Vacuum properties of mesons in a linear sigma model with vector mesons and global chiral invariance

Abstract

We present a two-flavor linear sigma model with global chiral symmetry and vector and axial-vector mesons. We calculate $\ensuremath{\pi}\ensuremath{\pi}$ scattering lengths and the decay widths of scalar, vector, and axial-vector mesons. It is demonstrated that vector and axial-vector meson degrees of freedom play an important role in these low-energy processes and that a reasonable theoretical description requires globally chirally invariant terms other than the vector-meson mass term. An important question for meson vacuum phenomenology is the quark content of the physical scalar ${f}_{0}(600)$ and ${a}_{0}(980)$ mesons. We investigate this question by assigning the quark-antiquark $\ensuremath{\sigma}$ and ${a}_{0}$ states of our model with these physical mesons. We show via a detailed comparison with experimental data that this scenario can describe all vacuum properties studied here except for the decay width of the $\ensuremath{\sigma}$, which turns out to be too small. We also study the alternative assignment ${f}_{0}(1370)$ and ${a}_{0}(1450)$ for the scalar mesons. In this case the decay width agrees with the experimental value, but the $\ensuremath{\pi}\ensuremath{\pi}$ scattering length ${a}_{0}^{0}$ is too small. This indicates the necessity to extend our model by additional scalar degrees of freedom.