Aharonov-Bohm Effect in Generalized Electrodynamics

The Aharonov-Bohm (AB) effect is considered in the context of Generalized Electrodynamics (GE) by Podolsky and Bopp. GE is the only extension to Maxwell electrodynamics that is locally {\normalsize{}U(1)}-gauge invariant, admits linear field equations and contains higher-order derivatives of the vector potential. GE admits both massless and massive modes for the photon. We recover the ordinary quantum phase shift of the AB effect, derived in the context of Maxwell electrodynamics, for the massless mode of the photon in GE. The massive mode induces a correction factor to the AB phase shift depending on the photon mass. We study both the magnetic AB effect and its electric counterpart. In principle, accurate experimental observations of AB the phase shift could be used to constrain GE photon mass.


Introduction
Maxwell equations of electromagnetism make room for the description of the electric field E and the magnetic field B in terms of the electric potential φ and the vector potential A [1].This is the case due to electrodynamics' invariance under the U(1) group of transformations, or phase transformations.This gauge freedom reduces the six degrees of freedom in the vectors {E, B} to the four degrees of freedom in the pair {φ, A} [2].It is thus said that electromagnetic interactions is the gauge theory for the U(1) group [3].
The equivalent description of electrodynamics in terms of the fields {E, B} or in terms of the gauge potentials {φ, A} raises the question of what option is more fundamental.To put it another way: Would the potential φ and A be a simple mathematical artifact to facilitate the computations in some applications of electromagnetism but with no concrete physical reality?Previous to the advent of quantum mechanics and gauge theory, the answer would most certainly be that E and B are the fundamental entities of the electromagnetic interaction [4].However, the theme proved itself much more subtle since QED also presents gauge invariance in a complemented way which includes vector states in the physical Hilbert space [5].
In fact, in 1959 Aharonov and Bohm proposed an experimental setup that could assess the effect of the potentials {φ, A} over charged particles in regions where the fields {E, B} would be zero [6].The idea was that the wavefunction associated to the charged particles would undergo a phase shift due to interaction with the potentials {φ, A}.This phase shift, if measured, would attribute fundamental physical role to the gauge potential {φ, A}. 1 This interpretation fits well within the gauge theory framework, which theoretically describes interaction through the minimal coupling prescription introduced by Weyl [8], used by Yang and Mills [9], and justified deductively by Utiyama [10].The minimal coupling prescription recommends the mapping of ordinary differential operators such as ∂ µ onto the covariant derivative ∇ µ = ∂ µ + A µ [11,12].Herein, notice the appearance of the four-potential A µ = {φ/c, A} (not of the fields {E, B} embedded in the fieldstrength F µν ).
As early as 1960, Chambers already claimed to have measured Aharonov-Bohm (AB) effect [13].This sparked debate in the community, which resisted to fully embrace the result if any due to experimental shortcomings.Arguably, Chambers' experiment presented accuracy issues [14].However, these challenges were overcame over the years and, by the time Tonomura and others published their findings in 1983 [15,16], the AB effect was already accepted beyond any criticism; moreover, it has since been measured by different experimental techniques, e.g.[17,18,19,20].The dual of AB effect was proposed by Aharonov and Casher in 1984 [21].Ref. [22] is an excellent introductory review of Aharonov-Bohm and Aharonov-Casher effects.
The original AB effect was proposed in the context of non-relativistic quantum mechanics.However, it has being generalized to the relativistic realm of Dirac equation [23].The AB effect was also made impact on the study of field theory on curved spacetime; in particular, people worked out the AB effect due to the action of cosmic strings and other topological defects, cf.e.g.Ref. [24].Recent literature include other applications too, e.g.Refs.[25].
One key ingredient for the original AB effect was the U(1) gauge symmetry exhibited by the electromagnetic interaction.One might think that the only possible consistent theoretical description of this interaction-one that is both relativistic compatible and U(1) gauge invariantis Maxwell electrodynamics.This is indeed the case if one restricts oneself to field equations containing up to second-order derivatives of the gauge potentials {φ, A} (first-order derivatives of the field-strength).If the latter possibility is relaxed then a new avenue opens up to describe electromagnetism.This was the avenue followed by Podolsky, starting from 1942 [27,28,29]see also the contribution by Bopp [30].Podolsky introduced a generalization to Maxwell field equations that includes fourth-order derivatives of the gauge potentials while, at the same time, keeping the linearity of the related differential equations.In terms in terms of B and E, these generalized field equation read: where with c = 1/ √ µ 0 ϵ 0 .Notice that Maxwell equations are recovered from (1) in the limiting case where Podolsky parameter a is negligibly small.Podoslsky's Generalized Electrodynamics (GE) has an additional atractive feature: It is U(1) gauge invariant.Indeed, the paper [31] demonstrated that Podolsky electrodynamics is the only U(1) gauge theory leading to field equation in terms of {φ, A} that linear and of fourth-order.There sure exist other alternative theories of electromagnetism besides GE, but they are non-linear [32,33]-e.g.Euler-Heisenberg electrodynamics [34] and Born-Infeld electromagnetism [35,36]-, or violate U(1) symmetry-e.g. de Broglie-Proca model [37,38,39,40,41,42].Podoslky original motivations for GE included the attempt to eliminate divergences of the static potential of the point charge2 and to help and cure divergences appearing in the attempt to build the quantum version of electrodynamics.Latter on Podolsky electrodynamics was abandoned in the face of the usual techniques of QED.However, the interest in GE was renewed partially because of its gauge invariance (which is broken in ordinary QED's upon imposition of the regularizing cutoff techniques) but also because of its feature of accommodating a massive mode for the photon.In fact, Podolsky parameter a = ℏ (cm γ ) −1 brings a massive mode to the photon alongside the regular massless mode typical of the regular Maxwellian description-as a quick check of Eq. ( 1) reveal.Constraining this mass is the subject of a number of papers, e.g.[45,46,47].There are applications applenty of GE in QED [51,52,53,54], classical topics of eletromagnetism [55,56,57,58,59,60], and even in gravitation-in conection to BH physics [61,62]-and cosmology [63].
The U(1) gauge invariant character of GE enables us to study AB effect in this context.This is the main goal of this paper.Besides the theoretical interest of the study of Ab effect in GE, there is in principle experimental application.The AB effect could serve as a tool to probe GE in the low-energy regime.Specifically, AB effect could be a means to constrain a and add a lower bound to Podolsky mass; this is a possibility we would like to assess here.
The paper rest of the paper is organized as follows.Section 2 develops the magnetic AB effect in the context of GE.The electric version of AB effect is covered in section 3. Finally, section 4 contains our final comments, discussion and perspectives.The Appendix contains an deeper discussion of the role of U(1) gauge invariance for AB effect in GE.

