Closed-Form Solutions of Zero Dimensional $\phi^4$-Field Theory Using Bessel Functions: A Non-Perturbative Approach

TL;DR

This paper derives exact closed-form solutions for the zero-dimensional $\,\phi^4$-field theory integral and its moments using Bessel functions, providing a non-perturbative analytical approach often used as a learning tool in quantum field theory.

Contribution

It presents a novel derivation of closed-form solutions for the integral and moments of the zero-dimensional $\,\phi^4$-field theory using Bessel functions, offering a non-perturbative perspective.

Findings

Closed-form solutions expressed via Bessel functions.

Explicit formulas for even positive integer moments.

Provides a non-perturbative analytical method for the integral.

Abstract

The integral $\int_{-\infty}^{\infty} e^{- x^2 - g x^4} dx $ is used as an introductory learning tool in the study of Quantum Field Theory and path integrals. Typically it is analysed via perturbation theory. Close form solutions have been quoted but it is not clear how they were derived. So I set about deriving the close form solution on my own and using the same methodology obtain closed form expressions for the even positive integer moments.