The Relativistic Cornell-type Mechanism of Exotic Scalar Resonances

Abstract

The formalism of the coupled $q\bar{q}$ and the $\phi \phi (\pi -\pi$, $K\bar{K},\pi K,...$) scalar channels is formulated, taking into account the ground and radial excited $q\bar{q}$ poles. The basic role is shown to be played by the transition coefficients $k^{(I)}(q\bar{q},|\phi \phi )$, which are calculated using the quark-chiral Lagrangian without free parameters. The resulting method, called the pole projection mechanism (PPM), ensures: 1) one resonance for each $\phi \phi$ channel from the basic $q\bar{q}$ pole, e.g. the $f_0(500)$ resonance in the $\pi \pi$ channel; 2) a possibility to have two $\phi \phi$ resonances, coupled to the same $q\bar{q}$ state, when the channel coupling is taken into account in the meson-meson channels, which yields $f_0(500)$ and $f_0(980)$ from the same $n\bar{n}$ pole around 1 GeV; 3) the strong pole shift down for special ($\pi \pi ,\pi K)$ channels due to large transition coefficients $k^{(I)}$, computed in this formalism without free parameters. The parameters of calculated complex poles are in reasonable agreement with the experimental data of the resonances $f_0(500)$, $f_0(980)$, $a_0(980)$, $a_0(1450)$, $K_0^{\ast }(700)$, $K_0^{\ast }(1430)$, $f_0(1370)$ and $f_0(1710)$.

v2: 22 pages, 5 figures; new explanations are added, several important insertions are made, the list of references is enlarged