Magnetic AB effect
We start off by calculating the magnetic field produced by a long solenoid in the context of Generalized Electrodynamics.A beam of quantum particles will be later considered travelling around the solenoid and will be affected by Podolsky's version of the gauge potential.The phase shift induced in the two branches of the splitted beam gives the AB affect in GE.

Magnetic field of a long solenoid in GE
According to Ref. [29], the GE version for Ampère-Maxwell equation is where the coupling constant a = 1/m γ (in natural units) has dimensions of length and is related to the photon mass m γ in Podolsky electrodynamics.In the stationary case, the fields E = E(r) and B = B(r) do not depend on the time coordinate t, and Eq. ( 3) reduces to: where J is the current density.This is equivalent to: under the definition which adds the Podolsky contribution (−a 2 ∇ 2 B) to the regular Maxwellian term.The later is recovered in the limit a → 0. Eq. ( 5) has exactly the same form of the traditional Ampère law (under the mapping B → B eff ).Therefore, its integral form will also be the familiar one: In Eq. (7) we see the definition of the current I enc puncturing through the surface S enclosed by the circuit C: I enc = ´S J • ndS.The solution to this equation for various currents configurations is the subject of study of electromagnetism courses.For instance, a long solenoid of radius R, length l ≫ R and number of loops N carrying a steady current I produces an internal uniform magnetic field equal to µ 0 nIẑ in the direction ẑ along its axis; n = N/l is the density of turns [2].Outside the solenoid the field is zero.This conclusion follows directly from Eq. ( 7), i.e.: Notice that the effective field B eff inside the solenoid in constant-it assumes the same value irrespective of the distance r from the solenoid's axis.The actual field B is calculated by substituting (8) into the definition (6) and, from this perspective, the effective magnectic field B eff works as the source of B.
In principle, the magnetic field would be a vector field of the type B (r, ϕ, z) = B r (r, ϕ, z) ŝ + B ϕ (r, ϕ, z) φ + B z (r, ϕ, z) ẑ in cylindrical coordinates.However, from symmetry arguments and the right-hand rule: This greatly simplifies the form of the Laplacian of B in cylindrical coordinates.In fact, by making use of the dimensionless coordinate the differential equation ( 6) takes on the from: where B eff = µ 0 nI for ζ < R/a and B eff = 0 for ζ > R/a, in accordance with Eq. ( 8).Outside the solenoid, the right-hand side of Eq. ( 11) is null, in which case this differential equation is recognized as the (homogeneous) modified Bessel equation of order zero-see e.g.Ref. [26], Eq. 9.6.1 with ν = 0. Its solution is in terms of the modified Bessel functions I ν (ζ) and K ν (ζ): This should be valid everywhere provided that r > R, including the limit of large r, i.e. large ζ.
In this regime, we see that [26]: Therefore, we must discard the term scaling with I 0 (z) if we want to avoid that the field B z diverges far from the solenoid, i.e. we are forced to choose: c 1 = 0.This is the first boundary condition that we impose to our physical configuration, in the face of which, Eq. ( 12) reduces to: The constant c 2 is still to be determined.This can be done by imposing continuity of B z at the solenoid's surface, where r = R, i.e. ζ = R/a.For this goal, we need to match (14) with its counterpart B z (ζ) valid inside the solenoid.So we naturally turn to the task of solving the nonhomogeneous Eq. (11).Since this is a second-order differential equation, its general solution is the sum  is a particular solution to the differential equation (11).The traditional procedure is to make an educated guess for the solution by choosing a form very similar to the non-homogeneity factor.In effect, we take B part z = B eff .Therefore, which should be valid for r < R, i.e. aζ < R, including at the axis of the solenoid where r = ζ = 0.At this point, K 0 (ζ) diverges.In fact [26], (16) where γ ≃ 0.57722 is the Euler-Mascheroni constant.In order to avoid unphysical B z 's divergences we must take: c 4 = 0.This imposition leads to: The last step in determining generalized electrodynamics magnetic field B is to glue the solutions ( 14) and ( 17) at the solenoid's surface.By demanding continuity of B z (ζ) at ζ = R/a, one gets: c 2 K 0 (R/a) = c 3 I 0 (R/a) + B eff .The imposition of continuity for the derivative of the field, Combining these equation enables us to determine both c 2 and c 3 .The calculation is facilitated by utilizing the Wronskian of I 0 and K 0 [26], namely The result is the final form of the GE magnetic field for a long, thin solenoid: where we have used ( 8) and (10).The general solution (18) can be specified in terms of elementary functions by recalling that GE coupling constant a should be small inasmuch the Podolsky contribution is vastly subdominant with respect to Maxwell's electrodynamics.Therefore, R ≫ a and we may use the asymptotic forms for large values of the argument [26] to write: In the first line, r > R ≫ a; consequently, we were allowed to approximate K 0 (r/a) by its functional form in Eq. ( 13).Also, I 0 (r/a) can be approximated by Eq. ( 13) as long as r ≫ a.This should be the case for all points except those very close to the solenoid's axis, where lim r→0 I 0 (r/a) = 1 and B z ≃ µ 0 nI.Hence, This form of B z (s) makes it explicit the continuity at r = R.In fact, both lines of ( 20) give (µ 0 nI) /2 at the solenoid's surface.This is half the value predicted by Maxwell electrodynamics: it is simply an artifact of the approximations used from (18) to (20).The approximations select the medium point between the discontinuous Maxwellian magnetic field, whose value (µ 0 nI) within the solenoid drops down to zero precisely at r = R and remains null outside the solenoid.The complete non-approximate solution (18) exhibits no such a feature, as shown in the plots of Fig. 1.B z (r) decreases smoothly as r increases; it begins with the value (µ 0 nI) at the solenoid's axis (r = 0), transits continuously through the solenoids surface and approaches zero as r → ∞.The precise value of B z at the solenoid's wall depends on the value of GE coupling constant: the smaller the a the closer the behaviour of B z to Maxwell's predictions.

