Low energy expansion of the four-particle genus-one amplitude in type II superstring theory (0801.0322v1)

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.

Analysis of the Low Energy Expansion of the Four-Particle Genus-One Amplitude in Type II Superstring Theory

This paper investigates the low-energy expansion of the four-particle genus-one amplitude in type II superstring theory. The work is focused on deriving the coefficients in the expansion of this amplitude, which involves both analytic and non-analytic components associated with different powers of the Mandelstam variables, labeled as $s$, $t$, and $u$. These variables are crucial in the kinematic analysis of the scattering processes within string theory.

The paper undertakes a detailed exploration of the modular integral computations necessary for evaluating these coefficients. The primary aim is to expand the understanding of these amplitudes up to order $s^6 R^4$, where $R$ represents the linearized super-curvature. This expansion is significant because it allows for analysis beyond the tree level, specifically at one loop, which is essential for understanding quantum corrections in string theory.

The authors employ techniques such as the Poisson resummation and transform the integral over the torus modulus to simplify the evaluation of the scalar propagator terms. They take care to separate analytic contributions from logarithmic threshold terms, with a particular emphasis on satisfying unitarity constraints. This separation is not trivial due to the intricacies of handling infrared effects and the precise calculation of discontinuities across branch cuts that arise in these amplitudes.

One notable aspect is their method of compactifying the amplitude on a circle of radius $r$. This compactification introduces new power-behaved terms in $r$, which serve as important boundary data for any non-perturbative expressions of the low-energy expansion. The authors provide detailed calculations to disentangle these terms, facilitated by the T-duality relations between type IIA and type IIB theories when compactified.

The results thus obtained correct and extend known perturbative results, providing insights into the analytic structure of the theory and contributing to the broader understanding of type II superstring amplitudes. Specifically, the work contributes by presenting systematic expansions of the modular functions involved and deriving results expressed as rational numbers multiplied by Riemann zeta values, hinting at a deeper mathematical structure inherent in string amplitudes.

Furthermore, the paper speculates on the intriguing possibility that these rational coefficients signify more profound relationships, potentially applicable across different genera and perturbative orders. Such insights could have significant implications, suggesting a certain rigidity and predictability in the structure of string amplitudes, and offering direction for future investigations into non-perturbative formulations.

Overall, this paper enriches the theory by expanding upon known results for string scattering amplitudes and revealing an underlying mathematical elegance through its complex computations and insightful theoretical deductions. The work not only adds to the specific knowledge of genus-one amplitudes in string theory but also provides a template for similar future studies aimed at uncovering the finer details of string interactions.