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

TL;DR

This work advances the explicit computation of the low-energy expansion of the genus-one four-particle amplitude in type II string theory, delivering analytic coefficients up to $s^6 R^4$ and partial results beyond. The authors develop a diagrammatic approach on the torus, separate analytic and nonanalytic (threshold) pieces, and systematically extract coefficients $J^{(p,q)}$ by integrating powers of propagators, while carefully handling large-$\tau_2$ degeneration with Eisenstein-polynomial subtractions. They show a consistent unitarity-driven structure for threshold terms in ten and nine dimensions, including explicit logarithmic pieces with scales $\mu_4,\mu_6$ fixed by matching degeneration limits, and they reveal a universal transcendental pattern where coefficients are rational combinations of products of Riemann zeta values. The nine-dimensional compactification, obtained via Poisson resummation and unfolding, preserves $r\to 1/r$ duality and yields a rich $r$-dependent analytic structure, including $r\log r^2$ terms that reproduce ten-dimensional thresholds upon resummation; these nine-dimensional boundary data constrain any proposed nonperturbative completion of the low-energy amplitude.

Abstract

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 s^6R^4 (where s is a Mandelstam invariant and R^4 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.

Figures (6)

• Figure 1: Unitarity equation relating the two-particle $s$-channel discontinuity of the amplitude to the square of the four-particle amplitude, integrated over the phase space of the intermediate particles and summed over the species of these particles.
• Figure 2: Degeneration limits of the genus-one string amplitude. (a) The limit in which all internal legs are massless propagators. This is the limit that gives the one-loop ten-dimensional supergravity contribution. (b) The triangle limit, in which there are three internal massless propagators. (c) The bubble limit in which two massless propagators are picked out. The blobs in (a) and (b) represent higher derivative contact interactions induced by stringy corrections.
• Figure 3: The two-vertex diagram with $\ell_{1}$ lines connecting the two vertices defines $\textrm{D}_{\ell_{1}}$. Each line is associated with a propagator on the torus with momentum $\textbf{p}_{i}=m_{i}+n_{i}\tau$
• Figure 4: The two possible three-vertex diagrams, with $\ell_{1}$, $\ell_{2}$ and $\ell_{3}$ lines joining the vertices defines $\textrm{D}_{\ell_{1},\ell_{2},\ell_{3}}$. Setting $l_3=0$ in (b) reduces the figure to the product of two two-vertex diagrams $\textrm{D}_{\ell_{1},\ell_{2},\emptyset}=\textrm{D}_{\ell_{1}}\times \textrm{D}_{\ell_{2}}$ shown in (a).
• Figure 5: The three possible non-degenerate four-vertex diagrams. (a) defines $\textrm{D}_{\ell_{1},\ell_{2},\ell_{3},\ell_{4}}$, (b) defines $\textrm{D}_{\ell_{1},\ell_{2},\ell_{3},\ell_{4};\ell_{5}}$, (c) defines $\textrm{D}_{\ell_{1},\ell_{2},\ell_{3};\ell_{4},\ell_{5},\ell_{6}}$.