Vector potential for a long solenoid in GE
GE is U (1) gauge-invariant [31]. 4Consequently, a vector potential A respecting the usual relation to the magnetic field may be introduced: Symmetry arguments stated in Section 2.1 led us to conclude that B = B z (r) ẑ -cf.Eq. ( 9).From this restriction and the right-hand rule, it follows: (A r = A z = 0).The curl of A in cylindrical coordinates [2] then reads: up to an integration constant (recall that A ϕ is a function of r only); this constant will be added later on.B z (r) is given by Eq. ( 18) and splits naturally in two different functions according to the interval of values assumed by r.
Outside the solenoid, the integral in (23) reads: The modified Bessel functions recurrence relations were used in the last step.In particular, the second equation in paragraph 9.6.26 of Ref. [26] leads to d dz [zK 1 (z)] = −zK 0 (z).Inserting ( 24) into (23): where is (aA out ) the integration constant omitted in (23).As a remark, notice the following.The asymptotic form of K 1 (z) for large z is obtained from Eq. 9.7.2 in Ref. [26]: so that lim r→∞ A ϕ (r) = 0.This shows that ( 25) is consistent with what is expected from ordinary Maxwellian electrodynamics [2].
Inside the solenoid, ˆrB z (r) dr = (µ 0 nI) ˆrdr − (R/a) K 1 (R/a) ˆrI 0 (r/a) dr Here, we have utilized Eq. ( 18) and the fact that d dz [zI 1 (z)] = zI 0 (z).This last statement is a consequence of the properties listed in the paragraph 9.6.26 of Ref. [26].Due to (26), Eq. ( 23) gives: The integration constant (aA in ) is determined by demanding consistency with the Maxwellian case -denoted by a superscript (M), [See e.g.Ref. [2], Eq. (5.70).]In effect, Eq. 9.6.10 in Ref. [26] gives the asymptotic form of I 1 (z) for small z: In order to avoid this divergence in (27) and make it completely consistent with (28), we must impose Furthermore, we demand continuity of A ϕ at the surface of the solenoid.Equating ( 25) and ( 27) at r = R, results: Figure 2: Behavior of the vector potential produced by a long-thin solenoid in the context of Generalized Electrodynamics.The horizontal axis is the distance from the solenoid's axis rnormalized by the solenoid's radius R. The vertical axis shows the vector potential A ϕ (r) weighted by its value at the solenoid's surface (µ 0 nI) (R/2).The definition A = a/R is used here.
Eqs. ( 29) and (30) put us in position to write down ( 25) and ( 27) as the complete solution for A ϕ in the context of GE: The first term in line one is precisely the vector potential outside the solenoid in Maxwell's case [2]: Just like in the magnetic field case, one can take into account that R ≫ a and approximate I 1 (R/a) and K 1 (R/a) in terms of elementary functions.Eq. ( 31) then reads: Wherever r is not too close to zero (axis of the solenoid), it is reasonable to take one more approximation, namely r ≫ a.In this case, This equation makes it clear that both the vector potential A ϕ (s) and its derivative is continuous at the solenoid's surface-see also Fig. 2 produced with Eq. (31).

Magnetic Aharonov-Bohm phase shift in GE
Consider the experimental configuration in Fig. 3(a).Therein, an electron beam splits off on the outside vicinity of a long thin solenoid; the two resulting branches of the original beam wrap around the solenoid in a ring shape path whose plane is perpendicular to the solenoid's axis.The two halves of the beam reunite in a single beam before hitting the detector.The detector measures the magnetic Aharonov-Bohm phase shift ∆g.The recipe for calculating AB phase g is provided by the path integral formulation in Chapter 2 of Ref. [1].Therein, we see that the interaction of the particle with the electromagnetic field is accounted for through the action integral (Einstein's sum convention admited).A α is the four-vector potential. 