Magnetic contributions in Bekenstein type models

In this work we analyze the spatial and time variation of the fine structure constant (\alpha) upon the theoretical framework developed by Bekenstein [1]. We have computed the field \psi related to \alpha at first order of the weak field approximation and have also improved the estimation of the nuclear magnetic energy and, therefore, their contributions to the source term in the equation of motion of \psi. We obtained that the results are similar to the ones published in Ref. [2] which were computed using the zero order of the approximation, showing that one can neglect the first order contribution to the variation of the fine structure constant. By the comparison between our theoretical results and the observational data of the E\"otv\"os-type experiments or the time variation of \alpha over cosmological time-scale, we set constrains on the free parameter of the Bekenstein model, namely the Bekenstein length.

In this work we analyze the spatial and time variation of the fine structure constant (α) upon the theoretical framework developed by Bekenstein [1]. We have computed the field ψ related to α at first order of the weak field approximation and have also improved the estimation of the nuclear magnetic energy and, therefore, their contributions to the source term in the equation of motion of ψ. We obtained that the results are similar to the ones published in Ref. [2] which were computed using the zero order of the approximation, showing that one can neglect the first order contribution to the variation of the fine structure constant. By the comparison between our theoretical results and the observational data of the Eötvös-type experiments or the time variation of α over cosmological time-scale, we set constrains on the free parameter of the Bekenstein model, namely the Bekenstein length.

I. INTRODUCTION
Both, theories that have been developed to try to unify the fundamental interactions (see, for example [3][4][5][6][7][8]) as Dirac's large numbers hypothesis (LNH) [9] predict that the fundamental constants might vary in space time. Given these theoretical motivations, there have been many attempts to look for observational and/or experimental data to corroborate such variations (see Refs. [10][11][12][13] and references therein) but almost all of them are until now consistent with a null variation. On the other hand, the analysis of the data performed in Refs. [14][15][16][17][18] suggests that the fundamental constants might have a different value than the present ones. A space variation of fundamental constants would produce a violation of the Weak Equivalence Principle (WEP) [19] which can be verified thanks to the energy conservation [20]. The Eötvös-type experiments are the most sensitive methods to verify the WEP since they are capable to detect the difference between the acceleration of bodies with different composition or structure [21][22][23][24][25][26]. For details on the variation of the fundamental constants see [27].
Bekenstein proposed a theory to study the fine structure constant variability based on first principles [28] and later, he claimed that no violations of WEP are perceived whereas a possible time variation of α can not be discarded with the existed observable bounds [1]. Kraiselburd & Vucetich [2] modified Bekenstein's original theory by replacing the classical model of charged particles by a quantum model of many bodies at the non-relativistic limit, but with classical fields and neglecting the contributions of the self-generated fields. They found that even considering the cancellation proposed by Bekenstein, between the electric field and the mass source term, quantum density fluctuations produce fluctuating electric currents which generate magnetic fields. These fields, are responsible for the appearance of an anomalous term in the acceleration of a system of particles that produces a violation of the WEP in Eötvös-type experiments. Later, Barrow & Magueijo [29,30] questioned Kraiselburd & Vucetich's interpretation of approaching the coupling of field ψ in the expressions D = ǫE = e −2ψ E and H = µ −1 B = e −2ψ B with the weak field approximation (e −2ψ ≈ 1).
The purpose of this work is twofold: 1. to show that nonlinear corrections to the zero order in Bekenstein's equations are negligible in the weak field regime, typical for experimental setups; 2. to analyze the prediction for the time variation of α due to the magnetic contributions in the semiclassical model and compare it with modern observational data.
This work is organized as follows. In Section II we briefly review the formalism of Bekenstein model and Kraiselburd & Vucetich approach. In Section III we introduce the improvements to the semiclassical model in order to study the spatial dependence, as well as the time variation of the fine structure constant in an expanding Universe. The observational available data on the variation of the α is shown in Section IV. In Section V we present the obtained bounds on the model parameters through the comparison between the observational data and the theoretical predictions. Finally in Section VI we draw the conclusions.

II. A REVIEW ON BEKENSTEIN'S FORMALISM AND THE SEMICLASSICAL APPROXIMATION
In this section we summarize the main aspects of Bekenstein model [1,28] and its semiclassical version presented in [2].

