[Paper Review] Investigation of meson loop effects in the Nambu-Jona-Lasinio model

Micaela Oertel|ArXiv.org|Dec 18, 2000

High-Energy Particle Collisions Research5 references5 citations

TL;DR

This paper investigates meson loop effects in the Nambu-Jona-Lasinio (NJL) model using three consistent approximation schemes—$φ$-derivable method, $1/N_c$-expansion, and one-meson-loop effective action—demonstrating that all preserve the Goldstone theorem and Gell-Mann–Oakes–Renner relation. The key result is that the $1/N_c$-expansion scheme successfully describes both pion and $ρ$-meson properties simultaneously, while meson loops are essential for modeling the $σ$-meson and enable the study of a first-order chiral phase transition at nonzero temperature.

ABSTRACT

The influence of mesonic fluctuations on quantities in the Nambu-Jona-Lasinio model is examined in different approximation schemes: an expansion in powers of 1/N_c and an expansion up to one-meson loop in the effective action formalism. It is explicitely proved that the Goldstone theorem as well as the Gell-Mann-Oakes-Renner relation hold within those schemes. The influence of meson-loop effects on the quark condensate, the pion mass, the pion decay constant and properties of rho- and sigma-meson are investigated. First we focus on the determination of a consistent set of parameters. In the 1/N_c expansion scheme it is possible to find a set of parameters which allows to simultaneously describe the quantities in the pion sector and those related to the rho-meson, whereas this turns out to be not possible within the expansion of the effective action. Besides, the relation of our model to hadronic models is discussed. In the last part of this thesis the behavior of the quark condensate at nonzero temperature is studied. In the low-temperature region agreement with the model independent chiral perturbation theory result in lowest order can be obtained. The perturbative 1/N_c expansion scheme does not allow for an examination of the chiral phase transition at higher temperatures, whereas this is possible within the meson-loop approximation scheme. A first order phase transition is found.

Motivation & Objective

• To systematically investigate the influence of mesonic fluctuations on low-energy hadronic properties within the Nambu-Jona-Lasinio model.
• To ensure consistency with chiral symmetry by preserving the Goldstone theorem and the Gell-Mann–Oakes–Renner relation in different approximation schemes.
• To determine whether the $1/N_c$-expansion or one-meson-loop scheme can simultaneously describe pion and $\rho$-meson properties.
• To study the role of mesonic intermediate states in the $\sigma$-meson sector and its implications for hadronic spectra.
• To analyze the behavior of the quark condensate and chiral phase transition at nonzero temperature using loop-improved approximations.

Proposed method

• Employing the $\Phi$-derivable method to resum ring diagrams, ensuring consistency with chiral symmetry and unitarity.
• Using the $1/N_c$-expansion to systematically include quantum corrections, with explicit proof that the Goldstone theorem and Gell-Mann–Oakes–Renner relation are preserved.
• Implementing a one-meson-loop expansion in the effective action formalism to include quantum fluctuations beyond mean field.
• Applying a consistent regularization scheme and evaluating two-loop diagrams for self-energies and vertex corrections.
• Deriving RPA-type meson propagators in medium using polarization functions with tensor structures in the rest frame of the heat bath.
• Using Matsubara formalism for finite-temperature calculations, converting four-dimensional momentum integrals into sums over fermionic Matsubara frequencies.

Experimental results

Research questions

• RQ1 Can meson loop effects be consistently incorporated into the NJL model while preserving the Goldstone theorem and the Gell-Mann–Oakes–Renner relation?
• RQ2 Which approximation scheme—$1/N_c$-expansion or one-meson-loop—allows for a simultaneous description of pion and $\rho$-meson properties?
• RQ3 How do mesonic intermediate states influence the properties of the $\sigma$-meson in the NJL model?
• RQ4 What is the behavior of the quark condensate at nonzero temperature, and does the model predict a chiral phase transition?
• RQ5 Can the meson-loop approximation scheme describe a first-order chiral phase transition, unlike the perturbative $1/N_c$-expansion?

Key findings

• The $1/N_c$-expansion scheme allows for a consistent parameter fit that simultaneously describes pion and $\rho$-meson properties, while the one-meson-loop scheme fails to do so.
• Mesonic fluctuations are essential for a proper description of the $\sigma$-meson, particularly through intermediate two-pion states.
• The one-meson-loop approximation scheme successfully preserves the Goldstone theorem and the Gell-Mann–Oakes–Renner relation, confirming its consistency with chiral symmetry.
• The $1/N_c$-expansion reproduces the low-temperature behavior of the quark condensate in agreement with chiral perturbation theory in lowest order.
• The meson-loop approximation scheme enables the study of the chiral phase transition at high temperatures and predicts a first-order phase transition.
• At zero temperature, the $\rho$-meson width is dynamically generated through two-pion intermediate states, consistent with vector dominance models.