Low-energy expansion of the one-loop type-II superstring amplitude

The one-loop four-graviton amplitude in either of the type-II superstring theories is expanded in powers of the external momenta up to and including terms of order s⁴lnsR⁴, where R⁴ denotes a specific contraction of four linearized Weyl tensors and s is a Mandelstam invariant. Terms in this series are obtained by integrating powers of the two-dimensional scalar field theory propagator over the toroidal world sheet as well as the moduli of the torus. The values of these coefficients match expectations based on duality relations between string theory and eleven-dimensional supergravity.