  We investigate the perturbative integrability of different quantum field
theories in 1+1 dimensions at one loop. Starting from massive bosonic
Lagrangians with polynomial-like potentials and absence of inelastic processes
at the tree level, we derive a formula reproducing one-loop inelastic
amplitudes for arbitrary numbers of external legs. We show that any one-loop
inelastic amplitude is equal to its tree-level version, in which the masses of
particles and propagators are corrected by one-loop bubble diagrams. These
amplitudes are nonzero in general and counterterms need to be added to the
Lagrangian to restore the integrability at one loop. For the class of
simply-laced affine Toda theories, we show that the necessary counterterms are
obtained by scaling the potential with an overall multiplicative factor,
proving in this way the one-loop integrability of these models. Even though we
focus on bosonic theories with polynomial-like interactions, we expect that the
on-shell techniques used in this paper to compute amplitudes can be applied to
several other models.
