Probing ALPs and the Axiverse with Superconducting Radiofrequency Cavities

Probing ALPs and the Axiverse with Superconducting Radiofrequency Cavities

Abstract

Axion-like particles (ALPs) with couplings to electromagnetism have long been postulated as extensions to the Standard Model. String theory predicts an “axiverse” of many light axions, some of which may make up the dark matter in the universe and/or solve the strong CP problem. We propose a new experiment using superconducting radiofrequency (SRF) cavities which is sensitive to light ALPs independent of their contribution to the cosmic dark matter density. Off-shell ALPs will source cubic nonlinearities in Maxwell’s equations, such that if a SRF cavity is pumped at frequencies and , in the presence of ALPs there will be power in modes with frequencies . Our setup is similar in spirit to light-shining-through-walls (LSW) experiments, but because the pump field itself effectively converts the ALP back to photons inside a single cavity, our sensitivity scales differently with the strength of the external fields, allowing for superior reach as compared to experiments like OSQAR while utilizing current technology. Furthermore, a well-defined program of increasing sensitivity has a guaranteed physics result: the first observation of the Euler-Heisenberg term of low-energy QED at energies below the electron mass. We discuss how the ALP contribution may be separated from the QED contribution by a suitable choice of pump modes and cavity geometry, and conclude by describing the ultimate sensitivity of our proposed program of experiments to ALPs.

1

Axions are well motivated new particles that have been proposed as a solution to the strong CP problem Peccei and Quinn (1977); Weinberg (1978); Wilczek (1978) (for a review of axions see Refs. Sikivie (2008); Kim and Carosi (2010); Hook (2018)). Additionally the only known consistent theory of quantum gravity, string theory, predicts a plethora of light ( eV) particles Svrcek and Witten (2006), some of which may couple to electromagnetism in a manner very similar to the axion. These particles have been termed axion-like particles (ALP) and the (possibly) large number of ALPs can been called the “axiverse” Arvanitaki et al. (2010). One or more of these species may be excellent dark matter (DM) candidates Preskill et al. (1983); Abbott and Sikivie (1983); Dine and Fischler (1983), and/or alleviate the hierarchy problem Graham et al. (2015a); Gupta et al. (2016); Hook and Marques-Tavares (2016); Davidi et al. (2018); Banerjee et al. (2018). In light of this strong motivation, there has been much experimental effort devoted towards looking for the axion and its cousins Graham et al. (2015b); Irastorza and Redondo (2018).

There are several general approaches for finding ALPs, roughly analogous to the multipronged approach of direct detection, indirect detection, and collider production for WIMP DM. If the ALP makes up the DM of the Universe, it may be detected in the laboratory by converting ALPs to electromagnetic energy (see Refs. Zhong et al. (2018); Du et al. (2018); Ouellet et al. (2018) for recent experimental results) or rotating the polarization of photons DeRocco and Hook (2018); Liu et al. (2018), or in radio telescopes by searching for conversion Pshirkov (2009); Hook et al. (2018); Huang et al. (2018); Safdi et al. (2018) or decay Caputo et al. (2018a, b) to photons in astrophysical environments. Another approach which does not require the ALP to be DM is colloquially known as light-shining-through-walls (LSW): a laser passes through a large magnetic field, some photons convert to ALPs, a wall blocks the remaining photons but not the ALPs, and after the wall another large magnetic field converts the ALPs back into detectable photons. LSW experiments have the advantage that both ALP production and detection are completely under experimental control. Such experiments are also “broadband” in the sense that they are simultaneously sensitive to a wide range of ALP masses, and indeed multiple species of ALPs.

In this Letter, we propose a new experiment along the lines of an LSW experiment that utilizes light-by-light scattering mediated by off-shell ALPs, with production and detection taking place inside the same superconducting radiofrequency (SRF) cavity. An ALP is a pesudoscalar defined by its mass and its coupling to electromagnetism . Its Lagrangian is

(1)

For processes involving photons with typical energy , an ALP with mass may be integrated out, giving an effective Lagrangian Evans and Rafelski (2018)

(2)

In other words, a off-shell ALP will induce small nonlinearities in electromagnetism, by which signal photons may be detected at a different frequency than the input photons. Note that this effect is local, and does not require the ALP to propagate to another spacetime point to be converted back to photons. Of course, we are also interested in ALPs which are very light. As such, we will extend the analysis of Evans and Rafelski (2018) to the case where . In this case the nonlinear effects are nonlocal, but we will show that detection of signal photons may still take place in the same spacetime region as the input photons.

