α′α′-Expansion of open string disk integrals via Mellin transformations

Open string disk integrals are represented as contour integrals of a product of Beta functions using Mellin transformations. This makes the mathematical problem of computing the α′α′-expansion around the field-theory limit similar to that of the ϵ  -expansion in Feynman loop integrals around the four-dimensional limit. More explicitly, the formula in Mellin space obtained directly from the standard Koba–Nielsen-like representation is valid in a region of values of α′α′ that does not include α′=0α′=0. Analytic continuation is therefore needed since contours are pinched by poles as α′→0α′→0. Deforming contours that get pinched by poles generates a set of (n−3)!(n−3)! multi-dimensional residues left behind which contain all the field theory information. Some analogies between the field theory formulas obtained by this method and those derived recently from using the scattering equations are commented at the end.