  Recently, infinite families of massive Feynman integrals were found to
feature an unexpected Yangian symmetry. In the massless case, similar
integrability properties are understood via the interpretation of individual
Feynman integrals as correlators in the massless fishnet theory introduced by
G\"urdo\u{g}an and Kazakov. Here we seek for an analogous interpretation of the
integrability of massive Feynman integrals. We contrast two approaches to
define simple massive quantum field theories in four dimensions. First, we
discuss spontaneous symmetry breaking in the massless bi-scalar fishnet theory.
We then propose an alternative route to a massive fishnet theory by taking a
double-scaling limit of N=4 SYM theory on the Coulomb branch. Both approaches
lead to a massive extension of the massless fishnet theory, differing in how
masses enter into the propagators. In the latter theory, planar off-shell
amplitudes are in one-to-one correspondence with precisely those massive
Feynman integrals that were shown to be invariant under the Yangian. This
suggests a re-investigation of Coulomb branch N=4 SYM theory with regard to
integrability. Finally, we demonstrate that in the case of spontaneous symmetry
breaking, the original conformal symmetry leads to soft theorems for scattering
amplitudes in the broken phase.