A. Bekenstein model
Bekenstein modifies Maxwell's electromagnetic theory to introduce a scalar field ψ that couples to charges generating a variation in α [1,28]. The model holds under the following hypothesis 1. Maxwell's theory must be recovered when α is constant; 2. the dynamic field ψ is responsible for changes in α; 3. the dynamics of the electromagnetic and the ψ fields can be obtained from a variational principle; 4. the theory must be gauge and time-reversal invariant, and also preserve causality; 5. it is not possible to have a length smaller than the Planck length ℓ P 1 .
From these hypothesis the only possible action is given by where the expressions for the modified Maxwell and ψ field actions are where F µν is the usual electromagnetic tensor, f µν = e ψ F µν . g is the metric while and c are the Planck constant and the speed of light respectively. ℓ B is the Bekenstein's fundamental length scale, a constant length introduced by the model. The Lorentz factor is represented by γ, m is the particle mass, e and u µ are its electric charge and 4 th velocity respectively. Finally, G N is the gravitational constant, R the Ricci scalar, and x i (τ ) stands for the particle trajectory of the particle in presence of the fields A µ and ψ. Following the previous statements, an elementary electric charge and the fine structure constant α are locally defined as where e 0 , α 0 are the current values of the electron charge and the fine structure constant. The equations that describe the dynamics of the electromagnetic and ψ fields are In the previous equation , σ is the energy density of matter [28], and stands for the covariant flat d'Alambertian.
Bekenstein [28] has analyzed the spatial ∇ 2 ψ and the temporal 1 The cosmological equation of motion for ψ is related to the time variation of α. In an isotropic and homogeneous Universe, with a Robertson-Walker metric, the time variation of α (after calling ǫ = e ψ and 2|ǫ/ǫ| = |α/α 0 |) is given by where a represents the Universe scale factor. As one can see, the only source term is the one that comes from matter, since the electromagnetic radiation of the Universe does not contribute. Considering that magnetic term can be neglected such that the contribution of the Coulombic energy density is much higher in ordinary matter and ǫ ∂σ ∂ǫ ∼ 2m EM c 2 , where ρ m is the total rest-mass density of electromagnetic interacting matter (m EM ) and ζ C is the fractional contribution of the Coulomb energy to the rest mass. Mosquera et al. [34] have analyzed the solution of the above equation for two different regimes of the Universe: when matter and radiation dominate, and when matter and cosmological constant do, matching both solutions with the appropriate boundary conditions. They have performed a statistical analysis to test the solutions using observational data from the early Universe and other bounds from the late Universe arriving to the conclusion that Bekenstein's hypothesis ℓ B > ℓ P should be relaxed. In Ref. [28], Bekenstein states that the mass dependent term of Eq. (4b) is almost cancelled with the term of the electric field, therefore ∂σ ∂ψ + e −2ψ E 2 4π ≈ 0. He justifies this statement with the fact that field ψ can be written as a function of the coulombic potential φ C , and working on this relationship, he derives that there exists an asymptotically a null value of ψ [1] 2 . Hence, an electrically charged particle (or a set of point charges) satisfies ψ ≈ 0, meanwhile for a static system of magnetic dipoles, ∇ 2 ψ ≈ |ζ m |ρ m c 2 . The factor ζ m is the fractional contribution of the magnetic energy to the rest mass. For this latter system, Bekenstein claims that no WEP violation can be detected in laboratory experiments. 2 If this statement is valid the restrictions found in Ref. [34] cease to make sense since ∂ ∂t (a 3 1 ǫ ∂ǫ ∂t ) = 0 considering only the Coulomb contribution.

