Axion Dark Matter Coupling to Resonant Photons via Magnetic Field
AAxion Dark Matter Coupling to Resonant Photons via Magnetic Field
Ben T. McAllister, ∗ Stephen R. Parker, and Michael E. Tobar † ARC Centre of Excellence for Engineered Quantum Systems,School of Physics, The University of Western Australia,35 Stirling Highway, Crawley 6009, Western Australia, Australia (Dated: October 10, 2018)We show that the magnetic component of the photon field produced by dark matter axions viathe two-photon coupling mechanism in a Sikivie Haloscope is an important parameter passed overin previous analysis and experiments. The interaction of the produced photons will be resonantlyenhanced as long as they couple to the electric or magnetic mode structure of the Haloscope cavity.For typical Haloscope experiments the electric and magnetic coupling is the same and implicitlyassumed in past sensitivity calculations. However, for future planned searches such as those athigh frequency, which synchronize multiple cavities, the sensitivity will be altered due to differentmagnetic and electric couplings. We define the complete electromagnetic form factor and discuss itsimplications for current and future high and low mass axion searches, including some effects whichhave been overlooked, due to the assumption that the two couplings are the same.
Axions are a type of Weakly Interacting Slim Parti-cle (WISP) originating from the Peccei Quinn solutionto the strong CP problem in QCD [1]. They can beformulated as highly motivated and compelling compo-nents of Cold Dark Matter (CDM) [2–5]. Cosmologicalconstraints provide upper and lower limits on the massof the axion [6], yet still leave a large area of parameterspace to be searched. One of the most mature and sen-sitive experiments is the Sikivie Haloscope [7, 8], whichexploits the inverse Primakoff effect whereby a magneticfield provides a source of virtual photons in order to in-duce axion-to-photon conversion via a two photon cou-pling, with the generated real photon frequency beingdictated by the axion mass. This signal is then reso-nantly enhanced by a cavity structure and resolved abovethe thermal noise of the measurement system. It hasbeen well established that in a Haloscope with an axialDC magnetic field the expected power due to axion-to-photon conversion is given by [7–9]P a = (cid:18) g γ απf a (cid:19) ρ a m a V B QC , (1)where g γ is a dimensionless model-dependent parameterof O (1) [10–12], α the fine structure constant, f a thePeccei-Quinn energy breaking scale which dictates theaxion mass and coupling strength, m a the axion mass, ρ a the local density of axions, V the cavity volume, B the applied magnetic field, Q the cavity quality factor(assuming the bandwidth is greater than the expectedspread of the axion signal) and C the Haloscope formfactor describing the overlap between the electric fieldcreated by the converted axions and the electric fieldstructure of the resonant mode in the cavity.To date Haloscope searches have excluded some areasof the parameter space [9, 13], with further experimentscurrently under way and future efforts in various stagesof planning [14–16]. All of this work fundamentally relieson Eq. (1) to set constraints on f a and hence the mass XXXXXXXX B0 z Ea z Ba θ z r θ Ec z Bc θ FIG. 1. Sketch of electromagnetic fields inside an axion Halo-scope. The solenoid (gold) produces a static magnetic field,B (cid:126) ˆ z , which interacts with axions to produce an electric field(purple), E a (cid:126) ˆ z Eq. (5), and a magnetic field (magneta), B a (cid:126) ˆ φ Eq. (6). For the TM mode the cavity supports an electricfield (aqua), E c (cid:126) ˆ z , and a magnetic field (blue), B c (cid:126) ˆ φ . of the axion and the strength of axion-photon coupling.In deriving Eq. (1) the coupling of an axion to a photonelectric field is explicitly considered in the form factor,C, and it is then assumed that the corresponding mag-netic field coupling is the same. Recent work describesthe design of a magnetometer detection experiment en-hanced by an LC circuit that utilizes the magnetic fieldcoupling [17]. In this work we consider the coupling ofthe photon magnetic field directly to a Haloscope cavity,and from this we are able to define the complete electro-magnetic form factor