Vertex Function, Coupling Constant, and Composite Particle

Abstract

A vertex function with only three kinds of strongly interacting particles is studied. As one of them becomes unstable, the structure singularities (one of which corresponds to the anomalous threshold) of this vertex function are shown to move out of the unphysical sheet and remain on the physical cut. In writing out its dispersion relation, we have to choose the correct Riemann sheet for its absorptive part. This choice is made by using a new simple method. A physical interpretation of this structure anomaly is given. With the help of this vertex function, we discuss a model of composite particles, stable or unstable. We obtain the trilinear scalar-type coupling constant in terms of the three masses by using an unsubtracted dispersion relation of this vertex function. A method to estimate the lifetime of the unstable particle is proposed. Another condition between the coupling constant and the masses is obtained from consideration of the charge structure of the composite particle. The two independently obtained conditions can be used to determine both the coupling constant and the binding energy simultaneously. As an explicit example, we consider $\Sigma$ as a bound state of $\Lambda$ and $\pi$ through a scalar-type $\Sigma \Lambda \pi$ coupling. The calculated mass of $\Sigma$ is in excellent agreement with its observed mass and the coupling constant is found to be $\frac{g^2}{4\pi }=1.4\mathrm{}$. A brief discussion on various problems concerning the present work is also given.