B. The semiclassical approach
In Bekenstein's theory particles are treated classically. This assumption is not accurate, especially in those cases where quantum effects become relevant, such as white dwarfs or condensed matter physics, or at high-energy scales (a small distance scales) since fermions have their own "natural length scale". For these reasons, Kraiselburd & Vucetich [2] have proposed a semiclassical model where particles are quantized and fields remain classic. Once matter is quantized, the main subject is to understand matter distributions movements inside a scalar field ψ, and the contributions of the electric and magnetic field of this interaction.
The full action and Lagrangian of a body (or an ensemble of point charges) have been analyzed in the non-relativistic limit for the charges, with the full expression for the electromagnetic field. If only the external ψ and gravitational fields are taken into account, meaning that the self-generated fields are neglected, the equation of motion for the scalar field ψ is being 4π c . Ψ represents the charged particle field, while the particle mass has been renormalized as m = m 0 + ∂m with m 0 independent of ψ. The expression ∂m can only be computed through the quantum electrodynamics renormalization since it arises from the quantum fluctuations of the electromagnetic field, ∂m ≈ 3 4π e 2ψ α 0 m 0 log Λ 2 m 2 0 , with Λ the ultraviolet cut-off frequency imposed by quantum electrodynamics. Under this scheme, it is not possible to determine if Bekenstein's cancellation statement (in this case, ∂(Ψ∂mΨ) ∂ψ + e −2ψ E 2 4π ≈ 0) is valid or not since it is necessary to quantify the fields [31].
Even taking Bekenstein's statement as valid, in a quantum model of matter, quantum fluctuations of protons and neutrons oscillating in anti-phase create a dipole moment which produces a variable current and magnetic field. Haugan and Will [32,33] have made estimations of these contributions to the magnetic energy from a minimal nuclear shell model roughly obtaining where R(A) = 1.2A 1/3 fm andÊ ∼ 25MeV are the nuclear radius and the giant dipole mean absorption energy respectively, and A is the atomic number. The integrated strength function σdE satisfies the Thomas-Reiche-Kuhn sum rule but for light nuclei, this approximation is no longer suitable. Consequently, in this article we use other estimates for nitrogen, hydrogen and beryllium (see Appendix A).
The magnetic energy density e m , which is concentrated near atomic nucleus, can be written in terms of the fractional contribution of the magnetic energy to the rest mass of nuclear specie is the local mass-weighted average of ζ m and n b (x) is the particle density of the nuclear specie b. Finally, ρ b (x) stands for the mass density of the nuclear specie b, and ρ(x) is the local mass density.
Ref. [2] is based specifically on the analysis of the magnetic contributions in the spatial part of a semiclassical Bekenstein model. Taking the spatial part of Eq. (7) and rewriting it using expression of Eq. (9) it becomes A solution for an arbitrary distribution of sources with spherical symmetry of the scalar field ψ under the weak field approximation (e −2ψ ∼ 1) can be found as, being R s the distribution radius and Then, using the Newtonian potential, φ N (r) = G N M/r, the field outside the distribution (r > x) can be written as In the last expression,ζ m is the mass-averaged value ofζ m . The Lagrangian of the theory, according to Ref. [2], is where M tot is the body mass, V CM is the velocity of the center of mass and R CM is the center of mass position.
Recalling the assumption of the cancellation between the electrostatic contribution and mass dependence on ψ, one can write the acceleration that suffers a body immersed in an external gravitational (g) and ψ fields as with ∇φ N (r) = g. The neglected terms in the last expression are of either tidal order, negligible in laboratory tests of WEP, or of higher order in ψ. The last term on the right-hand side of the above equation is an anomalous acceleration generated by the scalar field ψ. Consequently, the validity of the theory can then be tested with Eötvöstype experiments that measure the correlation between inertial mass and gravitational mass. From expressions of Eqs. (12) and (14) it is possible to compute the Eötvös parameter which associated with the differential acceleration of two bodies (A and B) of different composition where ζ S m , ζ A m and ζ B m are the magnetic energy fractions of the source and bodies A and B respectively. The bounds on ℓB ℓP 2 obtained in Ref. [2] using the data of the most accurate versions of the Eötvös experiment presented in Table I, are where the last equation correspond to the upper bound at 3σ level. From these results it is clear that even when the electric field does not contribute to the scalar field ψ, the quantum fluctuations contribute (through currents) in such a way that the upper limit found for the ℓ B /ℓ P parameter discards the theory.   In this section we present the computation of the field ψ by taking into account the following order in the series expansion in the equation of motion under the scheme proposed by Kraiselburd & Vucetich [2].
In order to carry out this calculation, we proceed to solve Eq. (10) with an iterative method inspired in the Picard method of solution for differential equations. Considering a weak field, e −2ψ ≈ 1−2ψ+O(ψ 2 ), we begin by introducing this approximation of the exponential in the equation of motion (see Eq. (10)). Then, we replace the scalar field ψ on its right-hand side (source term) with the solution given by Eq. (12), that is the zero order approximation ψ 0 (r), to find the next order solution ψ 1 (r), The solution is then where R is the radius and M is the mass, both of the source body. The previous integral can be computed as where, once again,ζ m stands for the mass average ofζ m . Therefore, the asymptotic behaviour of the field turns out to be For Eötvös-type experiments in which generally the source is the Earth or Sun, the factor 6ζ m φN (R) c 2 of the second term inside the bracket is of the order ∼ 2.37 × 10 −10 and ∼ 5.27 × 10 −14 respectively. Therefore, the contribution of this term would be negligible.