for the axion Haloscope. RevisingEq. (1) to incorporate the complete form factor has ma-jor repercussions for the sensitivity of planned searchestargeting high and low frequency axions, including thoseutilizing novel cavity geometries [18, 19].The modified Maxwell’s equations accounting for an a r X i v : . [ h e p - ph ] J un axion field, a , with no spatial dependence in the presenceof a DC magnetic field as shown in fig.1 are given by [7] ∇ · (cid:126)E = 0 ∇ · (cid:126)B = 0 ∇ × (cid:126)E = − ∂ (cid:126)B∂t (2) ∇ × (cid:126)B = 1 c ∂ (cid:126)E∂t − g αγγ (cid:126)B c ∂a∂t , (3)where g αγγ is the strength of axion-photon coupling(equal to g γ α / πf a ), a = a e − jω a t and (cid:126)B = B (cid:126) ˆ z (thesolenoid magnetic field). For this situation (cid:126)E = 0 and ∇ × (cid:126)B = 0, thus Eq. (3) becomes: ∇ × (cid:126)B a = 1 c ∂ (cid:126)E a ∂t , (4)where (cid:126)B a is the magnetic field component of the photonsproduced via the axion-photon coupling. The right handside of Eq. (3) defines the electric field generated by theaxions as (cid:126)E a = − g αγγ a c B (cid:126) ˆ z. (5)Applying Stokes Theorem inside the solenoid, fromEq. (4) and (5) we conclude that (cid:126)B a = − g αγγ c rB ∂a∂t (cid:126) ˆ φ, (6)where r is the distance from the centre of the solenoid inthe radial direction.We now consider a cylindrical cavity with resonantelectric and magnetic mode field structures defined as (cid:126)E c and (cid:126)B c respectively. The electric energy stored in thecavity mode is given byU e = 14 (cid:15) (cid:90) dV c | E c | , (7)while the magnetic energy stored in the cavity mode isgiven by U m = 14 1 µ (cid:90) dV c | B c | . (8)Now we consider the electric and magnetic energy con-verted from axions. From the effective axion electric field(Eq. (5)) the electric energy converted into the cavity isU a,e = 14 (cid:15) g αγγ a c B (cid:90) dV c (cid:126)E c · (cid:126) ˆ z, (9)while from the effective axion magnetic field (Eq. (6)) themagnetic energy converted into the cavity isU a,m = 14 1 µ g αγγ c B ∂a∂t (cid:90) dV c r (cid:126)B c · (cid:126) ˆ φ. (10) By equating the electric energy deposited into the cavityby the axions Eq. (9), with the electric energy storedin the cavity mode, Eq. (7), we can express the electricenergy in the cavity due to axion conversion asU a,e = 12 c (cid:15) g αγγ a B V C E , (11)where we have defined the cavity mode-dependent elec-tric form factor asC E = (cid:12)(cid:12)(cid:12)(cid:82) dV c (cid:126)E c · (cid:126) ˆ z (cid:12)(cid:12)(cid:12) V (cid:82) dV c | E c | . (12)This is the standard axion Haloscope form factor com-monly denoted by C throughout the literature (seeEq. (1)). It has been implicitly assumed that the mag-netic field contributes the same amount of energy and soa factor of two has been applied to Eq. (11) in the past tocalculate the sensitivity of Haloscope experiments. Now,in this work we explicitly consider the magnetic field con-tribution. Proceeding as before, by considering Eq. (8)and (10) we can express the magnetic energy in the cavitydue to axion conversion asU a,m = 12 1 µ g αγγ a B V C B , (13)where we define the cavity mode-dependent magneticform factor as C B = ω a c (cid:12)(cid:12)(cid:12)(cid:82) dV c r (cid:126)B c · (cid:126) ˆ φ (cid:12)(cid:12)(cid:12) V (cid:82) dV c | B c | . (14)It is worth emphasizing that r and φ refer to the solenoidcoordinates, not the cavity. The total electromagneticenergy stored in the cavity will be the sum of the elec-tric and magnetic contributions given by Eq. (11) andEq. (13)U a,em = U a,e + U a,m = 12 1 µ g αγγ B V a rms (C E + C B ) . (15)To obtain the signal power expected on resonance, wemultiply Eq. (15) by ω a Q L such thatP a = 1 µ g αγγ a rms ω a V B Q L C EM = g αγγ ρ a m a V B Q L C EM , (16)(see [20]) here the axion frequency, ω a , is equal to thecavity resonance frequency and C EM is the complete elec-tromagnetic form factor, (C E + C B ) /
