Two-loop corrections to large order behavior of φ⁴ theory

We consider the large order behavior of the perturbative expansion of the scalar φ⁴ 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(10³) distinct eigenvalues determined numerically with the corresponding eigenfunctions.