On Bulk Singularity Structures and all order ${\alpha }^{\mathrm{\prime }}$ Contact Terms of BPS String Amplitudes

The entire form of the amplitude of three SYM ( involving two transverse scalar fields, a gauge field) and a potential $C_{n-1}$ Ramond-Ramond (RR) form field is found out. We first derive $<V_{C^{-2}} V_{A^{0}} V_{\phi ^{0}} V_{\phi ^{0}}>$andthenstartconstructinganinfinitenumberof$t,s$ channel bulk singularity structures by means of all order ${\alpha }^{\mathrm{\prime }}$ corrections to pull-back of brane in an Effective Field Theory (EFT). Due to presence of the complete form of S-matrix, several new contact interactions as well as new couplings are explored. It is also shown that these couplings can be verified at the level of EFT by either the combinations of Myers terms, pull-back, Taylor expanded of scalar fields or the mixed combination of the couplings of this paper as well as employed Bianchi identities. For the first time, we also derive the algebraic and the complete form of the integrations for some arbitrary combinations of Mandelstam variables and for the most general case $\int d^{2}z\mathrm{\mid }1-z{\mathrm{\mid }}^a\mathrm{\mid }z{\mathrm{\mid }}^b(z-\bar{z}{)}^c(z+\bar{z}{)}^3$ on upper half plane as well.