Generalized Beth--Uhlenbeck entropy formula from the $Φ-$derivable approach

TL;DR

The paper addresses the entropy of dense fermionic systems with strong two-particle correlations, including bound and scattering states. It develops a conserving, self-consistent framework via the $\Phi$-derivable approach at the two-loop (sunset) level and derives a generalized Beth-Uhlenbeck formula for entropy, where the spectral weight is governed by a phase shift $\delta$ and exhibits a squared Lorentzian form near resonances. The main result is the entropy expression $S_b=2V\int \frac{d^3q}{(2\pi)^3} \int_0^{\infty} \frac{d\omega}{\pi} \sigma_b(\omega) \sin^2\delta(\omega,\mathbf{q}) \frac{\partial \delta(\omega,\mathbf{q})}{\partial \omega}$, with $\sigma_b(\omega)$ defined from the Bose-Einstein distribution and the derivative optical theorem connecting propagator phase to the self-energies. This framework captures Mott dissociation, respects Levinson's theorem, and provides a consistent description beyond low density, with potential applications to quark matter, nuclear matter, and related many-body systems.

Abstract

We derive a generalized Beth-Uhlenbeck formula for the entropy of a dense fermion system with strong two-particle correlations, including scattering states and bound states. We work within the $Φ-$derivable approach to the thermodynamic potential. The formula takes the form of an energy-momentum integral over a statistical distribution function times a unique spectral density. In the near mass-shell limit, the spectral density reduces, contrary to naïve expectations, not to a Lorentzian but rather to a "squared Lorentzian" shape. The relation of the Beth-Uhlenbeck formula to the $Φ$-derivable approach is exact at the two-loop level for $Φ$. The formalism we develop, which extends the Beth-Uhlenbeck approach beyond the low-density limit, includes Mott dissociation of bound states, in accordance with Levinson's theorem, and the self-consistent back reaction of correlations in the fermion propagation. We discuss applications to further systems, such as quark matter and nuclear matter.

Figures (3)

• Figure 1: The two-loop functional $\Phi$, where the solid line is the electron propagator and the dashed line the photon propagator.
• Figure 2: Self-energy diagrams derived from the $\Phi$ in Fig. \ref{['fig:Phi']}.
• Figure 3: Feynman diagram for the $\Phi-$ functional for scalar $\phi^3$ theory at two-loop order (sunset diagram).