# A novel approach to perturbative calculations for a large class of interacting boson theories

## TL;DR

The paper introduces a new method for perturbative calculations of the S-matrix in a broad class of boson theories, leveraging solutions to Diophantine equations, bypassing traditional Feynman diagrams, and applicable to various quantum field models.

## Contribution

It presents a novel algebraic approach linking Green's functions to Diophantine equations for perturbative expansions, extending calculability to arbitrary orders and models.

## Key findings

- Method computes S-matrix to any order for many boson theories.
- Avoids traditional Feynman diagram techniques.
- Demonstrated on Unruh-DeWitt detectors and connected to Hafnian enumeration.

## Abstract

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.