  We extend the study of the two-dimensional euclidean $\phi^4$ theory
initiated in ref. [1] to the $\mathbb Z_2$ broken phase. In particular, we
compute in perturbation theory up to N$^4$LO in the quartic coupling the vacuum
energy, the vacuum expectation value of $\phi$ and the mass gap of the theory.
We determine the large order behavior of the perturbative series by finding the
leading order finite action complex instanton configuration in the $\mathbb
Z_2$ broken phase. Using an appropriate conformal mapping, we then Borel resum
the perturbative series. Interestingly enough, the truncated perturbative
series for the vacuum energy and the vacuum expectation value of the field is
reliable up to the critical coupling where a second order phase transition
occurs, and breaks down around the transition for the mass gap. We compute the
vacuum energy using also an alternative perturbative series, dubbed exact
perturbation theory, that allows us to effectively reach N$^8$LO in the quartic
coupling. In this way we can access the strong coupling region of the $\mathbb
Z_2$ broken phase and test Chang duality by comparing the vacuum energies
computed in three different descriptions of the same physical system. This
result can also be considered as a confirmation of the Borel summability of the
theory. Our results are in very good agreement (and with comparable or better
precision) with those obtained by Hamiltonian truncation methods. We also
discuss some subtleties related to the physical interpretation of the mass gap
and provide evidence that the kink mass can be obtained by analytic
continuation from the unbroken to the broken phase.
