2008/01/02 by Michael B. Green, Jorge G. Russo, Pierre Vanhove · 1 citation
Physics and Astronomy · #hep-th
paper · pdf · doi:10.1088/1126-6708/2008/02/020
published as JHEP 0802:020,2008 · JHEP style. 6 eps figures. 50 pages
A diagrammatic expansion of coefficients in the low-momentum expansion of the genus-one four-particle amplitude in type II superstring theory is developed. This is applied to determine coefficients up to order s6R4 (where s is a Mandelstam invariant and R4 the linearized super-curvature), and partial results are obtained beyond that order. This involves integrating powers of the scalar propagator on a toroidal world-sheet, as well as integrating over the modulus of the torus. At any given order in s the coefficients of these terms are given by rational numbers multiplying multiple zeta values (or Euler--Zagier sums) that, up to the order studied here, reduce to products of Riemann zeta values. We are careful to disentangle the analytic pieces from logarithmic threshold terms, which involves a discussion of the conditions imposed by unitarity. We further consider the compactification of the amplitude on a circle of radius r, which results in a plethora of terms that are power-behaved in r. These coefficients provide boundary `data' that must be matched by any non-perturbative expression for the low-energy expansion of the four-graviton amplitude. The paper includes an appendix by Don Zagier.