5We also learn from Ref. [1] that the action S I contributes as an additional phase factor Φ I to the wavefunction Ψ. Eq. ( 35) follows from the Lagrangian formalism for particles together with the equation for the Lorentz force of electromagnetism.Now, the Lorentz force in not derived from Maxwell equation of electrodynamics.It is an additional piece of the theory.In GE, Maxwell equations are generalized to contain higher-order derivatives of the field strengths.However, the Lorentz force equation is unchanged in GE and so is the Lagrangian formalism involving the action integral.Therefore, Eqs. ( 35) and ( 36) should apply to GE's case.In fact, we have g = S I /ℏ, i.e.
where Γ is the integration path.In Aharonov-Bohm experimental sets, we measure the phase diference ∆g in closed paths, such as that represented in Fig. 3.In effect, we are interested in where is the Γ 2 (Γ 1 ) is the branch on the right (left) of the solenoid with respect to the direction of the beam in Fig. 3(a).The negative sign on the right hand side of Eq. ( 37) suggests that we integrate backwards along Γ 1 (whilst we integrate forward along Γ 2 ).This results in the loop integral: This is the expression we are seeking for calculating the AB effect in the context of GE.Therefore: where the three-vector A j is A = A ϕ (r) φ due to (22) and A 0 = −A 0 = −φ/c = 0 because there is no electric potential in the pure magnetic AB effect.Then, with dl = dr r + rdϕ φ + dz ẑ (41 is the line element in cylindrical coordinates.Let us use the first line of ( 33) into (40): Consider once again the setup in Fig. 3(a).One can orient the y-axis in the direction of the outgoing beam; so that the x-axis points outward the page.We call right (left) branch the part of the beam circling the solenoid in the positive (negative) region of the x-axis.Then, the angle goes from −π/2 to π/2 in the right branch of the beam and from 3π/2 to π/2 in the negative branch in the clockwise direction.Then, ˛dϕ = ˆπ/2 as it should be in any case.Hence, e −(r−R)/a r (2π) , which we will write in terms of the absolute value of the magnetic field inside the solenoid in Maxwell electrodynamics: and the solenoid's cross section S = πR 2 ; ( i.e.
is the flux of the magnetic field through the solenoid as predicted by Maxwell electrodynamics.Thus, The term in front of the square brackets is precisely the phase shift expected in Maxwellian electrodynamics-see Eq. ( 46): Therefore, ∆g ∆g with the phase shift displacement due to GE being: Eq. ( 45)-or Eq. ( 47)-give the phase shift ∆g for the magnetic Aharonov-Bohm effect in the context of Podolsky electrodynamics.The radial dependence of ∆g is a novelty of the AB effect in GE: in Maxwell electrodynamics is a constant, cf.Eq. (46).Inasmuch the GE typical effectiveness range a is a small constant, a ≪ R, the phase shift deviation δg is a tiny amount.Fig. 4 shows the plot of ∆g/∆g (M ) as a function of the radial distance s.
The perturbation δg (r) due to GE is exponentially suppressed with the increasing distance r > R from the exterior of the solenoid.The maximum value for the phase shift displacement is δg max (R) = a/R and occurs at r = R, the outside surface of solenoid.Therefore, the best case scenario for detection of the Aharonov-Bohm effect in Generalized Electrodynamics is to produce a beam splitting in which the right and left branches circle around the solenoid almost touching its exterior surface.In this case, Actually, one should achieve δr ∼ a to access the distance scales where GE is effective.Whether this is experimentally possible is still to be determined.Regardless, let us write  and write down δg for a beam skimming through the solenoid's external surface: If we keep up to first order terms in ϵ, we have This is the quantity that should be aimed at experimentally.

