[Paper Review] Exploring meson correlators in the model with four-quark interaction Lagrangian

С. В. Молодцов, M. K. Volkov+1 more|ArXiv.org|Dec 14, 2008

High-Energy Particle Collisions Research3 citations

TL;DR

This paper investigates meson correlation functions in a Keldysh-type model with a four-quark interaction Lagrangian, demonstrating that despite singularities in mean energy and quark condensate, meson observables remain finite and compatible with experimental energy scales. The model provides a tractable framework for studying nonequilibrium quark/hadron systems, with analytical results showing stable bound states across all quark momenta.

ABSTRACT

Meson correlation functions are studied in the model with four-fermion interaction Lagrangian. We demonstrate that despite the singular character of system mean energy and corresponding quark condensate found out the meson observables are finite, quite well identified and compatible with experimental energy scale. It allows one to use the similar model Hamiltonians for describing the nonequilibrium features of quark/hadron systems which reveal themselves in studying ultrarelativistic heavy ion collisions. The analytical results for meson correlation functions in the Keldysh model are given.

Motivation & Objective

• To explore the viability of a Keldysh-type four-quark interaction model for describing nonequilibrium dynamics in quark/hadron matter.
• To address the challenge of singular mean energy and quark condensate in such models by focusing on observable meson correlation functions.
• To establish whether the model yields finite, physically meaningful meson properties compatible with experimental energy scales.
• To compare the Keldysh model with the Nambu–Jona-Lasinio (NJL) model in terms of quasi-particle and meson observables.
• To validate the model's applicability for studying early-stage dynamics in ultrarelativistic heavy-ion collisions.

Proposed method

• Formulates a four-fermion interaction Lagrangian using a momentum-space formfactor resembling the Keldysh model, with spatially separated currents mediated by a formfactor.
• Applies the Bogolyubov–Hartree–Fock approximation to describe quarks as quasi-particles across a broad momentum range.
• Performs analytical continuation from Euclidean to Minkowski space to compute meson correlation functions and extract dispersion laws.
• Uses the x-representation method with Feynman parameterization to evaluate integrals involving propagators and poles in the complex plane.
• Evaluates meson energy dispersions for scalar, pseudoscalar, vector, and axial-vector mesons using the resulting self-energy corrections.
• Compares results with the NJL model, focusing on dynamical mass generation and meson mass spectra under identical parameter tuning.

Experimental results

Research questions

• RQ1 Can meson correlation functions remain finite and physically meaningful despite singularities in the mean energy and quark condensate in a four-quark interaction model?
• RQ2 How do the meson dispersion laws in the Keldysh model compare to those in the NJL model under similar dynamical mass generation?
• RQ3 What is the behavior of bound states in the Minkowski space formulation of the Keldysh model across all quark momenta?
• RQ4 To what extent does the formfactor structure (Keldysh vs. NJL) affect meson observables when quasi-particle properties are similar?
• RQ5 Can the Keldysh model serve as a reliable effective framework for describing nonequilibrium quark-gluon dynamics in heavy-ion collisions?

Key findings

• Meson correlation functions in the Keldysh model are finite and well-defined, despite the divergence of the mean energy and quark condensate.
• The scalar meson mass is given by $ P_{ ho}^{2} = 4E^{2} - 4rac{ ilde{G}}{E} \mathbf{p}^{2} $, and the pseudoscalar meson mass by $ P_{ ho}^{2} = 4E(E - ilde{G}) $, both compatible with experimental energy scales.
• Bound states of quark-antiquark pairs exist at all quark momenta, as shown by the dispersion laws in Minkowski space.
• The model exhibits degeneracy between $ \pi $ and vector mesons, and between $ \sigma $ and axial-vector mesons, when $ \tilde{G}_{\text{V}} = \tilde{G} $, indicating sensitivity to coupling strength tuning.
• With $ \tilde{G}_{\text{V}} = \tilde{G}/2 $, the meson spectrum shows realistic mass ordering, with $ \tilde{G}_{\text{V}} < \tilde{G} $, consistent with experimental expectations.
• Analytical continuation from Euclidean to Minkowski space is feasible and controlled in this one-dimensional model, enabling reliable extraction of physical observables.