Comment on "Axion Dark Matter Coupling to Resonant Photons via Magnetic Field"
aa r X i v : . [ h e p - e x ] J un Comment on “Axion Dark Matter Coupling to Resonant Photons via Magnetic Field”
Sangjun Lee,
1, 2
Sung Woo Youn, and Y. K. Semertzidis
1, 2 Dept. of physics, KAIST, Daejeon 34141 Republic of Korea Center for Axion and Precision Physics Research, IBS, Daejeon 34141 Republic of Korea (Dated: June 29, 2016)
A recent Letter [1] claims that for typical dark mat-ter axion search experiments using cylindrical haloscopes,the power gain depends on the relative position of a cav-ity with respect to the center of a solenoidal magneticfield due to different electric and magnetic couplings. Wereview this Letter and find a misinterpretation of the co-ordinate system. We correct this and see no dependenceof the coupling strength on the cavity location and theelectric and magnetic energies stored in a cavity modeare equal.From the modified Maxwell’s equations in the presenceof axion-photon coupling with the background axion field a = a e − iω a t , the electromagnetic field components ofphotons produced via the axion-photon conversion insidethe solenoid producing homogeneous magnetic field ~B = B ˆ z are obtained as ~E a = − g aγγ acB ˆ z, ~B a = − g aγγ c rB ∂a∂t ˆ φ, (1)where g aγγ is the axion-photon coupling constant and( r, φ ) is the polar coordinate system of the solenoid. Since ~E a and ~B a are determined by the boundary condition ofthe solenoid, they do not depend on the location of thecavity.Denoting by ~E c and ~B c the electric and magnetic fieldsof a resonant cavity mode, the electric and magnetic en-ergies stored in the cavity mode from the axion-to-photonconversion are given by U a,e = 14 ǫ g aγγ a c B V C E , (2) U a,m = 14 ǫ g aγγ a c B V C B , (3)with the electric and magnetic form factors being defined,respectively, as C E = (cid:12)(cid:12)(cid:12)R dV c ~E c · ˆ z (cid:12)(cid:12)(cid:12) V R dV c | E c | , C B = ω a c (cid:12)(cid:12)(cid:12)R dV c r ~B c · ˆ φ (cid:12)(cid:12)(cid:12) V R dV c | B c | , (4)where the integration is over the cavity volume.At a first glance, C B in Eq. 4 would produce differ-ent results when the cavity is no longer coaxial with the solenoid and has some offset e due to the appar-ent radial dependence ( C E remains unchanged since ~E a is homogeneous inside the solenoid). However, in thepresence of the offset, ~B c lies in the azimuthal direc-tion in the cavity coordinates, ˆ φ c , while r and ˆ φ inthe integrand of C B remain the same in the solenoidcoordinates since these are due to the ~B a field. Thuswe evaluate the dot product between ˆ φ c and ˆ φ , i.e.,ˆ φ c · ˆ φ = cos( φ + φ c ) = − [ r c − e cos φ c ] /r . Then C B inEq. 4 becomes C B = ω a c (cid:12)(cid:12)(cid:12)R dV c B c φc r c − e cos φ c (cid:12)(cid:12)(cid:12) V R dV c | B c | . (5)Recalling the corresponding expression in the Letter, C B = ω a c (cid:12)(cid:12)(cid:12)R dV c B c φ r − e cos φ (cid:12)(cid:12)(cid:12) V R dV c | B c | , (6)we find the radial and azimuthal coordinates are inter-preted in the solenoid system rather than the cavity sys-tem, which in turn results in C B varying with the offset.It must be emphasized that r c and φ c are the polar co-ordinates of the cavity. Now the cos φ c term in the inte-grand of Eq. 5 vanishes after integration over the cavityvolume. Therefore, it is found that C B has no position(offset e ) dependence in the solenoid and has the samevalue as C E , e.g., C B = C E = 0 .
69 for the TM mode.This work was supported by IBS-R017-D1-2016-a00/IBS-R017-Y1-2016-a00. We also thank S. H. Changat the Center for Theoretical Physics of the Universe(CTPU) of Institute for Basic Science (IBS) for his con-firmation of our finding. [1] B. T. McAllister, S. R. Parker, and M. E. Tobar, “AxionDark Matter Coupling to Resonant Photons via MagneticField”, Phys. Rev. Lett.116