Mass splittings and the finiteness problem of mass shifts in the type-II superstring at one-loop order

Abstract
The on-shell two-point amplitudes in the type-II superstring are constructed for some states in the NS-NS (where NS denotes Neveu-Schwarz) sector in the old covariant formalism. The result indicates that the second and higher mass levels split at one-loop corrections. Also it is noted that the integral representations do not readily give on-shell amplitudes and there is some difficulty in demonstrating the finiteness of the two-point amplitudes despite their on-shell modular invariance.