Solutions to Axion Electrodynamics in Various Geometries

Recently there has been a surge of new experimental proposals to search for ultra-light axion dark matter with axion mass, $m_a\lesssim1\,\mu$eV. Many of these proposals search for small oscillating magnetic fields induced in or around a large static magnetic field. Lately, there has been interest in alternate detection schemes which search for oscillating electric fields in a similar setup. In this paper, we explicitly solve Maxwell's equations in a simplified geometry and demonstrate that in this mass range, the axion induced electric fields are heavily suppressed by boundary conditions. Unfortunately, experimentally measuring axion induced electric fields is not feasible in this mass regime using the currently proposed setups with static primary fields. We show that at larger axion masses, induced electric fields are not suppressed, but boundary effects may still be relevant for an experiment's sensitivity. We then make a general argument about a generic detector configuration with a static magnetic field to show that the electric fields are always suppressed in the limit of large wavelength.


I. INTRODUCTION
Starting about 10 4 years after the Big Bang and lasting 10 10 years after, the gravitational evolution of the universe was driven mostly by Dark Matter (DM). But despite the wealth of compelling evidence for DM, we have not yet understood it on a particle level or determined how it fits next to the Standard Model (SM) of particle physics. In fact, the field of possible explanations for DM is so broad as to incorporate masses from ∼ 10 −22 eV to ∼ 100M .
One of the leading candidates to explain the DM abundance is the axion. It was originally proposed to solve the strong-CP problem in QCD [1][2][3], but its weak interaction strength with SM particles and an elegant production mechanism in the early universe make it a promising candidate to explain DM as well [4][5][6].
Unlike the more thoroughly constrained DM candidate, the Weakly Interacting Massive Particle (WIMP), the axion is expected to be extremely light with 10 −14 m a 1 eV (see [7][8][9] for a recent review). This implies that unlike WIMP DM, which would have a few particles per cubic meter, axion dark matter (aDM) would have a very high number density and behave like a coherent field. In this case, the DM energy density is better thought of as the kinetic and potential energy of a classical field rather than a dilute gas of individual particles.
If produced by the misalignment mechanism [4,5], the time evolution of the axion field is expected to be given by where the frequency of oscillation is equal to the axion mass ω a = m a and an arbitrary overall phase. If aDM is responsible for the observed DM density, we can relate a 0 = √ 2ρ DM /m a , where ρ DM is the local DM * ouelletj@mit.edu density of ∼ 0.3 GeV/cm 3 [10]. Though aDM is extremely cold, it is expected to have a very small velocity spread due to gravitational effects. In the potential well of the Milky Way we expect a typical local velocity spread of v DM ∼ 220 km/s. This results in a small spread in oscillation frequency due to Doppler shifting, ∆ω a /ω a ∼ v 2 DM ≈ 10 −6 , as well as small spatial gradients on the scale of the de Broglie wavelength, λ D = 2π/|k D |.
Experiments searching for aDM often leverage the fact that the axion couples to the photon and thus creates a small modification to electromagnetism. The axionor any axion-like particle (ALP) for that matter -will create a modification to the electromagnetic Lagrangian, that can be written in terms of the Maxwell field tensor F µν , electric current J µ e , and axion field a: WhereF µν = ε µνσρ F σρ , and g aγγ is an unknown, but small, coupling between the axion and photon. The aFF term can be treated as an axion-to-twophoton coupling which converts photons into axions and vice-versa, as in light shining through wall (LSW) [11] and axion helioscope [12,13] experiments. However, since aDM would imply a high occupation number for the field a, the Lagrangian can also be treated in the classical limit as a modification to Maxwell's equations [14]: create a strong static B-field and look for the small AC fields sourced by the aDM terms. As we will see below, the exact implications of these additional terms for an experiment will depend strongly on the relative size of the detector to the Compton wavelength of the axion λ a = 2π/ω a . Experiments like ADMX [15][16][17], HAYSTAC [18], and others [19][20][21] have built resonant cavities to probe axion masses in the range m a ∼ 10 −6 −10 −5 eV. In this range, the axion has Compton wavelengths of order 6 − 60 cm and comparable to the physical size of the detector. Practical considerations limit the range of masses that can be probed with detectors comparable in size to λ a . At shorter Compton wavelengths, ∼ 1 mm, experiments like MADMAX propose to manipulate electric fields using arrays of dielectric plates [22] to coherently add effects over many Compton wavelengths within their detector. Recently, several experiments have been proposed to search for aDM with with much lower masses of 10 −14 − 10 −6 eV and therefore Compton wavelengths much larger than the detector. These include experiments like ABRACADABRA [23], DM Radio [24], BEAST [25] and others [26][27][28][29].
In the limit of large λ a , the typical approach is to build a detector with a strong DC magnetic field and search for an induced AC B-field. Experiments like [23,24,26], take the magneto-quasistatic (MQS) approximation and assume the displacement current in Eqn. 3d to be small. The axion term can then be treated as an effective current, J eff that sources a real B-field, which can be detected. However, [25] proposes an alternate approach, keeping the displacement currents, to measure an induced AC field E = −g aγγ aB in a strong DC B-field. This has caused disagreement in the community about whether the axion induced electric field would be large enough to be observable or whether it is significantly suppressed -specifically, whether the electric field is given by E = −g aγγ aB, or whether it is suppressed by powers λ a . This has prompted new interpretations of the effect of the axion field in the presence of electromagnetic fields [30].
In section II, we explicitly solve the modified Maxwell's equations in the case of an infinite solenoid without assuming the MQS approximation and demonstrate explicitly that the electric field vanishes everywhere in the large λ a limit. In section III, we generalize this conclusion and show that for a broad class of detectors, the MQS approximation is always valid in the large λ a limit and that the vanishing of the electric field is a generic quality. From this we conclude that an experiment with a static B-field will always be more sensitive to axion induced magnetic fields over electric fields. Finally, in section IV, we address an alternate -but completely equivalent -approach outlined in [25,30] which can mislead one into thinking a measurable electric field is always generated.
As is common, we will assume that the spatial gradients of the axion field are negligible, ∇a ≈ 0. This is because the de Broglie wavelength is about three orders of magnitude larger than the Compton wavelength (λ D ≈ 10 3 λ a ), and thus spatial gradient terms are suppressed.