Famously, loop contributions from virtual electrons will also induce such nonlinearities in pure quantum electrodynamics (QED), which are parameterized by the Euler-Heisenberg (EH) Lagrangian Heisenberg and Euler (1936); Schwinger (1951). To lowest order in and , this is

(3)

valid for . Light-by-light scattering with real photons has recently been observed at GeV energies Aaboud et al. (2017), but Eq. (3) has never been probed with real photons at . Thus, an experiment that is designed to look for non-linearities induced by ALPs would, if sensitive enough, also have the guaranteed physics result of discovering light-by-light scattering at low energies for the first time ever! Crucially, the effects of ALPs and the EH Lagrangian are not exactly degenerate, as the ALP Lagrangian only contains , while the EH Lagrangian also contains . Thus, the two effects are linearly independent and may be disentangled with a suitable choice of field configurations.

Comparing Eqs. (2)–(3), we expect the ALP contribution to 4-photon processes to exceed the EH contribution when Bernard (1997); Evans and Rafelski (2018)

(4)

The best laboratory bounds on are from the OSQAR  Ballou et al. (2015) and PVLAS Della Valle et al. (2016) experiments, which constrain for . Surpassing these bounds with a radiofrequency experiment () would not require sensitivity to the EH Lagrangian, since the ALP term in the Lagrangian would be much larger. Under reasonable assumptions about solar physics, the bounds from the CAST experiment Anastassopoulos et al. (2017) constrain from thermal ALPs produced in the sun. (More stringent bounds can be obtained for from the absence of photon-ALP oscillations in galactic magnetic fields Brockway et al. (1996); Grifols et al. (1996); Payez et al. (2015); Marsh et al. (2017).) An experiment which surpasses these bounds would also be sensitive to the EH contribution, as the ALP contribution would be of similar size.

Taking these estimates as motivation, a very interesting proposal Brodin et al. (2001); Eriksson et al. (2004) suggested a setup for detecting the EH Lagrangian using SRF cavities. In this Letter, we extend the results of Ref. Eriksson et al. (2004) to include the contributions from the ALP Lagrangian (1). Such contributions have been considered before Bernard (1997), but in the context of colliding laser beams and momentum-space Feynman diagrams, which are not appropriate for the boundary conditions imposed by cavity experiments.

Furthermore, while it was demonstrated in Bernard (1997) that such contributions will likely never be sensitive to the so-called QCD axion which solves the strong-CP problem Peccei and Quinn (1977); Weinberg (1978); Wilczek (1978), renewed interest in the axiverse strongly motivates a re-examination of this result for general ALPs. Indeed, multiple ALPs will all contribute to nonlinearities in electromagnetism. In a standard on-shell ALP search, e.g. a LSW experiment, if all the couplings of the ALPs are of the same order , the strength of the effect scales as , where is the number of ALPs lighter than the typical energy scale of the photons in the experiment (e.g. 2.33 eV for OSQAR). By contrast, because our proposed experiment is sensitive to off-shell ALPs, our signal scales as , better than any previous experiment. Sufficiently strong bounds on this product may begin to constrain axiverse scenarios which predict large numbers of light ALPs, for example compactification manifolds with large numbers of nontrivial cycles Douglas and Kachru (2007).

Figure 1: Schematic of our proposed experiment, adapted from Ref. Eriksson et al. (2004). An SRF cavity is pumped at frequencies and . The signal mode (equal to in our examples), sourced either by ALPs or the EH Lagrangian, is detected in a filtering region and amplified. Note that extends through the bulk of the cavity, but the filtering geometry suppresses the pump fields in the detection region.

ALP-induced cavity source terms. Equation (1) implies that Maxwell’s equations are modified in the presence of nonzero  Sikivie (1983). Ignoring the EH terms for now, the modified equations of motion for ALPs with zero external charges or currents are

(5)
(6)
(7)

We will assume that the are small and use classical field perturbation theory. Equation (7) shows that regions of nonzero in a conducting cavity will source the field proportional to ; Eqs. (5)–(6) imply that will in turn source signal fields cubic in the cavity fields and proportional to . If all the are identical, the signal fields will be proportional to .

