  The scattering predictions of a web of theories including Yang-Mills (YM),
gravity, bi-adjoint scalar, the non-linear sigma model (NLSM),
Dirac-Born-Infeld-Volkov-Akulov (DBI-VA) and the special Galileon (sGal) form a
class of special objects with two fascinating properties: they are related by
the double-copy procedure, and they can be defined purely by on-shell
constraints. We expand on both of these properties. First we show that NLSM
tree-level amplitudes are fully determined by imposing color-dual structure
together with cyclic invariance and locality. We then consider how hard-scaling
can be used to constrain the predictions of these theories, as opposed to the
usual soft-scaling. We probe the UV by generalizing the familiar BCFW shift
off-shell to a novel single hard limit. We show that UV scalings are sufficient
to fully constrain: 1. Bi-adjoint doubly-ordered amplitudes, assuming locality;
2. NLSM and BI, assuming locality and unitarity; 3. Special Galileon, assuming
locality, unitarity, and a UV bound for the general Galileon vertex. We see how
potentially distinct aspects of this UV behavior can be understood and unified
via double-copy relations. Surprisingly, we find evidence that assuming
unitarity for these theories may not be necessary, and can emerge via UV
considerations and locality alone. These results complete the observations
that, like IR considerations, UV scaling is sufficient to fully constrain a
wide range of tree-level amplitudes, for both gauge, gravity, and effective
field theories.