II. AXION DARK MATTER AND THE INFINITE SOLENOID
The simplest geometry to consider is the case of the infinitely tall solenoid. Of course, in practice this geometry is not physically achievable. A physical solenoid will have a finite extent and thus returning fields outside the winds of the solenoid. But in many experimental setups, these fringe fields are small compared to the field inside the solenoid and lead to sub-dominant corrections. An infinite solenoid is a useful example on which to see the major effects. We will comment on this point later.
Assume we have an infinitely tall solenoid of radius R pointing along theẑ direction. The current density along the walls is such that the unmodified Maxwell's equations would lead to the solution See Fig. 1. Even with the modified form of Maxwell's equations this solution is correct to zeroth order in g aγγ . Further, let's assume that current cannot flow along the solenoid walls in theẑ direction. For instance, we can take this to be a densely packed set of current carrying loops that only carry current in theφ direction. We can take the time derivative of equation 3d, to get where we have taken advantage of the fact that with ∇ · E = 0 everywhere, and that J is constant in time.
Similarly, we could take the time derivative of Eqn. 3c, to get Discarding terms of O(g 2 aγγ ), we are left with two wave equations to solve: It is clear from these equations that the only non-trivial solutions will be for E z and B φ . The other components are not affected by the axion field at leading order. Since the axion field is nicely decomposable into frequency modes, we will move into frequency space and drop transient solutions. Because of the symmetry, we propose the solutions (10b)

A. The B-Field Solution
Plugging (10b) into (9b) and performing a change of variables to ρ = ω a ρ, we get the Bessel equation with a boundary condition at ρ = R: The solutions to this are Bessel functions of order 1, with a boundary conditions at ρ = 0 and ρ = ω a R: Here, we required that for ρ < ω a R the diverging N 1 (ρ ) solution is suppressed, and for ρ > ω a R an outward traveling wave given by the Hankel function, H + 1 (ρ ). (An inward traveling wave, H − 1 (ρ ), is also a correct solution, however would imply power flowing into the oscillating axion field from infinity rather than out of it. ) We can now find the full solution, by requiring continuity of B across the boundary, and a step discontinuity in ∂ψ B ∂ρ as required by the δ function. (Remember that we specified that current could not flow alongẑ.) This can then be solved further to yield where we have leveraged Abel's identity to simplify the Wronksian of Bessel functions as This fully specifies the solution of the B-field driven by the axion at leading order in g aγγ . Figure 2 shows the behavior of B φ for various values of R/λ a .

