RNS derivation of N-point disk amplitudes from the revisited S-matrix approach

In the past year, in arXiv:1208.6066 we proposed a revisited S-matrix approach to efficiently find the bosonic terms of the open superstring low energy effective lagrangian (OSLEEL). This approach allows to compute the ${\alpha'}^N$ terms of the OSLEEL using open superstring $n$-point amplitudes in which $n$ is very much lower than $(N+2)$ (which is the order of the required amplitude to obtain those ${\alpha'}^N$ terms by means of the conventional S-matrix approach). In this work we use our revisited S-matrix approach to examine the structure of the scattering amplitudes, arriving at a closed form for them. This is a RNS derivation of the formula first found by Mafra, Schlotterer and Stieberger in arXiv:1106.2645, using the Pure Spinor formalism. We have succeeded doing this for the 5, 6 and 7-point amplitudes. In order to achieve these results we have done a careful analysis of the kinematical structure of the amplitudes, finding as a by-product a purely kinematical derivation of the BCJ relations (for N=4, 5, 6 and 7). Also, following the spirit of the revisited S-matrix approach, we have found the $\alpha'$ expansions for these amplitudes up to ${\alpha'}^6$ order in some cases, by only using the well known open superstring 4-point amplitude, cyclic symmetry and tree level unitarity: we have not needed to compute any numerical series or any integral involving polylogarithms, at any moment.