Abstract
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′-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′ that does not include α′=0. Analytic continuation is therefore needed since contours are pinched by poles as α′→0. Deforming contours that get pinched by poles generates a set of (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.
Highlights
With za,b = za − zb, and sa,b = 2ka · kb are the usual Mandelstam variables
In general the KobaNielsen factor has singular behavior when approaching the boundaries of the integration domain of (1.5), which is responsible for the pole structure of the string amplitudes, but hinders a straightforward analytic evaluation
We are going to show that the application of the Mellin transformation (1.6) in the disk integrals gives rise to a new integral representation involving Beta functions only, valid for all non-singular kinematic configurations
Summary
We perform our analysis at 5 points, which is the first non-trivial case. We see that at 5 points, whatever the ordering of Parke-Taylor factor is, the singular behavior is only a result of collision among the first poles from the left and the right families, and so in order to fully resolve it we only need to deform the contour rightwards to extract one of or both of the two first right poles (depending on whether the two right families become identical in the limit, i.e., whether −θ3,4 − θ3,5 + 1 = 0), which are w1∗ = 0, w2∗ = 1 + s3,4 + s3,5,. For higher multiplicities we will inevitably encounter multidimensional contour integrations that complicate the contour deformation, the leading behavior of the disk integral in the α → 0 limit remains relatively simple, since it is still fully absorbed in the first poles picked up by the deformation This provides a new way of decomposing the field theory counterpart
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