  In integrable field theories in two dimensions, the Bethe ansatz can be used
to compute exactly the ground state energy in the presence of an external field
coupled to a conserved charge. We generalize previous results by Volin and we
extract analytic results for the perturbative expansion of this observable, up
to very high order, in various asymptotically free theories: the non-linear
sigma model and its supersymmetric extension, the Gross--Neveu model, and the
principal chiral field. We study the large order behavior of these perturbative
series and we give strong evidence that, as expected, it is controlled by
renormalons. Our analysis is sensitive to the next-to-leading correction to the
asymptotics, which involves the first two coefficients of the beta function.