Unless otherwise specified, we now restrict to the case of a single ALP, . We may use the Green’s function for the ALP field to write the signal fields solely in terms of the cavity fields, which we refer to as pump fields from now on. The appropriate Green’s function is the classical retarded Green’s function for the Klein-Gordon equation . The solution for is then

(8)

We will consider a cavity pumped simultaneously at resonant frequencies and , with associated modes and , shown schematically in Fig. 1. The total input electric field in the cavity is , where it is understood that the physical field is the real part of the complex field and that the correct phase relationships exist between and . From now on, we will drop the explicit time dependence of the pump modes.

The terms involving derivatives of on the right-hand side of Eqs. (5)–(6) can be interpreted as an effective ALP charge and current:

(9)

Note that since is quadratic in the pump fields, and are cubic in the pump fields, and will thus have frequency components , , , and . If and satisfy Maxwell’s equations, then and satisfy the continuity equation . Thus, using the solution for in Eq. (8), we may treat as a source for the cavity involving only the pump fields and , identical in formalism to a real current source involving moving charges.

Signal strength. To solve for the signal fields, we will use the general formalism of cavity Green’s functions Hill (2009). We assume that a signal mode is a resonant mode of the cavity which matches one of the frequency components of . Assuming a finite quality factor for this mode, the ALP-sourced field which develops in a cavity of volume is

(10)

where is dimensionless with normalization .

To estimate the size of the signal, we normalize the pump modes such that , and write where is dimensionless and has dimension . In the two limiting cases of and , we choose to be and , respectively. The number of photons in the signal field is

(11)

where we have defined the dimensionless cavity form factor

(12)

We make explicit the fact that and both depend on through the Green’s function , which affects the solution for in Eq. (8).

At this point, we do a brief comparison between the parametrics of a LSW experiment and our own proposal. In the low- limit, the number of signal photons per number of input photons, , in the two experiments scales like

(13)

where and refer to the two -fields used for production and detection of ALPs, respectively, and is the typical size of the experiment (for our setup, we assume ). Our experiment scales similarly to a LSW experiment except that our final number of photons has been enhanced by due to the cavity, and there is only a single field region rather than separate production and detection regions; instead of and , the input oscillating field does the conversion. A static field can be made about 40 times larger than an oscillating field, but this deficit is more than made up for by the large factor of SRF cavities, which can reach Romanenko et al. (2014, 2018).

It is worth noting that the scaling of a LSW experiment that utilizes cavities, e.g. ALPS-II, would also be enhanced by , but such cavities could not be made superconducting at -fields above a few T, and the largest achievable in copper cavities is about . Our cavity experiment would still have superior sensitivity to such an LSW experiment, unless the ALP were both produced and detected in high- SRF cavities, where the larger compensates for the smaller  Harnik (); Grassellino (). Furthermore, in our setup, the detection frequency is not a harmonic of the input frequency, which may help reduce backgrounds. In the large mass limit of a LSW experiment, , until and then ALP photon conversions cannot occur. In the large mass limit of our proposal, , giving a much better reach at large masses.

Cavity form factors: heavy and light ALPs. To understand the signal strength as a function of , we compute the cavity form factors in two limits: , where , and , where .

The first case corresponds to “integrating out” the ALP in the context of field theory. By assumption, for a heavy ALP, so we can solve algebraically for in terms of the pump fields. In this limit, the ALP-induced current is

(14)

For the case , the ALP Green’s function is identical to the retarded Green’s function familiar from electromagnetism:

(15)

where is the retarded time. In this case, responds nonlocally to changes in and , with a time delay given by . Since the ALP-mediated current , which may be computed from Eq. (15) using Eq. (9), is also nonlocal, there is no simple expression in terms of the pump fields.

As an example, we calculate the cavity form factor for a right cylindrical cavity with the mode choices and (with mode labeling conventions following Hill (2009)). We choose the signal mode to be , which is satisfied for a cavity of radius and height . For this mode combination, we find for the heavy ALP and for the light ALP (see Supplementary Material for details). The latter result assumes that both and components of the signal can be added in quadrature, which is appropriate for photon counting at the standard quantum limit.

This example demonstrates that there exist modes for which the cavity form factor is relatively insensitive to , and thus a single cavity can be used to probe a broad range of ALP masses. This is one of the strongest advantages of our setup over traditional resonant searches for ALP dark matter Du et al. (2018); Zhong et al. (2018); Sikivie et al. (2014); Jackson Kimball et al. (2017), which require careful tuning to match a resonance (e.g. a cavity mode or a Larmor frequency) to . By contrast, once the tuning is accomplished, no further tuning is required to set limits on for any .

