Unmixing sub-leading Regge trajectories of $\mathcal{N} = 4$ Super-Yang-Mills

We study the conformal field theory data (CFT-data) of planar 4D $\mathcal{N} = 4$ Super-Yang-Mills theory in the strong 't Hooft coupling limit. This regime explores the physics of massive short strings in the flat-space limit of the dual AdS. We focus on the CFT-data of the massive short strings exchanged in the operator product expansion (OPE) of the four-point function dual to the Virasoro-Shapiro amplitude. This CFT-data arranges itself into Regge trajectories in the flat-space limit. Using inputs from recent advances in the computation of the AdS Virasoro-Shapiro amplitude, integrability, and a stipulation based on analyticity of the CFT-data in spin, we are able to fix all the CFT-data on the four unique sub-leading Regge trajectories, at leading non-trivial order, as a function of the string-mass level. One of our predictions is that one of the four unique sub-leading Regge trajectories decouples from the OPE in the flat-space limit. This hints at an emergent selection rule in the flat-space limit, similar to our previous results in arXiv:2310.06041. Our procedure should be applicable in a variety of similar setups like for the AdS Veneziano amplitude or in ABJM.