Anomalous g-Factors for Charged Leptons in a Fractional Coarse-Grained Approach

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.


Introduction
The electron spin magnetic moment, also called electron spin g − f actor g e , is a dimensionless coupling parameter that appears in the spin-orbit interaction and leads to the splitting in the atomic energy levels, giving rises to the fine structure.
Its value should not be confused with the related gyromagnetic γ ratio. The factor g e appears in the Zeeman effect, as a coupling factor, in the description of the interaction of atoms with external magnetic fields.
values" [4] for this parameter is g e,exp = 2.00231930436153 (53) that is supported by quantum electrodynamics (QED) and is slightly different from that of Dirac's theoretical prevision by quantum mechanics (QM). A good review on the introduction and history of g − f actors, up to 2007, can be found in ref. [5]. For a nonperturbative approach, the reader can see the ref. [6]. An update of the Electron and Muon g-Factors can be found in ref. [3].
In this article, we investigated the fractional coarse-grained aspects of the electron anomalous magnetic g − f actor, showing the possibility of mapping the experimental g − f actor with some theoretical fractionality parameter, so that we take the viewpoint that fractionality may be responsible for the deviation from the g − 2 quantum-mechanical result.
The possible justification for the experimental difference value from the theoretical Dirac g e = 2 is given in the realm of complexity of interactions for the electrons, considered as pseudo-particle "dressed" with the interactions and the medium.
Here, we look at the dynamical system as an open system that can interact with the environment and we argue that fractional calculus can be an important tool to study open classical and quantum systems. [7] . Fractional Calculus (FC) is one of the generalizations of the classical calculus. It provides a redefinition of mathematical tools and it seems very useful to deal with anomalous [8,9,10,11,12,13] and frictional systems. Several applications of FC may be found in the literature [14,15,16]. Presently, areas such as field theory and gravitational models demand new conceptions and approaches which might allow us to understand new systems and could help in extending well-known results. Interesting problems may be related to the quantization of field theories for which new approaches have been proposed [17,18,19,20]. In conection with to our work, it is worthy to mention here that a fractional Riemann-Liouville Zeeman effect, an atempt to implement gauge invariance in fractional field theories and an angular momentum algebra proposed with the Riemann-Liouvile formalism here reported in the paper of Ref. [21]. Low-energy nuclear excitations was studied by the constuction of a fractional symmetric rigid rotor in order to study barionic excitations [22].
Here, we claim that the use of an approach of FC based on a sequential form of modified Riemann-Liouville (MRL) fractional calculus [23] is more appropriate to describe the dynamics associated with field theory and particle physics in the space of nondifferentiable solution functions, or in the coarse-grained space-time.
It seems that a reasonable way to probe the classical framework of physics is to highlight that, in the space of our real world, the generic point is not infinitely small (or thin), it rather has a thickness. In a coarse-grained space, a point is not infinitely thin, and here, this feature is modeled by means of a space in which the generic differential is not dx, but rather (dx) α , and likewise for the time variable t. It is noteworthy to highlight the ideas in the interesting work by Nottale [24], where the notion of fractal space-time is first introduced. Non-integer differentiability and randomness [25] are mutually related in their nature, in such a way that studies on fractals on the one hand, and fractional Brownian motion on the other hand, are often parallel in the ref. [24]. A function continuous everywhere, but nowhere integer-differentiable, necessarily exhibits random-like or pseudo-random features, in that various samplings of these functions, on the same given interval, will be different. This may explain the huge amount of literature extending the theory of stochastic differential equations to describe stochastic dynamics driven by fractional Brownian motion [23,26,27].
