Arbitrary four string scattering at high energy and fixed angle

We calculate, using the group theoretic approach to string theory, the tree and one loop scattering of four open and closed arbitrary bosonic string states. In the limit of high energy, but fixed angle, the multi-string vertex at tree and one loop levels that we find takes a very simple form. We propose, and present arguments for, a form for the high energy multi-string vertex at all loops; in particular, we give a path integral derivation of this vertex. Our results agree with those of Gross and Mende for tachyon scattering amplitudes, but those for any other string scattering are substantially different from that discussed in [D.J. Gross, High-energy symmetries of string theory, Phys. Rev. Lett. 60 (1988) 1229]. We also develop some of the technology used in the group theoretic method to compute loop corrections.