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A diagrammatic expansion of coefficients in the low-momentum expansion of thegenus-one four-particle amplitude in type II superstring theory is developed.This is applied to determine coefficients up to order s^6R^4 (where s is aMandelstam invariant and R^4 the linearized super-curvature), and partialresults are obtained beyond that order. This involves integrating powers of thescalar propagator on a toroidal world-sheet, as well as integrating over themodulus of the torus. At any given order in s the coefficients of these termsare given by rational numbers multiplying multiple zeta values (orEuler--Zagier sums) that, up to the order studied here, reduce to products ofRiemann zeta values. We are careful to disentangle the analytic pieces fromlogarithmic threshold terms, which involves a discussion of the conditionsimposed by unitarity. We further consider the compactification of the amplitudeon a circle of radius r, which results in a plethora of terms that arepower-behaved in r. These coefficients provide boundary `data' that must bematched by any non-perturbative expression for the low-energy expansion of thefour-graviton amplitude.  The paper includes an appendix by Don Zagier.

Low energy expansion of the four-particle genus-one amplitude in type II superstring theory