1 Stern et al. Cavity design for high-frequency axion dark matter detectors I. Stern,1 A.A. Chisholm,1 J. Hoskins,1 P. Sikivie,1 N.S. Sullivan,1 D.B. Tanner,1 G. Carosi,2 and K. van Bibber3 1Department of Physics, University of Florida, Gainesville, Florida, 32611, USA 2Lawrence Livermore National Laboratory, Livermore, California, 94550, USA 3Department of Nuclear Engineering, University of California, Berkeley, California 94720, USA In an effort to extend the usefulness of microwave cavity detectors to higher axion masses, above ~8 µeV (~2 GHz), a numerical trade study of cavities was conducted to investigate the merit of using variable periodic post arrays and regulating vane designs for higher-frequency searches. The results show that both designs could be used to develop resonant cavities for high-mass axion searches. Multiple configurations of both methods obtained the scanning sensitivity equivalent to approximately 4 coherently coupled cavities with a single tuning rod. I. BACKGROUND 0 < v < c/1000, making the spread of energy ~10-6 mc2. The axion haloscope uses the inverse Primakoff effect to search for The detection of the axion is of significant interest to CDM axions in our galactic halo. particle physics and to cosmology. The axion is a hypothetical In the Sikivie7 haloscope detector, a microwave cavity pseudoscalar particle which was originally postulated to permeated by a strong magnetic field leads to resonant explain, within the framework of the standard model of conversion of axions to photons. From Eq. (1), it can be particle interactions, why P (parity) and CP (charge shown that the coupling strength of the axion to a resonant conjugation times parity) are conserved by the strong 3 mode is proportional to ∫ d x B0 ⋅ Emnp(x), where Emnp is the interactions.1-3 If the axion mass were in the range of 10-6-10-3 4 electric field of the mode. The indices m, n, and p identify the eV, the axion is a natural cold dark matter (CDM) candidate. various modes. For cylindrical cavities, they correspond to the With recent diminutions of the available parameter space for 5 polar coordinates ϕ, ρ, and z, respectively. The integral is over the weakly interacting massive particle (WIMP), the axion is the volume of the cavity. now by far the most promising candidate for the constitution The power produced in the cavity for a particular mode is of cold dark matter and this has generated a proliferation of given by searches around the world. String theory further suggests the simultaneous presence of many ultra-light axion-like particles, ρ , (2) possibly populating each decade of mass down to the Hubble scale of 10-33 eV.6 where m is the mass of the axion, ρ is the local mass density Axion detection has proven to be extremely challenging. a a of the axion field, V is the volume of the cavity, and Q is the CDM axions have a lifetime vastly greater than the age of the quality factor of the cavity (assumed to be less than the kinetic universe,5 have exceptionally weak interactions with matter energy spread of the axion at the Earth).10 C is the and radiation, and were originally thought to be "invisible" to mnp normalized form factor describing the coupling of the axion to all detection techniques. However, in 1983 Sikivie7 proposed a a specific mode. It is given by method by which these axions plausibly could be detected. He showed the decay rate of axions to photons can be greatly increased within a strong magnetic field, through the inverse Primakoff8 effect. The Lagrangian for the axion-photon , (3) interaction is given by where (x) is the permittivity within the cavity normalized to , (1) vacuum. where g is the axion-photon coupling constant, proportional to the mass of the axion, a is the axion field, and E and B are the electric and magnetic fields, respectively. The coupling allows the axion to decay to two photons, as shown in Fig. 1(a). In a static external magnetic field, an axion may convert to a photon whose energy equals the total energy of the axion, as shown in Fig. 1(b). The B in Eq. (1) is effectively changed to the static magnetic field, B0. Thus, as the external magnetic field strength is increased, so is the conversion rate of the axion. This process is effective for relativistic or non- relativistic axions.9 (a) (b) The cold dark matter in the Milky Way halo has a velocity FIG. 1. Feynman diagrams for axion decay into photons. (a) The conversion much less than the speed of light (v ≪ c); if virialized, in vacuum. (b) The inverse Primakoff effect in a static magnetic field (B0). Citation: Review of Scientific Instruments 86, 123305 (2015); doi: 10.1063/1.4938164 2 Stern et al. The mass of CDM axions is constrained