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

Abstract

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 ${f}_{0}(980)$ having the largest $s\overline{s}$ mixture of 10%. The ${f}_{0}(1370)$ is the nodeless $s\overline{s}$ state, while the ${f}_{0}(1500)$ is a $n\mathrm{n\ifmmode \bar{}\else \={}\fi{}}=(u\ifmmode \bar{u}\else \={u}\fi{}+d\overline{d})/\sqrt{2}$ state with a single node. [The presence of that node accounts for the small width of the ${f}_{0}(1500).]$ The next state is a $n\mathrm{n\ifmmode \bar{}\else \={}\fi{}}$ state with two nodes at 1843 MeV. Thus, we identify the ${f}_{0}(1770)$ 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 ${f}_{0}(400--1200),$ 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 \ensuremath{\pi}\ensuremath{\pi} and $K\overline{K}$ channels. [For a full understanding of the ${f}_{0}(400--1200)$ one needs to also consider the role of t-channel \ensuremath{\rho} exchange.] The model used in this work is based upon weak quarkonium-glueball coupling. However, the four-pion decay of the ${f}_{0}(1370)$ and the ${f}_{0}(1500)$ 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 ${f}_{0}(980)$ 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.