The Strong‐Coupling Expansion and the Singularities of the Perturbative Expansion

We discuss the strong‐coupling expansion in the (λφ⁴)_d theory, quantum electrodynamics and also in different scalar models in a d‐dimensional Euclidean space, and the singularities of this perturbative expansion assuming the ultra‐local approximation. We analyse the analytic structure of the zero‐dimensional generating functions in the complex coupling constant plane for the case of (λφ⁴)_d and also quantum electrodynamics. We find a superposition between a branch point and essential singularity at g₀ = 0 in the scalar model and also in quantum electrodynamics. Further, still using the strong‐coupling expansion as a formal representation for the generating functional of complete Schwinger functions given by Z(h), we present an interacting field theory model that we call the Sinh‐Gordon model (V(g₁, g₂; φ) = g₁(cosh(g₂ φ) − 1)) where the ultra‐local generating functional given by Q₀(h) coincides with the generating functional of complete Schwinger functions. In other words, the ultra‐local approximation is exact, i.e. Z(h) ≡ Q₀(h). As a consequence, it is possible to calculate the vacuum energy per unit‐volume. From the distribution of the the zeros of the modified Bessel function of third kind it is possible to show that the model has a finite vacuum energy per unit‐volume. Finally we sketch how it is possible to obtain a renormalized generating functional for all Schwinger functions, going beyond the ultra‐local approximation.