Recently, we have also used FC to analyze the well-established canonical quantization symplectic algorithm [28]. The focus was to construct a generalized extension of that method to treat a broader number of mechanical systems with respect to the standard method. In this sense, we adopted the MRL prescription for fractional derivative. There we argued that there is a number of problems in considering classical systems besides the ones that involve the quantization of second-class systems. These problems encompass the so-called nonconservative systems. The peculiarity about them is that the great majority of actual classical systems is nonconservative but, in spite of that, the most advanced formalisms of classical mechanics deal only with conservative systems [29]. Following [28], dissipation, for example, is present even at the microscopic level. There is dissipation in every non-equilibrium or fluctuating process, including dissipative tunneling [30] and electromagnetic cavity radiation [31], for instance. There we argued that one way to suitably treat nonconservative systems is through Fractional Calculus, FC, since it can be shown that, for example, a friction force has its form stemming from a Lagrangian that contains a term proportional to the fractional derivative, which may be a derivative of any non-integer order [29].
Field theory aspects of non-linear dynamics are today an important subject of study in different physical and mathematical sub-areas, but the real success and a radically new understanding of non-linear processes has occurred over the past 40 years. This understanding was inspired by the discovery and insight of chaotic dynamics, where the randomness of some physical processes are considered; more precisely, when particle trajectories are indistinguishable for random process [32]. In a previous work, also based on this approach, we have worked out a suggested version of a fractional Schrödinger equation, with a lowest-order relativistic correction, obtained from a fractional wave equation [33] to which a mass term has been adjoined, to give us a fractional Klein-Gordon equation (FKGE). With the definition of some fractional operators, the McLaurin expansion and an ansatz for the plane wave solutions, we have obtained fractional versions of Bohmian equations to describe the particle dynamics associated with Bohmian mechanics, in the space of non-integer differentiable functions. We had also done a formulation for an anomalous dispersion relation and to a refraction index, related to massless particle in a coarse-grained media and a vacuum refractive index for a coarse-grained non-trivial optical medium. In Ref. [33], we have argued that the modeling of TeV-physics may demand an approach based on fractal operators and FC and we claimed that, in the realm of complexity, nonlocal theories and memory effects were connected to complexity and also that the FC and the nondifferentiable nature of the microscopic dynamics may be connected with time scales.
Using the MRL definition of fractional derivatives, we have worked out explicit solutions to a fractional wave equation with suitable initial conditions to carefully understand the time evolution of classical fields with fractional dynamics.
First, by considering space-time partial fractional derivatives of the same order in time and space, a generalized fractional D'Alembertian is introduced. By means of a transformation of variables to light-cone coordinates, an explicit analytical solution was obtained. Also, aspects connected with Lorentz symmetry were analyzed with two different approaches.
Here, to achieve our goals, we carefully build up a fractional Dirac equation from fractional Weyl equation that, on the other hand, was constructed with helicity and projection concepts. We begin by constructing the fractional angular momentum algebra in a coarse-grained scenario in order to justify the fractional spin operator eigenvalue, used to project the spin in the linear momentum direction. Also, a covariant fractional derivative is obtained by proposing a spinor field transformation that leads to a fractional potential vector field gauge transformation. Then, the minimal coupling to the electromagnetic field is naturally obtained by writing down a fractional Lagrangian density with an included term that takes into account the electronic current that, in turn, is based on the electromagnetic current. So, we are not proposing the matter transformation from the very start [21]; we rather get it from the matter-gauge coupling together with the gauge field transformation. Finally, by an non-relativistic approximation of the fractional Dirac equation, the fractional Pauli equation is obtained that leads to an explicit expression for the fractional g-factor. We point out that fractional Dirac equation has already been studied by several authors over the past decade [34,35,36]. Our paper is outlined as follows: In Section 2, we consider the mathematical background, with some expressions of the fractional coarse-grained calculus and the modified Riemann-Liouville fractional derivative. Section 3 contains the Fractional Angular Momentum Algebra, in a coarse-grained scenario. Section 4 is devoted to Fractional Field Equations: Weyl and Dirac and the results for g-factor for electron, muon and tau are presented. In Section 5 we present the Gordon Decomposition for Fractional Dirac Equation and also the Fractional Spin Current. Finally, in Section 6, we cast our Discussion and Conclusions.

