  In this paper, we explain how perturbative quantum field theory can be
formulated in terms of (a version of) vertex algebras. Our starting point is
the Wilson-Zimmermann operator product expansion (OPE). Following ideas of a
previous paper [arXiv:0802.2198], we consider a consistency (essentially
associativity) condition satisfied by the coefficients in this expansion. We
observe that the information in the OPE coefficients can be repackaged
straightforwardly into "vertex operators" and that the consistency condition
then has essentially the same form as the key condition in the theory of vertex
algebras. We develop a general theory of perturbations of the algebras that we
encounter, similar in nature to the Hochschild cohomology describing the
deformation theory of ordinary algebras. The main part of the paper is devoted
to the question how one can calculate the perturbations corresponding to a
given interaction Lagrangian (such as $\lambda \varphi^4$) in practice, using
the consistency condition and the corresponding non-linear field equation. We
derive graphical rules, which display the vertex operators (i.e., OPE
coefficients) in terms of certain multiple series of hypergeometric type.
