α'-Expansion of Open String Disk Integrals via Mellin Transformations

TL;DR

This work addresses the challenge of the $\alpha'$-expansion of open string disk integrals by mapping them to a Mellin-space representation built from Beta functions. The method uses Mellin transformations to disentangle moduli, transferring singularity structure to Mellin-space poles, and contour deformation to extract the leading order terms as $(n-3)!$ residues, providing a new non-local but systematic decomposition of the field-theory limit. The approach applies to generic multiplicities and orderings, yielding explicit residue formulas for low points and general Beta-function products for arbitrary $n$, with leading terms scaling as $\alpha'^{3-n}$. The results illuminate connections to field-theory limits, scattering-equations frameworks, and BCJ-type relations, and suggest potential diagrammatic interpretations and numerical applications.Wraps all mathematical notation in $...$ and highlights the central role of the $(n-3)!$ residue structure in organizing the zero-slope limit.

Abstract

Open string disk integrals are represented as contour integrals of a product of Beta functions by using Mellin transformations. This makes the mathematical problem of computing the α' expansion around the field theory limit basically identical to that of the εexpansion in Feymann 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' that does not include α'=0. Analytic continuation is therefore needed since contours are pinched by poles as α' approaches 0. Deforming contours that get pinched by poles generates a set of (n-3)! multidimensional residues left behind which contain all the field theory information. We end by drawing some analogies between the field theory formulas obtained by this method and those derived recently from using the scattering equations.