Mathematical Background
In the sequence we use the an alternative approach by considering fractional coarse-grained space-time instead of fractional space functions, meaning that nor the space nor the time are infinitely thine but have "thickness".
As the use of certain calculation rules are essential to our approach, we briefly comment on this point, before presenting these rules. It seems to us worthy to note that the chain rule, as well as the product rule of Leibniz, for the fractional approach used here, had their validity mathematically proven in view of the recent demonstration of the formal fractional Taylor expansion, in the context of the modified Riemann-Liouville (MRL) formalism [37]. We also point out to the fact that to the date prior to the publication of this statement, in its countersigned mathematical formal approach, doubts about the validity of the rules could reasonably exist, because of previous statements somewhat incomplete. However, given the existence of this new theoretical formal support, published in an international journal physica A, endorses the validity of the fractional Taylor expansion and therefore proves it. The fractional Taylor expansion is the mathematical basis for the validation of the chain rule and also of the Leibniz product rule, in the MRL context. We also emphasize that the rules so obtained can then be viewed as good approximations. Than, we point out that the Leibniz rule used here is a good approximation that comes from the first two terms of the fractional Taylor series development, that holds only for nondifferentiable functions [38,39] and are as good and approximated as the classical integer one. These approaches are quite similar to their counterparts made in the integer order calculus and therefore so good as. The well-tested definitions for fractional derivatives, so called Riemann-Liouville and Caputo have been frequently used for several applications in scientific periodic journals. In spite of its usefulness they have some dangerous pitfalls. For this reason, recently it was proposed an interesting definition for fractional derivative [23], so called modified Riemann-Liouville (MRL) fractional derivative, which is less restrictive than other definitions and its basic definition is Some advantages can be cited, first of all, using the MRL definition we found that derivative of constant is zero, and second, we can use it so much for differentiable as non differentiable functions. They are cast as follows [39]: (i) Simple rules: where f is α-differentiable and u is differentiable with respect to x and, For further details, the readers can follow the refs. [40,38,39] 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.
Here, another comment is pertinent: the fractional MRL approach for nondifferentiable functions has similar rules and has definition with a mathematical limit operation comparable to certain definitions of local fractional derivatives, as that introduced by Kolwankar and Gandal [41,42,43] with some studies in the literature. For example, the works of Refs. [44,45,46] or the approaches with Hausdorff derivative, also called fractal derivative [47,48], that can be applied to power-law phenomena and the recently developed α − derivative [49]. The MRL approach seems to us to be an integral version of the calculus mentioned above and all of them deserve to be more deeply investigated, under a mathematical point of view, in order to give exact differences and similarities respect to the traditional fractional calculus with Riemann-Liouville or Caputo definition and with local fractional calculus and even fractional q-calculus [50,15,51], as well as in the comparative point of view of physics [48,51,52], for the scope of applicability. Now that we have set up these fundamental expressions, we are ready to carry out the calculations of main interest.

Fractional Angular Momentum Algebra
Here we will derive the commutation algebra for spin − 1/2 particles in a coarse-grained medium.
Since in this approach MRL the chain and Leibniz rules holds, it is not difficult to obtain the commutation relation for position momentum and defining the fractional angular momentum components [22,21], we can writê Again, one can obtain the commutation relations using the above definitions and the chain(2.4), (2.5) and Leibniz (2.3) rules for MRL as To build up the algebra, we define some the operators, as in the following.
The square of fractional angular moment operator is defined as in ref. [22]: We also define the fractional angular momentum operator L α and the fractional raising and lowering operators, L α + and its hermitian conjugate L α − , respectively as It follows, the commutation the relations The above commutation relations indicates that the ordinary integer angular momentum algebra do not change. This implies that the raising and the lowering operators acting on a eigenstate |j, m , leads to an new state vector |j, m ± 1 but with eigenvalue Γ(α + 1) α M α , that is, it raises theL α z eigenvalue by the latter increment. Another important conclusion is that, for the ordinary Pauli spin matrices, the representation for the basis of L α 2 is the same as the integer case. This allows us to rewrite all usual relations from ordinary quantum mechanics spin algebra in a coarse-grained scenario. The only difference is the instead of ℏ we have to substitute in all relations an effective factor There is no fractional number of particles but, there exist an effective Planck constant.
