Color Relations and Off-Shell Double-Copy for Towers of Theories

Abstract: Only a handful of elementary color building blocks are known to participate in the double copy, most notably the structure constants of gauge theory. This work extends the double copy to include the infinite family of totally antisymmetric ``structure constants'' with arbitrarily many indices. We propose and motivate the Jacobi identities relating these $n$-index structure constants. For each $n$, we provide a derivatively-coupled scalar theory that is color-dual off-shell. The lowest level in this tower is two-dimensional Zakharov-Mikhailov theory, with one theory for each higher spacetime dimension. Each member of the tower exhibits a conserved current associated with color-kinematics duality, a soft theorem, an on-shell recursion relation, and classical conformal invariance. We also provide a tower of modified non-abelian Chern-Simons theories whose fields couple non-linearly through the $n$-index structure constants. The modified Chern-Simons theories are trivially topological and classically conformal but generally exhibit unusual features like kinetic mixing. A potentially unphysical gauge condition is identified that would ensure that the tower is color-dual off-shell.