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arXiv · 2108.07820

Amplitudes and the Riemann Zeta Function

Grant N. Remmen
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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 $ζ\left(\frac{1}{2} \pm iμ_n\right) = 0$. 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 $ζ$. Unitarity bounds from dispersion relations for the forward amplitude translate to positivity of the odd moments of the sequence of $1/μ_n^2$.

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Grant N. Remmen. 2021-12-08. Amplitudes and the Riemann Zeta Function. https://doi.org/10.1103/physrevlett.127.241602

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