[Paper Review] Sigma meson mass and width at finite density

M. J. Vicente Vacas, E. Oset|ArXiv.org|Apr 23, 2002

Quantum Chromodynamics and Particle Interactions17 references3 citations

TL;DR

This paper investigates the density dependence of the sigma meson mass and width using a chiral unitary approach that dynamically generates the sigma as a $π\pi$ resonance. By incorporating pion self-energy corrections from particle-hole and $2ph$ excitations in nuclear matter, the study finds the sigma mass drops to ~250 MeV and width to ~300 MeV at 1.5 times nuclear saturation density, indicating strong medium effects despite broadening from in-medium decay channels.

ABSTRACT

The sigma meson mass and width are studied at finite baryonic density in the framework of a chiral unitary approach which successfully reproduces the meson meson phase shifts and generates the f0 and sigma resonances in vacuum.

Motivation & Objective

• To study the in-medium properties of the $\sigma$ meson, particularly its mass and width, at finite baryonic density.
• To investigate whether the $\sigma$ meson, dynamically generated as a $\pi\pi$ resonance, becomes lighter and narrower in dense nuclear matter.
• To assess the impact of medium effects—especially pion self-energy corrections from particle-hole and $2ph$ excitations—on the $\sigma$ pole position in the complex energy plane.
• To determine whether the $\sigma$ meson could exhibit observable signals in $2\pi$ or $2\gamma$ invariant mass distributions in heavy-ion collisions or photoproduction experiments.

Proposed method

• The study employs a chiral unitary approach based on lowest-order chiral perturbation theory Lagrangians to describe $\pi\pi$ scattering in vacuum and in dense medium.
• The Bethe-Salpeter equation is solved in a coupled-channel framework including $\pi\pi$ and $K\bar{K}$ channels, with on-shell factorization and regularization via a cut-off ($\Lambda = 1.03$ GeV).
• Pion self-energy corrections in nuclear medium are calculated using the Lehmann representation, allowing analytical continuation into the complex energy plane.
• The $\pi\pi$ scattering amplitude is computed in the complex energy plane to locate the $\sigma$ pole position via pole search techniques.
• Medium effects are included through $p$-wave coupling of pions to particle-hole ($ph$) and $\Delta$-hole ($\Delta h$) excitations, with $2ph$ contributions added for higher-density accuracy.
• The effective potential for the $\sigma$ is parametrized as $V_{\rho} = a(\rho/\rho_0) + b(\rho/\rho_0)^2$, with $a = -358 - i108$ MeV and $b = 140 + i23.6$ MeV.

Experimental results

Research questions

• RQ1 How does the $\sigma$ meson mass evolve with increasing baryonic density in nuclear matter?
• RQ2 What is the role of in-medium $\pi$ self-energy corrections, particularly from $ph$ and $2ph$ excitations, in modifying the $\sigma$ width?
• RQ3 Does the $\sigma$ meson become narrower in dense matter, as predicted by some models, despite increased in-medium decay channels?
• RQ4 To what extent do the $K\bar{K}$ channel and higher-order density corrections affect the $\sigma$ pole position?
• RQ5 Can the observed medium effects in $A(\pi,2\pi)$ and $A(\gamma,2\pi)$ reactions be explained by the dynamical $\sigma$ resonance in dense medium?

Key findings

• The $\sigma$ meson mass decreases monotonically with increasing baryonic density, reaching approximately 250 MeV at 1.5 times nuclear saturation density ($1.5\rho_0$).
• The $\sigma$ width decreases moderately but remains relatively large, reaching about 300 MeV at $\rho_0$, despite the reduced phase space for $2\pi$ decay.
• The inclusion of $2ph$ pion self-energy contributions further reduces both the mass and width, indicating their importance at high densities.
• The $K\bar{K}$ channel has a negligible effect, altering the $\sigma$ mass by less than 1% and increasing the width by about 5%.
• The results are robust against the omission of the $\eta\eta$ channel and show qualitative agreement with other models, though the mechanism here is driven by $p$-wave pion-medium interactions rather than $f$-function reduction.
• The large changes in $\sigma$ mass and width suggest strong signals in $2\pi$ or $2\gamma$ invariant mass spectra in heavy-ion collisions, particularly in $A(\gamma,2\pi^0)$ reactions.