Electric AB effect
The electric version of AB effect is obtained in an experimental configuration schematically represented in Fig. 5.The problem is the dual of the magnetic AB effect-see Fig. 3. Now, the electric beam splits in two hemispherical branches that pass through two long tubes of charges before recombining and finally hitting the detector.In Maxwell electrodynamics there is no electric field inside the cylinders of charge, so that no effect would be expected upon the electrons in the beam branches; however, a phase shift is detected and explained on the basis of the influence of the non-null electric potential φ within the tubes-cf.the first term in Eq. ( 39) and A 0 = −φ/c .There will be an analog of the electric AB effect in the context of GE.In order to start modelling the problem, it is necessary to compute the electric field E predicted by GE inside a long tube of static charges.

Electric field of a long charged tube in GE
We begin by repeating the strategy of Section 2.1.Take the Gauss-Podolsky law, i.e. the first equation in (1): For an static uniforme distribution of charges on the surface of a thin tube of radius R, one does not expect E to depend on time t.Therefore, Eq. ( 50) reduces to: since the Laplacian operator commutes with the divergent of E. Eq. ( 51) which motivates the definition of an effective electric field: that respects Gauss's law in its traditional form: The integral form of this equation is: where is the charge enclosed by the Gaussian surface S of volume V and surface element da = nda (n is an unity vector pointing outward the closed surface).
Eq. ( 54) can be applied to one of the cylinders in Fig. 5.This is a textbook problem typical of the course Electromagnetism 101.In fact, take a Gaussian in a shape of a closed cylinder of radius r and length l.The axial symmetry of the problem imposes E eff = E eff (r) = E eff (r) r.Then, the left-hand side of ( 54) is: ¸S E eff • da = E eff (r) (2πr) (l).The right-hand side of (54) depends on the amount of charge enclosed by the Gaussian cylinder.If the Gaussian surface is within the tube of charges, it encloses none of the superficial charge: q enc = 0 for s < R. If the radius of the Gaussian cylinder is greater than the radius R of the tube of charges, then q enc = σ (2πR) (l) for s ⩾ R, where we have admitted a uniform surface charge distribution: σ = constant.Equating the previous results yields: Plugging this result into Eq.( 52), leads to the differential equation for the electric field in the context of GE: In the limit as a → 0, E = E eff and the Maxwellian result is recovered, as it should be.In this case, we see that the electric field points in the axial direction r; moreover, it depends only on the axial coordinate r.We will assume these features are shared by electric field of GE: Due to (58), the functional form of the Laplacian of the electric field reduces considerably and Eq. ( 57) reads: and Herein, use was make of the dimensionless coordinate ζ = r/a.
In the following we will solve Eqs. ( 59) and ( 60) one at a time.By comparing Eq. ( 59) with Eq. 9.6.1 of Ref. [26], we conclude that it as a modified Bessel equation with ν = 0.As such, it is solved by: The integration constants (b 1 , b 2 ) are constrained through imposition of boundary conditions.We demand that E s (ζ) is regular at the tube's axis, where ζ = 0. From the fact that lim z→0 K 0 (z) → ∞, we must choose b 2 = 0 (62) in order to avoid an unphysical divergent E r .The result is: The constant b 1 is unconstrained for the time being.Later it will be fixed by requiring continuity with the electric field outside the tube of charges.This leads us to our next task: to solve the inhomogeneous modified Bessel equation (60).We try a solution of the type where and, we have, Here alongside the Bessel functions: where z −ν H ν (z) is an integer function of z.In particular and for concreteness, the power series expansion of Struve function is given by: From ( 64), ( 65) and ( 66): We are interested in the the asymptotic behavior of the above function as ζ → ∞.Ref. [26] (Section 12.2.6)informs us: Let S ν (z) be defined as the right-hand side of the above equation.Hence, Accordingly, L 0 (z) ∼ I 0 (z) + S 0 (z).This is enough to show that2.1 since lim z→∞ I 0 (z) → ∞-cf.Eq. ( 13).Notice that S ν is finite in the asymptotic limit z → ∞.
