Massive double copy in the high-energy limit

The exploration of the massive double copy is still in its infancy, and only a few examples in limited contexts are known. In this paper, we analyze the newly discovered double copy for topologically massive theories beyond tree-level amplitudes in the high-energy (eikonal) limit. We find that contrary to the simple double copy relation that occurs in the massless d >= 4 case, the massive double copy requires information outside the eikonal limit to give correct results. However, a simple double copy relation arises for the phase shift. Additionally, we relate the eikonal amplitudes to the corresponding shockwave backgrounds and find a classical double copy relation that is manifest only for the appropriate choice of boundary conditions. This analysis hints towards the existence of a topologically massive double copy at all loop orders.