A. Magnetic contributions to the acceleration
In this section we compute and discuss the acceleration changes due to the addition of the new computed field ψ 1 . According to Eq. (20) Consequently, the acceleration that a body j in presence of a with the Newtonian gravitational and a scalar fields becomes (see Eq. (14)) One can observe again an anomaly, which is composed of two terms. The first of them agrees with the one described in the previous section (see Eq. (15)), while the second one is the result of the next order of the field ψ expansion described in this work.

B. Magnetic contributions to the time variation of α in an expanding Universe
In this section we analyze the time variation of the fine structure constant using the development carried out in Ref. [34] for an expanding Universe. This calculation is performed by taking into account the magnetic contribution of the baryon matter.
From Eq. (5), considering valid Bekenstein's cancellation in the semiclassical approach, we obtain where we have used ρ m = Ω b ρC a 3 (t) , ρ C = 3/(8πG N )H 2 0 and ℓ P = (G N /c 3 ) 1/2 , H 0 is the Hubble constant, Ω b is the Universe baryon matter density and t c is an integration constant that can be null. One can assume that the field variation is only generated by the magnetic energy of the cosmological baryon matter. If one consider that the primordial Universe is composed only of Hydrogen and Helium-4, the fractional contribution of the magnetic energy to the rest baryon mass is given by where E H m and E The scale factor has different expressions depending on the stage of the Universe evolution. In a flat Friedmann-Robertson-Walker metric (FRW) and assuming the scalar field contribution null, the equation that describes the Universe scale factor time evolution can be written as 1 a ∂a ∂t where the initial conditions are a(0) = 0 and a(t 0 ) = 1, being t 0 the age of the Universe, Ω m is the matter contribution to the density, Ω r is the radiation density and Ω Λ stands for the contribution of the cosmological constant to the density. In Ref. [34], the authors have performed a piece-wise approximate solution by solving the FRW equation for two different cases: a) radiation and matter (early Universe) that can be applied to nucleosynthesis and recombination of primordial hydrogen; and b) matter and cosmological constant (middle-age and present Universe) which is used for quasar absorption systems, geophysical data, and atomic clocks. Then, they have matched both solutions in the time that the regime changes (t 1 ) and considered that the scale factor is a smooth and continuous function of time. They have obtained • early Universe, dominated by radiation and matter (t < t 1 ) (26b) • middle-age Universe, dominated by matter and cosmological constant (t > t 1 ) Thereafter, we obtain the time variation of α (being ln ǫ(t) ǫ(t0) ≃ 1 2 ∆α α0 ) for each stage of the Universe by replacing the scale factor in Eq. (23); 1. early Universe dominated by radiation and matter (t < t 1 ) 2. middle-age Universe dominated by matter and cosmological constant (t > t 1 ) 3. present Universe dominated by matter and cosmological constant (t = t 0 )

IV. OBSERVATIONAL DATA ON THE TIME VARIATION OF THE FINE STRUCTURE CONSTANT
In this section we present the observational data on the time variation of the fine structure constant. To study this model of the time variation of α, we take into account not only the different stages in the Universe's evolution but also use three different sets of cosmological parameters [11,35]. Since the baryon density (Ω b h 2 ) can be determined by two independent analysis: i) by the comparison between theoretical calculation of primordial abundances and the observable data; or ii) by the study of the CMB data (Cosmic Microwave Background) [36][37][38][39], we used the results of the analysis of the CMB data. That is Ω b h 2 = 0.02212±0.00028, Ω b h 2 = 0.0231±0.0013 [35], Ω b h 2 = 0.0218±0.0004 [11].