2. For the stan-dard Haloscope design, where the cavity resonant TMmode is axially symmetric with respect to the centralaxis of the applied magnetic field, the electric and mag-netic form factors are equal (C E =C B ) and Eq. (16) re-verts to Eq. (1). However, due to the r dependence in F o r m F a c t o r (a) F o r m F a c t o r (b) F o r m F a c t o r (c) F o r m F a c t o r (d) FIG. 2. Electric form factor (green), magnetic form factor (orange), and electromagnetic form factor (blue) as a function ofthe offset of the cavity centre from the solenoid centre, normalized to the cavity radius (e/a). Results are presented for thefollowing modes: (a) TM , (b) TM , (c) TM and (d) TM . C B (Eq. (14)) it is clear that for more general experi-ments with cavities offset from axial symmetry, or withseparated electric and magnetic fields, the value of C E and C B would not be equal, and the complete electro-magnetic form factor would need to be calculated. Forexample, efforts to search for higher frequency axionscan involve power-summing multiple cavities within asolenoid [15, 21]; each of these nominally identical cav-ities will produce different amounts of power dependingupon their relative location within the solenoid. Nowwe explore the sensitivity of a single resonant cavity off-set from the centre of a solenoid, which obviously im-plies that the radius of the cavity is smaller than that ofthe solenoid. Such a setup is most applicable to higherfrequency axion searches, either with a single cavity ormultiple cavities. Considering the radial dependence ofthe magnetic form factor defined in Eq. (14), and un-derstanding that this relates to the radial value of thesolenoid’s natural cylindrical coordinate system, we mustperform a coordinate transform in order for the dot prod-uct of the axion and cavity fields to be physically mean-ingful. This gives the following expression for the mag- netic form factor,C B = ω a c (cid:12)(cid:12)(cid:12)(cid:82) dV c B c φ r − e cos φ (cid:12)(cid:12)(cid:12) V (cid:82) dV c | B c | , (17)where B c φ is the φ component of the cavity magneticfield, r and φ are the solenoid’s natural radial andazimuthal coordinates and e is the offset of the centreof the cavity from the centre of the solenoid. Wecould perform a similar transformation for the electricfield, but for the TM x mode family as employed byHaloscope experiments it is trivial as the z directions ofthe two systems are the same. Form factors for TM x modes were computed numerically as a function of e/a,where a is the cavity radius. Fig. 2 shows plots of themagnetic form factor, C B , the electric form factor, C E ,and the combined electromagnetic form factor, C EM (= C B +C E ), for various TM modes as a function ofe/a. While the traditional electric form factor remainsconstant for all modes as expected, the electromagneticform factor decreases with offset from the centre forthe TM mode, but for higher order modes we seepositions with an increased form factor. For a TM mode, as the cavity moves away from the centre ofthe solenoid, some of the φ direction magnetic fieldof the cavity, which was previously in phase with thesolenoid field is now opposing the magnetic field of thesolenoid (see fig. 1). The cancellation of fields leads toreduced mode overlap and lower sensitivity. However,for higher order modes with alternating in and outof phase magnetic field components, as the cavity isoffset from the centre there are regions where more ofthe cavity magnetic field is in the same direction asthe solenoid field (as it was previously in the oppositedirection when the cavity was central), thus achieving ahigher overlap and a higher form factor. The greatestimprovement can be seen in the TM mode, with asensitivity increase of 75%. Clearly, great care mustbe taken when positioning cavities within the solenoidto avoid reduced sensitivity to axions. Conversely, thisalso opens up the potential to increase the sensitivityof Haloscope axion searches, which use higher ordermodes. Whilst the axion magnetic coupling has beenconsidered before, the implication of this coupling formany current and future proposed axion searches hasnot been explicitly treated. Any cavity-based axiondetection scheme must consider this coupling, as evena slight deviation from perfect central placement ofcavities in a magnetic field will effect the completeelectromagnetic form factor. Furthermore, it is oftenassumed that metal tuning rods which are commonlyemployed in such searches do not significantly alter themode overlap - typically only the electric coupling isconsidered in such discussions and it is clear that themagnetic coupling will