  In this work, we investigate aspects of the electron, muon and tau
gyromagnetic ratios (g-factor) 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-factors
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-factor 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 effective interactions of the
different families with the medium.
