Shift symmetries, soft limits, and the double copy beyond leading order

In this paper, we compute the higher derivative amplitudes arising from shift symmetric-invariant actions for both the non-linear sigma model and the special galileon symmetries, and provide explicit expressions for their Lagrangians. We find that, beyond leading order, the equivalence between shift symmetries, enhanced single soft limits, and compatibility with the double copy procedure breaks down. In particular, we have shown that the most general even-point amplitudes of a colored-scalar satisfying the Kleiss-Kuijf (KK) and Bern-Carrasco-Johansson (BCJ) relations are compatible with the non-linear sigma model symmetries. Similarly, their double copy is compatible with the special galileon symmetries. We showed this by fixing the dimensionless coefficients of these effective field theories in such a way that the arising amplitudes are compatible with the double copy procedure. We find that this can be achieved for the even-point amplitudes, but not for the odd ones. These results imply that not all operators invariant under the shift symmetries under consideration are compatible with the double copy.