A note on the non-perturbative zero-dimensional λϕ4 model

We exhibit the partition function of the zero-dimensional λϕ4 model as a simple exact expression in terms of the Macdonald function for Re(λ)>0. Then by analytic continuation, we obtain an expression defined in the complex coupling constant plane λ, for |argλ|<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. This shows that at least in zero dimension the partition function can be defined for negative coupling constant (where the integral is formally divergent), provided it has a non-vanishing imaginary part. 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. Furthermore, a connection between the non-perturbative zero-dimensional solution and the ultralocal λϕ4 model in arbitrary dimension D is presented.