diagram. Next, classical, semiclassical (up to order \({{\hbar }^{3}}\)), and purely quantum (Beth-Uhlenbeck) virial expansions of the equation of state are studied. Various extrapolations of the virial equation of state are considered including the van der Waals form and the excluded volume models. We derive the expression for the second virial coefficient within the effective range approximation for the scattering amplitude. Comparison with purely quantum Beth–Uhlenbeck result employing experimental phase shifts shows approximate agreement for low temperatures, \(T \lesssim 20\) MeV. Then, we study the problem of the anomalously large empirical value of the nucleon-nucleon scattering length, appearing due to existence of the quasi-bound state in nucleon-nucleon scattering. The latter is destroyed already in a dilute matter due to the Pauli blocking. Subtracting the quasi-bound state term, we show that the second virial coefficient has the same low temperature dependence as one obtained within the Φ derivable approach in the Born approximation. Also, we discuss origins of differences arising beyond the region of the common validity of the Φ derivable and virial approaches.