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

As the number of loops goes to infinity Feynman $\alpha$-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 $\alpha$-parameter is integrated in the usual way and the dependence of the mean-values of the other $\alpha$-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 $\alpha$-parameters mean-values are solved numerically. Agreement with previous determinations of the intercept $\alpha (0)$ is obtained for $\alpha (0)
\gsim$\mathrm{0.3.}However,weareabletocalculate$\alpha (t/m^2)$for-3.6$\lsim$ $t\mathrm{/}{m}^2$ $\text{lsim}$ 1.8 and find that it is close to linear. We consider the massless limit of the theory and find that the $\alpha$-parameters mean-values and the trajectory $\alpha (t)$ have limits which are independent of the mass, a phenomenon which also occurs for renormalizable theories via the renormalization group equations.