Role of a four-quark and a glueball state in pion-pion and pion-nucleon scattering

We consider the two-flavor version of the extended linear sigma model (eLSM), which contains (pseudo)scalar and (axial-)vector quark-antiquark mesons, a scalar glueball [predominantly corresponding to $f_0\left(1710\right)]$, as well as the nucleon and its chiral partner. We extend this model by the additional light scalar meson $f_0\left(500\right)$, predominantly a putative four-quark state. We investigate various interaction terms of the four-quark and glueball states with the other particles, some of which preserve and some of which explicitly break the $\text{U}{\left(1\right)}_{\mathrm{A}}$ symmetry. We test our model by performing a global fit to masses and decay widths of the scalar resonances and pion-pion scattering lengths. We also discuss the influence of the scalar four-quark state and the glueball on the baryon sector by evaluating pion-nucleon scattering parameters. We find that the inclusion of $f_0\left(500\right)$ improves the description of pion-pion and pion-nucleon scattering lengths.

Figure 1

The dashed lines correspond to the pion, the wavy line to the $\rho$ meson, and the double-dashed line to the $\sigma$ or the $\chi$. (Each diagram with internal double-dashed lines occurs twice, once with the exchange of $\sigma$ and once with the exchange of $\chi$.)

Figure 2

Pion-nucleon scattering diagrams at tree level. The solid line represents the nucleon, the double line its chiral partner, the dashed line the pion, the double-dashed the scalar-isoscalars $H, S$, and $G^{\prime }$, and the wavy line the $\rho$ meson.