Covariant confinement model for the calculation of the properties of scalar mesons

We continue our studies of a relativistic quark model that features chiral symmetry, covariance, and confinement. In this work we apply our model to the study of scalar-isoscalar mesons. Several of the parameters of the model have been determined in our earlier work, so that only two new parameters are needed for our analysis. We find a good fit to the spectrum of the $f_0$ mesons, if we add a glueball with energy of about 1700 MeV. In this model we are rather close to “ideal mixing,” with the $f_0\mathrm{}\left(980\right)\mathrm{}$ having the largest $s\overline{s}$ mixture of 10%. The $f_0\mathrm{}\left(1370\right)\mathrm{}$ is the nodeless $s\overline{s}$ state, while the $f_0\mathrm{}\left(1500\right)\mathrm{}$ is a $n\mathrm{n̄}=(uū+d\overline{d})/\sqrt{2}$ state with a single node. [The presence of that node accounts for the small width of the $f_0\mathrm{}\left(1500\right).]$ The next state is a $n\mathrm{n̄}$ state with two nodes at 1843 MeV. Thus, we identify the $f_0\mathrm{}\left(1770\right)\mathrm{}$ as the state with the largest glueball component. It was found that the vacuum polarization functions that describe coupling to the two-meson and other continuum meson channels play an important role in achieving a good fit to the experimentally determined spectrum. In this work we use a Gaussian regulator in all our calculations of meson decay amplitudes. In the first part of our study we multiply the Gaussian regulator by a $P^2$-dependent factor that was chosen so as to modify the threshold behavior of our polarization functions. With that factor in place, we can study the spectrum of $f_0$ states without introducing the imaginary parts of the polarization functions that describe decay to the two-meson continuum. When we do introduce the imaginary parts, we use the vacuum polarization functions with unmodified threshold behavior. The use of the $P^2$-dependent factor helps to clarify the nature of the $f_0(400\mathrm{}–1200),$ which is seen, in part, to have its origin as a rather complex threshold effect associated with the rapid increase of the amplitudes for decay to the ππ and $K\overline{K}$ channels. [For a full understanding of the $f_0(400\mathrm{}–1200)$ one needs to also consider the role of t-channel ρ exchange.] The model used in this work is based upon weak quarkonium-glueball coupling. However, the four-pion decay of the $f_0\mathrm{}\left(1370\right)\mathrm{}$ and the $f_0\mathrm{}\left(1500\right)\mathrm{}$ suggests that these states may be strongly mixed with the glueball, which may have a large four-pion decay width. It is also possible that mixing of these states with the $f_0\mathrm{}\left(980\right)\mathrm{}$ may be important for understanding the four-pion decay widths. We provide a short discussion of quarkonium-glueball mixing in a schematic model. There is not enough information presently available to treat that problem in an unambiguous manner.