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