B. The E-Field Solution
Returning to the E z component, we can plug (10a) into (9a) and performing a change of variables get another Bessel equation: which has solutions Again, we have required that ψ E (ρ ) be finite at ρ = 0, and an outward traveling wave for ρ > ω a R.
Here, the boundary conditions require that E z and its derivative be continuous across the boundary. The former condition can be seen by integrating ∇ × E around a small contour just inside and outside of the solenoid; the latter can be seen by integrating Eqn. 16 between [ω a R − ε, ω a R + ε] as ε → 0.
We can again simplify this further to Where we have taken advantage of the Bessel function . These equations fully specify the E-field solution.
Putting these together with the solutions for the Bfield yields a nice compact form The solutions for E z and B φ are plotted together in Fig. 2 for various values of R/λ a . for several values of R/λa. The solid lines are the terms driven by the axion field (in phase for Ez and shifted by π 2 for B φ ). The only approximation are that these are to first order in gaγγ.

C. The Long Compton Wavelength Limit
The variable ρ , is actually the ratio of the radial coordinate scaled by the Compton wavelength of the axion ρ = 2πρ/λ a . Not surprisingly, this marks the Compton wavelength as the relevant length scale of the problem. If R λ a , we will get one type of behavior, as compared to R ∼ λ a or R λ a . This can be seen in Fig. 2. In the long Compton wavelength limit, R λ a (or equivalently ρ = ω a R 1), both sides the solenoid can be thought of as "oscillating in phase" and the fields add coherently over the relevant distance scales. This is the limit relevant for experiments like ABRACADABRA [23], DM Radio [24], BEAST [25] and other LC-resonator searches [26].
We can take the asymptotic limits of the Bessel functions to see how the field near the solenoid behaves. Equation 12 becomes with the coefficients given by inserting this and converting back to ρ, yields the radial behavior The factor of i simply indicates a π 2 -phase shift from the axion field. This is expected since the B-field in Eqn. 9b is driven by ∂a ∂t . Plugging this back into Eqn. 10b, we have our full solution for the axion induced B-field to first order in g aγγ and in the limit of ρ, R λ a : Here, we have summed over axion frequency modes ω a to convert iω a a 0 e iωat back into ∂a ∂t to make the solution true for arbitrary a(t).
It should be noted, that this is exactly the result that we would expect from taking the MQS approximation and treating the oscillating axion field in a magnetic field as an "effective current", J eff = g aγγ ∂a ∂t B 0ẑ as is done in [23,24,26].
Looking at the electric field behavior in the long wavelength limit, Eqn. 17 becomes (25) where γ is the Euler-Mascheroni constant, (γ ≈ 0.5772...). With the coefficients given by This implies that, to first order in g aγγ and for ρ, R λ a , we have Or more precisely, that in the limit of ρ, R λ a , electric fields are suppressed by R λa 2 . This behavior can be seen in Fig. 2. This is in direct contrast with the argument set forth in [25], which searches for an axion induced electric field in the long Compton wavelength limit. This conclusion is reached here using a particular geometry, but the conclusion is a lot more general, as we will show in the next section. It is worth noting that the E-field solution proposed in that work, E = −g aγγ aB, does appear in the solution to Maxwell's equations as the ρ independent term in Eqn. 17. But in the large λ a limit it is canceled by the other term in the full solution -given in Eqn. 26. In the short Compton wavelength limit, the field E = −g aγγ aB appears as an offset to the oscillating Bessel function: E z = (a E J(ω a ρ) − g aγγ a 0 B 0 ) e iωat . When the Bessel function has many oscillations within 0 < ρ < R, the spatial average approaches −g aγγ a 0 B 0 e iωat . This can be seen in the lower panel of Fig. 2 as the offset between the solid and dotted red lines.
An experimental setup with a capacitor inside the solenoid (similar to [25]) would in fact see charges displaced by the oscillating axion induced E-field. But this would only be a measurable effect in the R λ a limit (i.e. for frequencies ω a /(2π) 300 MHz). This is akin to the microwave cavity designs used by [15][16][17][18][19][20][21], but without the resonator cavity. Interestingly, there are other recent proposals for the R ∼ λ a regime using this type of detector, but with all resonant enhancement moved into electronics [31]. At shorter wavelengths still, other experimental techniques have been proposed which rely on manipulating the E-field with dielectric plates. [22]. These latter approaches, where R λ a , are not incompatible with the results presented here.

