Perturbative Results Without Diagrams

TL;DR

This paper introduces a numerical approach to compute higher-order perturbative contributions in quantum field theory without enumerating individual diagrams, demonstrated on polaron ground-state energy calculations.

Contribution

It applies a cumulant expansion method combined with extrapolation to evaluate high-order terms, avoiding the factorial complexity of traditional diagrammatic techniques.

Findings

Calculated 4- and 5-loop coefficients for polaron energy

Demonstrated the method's effectiveness with advanced multidimensional integration

Provided a new approach to higher-order perturbative calculations

Abstract

Higher-order perturbative calculations in Quantum (Field) Theory suffer from the factorial increase of the number of individual diagrams. Here I describe an approach which evaluates the total contribution numerically for finite temperature from the cumulant expansion of the corresponding observable followed by an extrapolation to zero temperature. This method (originally proposed by Bogolyubov and Plechko) is applied to the calculation of higher-order terms for the ground-state energy of the polaron. Using state-of-the-art multidimensional integration routines 2 new coefficients are obtained corresponding to a 4- and 5-loop calculation.