Amplitudes and the Riemann Zeta Function.
Amplitudes and the Riemann Zeta Function.
Grant N. Remmen
TLDR
Physical properties of scattering amplitudes are mapped to the Riemann zeta function, and a closed-form amplitude is constructed, describing the tree-level exchange of a tower with masses m_{n}^{2}=μ-2, where ζ(1/2±iμ_{n})=0.
Abstract
Physical properties of scattering amplitudes are mapped to the Riemann zeta function. Specifically, a closed-form amplitude is constructed, describing the tree-level exchange of a tower with masses m_{n}^{2}=μ_{n}^{2}, where ζ(1/2±iμ_{n})=0. Requiring real masses corresponds to the Riemann hypothesis, locality of the amplitude to meromorphicity of the zeta function, and universal coupling between massive and massless states to simplicity of the zeros of ζ. Unitarity bounds from dispersion relations for the forward amplitude translate to positivity of the odd moments of the sequence of 1/μ_{n}^{2}.