III. DEMONSTRATING THE MQS APPROXIMATION FOR A GENERIC DETECTOR
The argument in the previous section can be made much more general by directly demonstrating that the MQS approximation holds in the presence of an oscillating axion field in the large λ a limit. In the following argument, we will make two assumptions: 1. our detector is composed of a collection of timeindependent charges and currents, ρ e and J e ; 2. our detector fits into some box with a diagonal size L. Thus both the ρ e and J e used to create our primary fields and whatever apparatus we use to detect axion induced fields are contained within |x − x | < L.
The precise shape of the box in the second assumption is irrelevant -it only establishes a characteristic size for our detector. We make no assumptions about the configuration of the currents and charges within the box. We can split our E and B fields into terms of similar order in g aγγ where E 1 and B 1 will be proportional to g aγγ . We rewrite our wave equations as Notice that E 0 and B 0 are independent of time because of our first assumption above.
Focusing on Eqn. 30c, we can split E 1 into E 1 = E 1 − g aγγ a 0 B 0 , and get an equation for E 1 At this point, we can use the retarded Green's functions to solve for our fields.
Notice that the −1 in the Eqn. 32c came from solving for E 1 and substituting Eqn. 32b in for the offset term, −g aγγ a 0 B 0 . We point out that ρ e does not appear in our axion induced fields. This is because in the limit that ∇a is small, we cannot use static electric fields alone to detect axions -regardless of their shape. This is evident from Eqns. 3.
At this point our solution is very general. Up to now, we have only used the first assumption that our charges and currents are constant in time. We use the second assumption to examine what happens in the limit of L λ a . Notice that our solutions are completely in terms of charges and currents, which are completely contained within our box of size L -as opposed to fields, which can extend outside of the box.
If both x and x are within our box then |x − x | ≤ L. And now we examine the behavior of the axion induced electric fields by Taylor expanding Eqn. 32c in the limit of ω a L 1: Where the last step takes advantage of the fact that our current is contained and so equal to zero at the surface of the box, S. Hence, the electric field is suppressed by (L/λ a ) 2 . Of course, a similar process can be done for Eqn. 32d, but it is easy to see that the relevant difference between this equation and Eqn. 32c is the −1 in the numerator. The leading term remains and the result is not suppressed by an additional powers of λ a .
This conclusion is very general and does not depend on the precise details of our detector. We only assumed that 1) the currents and charges that drive our primary fields are constant in time; and 2) our detector is of characteristic size L λ a . Under these assumptions we have shown that axion induced electric fields are always suppressed. We have actually just showed that the MQS approximation continues to hold in the presence of an oscillating axion field with large λ a .
And interesting thing worth noting is that in this calculation we have neglected terms proportional to ∇a as they are suppressed factors of λ a /λ D ∼ 10 −3 . However, when L/λ a 10 −3 , it is possible for electric fields generated by the ∇a · B term in Eqn. 3a to dominate over the electric fields generated by the ∂a ∂t B term in Eqn. 3d. Finally, it is worth describing the behavior of E 1 and B 1 in the limit of ω a L 1. In this limit, the exponentials in Eqns. 32c and 32d oscillate very rapidly and, under reasonable assumptions about how quickly J e varies within L, will cause the integrals to average to zero. All that will remain is which is exactly the −g aγγ aB 0 term. This is completely consistent with what we saw in the case of the infinite solenoid where the coefficients in Eqn. 20 fall off with the Bessel functions at large values of ω a L as ∼ 1/ √ ω a L and all that remains is the −g aγγ aB 0 term.
From this, we conclude that if (1) our currents and charges are independent of time and (2) with reasonable assumptions about how rapidly our current distributions vary on length scales ∼ λ a L, the effect of the axion can be given by E a (x, t) = −g aγγ aB 0 . However, this is not the limit proposed for axion searches in the mass range m a 1 µeV.

