On The Ladder Bethe-Salpeter Equation

The Bethe-Salpeter (BS) equation in the ladder approximation is studied within a scalar theory: two scalar fields (constituents) with mass $m$ interacting via an exchange of a scalar field (tieon) with mass $\mu$. The BS equation is written in the form of an integral equation in the configuration Euclidean $x$-space with the kernel which for stable bound states $M<2m$ is a self-adjoint positive operator. The solution of the BS equation is formulated as a variational problem. The nonrelativistic limit of the BS equation is considered. The role of so-called abnormal states is discussed. The analytical form of test functions for which the accuracy of calculations of bound state masses is better than 1% (the comparison with available numerical calculations is done) is determined. These test functions make it possible to calculate analytically vertex functions describing the interaction of bound states with constituents. As a by-product a simple solution of the Wick-Cutkosky model for the case of massless bound states is demonstrated

Glueball as a bound state in the self-dual homogeneous gluon field

Using a simple relativistic QFT model of scalar fields we demonstrate that the analytic confinement (propagator is an entire function in the complex $p^2$--plane) and the weak coupling constant lead to the Regge behaviour of the two-particle bound states. In QCD we assume that the gluon vacuum is realized by the self-dual homogeneous classical field which is the solution of the Yang-Mills equations. This assumption leads to analytical confinement of quarks and gluons. We extract the colorless $0^{++}$ two-gluon state from the QCD generating functional in the one-gluon exchange approximation. The mass of this bound state is defined by the Bethe-Salpeter equation. The glueball mass is $1765~{\rm MeV}$ for $\alpha_s=0.33$ if the gluon condensate is $=0.012~{\rm GeV}^4$.Comment: 3 pages. Parallel talk given at the 5rd International Conference on Quark Confinement and the Hadron Spectrum (Confinement V), Gargnano, Italy, September 10-14, 2002. To appear in the proceeding