A. BBN constraints
During the first three minutes of the Universe the light elements, such as deuterium, 4 He, 3 He, 7 Li, were produced. In order to include the fine structure constant time variation upon cosmological time-scale, we have modified the numerical code developed by Kawano et al. [40,41] (for the modifications to the code see [34,[42][43][44][45] and references therein). As said before, the baryon density was fixed at three different values.
The possible variation of α has been calculated through the comparison of the observable data and the theoretical results using a χ 2 -test. For the deuterium, we have considered the observable data reported in Refs. [46][47][48][49][50][51][52][53][54]. For 4 He we used the data of Refs. [55][56][57][58][59][60][61][62][63], and the results of Refs. [64][65][66][67][68][69][70] for 7 Li. In order to check the consistency of the data we have followed the analysis of Patrignani et al. [71] and increased the observational errors by a factor Θ D = 1.28, Θ4 He = 2.79 and Θ7 Li = 2.04. In Table II we present the results. The inclusion of the lithium data gives a rather poor fit for all the models, meaning for the three values of Ω b h 2 . If the middle value of the baryon density is used (Model II), the fit is excellent and the variation of α is consistent with a null-variation at 1σ, if one excludes the 7 Li data in the analysis. The fits obtained with the largest (Model III) or the smallest (Model I) value for Ω B h 2 considered in this work, are good fits, however, ∆α α is consistent with zero at 2σ if the lithium data is removed in the statistical test.

Model
Data

B. CMB constraints
The recombination process, when the first hydrogen atoms formed and the Universe became transparent to photons, occurred approximately 300000 years after the Big Bang. The photons released during this process are the Cosmic Microwave Background radiation (CMB). This radiation corresponds to a black body radiation at temperature T 0 = 2.725 0 K with small anisotropies of the order of 10 −6 0 K.
The CMB formation is totally mediated by electromagnetic processes. Thus, a variation in the coupling constant of this interaction (that is α) involves modifications in the interactions between photons and electrons as well as in the recombination scenario and consequently in the spectrum of CMB fluctuations [11,35,72]. In Table III we show the bounds obtained in Ref. [11,35] using WMAP7, WMAP9 and Planck satellites data.

C. Quasars absorption spectra
Spectroscopic observations of extragalactic objects such as quasars can be used to study the time variation of the fine structure constant. High redshift quasar spectra present absorption resonant lines of alkaline ions whose splits are proportional to α 2 . Therefore, studying the separation of the doublets, one can obtained bounds for the variation of α. This analysis (called AD) have been applied, in the literature, to different doublet absorption lines systems at different redshifts [10,73,74]. The results are consistent with a null variation of the constant.
The Many Multiplet Method (MMM) which compares transitions of different species with widely differing atomic masses together with different transitions of the same species, improves the sensitivity, but its systematic errors are generating great controversies 3 . In Tables IV, V  OH conjugated lines 025 0.5 ± 1.26 [80] Molecular and radio lines 0.24 −0.10 ± 0.22 [81] Molecular and radio lines 0.69 −0.08 ± 0.27 [81] Molecular and radio lines 0.765 < 0.67 [82] OH conjugated and radio lines 0.   Table VI: Bounds on the time variation of the fine structure constant from different quasars [78].