IV. ALTERNATE APPROACH USING POLARIZATION
In this section, we address the approach laid out in [25,30] and show the way this approach can naively lead one to expect a large electric field when one is not present. But when boundary conditions are correctly applied, this is a perfectly valid approach.
Following [25,30], we can reformulate Eqn. 3 in terms of the macroscopic fields D and H: Since we are solving these equations in free space (where there are no bound charges), these equations are completely equivalent to Eqn. 3 and should yield the exact same solutions. However, in this approach we can rewrite this in terms of a set of modified fields with which we can write an analogous set of macroscopic Maxwell's equations with no axion modification terms In four-vector notation, what we have done here is to envelope the axion current J µ a = g aγγ (B · ∇a, −E × ∇a + ∂ t B) into a redefinition of the electromagnetic field tensor We can see from a straight forward application of Noether's theorem that P µν a should be given by P µν a = g aγγ aF µν (39) And of course, the continuity equation follows trivially from the fact that because the derivatives are symmetric under interchange of µ and ν andF µν is anti-symmetric. This entire approach is completely analogous to the way the macroscopic form of Maxwell's equations splits the electric current into J µ bound and J µ free and attaches the former into a redefinition of F µν → G µν = F µν − P µν bound , where for a material polarization P and magnetization M, such that ∂ µ P µν bound = J ν bound . In each of these steps, our equations of motion remain completely unchanged and the continuity equation is always satisfied. We are simply moving terms around.
This appears to be a tidy reformulation of Eqns. 3, however, it must be emphasized that the physics is completely unchanged. Further, great care has to be taken when using these D a and H a fields, as the simplicity of Eqns. 37 can be deceptive. The reason is that the Lorentz force has not been changed, f = ρ e E + J e × B. In other words, charges and currents still rearrange themselves in response to E and B-fields. Therefore boundary conditions must still be placed on E and B rather than on D a and H a .
The purpose of this approach, however, is to continue the analogy, and to write a set of axion polarization and magnetization fields: But this is where the subtleties become critical. We must keep in mind that In other words, in the limit of small spatial gradients in a, the axion "bound charge density" is suppressed. Substituting this into Eqn. 3a tells us that P a does not create an E-field directly. We often intuitively think that an electrically polarized material has an associated electric field. However, this field comes from bound surface charges at the edge of the polarized material. But no such boundary can ever exist for P a . So while it might naively appear that an electric field must be present due to the axion polarization, it is not. Instead a time-varying P a generates a time varying magnetic field and that time-varying magnetic field can generate time-varying electric fields. Stepping back to our example of the infinite solenoid, we can easily calculate the polarization and magnetization to first order in g aγγ (neglecting terms proportional to ∇a): The intuition would be to view this as a time varying electric field inside our solenoid. But there is no divergence in P to generate such an electric field. Instead we note that P a varies with a and plug these values into Eqn. 37d and recover Eqn. 3d. This will recover the result in Sec. II. This underlines the fact that an axion polarization with no space-time derivatives cannot have any physical manifestations. This is also evident in the Lagrangian because the aFF terms becomes a complete derivative in the limit that ∂ µ a = 0. An analogous argument can be made about magnetization induced magnetic fields.
Despite our intuition underwise, the magnetization, M a , alone cannot generate a physically observable magnetic field, only when ∇ × M a = 0.
The approach of calculating axion induced polarization and magnetization is completely equivalent to the approach outlined in the first part of this paper. But great care must be taken when using this approach, because subtleties in the application of boundary conditions and physical intuition can conspire to produce physical effects where they should be suppressed.

V. CONCLUSION
In this work, we have stepped through the calculation of the axion induced E and B-fields in the presence of a strong magnetic field in an infinite solenoid. We showed that the solution E = −g aγγ aB is part of the full solution of the modified Maxwell's equations, however by itself it does not satisfy the required boundary conditions. In the large λ a limit, the full solution suppresses the electric fields everywhere by R λa 2 .
We then laid out the generic derivation of the MQS approximation in the presence of an axion field. And demonstrated that in any experimental setup with a time-independent charge and current distribution, the axion induced E-fields are always suppressed relative to the axion induced B-fields in the large λ a limit. The con-clusions of this work directly contradict the arguments outlined in [25,30], and this implies that the limits shown in [25] are too low by ∼6.5 orders of magnitude.
Unfortunately, this highlights the fact that it is extremely difficult to detect axion induced electric fields in the large λ a limit in any experimental setup with static primary fields. This should be kept in mind for future experimental proposals for axion haloscopes in the ultralight mass regime.
Finally, it should be noted that boundary conditions at the edge of the B-field shape the behavior of the fields inside. This also underscores the need for all axion haloscopes to carefully analyze the effect of the boundaries of their fields and not necessarily assume that the approximations of an infinite field are valid.