We present a two-flavor linear sigma model with global chiral symmetry and vector and axial-vector mesons. We calculate $\pi \pi$ 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 $\sigma$ 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 $\sigma$, 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 $\pi \pi$ scattering length $a_0^0$ is too small. This indicates the necessity to extend our model by additional scalar degrees of freedom.

• Received 26 March 2010

DOI:https://doi.org/10.1103/PhysRevD.82.054024