Integrable Field Theories with an Interacting Massless Sector

We present the first known integrable relativistic field theories with interacting massive and massless sectors. And we demonstrate that knowledge of the massless sector is essential for understanding of the spectrum of the massive sector. Terms in this spectrum polynomial in the spatial volume (the accuracy for which the Bethe ansatz would suffice in a massive theory) require not just L\"uscher-like corrections (usually exponentially small) but the full TBA integral equations. We are motivated by the implications of these ideas for AdS/CFT, but present here only field-theory results.