D. Oklo constraint
Oklo is a natural uranium-fusion nuclear reactor that operated 1.8 × 10 9 years ago in Gabon, Africa. The operating conditions of the reactor can be recovered from the analysis of nuclear and geochemical data to calculate the thermal neutron capture cross sections of several nuclear species including 149 Sm, 151 Eu and 155 Gd. From these cross sections it is possible to estimate the value of the resonance energy of the fundamental level at the time of the reaction. A variation of this energy (from the time of the reaction to the present) would reflect a variation in time of α [12,[88][89][90]. The time variation of the fine structure constant obtained with this natural reactor is [90] ∆α α 0 = (45 ± 15) × 10 −9 .
E. Long-lived β decayers constraints Thanks to laboratory measurements and/or by comparison with the age of meteorites, it is possible to determine the half-life of long-lived β decayers. Sisterna & Vucetich [91] have analyzed the dependence between the half-life of these decayers and the fundamental constants α, Λ QCD and G F so, any shift in the half life may be generated by a variation in these constants at the age of the meteorites and their value now. According to these authors, there is a linear relationship between the shift in the half-life of the β decayer of 187 Re from the solar system formation up today and the variation of α. The transition analyzed was Re →Os and it is assumed that this event has occurred 4.6 × 10 9 years ago. According to Ref. [91], the time variation of the fine structure constant is F. Atomic clocks constraints The time variation of the fine structure constant can be measured by comparing frequencies of atomic clocks with different atomic numbers during time intervals that range from one hundred days to two years. This is possible thanks to the development of very stable frequency oscillators based on hyperfine transitions. These hyperfine levels are determined by the interaction of the nuclear magnetic moment with the magnetic moment of a valence electron.
It is possible to restrict the variation of α by comparing the frequencies of two different atoms because the relativistic contribution to the splitting of the hyperfine levels ∆ grows with the atomic number as ∆ ∝ (Zα) 2 . Table VII shows the bounds for the time variation of the fine structure constant presented in the literature.

V. RESULTS
In this section we analyze and discuss the predictions that emerge from the improved Bekenstein model in the semiclassical regime and compare it with experimental and observational data in order to verify the validity of the theory. We analyze separately the spatial and temporal aspects of the theory.
A. WEP experiments for testing the spatial variation of α As we have explained in Section II B, the Eötvös parameter can be obtained from the difference of the acceleration of two different bodies of different composition. In order to determine and/or limit the validity of the theory, we use the data from Eötös-type experiments presented in Table I. Using Eq. (22), the theoretical prediction of the Eötvös parameter for the semiclassical Bekenstein model is where A and B are constant that depend on the composition of the bodies and do not depend on the theoretical parameter of the model. Note that the Eq. (15) can be written as η = A ℓB ℓP 2 . We perform a χ 2 -test to set constrains on the free parameter of the theory, namely ℓ B /ℓ P , by the comparison of our theoretical results with the observational data. In Table VIII we show our results and we also compared, in the same table, the results obtained using the approximation of Ref. [2] or the Eq. (15). The differences between this new limit and the one presented in Eq. (16) are due to the improvement in the estimation of the magnetic energy (see Appendix A).
As one can noticed, the results for both cases are the same, since the contribution from the "next order" (first order) in the series expansion of the field ψ is very small and can be neglected. Consequently, it is valid the use of the zero order approximation with the change on the estimation of the magnetic energy.
Since at 3σ-level, the upper bound of the Bekenstein length scale, for both cases, is ℓ B < 0.05ℓ P , it seems naturally to discard the theory.
In December 2017, the MICROSCOPE collaboration published an estimation of η ∼ 10 −15 consistent with zero [98]. If this result is used, the ℓB ℓP bounds would be even smaller.

B. Constrains from the time variation of α
In the Section III B we have computed the theoretical predictions for the variation of the fine structure constant in different stages of the Universe. Eqs. (28a), (29a) and (30) can be written as where the constantsȂ,B andC depend only on the cosmological parameters. The constraints on the free parameters of the theory, that is ℓB , are obtained by a least squares linear regression using the observational data described in Section IV. We have performed a discrimination into different groups of the available data, according to when the physical phenomena took place, that is: i) early Universe (CMB and BBN bounds); ii) middle-age Universe (Quasars, Long-live β decayers and Oklo reactor bounds); and iii) present Universe (bounds from atomic clocks). As we have mentioned before, we use three different sets of cosmological parameters.
In Table IX we present our results for the constraints on the free parameter for the different models (or cosmological parameters) described in Section IV A. The value for the variation of the fine structure constant during BBN used in the lineal regression correspond to the obtained in Section IV A when the lithium data was discarded in the statistical test. It can be noticed that in the three models, in the cases where the data corresponding to the early Universe are subtracted, the value of H 0 t c ℓB ℓP 2 and its uncertainty increase considerably. In addition, it can be seen that at one standard deviation, Bekenstein length parameter ℓ B becomes imaginary for all the cases with the exception of the analysis performed when the middle-age Universe data is excluded.
An important difference arises from the results for the different groups of cosmological parameters. The estimates for H 0 t c ℓB ℓP 2 are negative when we use the results of the cosmological parameters of Models I and II, whereas the estimates using the parameters of Model III are positive (except the case where BBN and CMB data are excluded).
Since several authors consider the integration constant t c null, we have also performed an estimation of parameter    Tables II and III).

