${\mathrm{w}}_{1+\mathrm{\infty }}$ in 4D Gravitational Scattering

In four-dimensional asymptotically flat spacetimes, an infinite tower of soft graviton modes is known to generate the symmetry algebra of ${\mathrm{w}}_{1+\mathrm{\infty }}$ at tree-level. Here we demonstrate that the symmetry action follows from soft graviton theorems and acts non-trivially on massive scalar particles. By generalizing previous analyses that were specifically tailored to the scattering of massless particles, our results clarify that ${\mathrm{w}}_{1+\mathrm{\infty }}$ symmetry is a universal feature of tree-level gravitational scattering in four-dimensional asymptotically flat spacetimes and originates from minimally-coupled gravitational interactions. In addition, we show that the ${\mathrm{w}}_{1+\mathrm{\infty }}$ symmetry acts non-diagonally on massive states by mixing an infinite number of conformal families. We also present a concrete example of non-local behavior on the celestial sphere in the presence of massive scattering states.