Expected sensitivity to ALPs. In order to estimate the sensitivities in the light and heavy mass limits, we compute the expected number of photons . From Eq. (11) we have

(16)

To measure the signal, we imagine a filtering geometry as suggested in Ref. Eriksson et al. (2004) (and shown in Fig. 1) where at some point in the geometry the pump fields are suppressed compared to the signal field, which is possible as long as . At this location, the signal can be measured without contamination from the pump modes. Because we know the input fields and their phases, we can calculate the waveform of the signal as a function of time. We naively estimate the signal-to-noise ratio (SNR) by using the Dicke radiometer equation and neglecting any information about the field phase:

(17)

where is the signal power, is the total measurement time, is the signal bandwidth, is the length of the cavity, and is the number of thermal photons at the signal frequency (valid for temperatures , and assuming thermal noise dominates). A detailed sensitivity calculation exploiting phase-sensitive amplifiers will be presented in a future work.

Our expected sensitivity to is then

(18)

In an actual experimental implementation, a cavity should be designed specifically to maximize the figure of merit in Eq. (18) while minimizing issues such as multipacting, dark currents, field emission, and surface nonlinearities Padamsee (2001, 2009).

For a fixed choice of modes and cavity size (and hence fixed and ), the reach is constant at small and degrades linearly at large . There is in principle some dependence of on , but as our example above demonstrates, with a suitable choice of modes this dependence is extremely mild. For a large number of light ALPs, all with , the limit in Eq. (18) should be interpreted as a limit on .

Figure 2: Projected sensitivities of our proposed Phases 1, 2, and 3 (approximations for shown in dashed lines), along with existing constraints from OSQAR Ballou et al. (2015), PVLAS Della Valle et al. (2016), and CAST Anastassopoulos et al. (2017), and the parameter space for the QCD axion. In particular, Phase 2 could simultaneously set world-leading limits on while also measuring the EH contribution to light-by-light scattering below the electron mass for the first time.

Phase 1: Conservative projected reach. We envision our experiment progressing in three stages, each building on current technology. For Phase 1, we take the following parameters: cavity temperature  K; a right cylindrical cavity of radius m and the mode combination giving height m and ; cavity volume ; as calculated above; pump field strength ; and a cavity bandwidth of Hz, corresponding to . This is much smaller than what typical high-performance SRF cavities can achieve, but a wide cavity response function for Phase 1 allows the frequency-matching condition to be approximately satisfied even if vibrational distortions of the cavity geometry shift by . Note also that our mode combination satisfies , making it amenable to filtering; to model this, we assume the total cavity length is twice the cavity height, m.

At 1.5 K, the thermal noise in the signal mode is photons. A lower operating temperature would be desirable, but the cooling power requirements are substantial: assuming that the pump modes have , characteristic of the best achieved in SRF cavities Romanenko et al. (2014, 2018), the cavity lifetime for the pump modes is and the power dissipated is . Since dilution refrigerators have a cooling capacity of (mW), the cavity must be operated at liquid helium temperatures.

We first consider the case where the injected pump bandwidth is comparable to the cavity bandwidth, Hz. For light ALPs a SNR of 5 can be achieved at , surpassing the OSQAR bound by nearly an order of magnitude in a single cavity lifetime of  s. Integrating the signal over a time 1 day, we can obtain a Phase 1 reach of . For , we can get the limits by using .

One could also pump the cavity with a bandwidth narrower than the cavity itself, for example by locking the pump tones to an atomic clock. Taking , the narrowest allowed bandwidth for a given measurement time , a bound of could be reached in a day. In the case , the signal to noise scales linearly with time, so the limit on scales as . The two bandwidth choices for Phase 1 are shown in Fig. 2; we have not explicitly calculated the reach for (shown as dashed lines), but we expect the light and heavy mass limits to be excellent approximations away from this region.

Phase 2: Detecting the Euler-Heisenberg contribution. As we have discussed, there is an irreducible contribution to cubic nonlinearities in Maxwell’s equations from the EH Lagrangian, see Eq. (3). The effective EH charge and current are Soljacic and Segev (2000); Brodin et al. (2001)

(19)

with and . The number of photons from the EH signal can be estimated similarly to the ALP case.

