RenormalizedG-convolution ofN-point functions in quantum field theory: Convergence in the Euclidean case I

Abstract

The notion of Feynman amplitude associated with a graphG in perturbative quantum field theory admits a generalized version in which each vertexv ofG is associated with ageneral (non-perturbative)n _v-point functionH ⁿ _v,n _v denoting the number of lines which are incident tov inG. In the case where no ultraviolet divergence occurs, this has been performed directly in complex momentum space through Bros-Lassalle'sG-convolution procedure.

In the present work we propose a generalization ofG-convolution which includes the case when the functionsH ⁿ _v arenot integrable at infinity but belong to a suitable class of slowly increasing functions. A “finite part” of theG-convolution integral is then defined through an algorithm which closely follows Zimmermann's renormalization scheme. In this work, we only treat the case of “Euclidean”r-momentum configurations.

The first part which is presented here contains together with a general introduction, the necessary mathematical material of this work, i.e., Sect. 1 and appendices A and B.

The second part, which will be published in a further issue, will contain the Sects. 2, 3 and 4 which are devoted to the statement and to the proof of the main result, i.e., the convergence of the renormalizedG-convolution product.

The table of references will be given in both parts.