We show that there is a special choice of parameters for which the Galileon theory is invariant under an enhanced shift symmetry whose nonlinear part is quadratic in the coordinates. This symmetry fixes the theory to be equivalent to one with only even powers of the field, with no free coefficients, and accounts for the improved soft-limit behavior observed in the quartic Galileon S-matrix.
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Hinterbichler et al. (2015) studied this question.
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