  Various groups recently argued that self-interacting vector field theories
lack a well-defined time evolution when the field grows to large amplitudes,
which has drastic consequences for models in gravity and high energy theory.
Such field amplitudes can be a result of an external driving mechanism, or
occur intrinsically, due to large values of the field and its derivatives in
the initial data. This brings a natural question: is small amplitude initial
data guaranteed to evolve indefinitely in these theories in the absence of an
outside driving term? We answer this question in the negative, demonstrating
that arbitrarily low amplitude initial data can still lead to the breakdown of
the theory. Namely, ingoing spherically symmetric wave packets in more than one
spatial dimensions grow as an inverse power of the radius, and their amplitudes
can generically reach high enough values where time evolution ceases to exist.
This simple example further establishes the pervasiveness of the pathology of
self-interacting vector field theories.
