Abstract
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.
Highlights
In recent years, there has been a resurgent interest in exploring the infrared behavior of field theories and its implications
We have constructed the higher derivative Lagrangians for both the nonlinear sigma model and the special Galileon by using building blocks given by the coset construction
We found that the leading-order 5-point amplitude which satisfies KK and BCJ relations does not arise from a theory invariant under the UðNÞ-nonlinear sigma model (NLSM) symmetries and parity
Summary
There has been a resurgent interest in exploring the infrared behavior of field theories and its implications (see e.g., [1] and references therein). A third approach towards computing the higher derivative operators invariant under the special Galileon symmetry was followed in [68]; the invariant Lagrangian was constructed up to quartic order in the Galileon field through a brane construction similar in spirit to [69,70] From these results, it is clear that the definitions of the exceptional scalar theories through their enhanced single soft limits, through their symmetries, or through the double copy, are only equivalent at leading order, and that this equivalence breaks down when including higher-order operators.
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