Analyticity of Off-shell Green's Functions in Superstring Field Theory

We consider the off-shell momentum space Green's functions in closed superstring field theory. Recently in \cite{LES2019}, the off-shell Green's functions -- after explicitly removing contributions of massless states -- have been shown to be analytic on a domain (to be called the LES domain) in complex external momenta variables. We extend the LES domain further to a larger domain within the primitive domain (where the analyticity of off-shell Green's functions in local QFTs without massless states is a well known result). The LES domain can be extended up to the union of certain convex tubes (i.e. primitive tubes) using Bochner's tube theorem and the fact that under complex Lorentz transformations the off-shell Green's functions retain their analyticity property. Up to the four-point function, we obtain such tubes analytically, e.g. for the four-point function there are 32 possible primitive tubes and all of them are obtained in this article. For the five-point function, out of 370 primitive tubes we are able to obtain 350 of them fully. For each of the remaining 20 tubes, it is difficult to show analytically that the application of Bochner's theorem yields the full tube. And this feature occurs for higher point functions as well.