We remark here that, if in the definition o fractional moment operator, eq. (4.5), the complex i were redefined as i α , the algebra probably could have completely changed.

Fractional Fields Equations: Weyl and Dirac
The Weyl Equation is a relativistic wave equation for describing massless spin − 1/2 particles.
Remembering that the helicity is the projection of the spin onto the direction of momentum, we proceed to pursue by this way to achievement of the fractional Weyl equations in the following.
We write the projection of spin onto its linear momentum as with the spin vector, − → S α , obtained from the angular momentum algebra, given by Here the Pauli matrices are the same since the structure of the algebra are not modified, as shown in section 3.
For a mass less particle, the relativistic energy-momentum expression may be [53] Considering a 2 × 1component spinorial field χ L,α belonging to a group representation . In term of this field and the above equations, we can write In order to proceed with the adequate quantization, we have carry on the correspondence principle. To this intention we propose the fractional operators energy and momentum as where the constant M x,α is included for dimensional reasons. By correspondence principle, the eq. (4.4) can be rewritten as here we have considered the space and time with the same fractionality.
µ χ L,α = 0 that leads to the propagating fractional wave equation for left helicity fermion i ( ) α (α) χ L,α = 0. The notation for the box symbol it is not to be confused with the fractional power operator in distribution theory. Here the the box symbol is defined as

Right Helicity:
A similar procedure can be used to construct a projection for right helicity spin, using in eq.(4.1) λ α = −1.
Now the spinor field is noted as ξ R and belongs to a group representation such as ξ R ∈ (0; 1/2). Following the approach sequence, the second fractional Weyl equation reads that also conduct us to the equation (4.13) In compact notation, the fractional derivative is given by∂ α µ = ( 1 c α ∂ α t ; M x,α ∇ α ) The two Weyl equations can be written in a more compact form in terms of a four dimensional spinorial field Ψ α as where γ µ are usual the Dirac gamma matrices. We proceed now by introducing a mass parameter that mixes the two quiral components to obtain an equations capable to describe the dynamics of a massive particle. To this intention we write the two fractional Weyl equations as where u α andũ α are parameters which, as we shall show below, will be the mass of the charged fermions. Imposing that the above equations be compatible with the fractional energy-momentum relation [53] E 2α = p 2α c 2α + m 2α c 4α . From the second of eq.(4.15), we obtain that, substituted in the first of those equations and rewriting, yelds The above equation indicates by comparing with a fractional Klein Gordon [53] equation that u α =ũ α = −m α .With theses observations, the fractional Dirac equation can now be written from (4.15) in the general case α = 1 as

Minimal Coupling, Field Transformation and the Covariant Fractional Derivative
The conjugated Dirac equation is Now, multiplying eq. (4.20) by Ψand eq. (4.21) by Ψ and subtracting we obtain The quantity in the bracket can be identified as a conserved current and the equation is a fractional continuity equation. From standard electromagnetism, we now that j µ A µ is the electric current coupled to electromagnetic field by a potential tensor A µ , where j µ is of form j µ = eΨγ µ Ψ.
