[Paper Review] FRG Approach to Nuclear Matter at Extreme Conditions

Péter Pósfay, G. G. Barnaföldi+1 more|arXiv (Cornell University)|Oct 16, 2015

Quantum, superfluid, helium dynamics1 citations

TL;DR

This paper applies the functional renormalization group (FRG) method within the local potential approximation (LPA) to derive the equation of state for Walecka-type models of nuclear matter under extreme conditions, incorporating quantum fluctuations. The key result is the successful realization of a Maxwell construction in the FRG framework, indicating a first-order phase transition, which is essential for modeling compact stars with mixed phases.

ABSTRACT

Functional renormalization group (FRG) is an exact method for taking into account the effect of quantum fluctuations in the effective action of the system. The FRG method applied to effective theories of nuclear matter yields equation of state which incorporates quantum fluctuations of the fields. Using the local potential approximation (LPA) the equation of state for Walecka-type models of nuclear matter under extreme conditions could be determined. These models can be tested by solving the corresponding Tolman--Oppenheimer--Volkov (TOV) equations and investigating the properties (mass and radius) of the corresponding compact star models. Here, we present the first steps on this way, we obtained a Maxwell construction within the FRG-based framework using a Walecka-type Lagrangian.

Motivation & Objective

• To incorporate quantum fluctuations into the equation of state of nuclear matter using the functional renormalization group (FRG) method.
• To test the validity of the FRG approach in the context of Walecka-type models at extreme densities and low temperatures.
• To explore the emergence of phase transitions, particularly the Maxwell construction, in the FRG framework.
• To assess the applicability of the semi-finite temperature approximation for zero-temperature nuclear matter, relevant for compact star physics.
• To lay the groundwork for solving the Tolman–Oppenheimer–Volkov (TOV) equations using the FRG-derived equation of state.

Proposed method

• The FRG method is used to compute the scale-dependent effective action via the Wetterich equation, integrating from a UV cutoff scale Λ to k = 0.
• The local potential approximation (LPA) is applied, assuming spatially slow-varying propagators and treating the effective potential U_k(φ) as a function of the field φ rather than a functional.
• The effective potential evolution is governed by a differential equation derived from the Wetterich equation, involving spectral functions and Fermi–Dirac/Bose–Einstein distributions at finite temperature.
• A semi-finite temperature approximation is introduced, assuming zero-temperature running of the potential approximates finite-temperature behavior at low T, validated by comparison with direct finite-T integration.
• The Walecka-type Lagrangian includes σ, ω, and π mesons, with the ω field treated in mean-field approximation, and the effective potential evolved numerically.
• The resulting potential is analyzed for phase structure, particularly the appearance of a flat region signaling a Maxwell construction.

Experimental results

Research questions

• RQ1 Can the FRG method with LPA reproduce a first-order phase transition in nuclear matter via a Maxwell construction?
• RQ2 How accurately does the semi-finite temperature approximation capture the zero-temperature behavior of the effective potential in dense nuclear matter?
• RQ3 Does the FRG-based effective potential for a Walecka-type model yield a nuclear mass consistent with experimental values?
• RQ4 To what extent do quantum fluctuations, as included in the FRG framework, alter the equation of state compared to mean-field approaches?
• RQ5 Can the FRG-derived equation of state be used to model compact stars via the TOV equations?

Key findings

• The FRG method successfully generates a Maxwell construction in the effective potential of a Walecka-type model, indicating a first-order phase transition.
• The semi-finite temperature approximation is validated as accurate for low temperatures, with pressure solutions from zero-temperature running closely matching finite-temperature results at β ≥ 0.1.
• The evolved potential at k = 0 shows a flattening region in the effective potential, characteristic of a first-order transition, consistent with the Maxwell construction.
• The model parameters (m² = 1.2 GeV², λ = 7.4, Λ = 1.3 GeV) reproduce the nucleon mass, confirming the physical consistency of the approach.
• The effective potential evolution shows a decreasing expectation value of the σ field as the scale k decreases, indicating spontaneous symmetry breaking.
• The FRG-based equation of state incorporates quantum fluctuations and provides a foundation for studying compact star properties through the TOV equations.