Therefore, E r (ζ) will diverge unless both the first and third term in (70) subtract to a finite value.The combination of these terms is only possible through the choice In the face of this, Eq. ( 70) reduces to depending on the single integration constant b 4 .
The complete solution for the electric field is assembled by considering the inside solution ( 63) and the outside solution (75) : The integrations constants b 1 and b 4 should be determined after suitable boundary conditions are applied.
The first boundary condition is given by demanding continuity of the electric field on the tube's surface at r = R, i.e. ζ = R/a: E in r (R/a) = E out r (R/a).From (76), this maps to the equation: The next boundary condition is to demand that continuity of the first-order derivative of the electric field at the tube: . The derivatives of the Bessel functions are given by dI 0 dζ = I 1 (ζ) and dK 0 dζ = −K 1 (ζ); besides: This comes from a result involving the integral of L 1 (t); for details we point the reader to Section 12.2.9 of [26].Also, the paragraph 12.2.1 of the same reference, makes it clear that L 1 (0) = 0. Thus, using the expression of the electric field (76), one finds:  This is the desired solution for the electric field produced by a long tube of static charges in the context of GE.Fig. 6 displays curves for E r (ζ) for multiple values of A ≡ a/R as a function of the the distance from the tube's axis in units of its radius, r/R.Notice that the plots in Fig. 6 could have being built directly from the expression (81) for E r (ζ) and the power series expansion for the modified Struve function L ν (z), Eq. (79). 6However, that was not the way we plotted the curves in Fig. 6.Instead, we considered Eqs.( 71) and (72) to express L ν (z) in terms of S ν (z).This is a fare strategy because R ≫ a (for physical consistency), so that the condition for large z is satisfied and is actually valid.Then, L ν (z) ∼ S ν (z) + I −ν (z) and we can take: L 0 (z) = S 0 (z) + I 0 (z) and L 1 (z) = S 1 (z) + I 1 (z).Here, the result I −ν (z) = I ν (z) for (ν integer) was utilized.In this way, the coefficients of the functions I 0 (ζ) and K 0 (ζ) appearing in Eq. (81) simply, leading to: where ζ = r a and R ≫ a is assumed.It was Eq. (83) that produced the curves in Fig. 6.That Eq. ( 83) is the correct solution can be verified in a number of ways, including: (i) by checking continuity of E r at r = R; and (ii) by computing the asymptotic limits to Maxwell ordinary predictions.We leave the proof of (i) for the interested reader.Checking (ii) is much more interesting from the physical point of view.In fact, we use the asymptotic forms from paragraphs 9.7.1 and 9.7.2 of Ref. [26], and The latter result stems from keeping the leading terms in the sum defining S ν (z)-see Eq. ( 82).The approximations (84) and (85) are useful within the square brackets in both lines of Eq. ( 83) because the argument of I 0 , I 1 , K 0 , K 1 , S 0 and S 1 therein is R a ≫ 1.Under (84) and (85), Eq. (83) reduces to: The Maxwell limit can be obtained for the interior solution for ζ ≈ 0, while in the exterior it is necessary to take ζ → ∞.According to paragraph 9.6.7 of Ref. [26]: which is damped down to zero since R ≫ a for the reasonable physical situation where Podolsky is only a small correction to Maxwell.The exterior solution far from the tube will be given by the second line of ( 86) and ( 84): Herein, the first term is the Podolsky correction, which decays from the surface on, and the last term is exactly the Maxwell solution. 7

Electric potential for a long charged tube in GE
As we have said before (see the beginning of Section 2.2), GE is U(1)-gauge invariant.Therefore, the electric potential φ can be calculated via just like in Maxwell electrodynamics.Indeed, Eq. ( 89) follows from GE field equations (1); in particular, it stems from the second and third equations in (1).
Let us calculate the electric potential φ from the expression of electric field in the previous section.Recall that the vector potential A is null for the configuration at hand: a long tube of static charges.Then, Eq. ( 89) is simply: (90) The last step comes from the fact that E = (E r , 0, 0), cf.(58).Integrating (90) gives: where The potential inside the tube is computed from Eqs. (91), the first line in (92), and (93).Analogously, the potential outside the tube comes from Eqs. (91), (92) line two, (94), and (95).The complete expression for the electric potential will be: where b 1 and b 4 are given by cf. explained in section 3.1 below Eq. (80).