For the fractional case, we can think of a minimal coupling term as e α Ψ α γ µ Ψ α A α µ , where A α µ = (φ α ; − A α ) is the fractional potential tensor transform under Gauge transform as (A α µ ) ′ = A α µ + ∂ α µ χ Following the integer model of Lagrangian, we can write a fractional Lagrangian with electromagnetic field coupled as that can be written as In order to the theory remain covariant, we assume a spinor field transformation of form where R(χ) have the unitary property and its explicit form will be determined in the sequence. The Lagrangian density with the electromagnetic coupling term lead us to define a fractional covariant derivative of form where the tensor field A α µ is considered to have a fractional Gage symmetry and transforms by The fractional covariant derivative of the field Ψ α will have to obey the same field transform as the fields, or In more details we can write which results in a fractional differential equation of form The solution of the above equation is This can be easily proven as follows. Fractionally deriving the above equation, with the use of eq.(2.5) results in which proves the assertion. The fractional Dirac equation, in a coarse-grained scenario may now be written with the minimal coupling as Separating spacial and time terms, multiplying by γ 0 and using the properties of gamma Dirac matrices, we obtain Using the correspondence principle, we have Now we write the Dirac spinor as 4 × 1 matrix in a coarse-grained scenario as where we have named the strong(s) and weak(w) components as a 2 × 1 matrices as ψ α,s = ψα,1 ψα,2 , ψ α,w = ψα,3 ψα,4 , respectively. In a sympletic form we can write (4.39)

Non-Relativistic Limit of Fractional Dirac Equation and the Fractional g-factor
In order to proceed with the non-relativistic limit of the fractional Dirac Equation, we have to consider the dominant term in the Hamiltonian as the rest energy given by m α c 2α . We then propose an ansatz for the solution to the fractional Dirac Equation as that furnish two equations as follows Considering now that the mass terms is dominant over the electrostatic one, that is, e α φ α ≪ m α c 2α (1 + Γ(α + 1)), and that the weak component fields ψ α,w has slow evolution, when compared to the rest energy, i α 1∂ α t ψ α,w < m α c 2α (1 + Γ(α + 1))ψ α,w . With these approximations, we can write for the second equation that leads to the relations between weak and strong field components as (4.44) The above expression gives immediately that the weak component is, in the scope of adopted approximations, very lower that the strong one, ψ α,w ≪ ψ α,s . Now inserting this result into the first equation (4.42), results that one equation for the strong component written as Defining the fractional momentum operator as − → π α ≡ − → p α − eα c α − → A α , and using the well known propriety of Pauli matrices (4.46) The eq. (4.46) is the fractional version of the Pauli equation in a coarse-grained scenario.
Note the when α = 1, g f rac = g = 2 and the equation becomes the usual Pauli equation. We can map a fractional parameter with the CODATA [4] known value of g exp for electrons and muons and also ref. [54] for taus particles. The mapping can be done by solving numerically the equation g f rac = 4 (1 + Γ(α + 1))Γ(α + 1) = g exp . (4.49) In our Discussion and Conclusions (Section 6), we shall discuss in more details this result and how we make use of it to fit the g − f actors of the charged leptonic particles.

The Gordon Decomposition for Fractional Dirac Equation:The Fractional Spin Current
From eq.(4.22) we can define the fractional current density as The above equation may be rewritten as From the Dirac equation eq. (4.20) and its conjugated (4.21) we may write, respectively, 3) Now, inserting these results in the first and second terms of the eq. (5.2), respectively, we can write for the fractional For the gamma matrix we can write that and and η µν is the metric tensor. Using the above definitions and inserting into the eq. (5.5), we obtain for the current a more decoupled form Defining σ µν = iΣ µν , the second term in the above equation may be identified with the spin contribution to the fractional current and reads as Note that the above decomposition is performed in the configuration space instead of the Fourier space, because we are not dealing with generalized functions, but with non differentiable functions.

