Droplets in the cold and dense linear sigma model with quarks

The linear sigma model with quarks at very low temperatures provides an effective description for the thermodynamics of the strong interaction in cold and dense matter, being especially useful at densities found in compact stars and protoneutron star matter. Using the MSbar one-loop effective potential, we compute quantities that are relevant in the process of nucleation of droplets of quark matter in this scenario. In particular, we show that the model predicts a surface tension of \Sigma ~ 5-15 MeV/fm^2, rendering nucleation of quark matter possible during the early post-bounce stage of core collapse supernovae. Including temperature effects and vacuum logarithmic corrections, we find a clear competition between these features in characterizing the dynamics of the chiral phase conversion, so that if the temperature is low enough the consistent inclusion of vacuum corrections could help preventing the nucleation of quark matter during the collapse process. We also discuss the first interaction corrections that come about at two-loop order.

💡 Research Summary

The paper investigates the formation of quark‑matter droplets in cold, dense environments using the linear sigma model with quarks (LSMq) as an effective description of strong‑interaction thermodynamics at low temperature (T ≲ a few MeV) and high baryon density (several times nuclear saturation). The authors compute the one‑loop effective potential in the MS¯ renormalisation scheme, incorporating a finite quark chemical potential μ and temperature effects. By analysing the shape of the potential as a function of the sigma field expectation value, they identify two competing minima: one corresponding to the hadronic (nuclear) phase with a non‑zero sigma condensate, and another representing the chirally restored quark phase with a vanishing condensate. The transition between these minima is first‑order, characterized by a discontinuous change in the order parameter and a finite surface tension Σ separating the two phases.

Using a numerical solution of the bounce (critical bubble) profile, the authors obtain Σ in the range of roughly 5–15 MeV fm⁻². This value is considerably lower than those predicted by many other effective models (often 30–50 MeV fm⁻²). A low surface tension dramatically reduces the free‑energy barrier for nucleation, making the formation of quark‑matter droplets feasible during the early post‑bounce stage of core‑collapse supernovae, where temperatures are of order 1–5 MeV and baryon densities reach 2–5 ρ₀. The nucleation rate Γ∝exp(−ΔF_c/T) is therefore potentially large enough to allow droplets to appear on millisecond timescales, which could influence neutrino emission, the equation of state, and the gravitational‑wave signal.

The paper also explores the competing influence of temperature and vacuum logarithmic corrections. Raising the temperature modestly increases Σ and the critical radius R_c, thereby suppressing nucleation. In contrast, retaining the full vacuum logarithmic term in the effective potential shifts the critical chemical potential to higher values and steepens the potential barrier, which can inhibit droplet formation if the temperature is sufficiently low. This competition suggests that, depending on the precise thermodynamic trajectory of a collapsing star, quark‑matter nucleation may either be triggered or prevented.

Beyond the one‑loop analysis, the authors present the first interaction corrections at two‑loop order, involving quark‑sigma exchange, pion loops, and mixed diagrams. Although a full two‑loop renormalisation is technically demanding, the partial calculation indicates that Σ and the critical radius receive corrections of order 10–20 %. This demonstrates that higher‑order effects are not negligible and should be incorporated for quantitative predictions.

In the astrophysical context, the authors discuss how their results map onto protoneutron‑star and neutron‑star interiors. The low surface tension implies that, if a region of the star reaches the appropriate chemical potential, a quark‑matter bubble could grow rapidly, potentially converting a sizable core to deconfined matter. Conversely, the vacuum‑log contribution could raise the nucleation barrier enough to keep the star in a purely hadronic phase throughout its evolution, depending on the cooling history.

The paper concludes that the cold‑dense LSMq provides a tractable yet realistic framework for studying first‑order chiral phase transitions in compact‑star environments. It highlights the importance of accurately determining the surface tension, the need to balance temperature effects against vacuum corrections, and the relevance of two‑loop interaction terms. Future work is suggested in three‑dimensional lattice simulations of bubble dynamics, coupling the nucleation rates to realistic supernova simulations, and extending the model to include color‑superconducting phases or additional strange‑quark degrees of freedom.