For Phase 2, we assume the cylindrical cavity geometry from Phase 1, but with . Indeed, in tuned SQUID magnetometers, a feedback circuit may be used to broaden the bandwidth without sacrificing Myers et al. (2007); such a scheme may be possible here. We find that

(20)

with and , with defined analogously to . This signal strength is roughly consistent with Ref. Eriksson et al. (2004) given our different choices of parameters and modes. Therefore, assuming , the EH signal can be detected within 20 days of running. The corresponding sensitivity to light ALPs for the same integration time is ; this is shown in Fig. 2. This would surpass the CAST bound of and would also be competitive with recent proposals to search for ALP DM at low masses such as ABRACADABRA Kahn et al. (2016); Ouellet et al. (2018).

If a positive signal were detected, the ALP nature of the signal could be verified with a second cavity with different mode combinations and checking if the size of the second signal matches the expectation from Eq. (16). If the ALP is heavier than , the combination of the two measurements would suffice to determine both and .

Naively, the sensitivity of this proposal to probe ALPs becomes limited when and the EH signal becomes an irreducible background. In principle, one can search for the ALP signal on the top of the thermal and EH backgrounds, but as with the “neutrino floor” in WIMP direct detection experiments, the SNR will grow much slower than . However, using a suitable choice of modes with a slightly different cavity design, the EH contribution can be removed, leaving behind only the ALP signal. The idea is to pump an additional mode degenerate with but with a different field configuration. By tuning the three different pump amplitudes, we can arrange to have with . For these special pump amplitudes, the EH contribution to light-by-light scattering vanishes at amplitude level, and there is no interference with the ALP amplitude.

We demonstrate the idea by considering a rectangular cavity of dimensions and the limit of . The three pump modes (labeled 1, , and 2) are and the signal mode is . The matching condition is satisfied for and . The total pump field is (where , , and are dimensionless) and similarly for . We find

(21)

Therefore, for we get and .

In the above example, we chose a rectangular cavity for simplicity because TE and TM modes with the same mode numbers are automatically degenerate, and because there is an additional free parameter in the cavity geometry which permits the correct configuration of form factors. We note that this idea can also be implemented with an elliptical cavity, which avoids the large field gradients present at the corners of rectangular cavities, and which may also be used for the Phase 1 search described above.

Phase 3: Probing the axiverse. As the optimistic endpoint of this proposed program of experiments, consider a large cylindrical cavity with , radius  m, and height  m, giving  MHz, with the same mode combinations as considered in Phases 1 and 2. We suppose a cavity geometry can be developed which permits with the EH contribution tuned away to sufficient precision as described in Phase 2, and a compact filtering geometry with length  m. Using the sensitivity scaling in Eq. (18), assuming the same maximal pump strength of 45 MV/m and integrating for a total time 1 year with , we find a maximum sensitivity at low masses of , shown in Fig. 2.

Revisiting the axiverse scenario, suppose that ALPs all had decay constants at the string scale, which we conservatively take to be as large as possible, namely the renormalized Planck scale of . These string ALPs would have photon couplings of , and our experiment would be sensitive to . The Phase 3 setup would be able to bound the number of string-scale ALPs with masses less than by . While this is still (much) larger than typical expectations from string theory, one could still imagine placing constraints on particular compactification geometries which contain large numbers of nontrivial cycles, allowing low-energy SRF cavity experiments to offer a fascinating probe into the ultra-high-energy regime of quantum gravity and the landscape of string theory vacua.

Acknowledgments. — AH, YK, and YS thank Ben Safdi and the University of Michigan Slack channel for facilitating discussion in the early stages of this work. YK thanks Prateek Agrawal, M.C. David Marsh, and the participants of the workshop “Axions in Stockholm – Reloaded” for discussions about the axiverse, and James Halverson for discussions about axions in string compactifications. We thank Daniel Bowring, Aaron Chou, Anna Grassellino, Roni Harnik, Kent Irwin, Jonathan Ouellet, Sam Posen, Alexander Romanenko, and Slava Yakovlev for enlightening discussions regarding cavity design and photon readout. We thank Junwu Huang, Gilad Perez and Jesse Thaler for helpful comments on the draft, and Lindley Winslow for support in the early stages of this project. AH is supported in part by the NSF under Grant No. PHY-1620074 and by the Maryland Center for Fundamental Physics (MCFP). The work of ZB is supported by the National Science Foundation under grant number NSF PHY-1806440. This work was supported in part by the Kavli Institute for Cosmological Physics at the University of Chicago through an endowment from the Kavli Foundation and its founder Fred Kavli.

