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 à , l à and E à , we compute the Lamb shift in the hydrogen atom and constrain the multiscale correction to the ordinary result, getting the absolute upper bound t à < 10 −23 s. For the natural choice α 0 ¼ 1=2 of the fractional exponent in the measure, this bound is strengthened to t à < 10 −29 s, corresponding to l à < 10 −20 m and E à > 28 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 à < 10 −13 s and E à > 35 MeV. For α 0 ¼ 1=2, the Lamb shift alone yields t à < 10

New Standard Model constraints on the scales and dimension of spacetime

“…Since we will take the absolute value, this difference is irrelevant. As we stressed above, the masses of the W boson and of the quarks are constant in both frames [11], and so is the kaon mass M K . Regarding the mass-splitting formula (3.10), in the integer frame (ordinary Standard Model) f K is defined as the correlator 0|j µ |K = f K q µ , where |0 is the leptonic state.…”

“…In the fractional picture,α qed = α qed v(t) [10], and one gets the upper bounds are the characteristic scale of the quantum electrodynamics processes. Noting that the 2σ-level relative error is 2δα qed /α qed ≈ 6.6 × 10 −10 , one gets [11] t * < 7 × 10 −26 s , ℓ * < 2 × 10 −17 m , E * > 10 GeV , α 0 ≪ 1 2 , (6.1) t * < 4 × 10 −35 s , ℓ * < 10 −26 m , E * > 2 × 10 10 GeV , α 0 = 1 2 . (6.2)…”

“…When also the spectral dimension d S is scale-dependent, modified dispersion relations arise from exotic kinetic terms φKφ in the action. The integral structure given by the measure weight v(x) and the symmetries imposed on the Lagrangian determine the differential structure of the derivative operator K. One specific multi-scale scenario is the multi-fractional theory with weighted derivatives [11,[22][23][24][25][26]. Instead of going through yet another review on the subject [4,27], we summarize here the main features of the theory.…”

“…The Standard Model of electroweak and strong interactions was developed in [11] for this theory. All masses are constant in both frames, including the vector boson masses M W and M Z , the Higgs mass and the quark masses.…”

“…In the specific case of the electroweak sector, the gauge couplings g and g ′ in the fractional frame are measuredependent, while their counterpartsg = g/ √ v andg ′ = g ′ / √ v in the integer frame are constant. In particular, the Fermi coupling in the integer picture is [11]…”