Chiral-scale effective field theory for dense and thermal systems

TL;DR

By formulating a chiral-scale EFT for nuclear matter and introducing the chiral-scale density counting (CSDC), this work provides a controlled expansion for dense and thermal NM based on the bsHLS framework. The expansion systematically organizes contributions as LO free Fermi gas, NLO one-boson-exchange, and higher-order multi-meson couplings, extended to finite temperature via RMF and the Laudau potential, with the organization in powers of the characteristic scale $k_c$: LO $O(k_c^{4})$, NLO $O(k_c^{6})$, N$^2$LO $O(k_c^{8})$, N$^3$LO $O(k_c^{10})$, and N$^4$LO $O(k_c^{12})$. The authors demonstrate that appropriate choice of CSDC order yields realistic EOS and bulk properties around saturation density and a finite-$T$ liquid-gas phase transition, while the high-density behavior remains consistent with chiral nuclear forces; they also reveal a scale-symmetry pattern with restoration at low density and re-breaking at high density, producing kink behavior in the dilaton condensate and in the sound velocity. The results underscore the importance of quantum corrections and QCD symmetry constraints in shaping the equation of state of dense NM and point to the need for beyond-RMF treatments and density-dependent couplings. The study sets the stage for future refinements, including RHF implementations and Bayesian exploration of parameter space, and applications to proto-neutron stars to illuminate the EOS across broader density and temperature ranges.

Abstract

We established a new power counting scheme, chiral-scale density counting (CSDC) rules, for the application of the chiral-scale effective field theory to nuclear matter at finite densities and temperatures. Within this framework, the free fermion gas is at the leading order, while one-boson-exchange interactions appear at the next-to-leading order, and the multi-meson couplings are at higher orders. Then, we applied the CSDC rules to study the nuclear matter properties, and estimated the valid regions of the CSDC rules. It was found that the zero temperature symmetric nuclear matter properties around saturation density and the critical temperature of liquid-gas phase transition can be captured by an appropriate choice of CSDC orders, and the results beyond these regions are align with the chiral nuclear force. Moreover, the evolution of scale symmetry was found to be consistent with previous studies. The results of this work indicate that the quantum corrections may be crucial in the studies of nuclear matter in a wide density region.

Figures (9)

• Figure 1: The diagrams for effective four-Fermion terms of the RHF method in the OBE model \ref{['eq:HOBE']}.
• Figure 2: The complete propagators of the nucleons in the medium defined in Eq. \ref{['eq:Dirac']}.
• Figure 3: The effective Feynman diagrams for three-meson terms in Hamiltonian. The red points indicate the possible three-meson interaction terms, e.g., $\sigma^3$, $\sigma\omega\omega$, $\sigma\rho\rho$ and $\sigma\pi\pi$.
• Figure 4: The most contributed integral kernel in Eq. \ref{['eq:rhoT']} with perturbation assumptions for $\mu_i^*>m_N^*$ (upper panel) and $\mu_i^*<m_N^*$ (lower panel). The blue shaded region indicates the kernel at $T=0$. The orange dashed lines is the shift of the integral kernel at finite $T$.
• Figure 5: Energy per nucleon as a function of nucleon number density for PNM. FG refers to free Fermi gas, the leading order in CSDC, $\mathcal{O}(k_{\rm{c}}^4)$. The blue shaded region represents the results from the leading-order chiral nuclear-force Drischler:2021kxf.