Scattering on the Worldvolume: Amplitude Relations in Brower-Goddard String Models

We investigate the Brower-Goddard extension of the Veneziano and Virasoro-Shapiro four-point amplitudes obtained by generalizing the Koba-Nielsen integrals to $d$-dimensional conformally invariant integrals. The amplitudes derived from this framework exhibit polynomial residues and can be shown to adhere to polynomial bounds at high energies. In odd dimensions, the amplitudes decompose into sums of three partial amplitudes, enabling the formulation of general amplitude relations that subsume the Kawai-Lewellen-Tye (KLT) formula as a particular case. The amplitudes contain multiple tachyons in their spectra. Still, we demonstrate that their residues comply with the positivity conditions mandated by unitarity for spacetime dimensions at or below critical values $D_\text{crit}(d)$, where $D_\text{crit}(6)=26$ and $D_\text{crit}(\infty)=10$. In closing, we contemplate physical applications for membranes and potential extensions of the formalism.