A Non-perturbative Solution of the Zero-Dimensional lambda phi^4 Field Theory

Abstract

We have done a study of the zero-dimensional $λϕ^{4}$ model. Firstly, we exhibit the partition function as a simple exact expression in terms of the Macdonald's function for $Re(λ)>0$. Secondly, an analytic continuation of the partition function for $Re(λ)<0$ is performed, and we obtain an expression defined in the complex coupling constant plane $λ$, for $|arg λ|<π$. Consequently, the partition function understood as an analytic continuation is defined for all values of $λ$, except for a branch cut along the real negative $λ$ axis. We also evaluate the partition function on perturbative grounds, using the Borel summation technique and we found that in the common domain of validity for $Re(λ)>0$, it coincides precisely with the exact expression.