Amplitudes and the Riemann Zeta Function

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 $m_n^2={\mu }_n^2$, where $\zeta \left(\frac{1}{2}\pm i{\mu }_n\right)=0$. 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 $\zeta$. Unitarity bounds from dispersion relations for the forward amplitude translate to positivity of the odd moments of the sequence of $1/{\mu }_n^2$.

5 pages, 2 figures

References in corpus (3)

• Causality, Analyticity and an IR Obstruction to UV Completion
• The EFT-Hedron
• On-shell constructibility of tree amplitudes in general field theories

Cited by in corpus (2)

• Spinning Sum Rules for the Dimension-Six SMEFT
• Analyzing Riemann's hypothesis