Electric Aharonov-Bohm phase shift in GE
Consider again the setup in Fig. 5. Let the upper-left (bottom-right) cylinder be called tube 1 (tube 2) kept at an electric potential φ 1 (φ 2 ).The beam of charged particles is split into two branches before passing through the tubes; branch 1 (branch 2) is approximately linear when passing through tube 1 (tube 2) and is regarded parallel to its axis, although it might not coincide with it.In fact, for generality we shall take branch 1 (branch 2) at a distance r 1 (distance r 2 ) of the axis of tube 1 (tube 2).This detail in the preparation of the beam splitting will be crucial for the detection of the Aharonov-Bohm effect in GE as opposed to the same effect in regular Maxwell electrodynamics.
The electric potential difference ∆φ is built from Eq. (96).Actually, only the first line of this equation (potential internal to the tube) will matter for computing the electric AB effect in GE since the beam traverse the inwards of the tubes.Then, Notice that ∆φ is a function of the distance ζ = r/a separating each beam branch of its corresponding tube's axis; it also depends on the radii R 1 and R 2 of tube 1 and tube 2, and of the surface charge density σ for each tube.In principle, R 1 ̸ = R 2 , σ 1 ̸ = σ 2 , and so on; however, the Podolsky coupling a remains the same in both tubes for obvious reasons.Eq. ( 99) has the form ∆φ = ∆φ (M ) + ∆φ (P ) .
The regular Maxwellian contribution is where φ in 0,i is the constant Maxwell potential in the interior of the i-th tube.Podolsky contribution to the potential difference is dubbed ∆φ (P ) ; it is given in terms of the hypergeometric function 1 F 2 and of Eq. (97) for b 1 .In order to simplify the latter expression, let both tubes be identical, i.e., with same radius R 1 = R 2 = R and same surface charge density σ 1 = σ 2 = σ; even so, we keep the beam branches at different positions with respect to the axis of their respective tubes (r 1 ̸ = r 2 ).Then, Podolsky correction to the electric potential difference reads: Eq. ( 39) gives the phase shift where ∆φ is given by (100), ( 101) and (102).Therefore, the plots in Fig. 7 inform directly the behavior of the complete phase shift ∆g, given a time interval t (and the charge q of the particles in the beam).
As before, let A = a/R where A ≪ 1 for assessing small Podosky corrections to Maxwell electrodynamics.The expression (102) presents numerical problems when one tries to plot the function ∆φ (P ) (r 1 , r 2 ) in the range r/R ∈ {0, 1} for A ≪ 1.In order to bypass this issue, we perform a power expansion of the electric field close to the inner radius of the tube: Hence, from (91), ( 104) and (105): Now, the electric potential difference (102) between the two identical tubes becomes: Using the asymptotic limit given in Eq. ( 86) for R ≫ a, we have Fig. 7 shows the plots for ∆φ (P ) in Eq. ( 108) with S 1 = 0 (the beam branch 1 travels along the axis of tube number 1) and S 2 = S = r/R.As expected, the smaller the A, the closer to Maxwell predictions.
In the limit R ≫ a, we could even use Eq. ( 87) for E in r : where Herein we considered equal-sized tubes (R 1 = R 2 = R) equipped with the same charge distribution (σ 1 = σ 2 = σ); the regular Maxwell contribution is denoted by ∆φ (M ) ≡ φ in 0,2 − φ in 0,1 .It is important to note that, in the case of equally charged identical tubes with beams passing through the same distance off the center (r 1 = r 2 ), the additional Podolsky term vanishes.Also, if the beam branches travel exactly along the center of each tube, there will be no contribution from Generalized Electrodynamics to the electric Aharonov-Bohm effect.