Probing ALPs and the Axiverse with Superconducting Radiofrequency Cavities

Supplementary Material

Zachary Bogorad, Anson Hook, Yonatan Kahn, Yotam Soreq

In this Supplementary Material, we give further details about the example mode choices we have used to calculate the cavity form factors, and discuss the choice of modes for optimizing our reach for both light and heavy ALPs.

Mode functions and signal currents

Here we explicitly calculate the axion current and overlap for the mode choices TE, TM, and TM in a cylindrical cavity of height and radius , using mode conventions from Ref. Hill (2009). The (un-normalized) -field of the signal mode only has a -component:

(S1)

where is the Bessel function of order 0 and is its second zero. Thus the form factor integrand (12) only receives a contribution from . The (un-normalized) pump fields are

(S2)
(S3)
(S4)
(S5)

where is the first zero of and is the first zero of . The component of with frequency will contain two mode 1 fields and one mode 2 field, i.e. terms like .

For the heavy mass case , inspecting Eq. (14) and keeping track of the time dependence, we have that the component of quadratic in mode 1 and linear in mode 2 is

(S6)

where and .

At this point, has frequency components , , and . We now wish to isolate the frequency component at . To do this, we note that terms appear such as

(S7)

so to isolate the desired frequency component, we make the replacement

(S8)

Similarly, for the other two terms we have

(S9)

where in all three cases only the phase component appears (i.e. there is no term). Thus the component of oscillating at the signal frequency is

(S10)

Plugging (S2)–(S5) into Eq. (S10), we see that the -dependence of all terms is , and there is no dependence. Evaluating Eq. (S10) with and which satisfies the frequency-matching condition for the third mode at , we obtain , which is plotted in Fig. S1. For comparison, we also plot with both profiles normalized to 1 at , showing that the shape of these functions is fairly similar and we expect a large overlap.

As noted in the main text, the current in the light mass case is nonlocal, so there is no simple analytic expression in terms of the pump fields. Nonetheless, we can evaluate the spatial integrals in Eq. (12) numerically, and isolate the frequency components as described above. Unlike the heavy mass case, both phase components and are present. The two phase components of are also shown in Fig. S1; note that the component which is in phase with the pump modes vanishes at the boundary , while the other phase component does not.

Figure S1: Signal mode and ALP-induced effective currents and for the TE/TM/TM mode combination. There is no dependence in either the signal mode or the effective currents; the mode profiles are evaluated at , normalized to 1 at , and plotted as a function of the remaining variable . The similar profiles to the signal mode for and both phases of leads to a large form factor for this mode choice.

Characteristics of light and heavy form factors

In order to test the feasibility of our proposed method, we searched through a number of cylindrical cavity mode combinations and calculated the expected coupling for each. Since there are, in principle, infinitely many possible mode combinations, we restricted to the six smallest non-trivial mode numbers for each field and mode number. We also took advantage of three selection rules for modes TE and TM:

  • Either (including all sign combinations) or .

  • Either (including all sign combinations) or .

  • If and are both TE modes or both TM modes, then and must have the same dependence as and , rather than .

We found several modes with in the range . We then chose five of these with generally smaller mode numbers and calculated for each in order to test whether the same cavity dimensions would allow for effective searches of both high- and low-mass ALPs. As noted above, because contains both phase components and , we compute by summing the form factors in quadrature for the two phase components. As described in the main text, this is appropriate for photon counting at the standard quantum limit, but in future work we will explore the benefits of phase-sensitive amplifiers, in which case one quadrature may dominate. The values of and for each of these five modes are given in Table S1.

TE TM TM 0.18 0.24
TE TM TM 0.12 0.080
TE TM TM 0.13 0.075
TE TM TM 0.11 0.079
TE TM TM 0.12 0.062
Table S1: The values of and for five mode combinations with relatively large values of and small mode numbers. Note that, for all five mode combinations, is within a factor of 2 of , making it reasonable to use the same cavity to search for both high- and low-mass ALPs.

We conclude that a precise choice of modes is not necessary for achieving a large cavity form factor for both heavy and light ALPs, though we find that the largest form factors come from and and being TM modes. These general properties are easy to reproduce in an elliptical cavity, which will likely be the basis for a realistic design.

Footnotes

  1. preprint: CERN-TH-2019-009

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