Non-Borel summable Φ4 theory in zero dimension: A toy model for testing numerical and analytical methods

The exact analytical solution for the generating functional of the zero-dimensional Φ4 theory with degenerate minima is obtained in the whole complex coupling parameter plane for testing purposes. The efficiency and precision of different computing tools, proposed in non-Borel summable field theories to obtain approximate solutions in both perturbative and nonperturbative regimes, are analyzed. Furthermore, a new resummation approach is proposed in order to successfully deal with factorially divergent series. It provides a representation of the generating function in terms of an unambiguously defined Laplace–Borel integral. On the other hand, a recent approach called the generalized Borel transform is shown to be an accurate and robust technique to capture non perturbative contributions in the coupling parameter. An extension of this approach to path integrals is proposed.