Logarithmic expansion of field theories: higher orders and resummations

A formal expansion for the Green’s functions of a quantum field theory in a parameter δ that encodes the “distance” between the interacting and the corresponding free theory was introduced in the late 1980s (and recently reconsidered in connection with non-hermitian theories), and the first order in δ was calculated. In this paper we study the O(δ2) systematically, and also push the analysis to higher orders. We find that at each finite order in δ the theory is non-interacting: sensible physical results are obtained only resorting to resummations. We then perform the resummation of UV leading and subleading diagrams, getting the O(g) and O(g2) weak-coupling results. In this manner we establish a bridge between the two expansions, provide a powerful and unique test of the logarithmic expansion, and pave the way for further studies.