  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 $\alpha (0) \
\gsim$ 0.3. However, we are able to calculate $\alpha (t/m^2)$ 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
$\alpha$-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.