Discussion and Conclusions
With the help of the eq. For the tau (QED, see eg. [54]): g τ = 2.00235442, α τ = 0.998142055249517567. The higher is the g − f actor, the farther is the alpha-parameter from α = 1 (g-fator is a decreasing function with alpha, as can be seen numerically from eq. (4.48)). Note that a bigger alpha means fractionality closer to 1. Moreover, if this parameter deviates from α = 1, it may indicate that the system is more sparse (roughness is larger), ie, there is a smaller range about the different types of interactions (particles and fields) in the vicinity of pseudo particule. Therefore, if the particle interacts less or have a lower mean life than other, the alpha-parameter would be a little more distant from one (in the downward direction), indicating a lower complexity of interactions. The closer α is to 1 indicates a greater complexity of these interactions (a particle interacting more, more structures can interact around). Alpha equal to 1 means one type of total mixing memory (providing an idealized integer model, which is not natural), with the pseudo partícules feeling the interactions of all kinds, so that, on average, these fluctuations are nullify (a statistical average).
In terms of mean life, muon has 10 −6 s, tau10 −15 s, while electrons are stable. The relation with the alpha-parameter can be understood as follows: The lower is the mean life means that the particle does not have sufficient time to interact with the surrounding environment and with other particles, implying that the alpha-parameter is more distant from 1.
The interaction universe seen by those particles is limited to the closer interacting particles. Stable particles like electrons can "see" the entire environment and interact more, justifying a fractional parameter closer to 1 than the other particles.
For the electron, the results may be indicating that the complexity of the interactions taken into account in the QED calculation may be lower than it would be in the in the reality seen by the experiment. That is, the electron, as a pseudo-particle, keeps hidden other interactions that are not well described or are incomplete in the SM description. Thus, the results are different for QED compared to the experimental results. In the experimental reality, the complexiy is greater than that considered in the QED calculations based on SM. That is, the fractionality should indicate that the SM, although very good, may not be providing all information necessary to describe the interactions in a more complete view, either at high energies or in granular or fractal space-time. Thus, the FC may gives evidences that the SM could require corrections (or higher order calculations by QED).
For the muon, an opposite behaviour in terms of fractional parameter could be observed and indicated that the QED calcultions might be taken into account more interactions and consequnt complexity than the particle realy experiment and, again, signaling that the SM may not be complete to describe the whole interaction scenario.
In summary, in the present work we have built up a fractional Dirac equation in a coarse-grained scenario by taking into account a fractional Weyl equation, a fractional angular momentum algebra, introducing a mass parameter and imposing that the equations be compatible with the fractional energy-momentum relation. Considering then a minimal coupling in the Lagrangian and introducing a field transformation to reveal a covariant fractional derivative, the free Lagrangian density with the electromagnetic coupling term leads us to define a fractional covariant derivative. In the sequel, we proceeded by a non-relativistic limit of the Dirac equation to obtain a fractional version of the Pauli equation. We than have investigated the anomalous magnetic moment for the charged leptons, electrons, muons and taus. By the a fractional approach we have obtained, In each the cases of study, that the results agrees with standard integer order in the convenient limits.
We have shown that a mapping of the anomalous magnetic g factor is possible in terms of a fractional parameter.
Defining a fractional current density we have also performed a fractional Gordon decomposition and identified the spin contribution to the fractional current density.
We suggested that the understanding of the results comparatively for electrons, muons e taus may be thought in the realm of complexity, mean life and pseudo-particles concepts and we had shown that α(electron) > α(muon) > α(tau).
We established a connection between mean life of the particle and fractionality, showing that small mean life can leads to a fractional parameter more distant from α = 1 than for particles with more stability and consequently greater mean life than the other one.
The complexity of the interactions involved may hide the detailed knowledge on its dynamical aspects, since the nonlocal characteristics of the interactions. Therefore, a detailed description may be very difficult to be obtained or even be impossible. This can explain the lacking of a precise formulation of theories. The theory may has beyond the standard model, since we think that the interaction aspects due complexity has to be taken into account. Therefore we suggest that this can be done with an effective theories, capable to give some indicative evidences for the existence of complex interactions. Here we think that the fractional calculus may be a good candidate for this effective theories.
Cresus F. L. Godinho is acknowledged for the discussions at an early stage of this work. The authors wish to express their gratitude to FAPERJ-Rio de Janeiro and CNPq-Brazil for the partial financial support.