Particle-physics constraints on multifractal spacetimes

Abstract

We study electroweak interactions in the multiscale theory with $q$-derivatives, a framework where spacetime has the typical features of a multifractal. In the simplest case with only one characteristic time, length and energy scale $t_*$, $\ell_*$, and $E_*$, we consider (i) the muon decay rate and (ii) the Lamb shift in the hydrogen atom, and constrain the corrections to the ordinary results. We obtain the independent absolute upper bounds (i) $t_* < 10^{-13}{\rm s}$ and (ii) $E_*>35\,\text{MeV}$. Under some mild theoretical assumptions, the Lamb shift alone yields the even tighter ranges $t_*<10^{-27}\,{\rm s}$, $\ell_*<10^{-19}\,{\rm m}$, and $E_*>450\,\text{GeV}$. To date, these are the first robust constraints on the scales at which the multifractal features of the geometry can become important in a physical process.