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