Standard Model in multiscale theories and observational constraints

Abstract

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 $t_{\ast }$, ${\ell }_{\ast }$ and $E_{\ast }$, we compute the Lamb shift in the hydrogen atom and constrain the multiscale correction to the ordinary result, getting the absolute upper bound $t_{\ast }<10^{-23}\mathrm{s}$. For the natural choice ${\alpha }_0=1/2$ of the fractional exponent in the measure, this bound is strengthened to $t_{\ast }<10^{-29}\mathrm{s}$, corresponding to ${\ell }_{\ast }<10^{-20}\mathrm{m}$ and $E_{\ast }>28\mathrm{TeV}$. 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 $t_{\ast }<10^{-13}\mathrm{s}$ and $E_{\ast }>35\text{MeV}$. For ${\alpha }_0=1/2$, the Lamb shift alone yields $t_{\ast }<10^{-27}\mathrm{s}$, ${\ell }_{\ast }<10^{-19}\mathrm{m}$ and $E_{\ast }>450\text{GeV}$.

25 pages. v2: authors' metadata corrected; v3: references added, new material added including a comparison with varying-couplings and effective field theories, a section on predictivity and falsifiability of multiscale theories, a discussion on classical CPT, expanded conclusions, and new QED constraints from the fine-structure constant; v3: minor typos corrected to match the published version

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Cited by in corpus (13)

• Dimension and Dimensional Reduction in Quantum Gravity
• Multifractional theories: an unconventional review
• Multifractional theories: an updated review
• Cosmic microwave background and inflation in multi-fractional spacetimes
• Lorentz violations in multifractal spacetimes
• ABC of multi-fractal spacetimes and fractional sea turtles
• Dimensional flow and fuzziness in quantum gravity: emergence of stochastic spacetime
• Deformed symmetries in noncommutative and multifractional spacetimes
• Black holes in multi-fractional and Lorentz-violating models
• Gravitational potential and galaxy rotation curves in multi-fractional spacetimes
• Black Hole Remnants from Dynamical Dimensional Reduction?
• Multifractional spacetimes from the Standard Model to cosmology
• On observational signatures of multi-fractional theory