  We consider the large order behavior of the perturbative expansion of the
scalar $\varphi^4$ field theory in terms of a perturbative expansion around an
instanton solution. We have computed the series of the free energy up to
two-loop order in two and three dimension. Topologically there is only an
additional Feynman diagram with respect to the previously known one dimensional
case, but a careful treatment of renormalization is needed. The propagator and
Feynman diagrams were expressed in a form suitable for numerical evaluation. We
then obtained explicit expressions summing over $O(10^3)$ distinct eigenvalues
determined numerically with the corresponding eigenfunctions.