Final Comments
This paper dealt with the analysis of the magnetic Aharonov-Bohm effect and the electric Aharonov-Bohm effect in the contest of Bopp-Podolsky's Generalized Electrodynamics.The phase shifts due to the interaction of the beam of particles with the magnetic vector potential and the electric potential were computed and the diferences with their Maxwellian counterparts stressed.
In order to do so, we followed a similar route in both the magnetic case and the electric case.Specifically, the magnetic field B (electric field E) was computed by solving Ampére-Maxwell-Podolsky law (Gauss-Podolsky law) for a stationary current (static charge sistribution) in a long solenoid (a long cylinder)-see Eq. (1).Faraday law and the equation for the absence of magnetic monopoles are the same in GE as in Maxwell electrodynamics; this is the case because GE is also U(1) gauge invariant [31].Therefore, the definition of magnetic field B (of the electric field E) in terms of the the vector potential A (of the electric potential φ and of A) in the contexts of GE and of Maxwell electrodynamics is the same, i.e.B = ∇ × A E = −∇φ + ∂A ∂t .The limits of the fields and potentials from GE to Maxwell theory were verified each step of the way.
Once A and φ for GE were determined by direct integration of the field equations, we moved on to compute the phase shifts ∆g according to the path integral technique summarized by Felsager [1].It was proven that Generalized Electrodynamics produces modification in ∆g with respect to the Maxwellian prediction ∆g (M ) .In fact, in the limit as R ≫ a, the magnetic case is described by Eqs. ( 47) and ( 48 while in the electric case is given by Eq. ( 109), The new contributions from GE are present in the last terms of Eqs. ( 111) and (112).We see that these terms are exponentially suppressed since R ≫ a, the radius of the macroscopic cylinders are much larger than GE parameter a.The latter should be submicroscopic in order to Podolsky's corrections to Maxwell theory be small, as required by extensive tests of the standard electrodynamics [41,42].In fact, Podolsky mass m γ = ℏ (ca) −1 for the massive mode of the photon was constrained to 370 GeV [47].According to this energy value, Podolsky's length parameter a = ℏc (m γ c 2 ) −1 should be ≈ 0.53 × 10 −18 m or less.This value agrees with the bound given by Carley et al. in Ref. [48], which argues that precision measurements of the Lymann-α line in the Schrödinger spectrum of a hydrogen atom suggest that Bopp-Landé-Thomas-Podolsky length a must be smaller than ≈ 10 −18 m.9This is at least three orders of magnitude smaller than the empirical proton radius (about 0.85 fm) [49].
In the face of what was said, the perspective of constraining a (an atto-metre length scale parameter) via Aharonov-Bohm effect measurements is not achievable with the current technology.This is true also in the best-case scenario, where a could be assessed in a the beam flyby version of the magnetic AB effect, cf.discussed at the end of section 2.3.Even so, the present work is satisfying from the theoretical point of view; it analyses a key effect in the interface of electrodynamics, gauge theory and quantum mechanics.At the same time, it adds an application to Generalized Electrodynamics, an alternative theory that has been intensely explored in the recent years [43,44,45,46,47,48,50,51,52,53,59,60,61,62,63].

z,
for ζ ⩽ R/a.Here, B hom z indicates the solution to the homogeneous modified Bessel equation; thus: B hom z (ζ) = c 3 I 0 (ζ) + c 4 K 0 (ζ).(We put in other constants, c 3 and c 4 , for generality.)The term B part z

Figure 1 :
Figure 1: Behavior of the magnetic field produced by a long-thin solenoid in the context of Generalized Electrodynamics as a function of the axial distance r from the solenoid's axis.The horizontal axis presents the radial distance r from the solenoid's axis normalized by its radius R. The vertical axis exhibits the magnitude of the field B in units of B (M) in = (µ 0 nI) the magnitude of B inside the solenoid from Maxwell's electromagnetism.Each curve corresponds to a different choice of value for Podolsky's parameter a in units of the solenoid's radius R, i.e.A = a/R.The smaller the value of A, the closer the behavior of B to Maxwell's predictions.

Figure 3 :
Figure 3: The beam of electrons splits before circling the solenoid.The trajectory of each half of the beam is a semi-circle.Both branches from a ring-shaped path.

Figure 4 :
Figure 4: Relative phase shift in the magnetic Aharonov-Bohm effect in the context of Generalized Electrodynamics.The correction δg (r) changes Maxwellian prediction in the exterior vicinity of the solenoid.The more increases the distance to the solenoid the smaller the effectiveness of GE on the phase shift.

Figure 5 :
Figure 5: The beam of electrons splits in two branches that enter two long cylinders of charges.The recombined branches hit the detector located in the upper-right corner where the AB phase shift is measured.
) The solution to the system formed by Eqs.(77) and (80) provide the integration constants b 1 and b 4 .(While computing b 1 and b 4 we made use of the Wronskian W (R/a) = I 0 (R/a) K 1 (R/a)+ I 1 (R/a) K 0 (R/a) = a R .)Plugging the thus found (b 1 , b 2 ) into Eq.(76) yields:

Figure 6 :
Figure 6: Behavior of the electric field produced by a infinite hole tube in the context of Generalized Electrodynamics.

Figure 7 :
Figure 7: Behavior of the electric potential difference produced by two infinite hole tubes in the context of Generalized Electrodynamics.

2 e
−(r−R)/a (magnetic AB in GE) , is the modified Struve function-see e.g.Ref. [26], Section 12.2.It is defined in term of the Struve function H ν (z) which composes the general solution of the differential equation