Virtues of a symmetric-structure double copy

We demonstrate a physical motivation for extending color-dual or BCJ double-copy construction to include theories with kinematic numerators that obey the same algebraic relations as symmetric structure constants, $d^{abc}=\text{Tr}[\{T^{a},T^{b}\}T^c]$. We verify that $U(N_c)$ nonlinear sigma model (NLSM) pions, long known to be color-dual in terms of antisymmetric adjoint factors, $f^{abc}$, are also color-dual in the sense of symmetric color structures, $d^{abc}$, explicitly through six-point scattering. This reframing of NLSM pion amplitudes complements our compositional construction of $d^{abc}$ color-dual higher derivative gauge operators. With adjoint and symmetric color-dual kinematics, we can span all four-point effective photon operators via a double-copy construction using amplitudes from physical theories. We further comment on a tension between locality and adjoint effective numerators, and the implications for spanning gravitational effective operators with non-adjoint kinematics.