  We study the resurgent trans-series for the free energy of the
two-dimensional O(4) sigma model in a magnetic field. Exploiting integrability,
we obtain very high-order perturbative data, from which we can explore
non-perturbative sectors. We are able to determine exactly the leading
real-valued exponentially small terms, which we check against the direct
numerical solution of the exact integral equation, and find complete agreement.
