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We investigate the Lagrangian formulation of the double-copy correspondencebetween gauge theories and gravity, up to the cubic order. Building on thedefinition of the double-copy field as a convolution of two vectors, we obtainfree gravitational Lagrangians as products of two Yang-Mills Lagrangians, in aform amenable to be easily extended to the massive case. We discuss the originof these results from tensionless strings and show the existence of gaugefixings that mix the two spin-one sectors and lead to an alternative,especially simple, version of the free Lagrangian. We then construct cubicvertices for the full double-copy multiplet, comprising a graviton, a two-formand a scalar particle, by means of the Noether procedure. Both at the free andat the cubic level the result gets uniquely fixed only upon imposing, on top ofgauge invariance, a left-right Lorentz symmetry ruling contraction of indicesamong double-copy fields. Whereas the outcome nicely matches the cubicinteractions of $\mathcal{N}=0$ supergravity, including the gauge-invariantcoupling between the scalar particle and the two-form, such a twofold Lorentzsymmetry seems to conflict with the perturbative reconstruction of space-timegeometry.

On the Lagrangian formulation of the double copy to cubic order