  In this letter, we study tree-level scattering amplitudes of scalar particles
in the context of effective field theories. We use tools similar to the soft
bootstrap to build an ansatz for cyclically ordered amplitudes and impose the
Bern-Carrasco-Johansson (BCJ) relations as a constraint. We obtain a set of
BCJ-satisfying amplitudes as solutions to our procedure, which can be thought
of as special higher-derivative corrections to SU(N) non-linear sigma model
amplitudes satisfying BCJ relations to arbitrary multiplicity at leading order.
The surprising outcome of our analysis is that BCJ conditions on higher-point
amplitudes impose constraints on lower-point amplitudes, and they relate
coefficients at different orders in the derivative expansion. This shows that
BCJ conditions are much more restrictive than soft limit behavior, allowing
only for a very small set of solutions.