by cosmology (ma 11-13 14 ≳ μeV and astronomical observations (ma ≲ 30 meV). The axion-photon coupling is model-dependent. In the KSVZ R (Kim-Shifman-Vainshtein-Zakharov)15,16 model, the coupling is . (4) eV d 17,18 In the DFSZ (Dine-Fischler-Srednicki-Zhitnitskii) model, . (5) eV In all models, including the two above, the coupling is proportional to the axion mass but the numerical constant that appears generally differs by similar factors among the models. In all models where the Standard Model gauge group is grand- unified into a simple gauge group, the coupling is the same as in the DFSZ model. FIG. 2: Schematic of ADMX microwave cavity. The tuning rods are shown in the outmost configuration and move along the circular arcs. R is the radius of the cavity and d is the height. II. AXION DARK MATTER EXPERIMENT (ADMX) The largest and most sensitive microwave cavity axion increase the curvature of Ez, adding the conducting rods search to date is the Axion Dark Matter eXperiment (ADMX). increases the frequency. ADMX searches for CDM axions using a 7.6 Tesla The frequency of the TM010 mode increases as the rods superconducting solenoid and a ~0.15 m3 cylindrical approach the center of the cavity. The mechanical system microwave resonator.19 The magnetic field of the solenoid is moves the rods in a circular arc from near the wall to near the aligned with the cylinder axis of the cavity (defined as the z- center. The experiment continually tunes the resonator in axis). increments of ~ f /5Q. Figure 2 shows a schematic of the From Eq. (3), it can be seen that a signal will be produced cavity. only when the resonant mode’s electric field integrates to a ADMX has conducted searches for a number of years, total nonzero z-component. Thus, all transverse electric modes successfully excluding axions with KSVZ coupling strength in the mass range of 1.9–3.5 µeV (460–860 MHz).21-24 Figure 3 (TE and TEM) will yield no detectable signal; only TM0n0 shows the published limits for the program. ADMX began modes will couple to axions. Further, the TM010 mode has the searching the TM mode range in the year 2014, extending largest form factor, with higher mode (TM020, TM030, TM040, 020 etc.) form factors decreasing as ~1/ f 2 for a simple cylindrical their search up to approximately 6.2 µeV (1.5 GHz). However, cavity, with f being the resonant frequency of the mode.20 the cavity design using two tuning rods does not provide for successful scanning above ∼1.5 GHz. Increasing the diameter For ADMX, the empty-cavity frequency ( f0) for the TM010 mode is ~550 MHz, and the theoretical unloaded quality factor of the tuning rods can drive the TM0n0 frequencies above 1.5 GHz, but the form factor (C) of the modes and scanning Q0 500,000 at 4 K. The frequency of the mode excited by the axion-to-photon range of the cavity decrease significantly. New resonant cavity 2 technologies need to be developed to enable searching for conversion is f ma c /h, where h is Planck’s constant Since the axion mass is unknown, ADMX must tune the cavity axions of greater mass. resonant frequency to carry out its search. ADMX adjusts the The ADMX High-Frequency (ADMX-HF) is a similar but separate detector program that began operation in 2015. It uses frequency of the modes using two conducting rods that run -3 3 20 parallel to the longitudinal axis of the cavity. The rods are a cavity with a volume of ~ 2x10 m , and a single tuning rod independently repositioned to various radial distances from the with a radius of r = R/2. The detector search for axions at centerline of the resonator, altering the boundary conditions frequencies approximately one order of magnitude higher than and changing the electromagnetic field solutions. This tuning ADMX. The TM010 empty-cavity frequency for ADMX-HF is can be shown by solving the two-dimensional wave equation ~2.3 GHz and the theoretical unloaded Q0 200,000 at 4 K. of a resonant cavity . (6) III. CAVITY DESIGN STUDY The following relationships illustrate the dilemma faced by For TM0n0 modes, = Ez and the axion search. The frequency of the exited mode scales , (7) approximately as the axion mass. The TM010 frequency scales inversely with the cavity radius; the volume as the cube of the where µ0 and are the permeability and permittivity within the radius and Q0 as the square root of the radius (at room cavity, respectively. Because the Laplacian is a measure of temperature) when the ratio of the height to the radius, d/R, is scalar curvature and the additional boundary conditions maintained (which is desired to minimize mode crowding).20 Citation: Review of Scientific Instruments 86, 123305 (2015); doi: 10.1063/1.4938164 3 Stern et al.