VI. SUMMARY AND CONCLUSIONS
In this work we have analyzed the magnetic contributions in the semiclassical Bekenstein model for the variation of the fine structure constant and show that the magnetic field produces an anomalous term in the acceleration. Based on previous works of Refs. [2,31], we have solved the equation of motion the scalar field ψ to first order (in the mentioned publications this calculation was made at zero order). We have also improved the estimation of nuclear magnetic energy, according to Refs. [100][101][102]. To set constrains on the Bekenstein's theory parameter, ℓ B , we have analyzed not only the spatial contribution of the equation of motion of the field related to the fine structure constant, but also its time variation. To perform the last computation we have considered the expansion rate of the Universe in different evolutionary stages.
We have examined possible violations of the Weak Equivalence Principle due to the incorporation of the scalar field. In order to constrain the Bekenstein length, we have compared our theoretical results with the experimental data from Eötvös-type experiments. We conclude that the first term in the series of the weak field approximation is the one that contributes the most and the following ones can be neglected. Our results are of the same order as those obtained in Ref. [2], and the small differences between them are due only to the change in the estimation of the magnetic energy. Consequently, it can be deduced that the semiclassical model for the spatial variation would be discarded.
We have also analyzed the time variation of the fine structure constant from the semiclassical model of Bekenstein in an expanding Universe. To determine the free parameters of the theory, ℓB ℓP 2 and ℓB ℓP 2 H 0 t c , we have made a comparison between the observational data for different times with our theoretical predictions. At 3σ level, we found for the different cosmological parameters used, that is the first constrain correspond to the Model I, the second to the Model II and the last one to Model III. Note that, for the first expression, Bekenstein length turns out to be a non-real number. These results rule out the theory. The removal or not of the early Universe data (that is BBN and CMB constrains on the variation of the fine structure constant) does not affect the results. The same constrains are obtained if the data removed are those from the present Universe (that is bound obtained from atomic clocks). However, in the case where the data of the middle-age Universe are not taken into account in the fit, the 3σ level upper bound does not allow to discard the theory since ℓ B ∼ ℓ P .
The 3σ level upper bound results for the parameter ℓB ℓP 2 H 0 t c are (in the same order as before) We obtained the same results when we excluded data from the middle-aged Universe and/or the present Universe. On the contrary, when the early Universe data are not taken into account the bounds are much less restrictive, that is Comparing our results with those obtained in Ref. [34] where possible time variations of α due to the Coulomb contributions are analyzed, it is observed that for the parameter ℓB their bounds are more tighten (around an order of magnitude or two smaller). Finally, the results for the Bekenstein length when t c = 0 indicate that only at one standard deviation the theory can be discarded in all cases for the three models except when the data from the middle-aged Universe are not considered in the statistical analysis. Contrasting these results with those obtained from the analyses of the planet's thermal flow (data from the late Universe) [31,99], their 3-σ upper bounds are of an order of magnitude more strict than ours using data of the whole evolution of the Universe.
It is important to note that almost all the mean values obtained for the parameters ℓB ℓP 2 and ℓB ℓP 2 H 0 t c from the analysis of the time variation of α are negative, producing an incongruity in the semiclassical model since Bekenstein length must be real, positive and greater than Planck length. This fact gives us a hint of the serious problems that the theory faces although the given bulging uncertainties.
In conclusion, it has been shown that even taking as valid Bekenstein's statement (about the cancellation in the source term between the contributions of electric field and the mass) in a semiclassical model, the magnetic contributions produce observable anomalous accelerations and time variations of the fine structure constant which are not compatible with the observable data.