  We study the free energy of integrable, asymptotically free field theories in
two dimensions coupled to a conserved charge. We develop methods to obtain
analytic expressions for its trans-series expansion, directly from the Bethe
ansatz equations, and we use this result to determine the structure of its
Borel singularities. We find a new class of infrared renormalons which does not
fit the traditional expectations of renormalon physics proposed long ago by 't
Hooft and Parisi. We check the existence of these new singularities with
detailed calculations based on the resurgent analysis of the perturbative
expansion. Our results show that the structure of renormalons in asymptotically
free theories is more subtle than previously thought, and that large $N$
estimates of their location might be misleading.
