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Standard Model in multiscale theories and observational constraints

We construct and analyze the Standard Model of electroweak and strong interactions in multiscale spacetimes with (i) weighted derivatives and (ii) $q$-derivatives. Both theories can be formulated in two different frames, called fractional and integer picture. By definition, the fractional picture is where physical predictions should be made. (i) In the theory with weighted derivatives, it is shown that gauge invariance and the requirement of having constant masses in all reference frames make the Standard Model in the integer picture indistinguishable from the ordinary one. Experiments involving only weak and strong forces are insensitive to a change of spacetime dimensionality also in the fractional picture, and only the electromagnetic and gravitational sectors can break the degeneracy. For the simplest multiscale measures with only one characteristic time, length and energy scale $t_*$, $\ell_*$ and $E_*$, we compute the Lamb shift in the hydrogen atom and constrain the multiscale correction to the ordinary result, getting the absolute upper bound $t_*<10^{-23}\,{\rm s}$. For the natural choice $α_0=1/2$ of the fractional exponent in the measure, this bound is strengthened to $t_*<10^{-29}\,{\rm s}$, corresponding to $\ell_*<10^{-20}\,{\rm m}$ and $E_*>28\,{\rm TeV}$. Stronger bounds are obtained from the measurement of the fine-structure constant. (ii) In the theory with $q$-derivatives, considering the muon decay rate and the Lamb shift in light atoms, we obtain the independent absolute upper bounds $t_* < 10^{-13}{\rm s}$ and $E_*>35\,\text{MeV}$. For $α_0=1/2$, the Lamb shift alone yields $t_*<10^{-27}\,{\rm s}$, $\ell_*<10^{-19}\,{\rm m}$ and $E_*>450\,\text{GeV}$.