be altered in a different way.Any experiments which introduce dielectric media intothe cavity volume will also need to consider the impactof these dielectrics on the electric and magnetic formfactors explicitly. Finally, detection of axions via themagnetic coupling has been proposed through the useof LC circuits as the readout mechanism [17] - as wehave now defined the complete electromagnetic formfactor for cavity-based experiments, we have openedthe possibility of low mass, preinflationary [22] axiondetection via 3D lumped LC resonators, commonlyknown as re-entrant cavities [23–26]. In conclusion, anaxion in the presence of a magnetic field will convertinto a real photon with electric and magnetic fieldcomponents. We have shown how the generated mag-netic field component interacts with a resonant cavitystructure, such as those utilized in axion Haloscopesearches. In doing so we have defined the complete elec-tromagnetic form factor for axion Haloscopes, which hasmajor repercussions for the sensitivity of future axionexperiments searching for both low and high mass axions.The authors thank Ian McArthur for useful discus-sions. This work was supported by Australian ResearchCouncil grants CE110001013 and DP130100205. ∗ [email protected] † [email protected][1] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. , 1440(1977).[2] L. Abbott and P. Sikivie, Physics Letters B , 133(1983).[3] J. Preskill, M. B. Wise, and F. Wilczek, Physics LettersB , 127 (1983).[4] M. Dine and W. Fischler, Physics Letters B , 137(1983).[5] J. Ipser and P. Sikivie, Phys. Rev. Lett. , 925 (1983).[6] L. J. Rosenberg, Proceedings of the Na-tional Academy of Sciences , 1415 (1983).[8] P. Sikivie, Phys. Rev. D , 2988 (1985).[9] J. Hoskins, J. Hwang, C. Martin, P. Sikivie, N. S. Sul-livan, D. B. Tanner, M. Hotz, L. J. Rosenberg, G. Ry-bka, A. Wagner, S. J. Asztalos, G. Carosi, C. Hagmann,D. Kinion, K. van Bibber, R. Bradley, and J. Clarke,Phys. Rev. D , 121302 (2011).[10] J. E. Kim, Phys. Rev. Lett. , 103 (1979).[11] M. Shifman, A. Vainshtein, and V. Zakharov, NuclearPhysics B , 493 (1980).[12] M. Dine, W. Fischler, and M. Srednicki, Physics LettersB , 199 (1981).[13] S. J. Asztalos, G. Carosi, C. Hagmann, D. Kinion, K. vanBibber, M. Hotz, L. J. Rosenberg, G. Rybka, J. Hoskins,J. Hwang, P. Sikivie, D. B. Tanner, R. Bradley, andJ. Clarke, Phys. Rev. Lett. , 041301 (2010).[14] O. K. Baker, M. Betz, F. Caspers, J. Jaeckel, A. Lindner,A. Ringwald, Y. Semertzidis, P. Sikivie, and K. Zioutas,Phys. Rev. D , 035018 (2012).[15] T. M. Shokair, J. Root, K. A. Van Bibber, B. Brubaker,Y. V. Gurevich, S. B. Cahn, S. K. Lamoreaux, M. A.Anil, K. W. Lehnert, B. K. Mitchell, A. Reed, andG. Carosi, International Journal of Modern Physics A , 1443004 (2014).[16] L. H. Nguyen, D. Horns, A. Lobanov, and A. Ringwald,“Wispdmx: A haloscope for wisp dark matter between0.8-2 µ ev,” (2015), arXiv:physics.ins-det/1511.03161.[17] P. Sikivie, N. Sullivan, and D. B. Tanner, Phys. Rev.Lett. , 131301 (2014).[18] R. Seviour, I. Bailey, N. Woollett, and P. Williams, Jour-nal of Physics G: Nuclear and Particle Physics , 035005(2014).[19] G. Rybka, A. Wagner, K. Patel, R. Percival, K. Ramos,and A. Brill, Phys. Rev. D , 011701 (2015).[20] E. J. Daw, A search for halo axions , Ph.D. thesis, MIT(1998).[21] D. S. Kinion,
First results from a multiple microwave cav-ity search for dark matter axions , Ph.D. thesis, UC, Davis(2001).[22] E. Berkowitz, M. I. Buchoff, and E. Rinaldi, Phys. Rev.D , 034507 (2015).[23] Y. Fan, Z. Zhang, N. Carvalho, J.-M. Le Floch, Q. Shan,and M. Tobar, Microwave Theory and Techniques, IEEETransactions on , 1657 (2014).[24] J.-M. Le Floch, Y. Fan, M. Aubourg, D. Cros,N. C. Carvalho, Q. Shan, J. Bourhill, E. N. Ivanov,G. Humbert, V. Madrangeas, and M. E. Tobar, Review of Scientific Instruments , 125114 (2013),http://dx.doi.org/10.1063/1.4848935.[25] N. C. Carvalho, Y. Fan, J.-M. Le Floch, and M. E. To-bar, Review of Scientific Instruments , 104705 (2014),http://dx.doi.org/10.1063/1.4897482.[26] M. Goryachev and M. E. Tobar, New Journal of Physics , 023003 (2015).[27] A. Wagner, G. Rybka, M. Hotz, L. J. Rosenberg, S. J.Asztalos, G. Carosi, C. Hagmann, D. Kinion, K. vanBibber, J. Hoskins, C. Martin, P. Sikivie, D. B. Tan-ner, R. Bradley, and J. Clarke, Phys. Rev. Lett.105