Inverse of the String Theory KLT Kernel

The field theory Kawai-Lewellen-Tye (KLT) kernel, which relates scattering amplitudes of gravitons and gluons, turns out to be the inverse of a matrix whose components are bi-adjoint scalar partial amplitudes. In this note we propose an analogous construction for the string theory KLT kernel. We present simple diagrammatic rules for the computation of the $\alpha'$-corrected bi-adjoint scalar amplitudes that are exact in $\alpha'$. We find compact expressions in terms of graphs, where the standard Feynman propagators $1/p^2$ are replaced by either $1/\sin (\pi \alpha' p^2/2)$ or $1/\tan (\pi \alpha' p^2/2)$, as determined by a recursive procedure. We demonstrate how the same object can be used to conveniently expand open string partial amplitudes in a BCJ basis.