Recursive method to obtain the parametric representation of a generic Feynman diagram

TL;DR

This paper introduces a recursive algebraic method to derive the Feynman or Schwinger parametric representation of complex scalar field theory diagrams, facilitating computational implementation.

Contribution

It presents a novel recursive algorithm that constructs parametric representations directly from an Initial Parameters Matrix for scalar Feynman diagrams.

Findings

Provides a systematic recursive procedure for parametric representation

Applicable to diagrams with arbitrary loops and external lines

Easily implementable in programming and symbolic computation languages

Abstract

A recursive algebraic method which allows to obtain the Feynman or Schwinger parametric representation of a generic L-loops and (E+1) external lines diagram, in a scalar $\phi ^{3}\oplus \phi ^{4}$ theory, is presented. The representation is obtained starting from an Initial Parameters Matrix (IPM), which relates the scalar products between internal and external momenta, and which appears explicitly when this parametrization is applied to the momentum space representation of the graph. The final product is an algorithm that can be easily programmed, either in a computer programming language (C/C++, Fortran,...) or in a symbolic calculation package (Maple, Mathematica,...).