Perturbative Self-Interacting Scalar Field Theory: A Differential Equation Approach

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 formul\ae, 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.