Stringy Completions of the Standard Model from the Bottom Up

We study a class of tree-level ansatzes for 2→2 scalar and gauge boson amplitudes inspired by stringy UV completions. These amplitudes manifest Regge boundedness and are exponentially soft for fixed-angle high energy scattering, but unitarity in the form of positive expandability of massive residues is a nontrivial consistency condition.  In particular, unitarity forces these ansatzes to include graviton exchange.  In the context of gauge boson scattering, we consider gauge groups SO(N) and SU(N).  In four dimensions, the bound on the rank of the gauge group is 24 for both groups, and occurs at the maximum allowed value of the gauge coupling:  g_YM^2 = 2 M_s^2/M_P^2 . In integer dimensions 5 ≤ D ≤ 10 , we find evidence that the maximum allowed rank r of the gauge group agrees with the swampland conjecture r ≤ 26-D.  The bound is surprisingly identical for both SO(N) and SU(N) in integer spacetime dimensions. We also study the electroweak sector of the standard model via 2→2 Higgs scattering and find interesting constraints relating standard model couplings, the putative string scale, and the Planck scale.