Anomalousg-Factors for Charged Leptons in a Fractional Coarse-Grained Approach

Abstract

In this work, we investigate aspects of the electron, muon and tau gyromagnetic ratios (g − f actor) in a fractional coarse-grained scenario, by adopting a Modified Riemann-Liouville (MRL) fractional calculus. We point out the possibility of mapping the experimental values of the specie's g − f actors into a theoretical parameter which accounts for fractionality, without computing higher-order QED calculations. We wish to understand whether the value of (g − 2) may be traced back to a fractionality of space-time.The justification for the difference between the experimental and the theoretical value g = 2 stemming from the Dirac equation is given in the terms of the complexity of the interactions of the charged leptons, considered as pseudo-particles and "dressed" by the interactions and the medium. Stepwise, we build up a fractional Dirac equation from the fractional Weyl equation that, on the other hand, was formulated exclusively in terms of the helicity operator. From the fractional angular momentum algebra, in a coarse-grained scenario, we work out the eigenvalues of the spin operator. Based on the standard electromagnetic current, as an analogy case, we write down a fractional Lagrangian density, with the electromagnetic field minimally coupled to the particular charged lepton. We then study a fractional gauge-like invariance symmetry, formulate the covariant fractional derivative and propose the spinor field transformation. Finally, by taking the non-relativistic regime of the fractional Dirac equation, the fractional Pauli equation is obtained and, from that, an explicit expression for the fractional g − f actor comes out that is compared with the experimental CODATA value. Our claim is that the different lepton species must probe space-time by experiencing different fractionalities, once the latter may be associated to the dressing of the particles and, then, to the effective interactions of the different families with the medium.Some authors argue that the anomalous magnetic moment (AMM) of the muon can be considered one of the most promising observables that can give rises to a rich new physics, specially in particle physics [1]. Other authors claim that the muons AMM is one of the most precisely measured quantities in particle physics, reaching a precision of 0.54ppm [2]. This argumentation is also reinforced because of the relatively large value of the muons mass [3].The tau anomaly is not very well known experimentally and, by this reason do not provide a good test of the StandardThe non-relativistic quantum mechanics predict by the reduction of the Dirac equation to the Pauli equation that g e = 2. Experimentally, the electron g − f actor has been measured with high precision, and the "2010 CODATA recommended 1

Axiomatic Local Metric Derivatives for Low-Level Fractionality with Mittag-Leffler Eigenfunctions

“…Certain systems in Nature seem to be well-described by some fractional parameter in a low-level limit of fractionality. Examples of such low-level systems are the anomalous magnetic g-factor of the charged leptons of the Standard Model, where the low level of fractionality seems to be consistent with the Quantum Electrodynamics corrections [25]. The possibility of constraining fractal space dimensionality from astrophysics and other areas [26], also points to a fractal dimension very close to an integer value, but still noninteger.…”

Generalized Maxwell Relations in Thermodynamics with Metric Derivatives

“…Additionally, as an outlook for possible research, Reference [40] and some approximate approaches in References [41][42][43], there are indications that the use of MD and variational principles, that perspectives of applications seem to open for modeling quantum systems, including the representation of quantum mechanics by hydrodynamic models, including those with stochastic processes [44].…”

On the Zitterbewegung Transient Regime in a Coarse-Grained Space-Time

“…Following the MRL definition, we find that derivative of a constant is zero, and second, we can use it for differentiable and non-differentiable functions as well. For further details on MRL formalism, the reader can follow [17,32,33,35], which contain all the basic for the formulation of a fractional differential geometry in coarse-grained space, and refers to an extensive use of coarse-grained phenomenon.…”

“…Our efforts to justify the use of FC in fractional systems with dissipative systems and the relationship with complex systems, coarse-grained medium, limit scale energy for the interactions and and non-integer dimensions may be found in the papers of [15,16,17,18,19].…”

“…Some backgrounds on coarse-grained media, references on fractal space-time and the efforts to build up a solid foundation for the construction of a geometry and field theory in fractional spaces may be found in [15,16,17,18,19] and references therein.…”