New Standard Model constraints on the scales and dimension of spacetime

Abstract

Using known estimates for the kaon--antikaon transitions, the mean lifetime of the muon and the mean lifetime of the tau, we place new and stronger constraints on the scales of the multi-fractional theories with weighted and $q$-derivatives. These scenarios reproduce a quantum-gravity regime where fields live on a continuous spacetime with a scale-dependent Hausdorff dimension. In the case with weighted derivatives, constraints from the muon lifetime are various orders of magnitude stronger than those from the tau lifetime and the kaon--antikaon transitions. The characteristic energy scale of the theory cannot be greater than $E_{\ast }>3\times 10^2\mathrm{TeV}$, and is tightened to $E_{\ast }>9\times 10^8\mathrm{TeV}$ for the typical value $\alpha =1/2$ of the fractional exponents in the spacetime measure. We also find an upper bound $d_{\mathrm{H}}<2.9$ on the spacetime Hausdorff dimension in the ultraviolet. In the case with $q$-derivatives, the strongest bound comes from the tau lifetime, but it is about 10 orders of magnitude weaker than for the theory with weighted derivatives.

Comments: 1+21 pages, 3 figures, 1 table. v2: reference added