Superstring Amplitudes, Unitarity, and Hankel Determinants of Multiple Zeta Values

TL;DR

The paper analyzes how unitarity and analyticity constrain the low-energy expansion of four-particle open and closed superstring tree amplitudes. By expressing the amplitudes as sums over infinite s- and u-channel poles and applying dispersion relations, the authors show that the expansion coefficients form Stieltjes moment sequences, forcing Hankel determinants built from these coefficients to be nonnegative. In the open string, coefficients are rational polynomials in zeta values, yielding positivity constraints on determinants built from ζ-values; in the closed string, coefficients are organized via Z(r,q) combinations of MZVs, with even zetas canceling in the full amplitude, while the s-channel part still satisfies Hankel positivity. The work connects unitarity-induced positivity to the mathematical structure of multiple zeta values, providing new inequalities and a framework for exploring higher-point and higher-genus amplitudes.

Abstract

The interplay of unitarity and analyticity has long been known to impose strong constraints on scattering amplitudes in quantum field theory and string theory. This has been highlighted in recent times in a number of papers and lecture notes. Here we examine such conditions in the context of superstring tree-level scattering amplitudes, leading to positivity constraints on determinants of Hankel matrices involving polynomials of multiple zeta values. These generalise certain constraints on polynomials of single zeta values in the mathematics literature.