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We present a recursive method to calculate the $\alpha'$-expansion of diskintegrals arising in tree-level scattering of open strings which resembles theapproach of Berends and Giele to gluon amplitudes. Following an earlierinterpretation of disk integrals as doubly partial amplitudes of an effectivetheory of scalars dubbed as $Z$-theory, we pinpoint the equation of motion of$Z$-theory from the Berends-Giele recursion for its tree amplitudes. A computerimplementation of this method including explicit results for the recursion upto order $\alpha'^7$ is made available on the websitehttp://repo.or.cz/BGap.git

Non-abelian $Z$-theory: Berends-Giele recursion for the $\alpha'$-expansion of disk integrals