Perturbation Theory Without Diagrams: The Polaron Case

TL;DR

This paper introduces a numerical approach to compute higher-order perturbative corrections in quantum field theory, specifically applied to the polaron problem, avoiding the complexity of diagrammatic calculations.

Contribution

It presents a method using cumulant expansion and extrapolation to evaluate higher-order terms without diagrammatic enumeration, demonstrated on the polaron ground-state energy.

Findings

Calculated new four- and five-loop coefficients.

Developed analytical and numerical techniques for reliable results.

Applied multidimensional integration routines successfully.

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 two new coefficients are obtained corresponding to a four- and five-loop calculation. Several analytical and numerical procedures have been implemented which were crucial for obtaining reliable results.