# A novel approach to perturbative calculations for a large class of interacting boson theories. (arXiv:1703.02153v3 [hep-th] UPDATED)

We present a method of calculating the interacting S-matrix to an arbitrary

perturbative order for a large class of boson interaction Lagrangians. The

method takes advantage of a previously unexplored link between the $n$-point

Green's function and a certain system of linear Diophantine equations. By

finding all nonnegative solutions of the system, the task of perturbatively

expanding an interacting $S$-matrix becomes elementary for any number of

interacting fields, to an arbitrary perturbative order (irrespective of whether

it makes physical sense) and for a large class of scalar boson theories. The

method does not rely on the position-based Feynman diagrams and promises to be

extended to many perturbative models typically studied in quantum field theory.

Aside from interaction field calculations we showcase our approach by expanding

a pair of Unruh-DeWitt detectors coupled to Minkowski vacuum to an arbitrary

perturbative order in the coupling constant. We also link our result to Hafnian

as introduced by Caianiello and present a method to list all (2n-1)!! perfect

matchings of a complete graph on 2n vertices.