Five-point Type IIB String Amplitudes at One Loop

Massless type IIB superstring amplitudes are organized according to the number of external states and their ${\mathrm U}(1)$ charge under the R-symmetry of type IIB supergravity. In this work, we analyze the low-energy expansion of one-loop five-point amplitudes in all charge sectors, focusing on the representative processes involving five gravitons and four gravitons with one dilaton. We compute the one-loop contributions to the moduli-dependent couplings in the type IIB effective action up to the $D^{12}R^5$ and $D^{14}\phi R^4$ interactions. The results are consistent with $S$-duality constraints in every charge sector and exhibit rich arithmetic structure, including single-valued multiple zeta values, affine linear combinations of logarithmic derivatives of the Riemann zeta function at odd integers, and a new constant of currently unknown nature.