Stringy Completions of the Standard Model from the Bottom Up

We study a class of tree-level ans\"atze for $2\to 2$ 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 ans\"atze to include graviton exchange. In the context of gauge boson scattering, we study 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 value of the gauge coupling $g_{YM}^2 = \frac{2M_s^2}{ M_P^2}$. In integer dimensions $5\leq D \leq 10$ , we find evidence that the maximum allowed allowed rank $r$ of the gauge group agrees with the swampland conjecture $r < 26-D$. The bound is surprisingly identical for both $SU(N)$ and $SO(N)$ in integer spacetime dimensions. We also study the electroweak sector of the standard model via $2 \to 2$ Higgs scattering and find interesting constraints relating standard model couplings, the putative string scale, and the Planck scale