[Paper Review] O(N) linear sigma model beyond the Hartree approximation at finite temperature

J. Baacke, Stefan Michalski|ArXiv.org|Dec 2, 2003

Quantum Chromodynamics and Particle Interactions1 references3 citations

TL;DR

This paper extends the O(N) linear sigma model beyond the Hartree approximation using the two-particle point-irreducible (2PPI) effective action formalism with a two-loop approximation, including the sunset diagram. It finds a second-order phase transition at finite temperature—contrary to the first-order transition found in the Hartree approximation—while preserving Goldstone's theorem through proper definition of physical masses via the second derivatives of the 1PI effective potential.

ABSTRACT

We study the O(N) linear sigma model with spontaneous symmetry breaking at finite temperature in the framework of the two-particle point-irreducible (2PPI) effective action. We go beyond the Hartree approximation by including the two-loop contribution, i.e., the sunset diagram. A phase transition of second order is found, whereas it is of first order in the one-loop Hartree approximation. Furthermore, we show the temperature-dependence of the variational mass parameters and comment on their relation to the physical sigma and pion masses.

Motivation & Objective

• To investigate the order of the phase transition in the O(N) linear sigma model at finite temperature beyond the Hartree approximation.
• To resolve discrepancies between Hartree approximation (first-order transition) and established expectations (second-order transition) using a higher-order approximation.
• To examine the behavior of variational mass parameters and their relation to physical masses in the broken and restored symmetry phases.
• To ensure consistency with Goldstone's theorem by defining physical masses through the second derivatives of the 1PI effective potential.

Proposed method

• Employing the 2PPI effective action formalism to compute the temperature-dependent effective action, including one-loop log det and two-loop sunset diagrams.
• Using a variational mass matrix decomposition into sigma and pion components to solve the gap equations in the O(N)-symmetric form.
• Solving the gap equations numerically for N=4, with λ=1 and μ²=v², to obtain temperature-dependent mass parameters Mσ² and Mπ².
• Defining physical masses as eigenvalues of the Hessian matrix of the 1PI effective potential, ensuring consistency with Goldstone's theorem.
• Numerically fitting the effective potential to extract the physical sigma mass from the second derivative at the minimum.
• Comparing results with the Hartree approximation to assess the impact of higher-order corrections on phase transition order.

Experimental results

Research questions

• RQ1 Does including two-loop corrections in the 2PPI formalism change the order of the phase transition in the O(N) linear sigma model at finite temperature?
• RQ2 How do the variational mass parameters Mσ² and Mπ² behave with temperature, and do they violate Goldstone's theorem?
• RQ3 Are the physical masses—defined as eigenvalues of the Hessian of the 1PI effective potential—consistent with the expectations of spontaneous symmetry breaking?
• RQ4 Can the 2PPI formalism at two-loop order reproduce a second-order phase transition, resolving the first-order artifact of the Hartree approximation?

Key findings

• The two-loop 2PPI approximation yields a second-order phase transition, in contrast to the first-order transition found in the one-loop Hartree approximation.
• The minimum of the effective potential φ₀(T) vanishes continuously at T > Tcrit, signaling a second-order phase transition.
• The variational pion mass parameter Mπ² remains non-zero even at zero temperature, indicating a symmetry-breaking artifact of the truncation.
• The physical pion mass is exactly zero at all temperatures due to the Hessian of the 1PI effective potential, preserving Goldstone's theorem.
• The physical sigma mass decreases with increasing temperature and vanishes at T = Tcrit, consistent with critical behavior.
• The results are robust for different λ values (λ=1 and λ=0.1), confirming the second-order nature of the transition across coupling strengths.