Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM

The tree-level S-matrix of Einstein's theory is known to have a representation as an integral over the moduli space of punctured spheres localized to the solutions of the scattering equations. In this paper we introduce three operations that can be applied on the integrand in order to produce other theories. Starting in $d+M$ dimensions we use dimensional reduction to construct Einstein-Maxwell with gauge group $U(1)^M$. The second operation turns gravitons into gluons and we call it "squeezing". This gives rise to a formula for all multi-trace mixed amplitudes in Einstein-Yang-Mills. Dimensionally reducing Yang-Mills we find the S-matrix of a special Yang-Mills-Scalar (YMS) theory, and by the squeezing operation we find that of a YMS theory with an additional cubic scalar vertex. A corollary of the YMS formula gives one for a single massless scalar with a $\phi^4$ interaction. Starting again from Einstein's theory but in $d+d$ dimensions we introduce a "generalized dimensional reduction" that produces the Born-Infeld theory or a special Galileon theory in $d$ dimensions depending on how it is applied. An extension of Born-Infeld formula leads to one for the Dirac-Born-Infeld (DBI) theory. By applying the same operation to Yang-Mills we obtain the $U(N)$ non-linear sigma model (NLSM). Finally, we show how the Kawai-Lewellen-Tye relations naturally follow from our formulation and provide additional connections among these theories. One such relation constructs DBI from YMS and NLSM.