The gaussian propagator formalism and the determination of the leading Regge trajectory for phi^3 field theory

Abstract

As the number of loops goes to infinity Feynman $α$-parameters undergo a fixing mechanism which entails a gaussian representation for propagators in scalar field theories. Here, we describe this mechanism in the fullest detail. The fixed values are in fact mean-values which can be determined via consistency conditions. The consistency conditions imply that one $α$-parameter is integrated in the usual way and the dependence of the mean-values of the other $α$-parameters on it must be determined. Here we present a method for doing this exactly which requires the solution of an equation system. We present an analytic solution for this equation system in the case of the ladder-graph topology. The Regge behaviour is obtained in a simple way as well as an analytic expression for the leading Regge trajectory. Then, the consistency equations for the two (in the ladder case) independent $α$-parameters mean-values are solved numerically. Agreement with previous determinations of the intercept $α(0)$ is obtained for $α(0) \ \gsim$ 0.3. However, we are able to calculate $α(t/m^2)$ for - 3.6 $\lsim$ $t/m^2$ $\lsim$ 1.8 and find that it is close to linear. We consider the massless limit of the theory and find that the $α$-parameters mean-values and the trajectory $α(t)$ have limits which are independent of the mass, a phenomenon which also occurs for renormalizable theories via the renormalization group equations.