Perturbative self-interacting scalar field theory: a differential equation approach

TL;DR

This paper analyzes the perturbative expansion of the partition function in -scalar field theory using a differential equation approach, deriving solutions in various dimensions and revisiting analytical calculations with Stirling's formula.

Contribution

It introduces a differential equation method to compute the first perturbative coefficient of the partition function in -scalar field theory, including solutions in multiple dimensions.

Findings

Derived a modified Bessel equation for the first expansion coefficient.

Solved the equation in 1, 4, and 11 dimensions.

Provided analytical and approximate solutions for the coefficients.

Abstract

We revisit the investigation about the partition function related to a \phi^4-scalar field theory on a n-dimensional Minkowski spacetime, which is shown to be a self-interacting scalar field theory at least in 4-dimensional Minkowski spacetime. After rederiving the analytical calculation of the perturbative expansion coefficients and also the approximate values for suitable limits using Stirling's formulae, which consists of Witten's proposed questions, solved by P. Deligne, D. Freed, L. Jeffrey, and S. Wu, we investigate a spherically symmetric scalar field in a n-dimensional Minkowski spacetime. For the first perturbative expansion coefficient it is shown how it can be derived a modified Bessel equation (MBE), which solutions are investigated in one, four, and eleven-dimensional Minkowski spacetime. The solutions of MBE are the first expansion coefficient of the series associated with the partition function of \phi^4-scalar field theory. All results are depicted.