Quantum Field Theory Applications of Heun Type Functions

TL;DR

This paper explores the application of Heun type functions to quantum field theory, specifically in solving eigenvalue equations related to instantons and scalar models, highlighting their natural occurrence and limitations in polynomial solutions.

Contribution

It demonstrates that solutions to certain quantum field theory eigenvalue problems are naturally expressed in terms of Heun functions without modifications, and shows the absence of polynomial solutions with discrete eigenvalues.

Findings

Eigenvalue equations in QCD instanton calculations are solvable via Heun functions.

Modified equations previously used elementary functions, but original forms involve Heun functions.

Polynomial solutions with discrete eigenvalues are not possible in the original equations.

Abstract

After a brief introduction to Heun type functions we note that the actual solutions of the eigenvalue equation emerging in the calculation of the one loop contribution to QCD from the Belavin-Polyakov-Schwarz-Tyupkin instanton and the similar calculation for a Dirac particle coupled to a scalar $CP^1$ model in two dimensions can be given in terms of confluent Heun equation in their original forms. These equations were previously modified to be solved by more elementary functions. We also show that polynomial solutions with discrete eigenvalues are impossible to find in the unmodified equations.