The Hierarchical ${\varphi }^4$ - Trajectory by Perturbation Theory in a Running Coupling and its Logarithm

We compute the hierarchical ${\varphi }^4$-trajectory in terms of perturbation theory in a running coupling. In the three dimensional case we resolve a singularity due to resonance of power counting factors in terms of logarithms of the running coupling. Numerical data is presented and the limits of validity explored. We also compute moving eigenvalues and eigenvectors on the trajectory as well as their fusion rules.