  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.
