New method of probing an oscillating EDM induced by axionlike dark matter using an rf Wien filter in storage rings

A hypothetical pseudoscalar particle axion, which is an immediate result of the Peccei-Quinn solution to the strong $CP$ problem, may couple to gluons and lead to an oscillating electric dipole moment (EDM) of fundamental particles. This paper proposes a novel method of probing the axion-induced oscillating EDM in storage rings, using a radio frequency (rf) Wien filter. The Wien filter at the frequency of the sidebands of the axion and $g-2$ frequency, $f_{\text{axion}}\pm f_{g-2}$, generates a spin resonance in the presence of an oscillating EDM, as confirmed both by an analytical estimation of the spin equations and independently by simulation. A brief systematic study also shows that this method is unlikely to be limited by Wien filter misalignment issues.

Figure 1

Vertical spin component versus time, with only the dc EDM of ${\eta }_{\mathrm{dc}}={10}^{-6}$ (blue), and only the ac EDM of ${\eta }_{\mathrm{ac}}={10}^{-6}$ (orange).

Figure 2

Fourier spectra of the vertical spin component, with only the dc EDM of ${\eta }_{\mathrm{dc}}={10}^{-6}$ (blue) and only the ac EDM of ${\eta }_{\mathrm{ac}}={10}^{-6}$ (orange). In the simulation the blue peak is located at $f_{g-2}\approx 120\mathrm{kHz}$ and the orange peaks are located at $f_{g-2}\pm f_{\text{axion}}$ where $f_{\text{axion}}=180\mathrm{kHz}$.

Figure 3

Vertical spin component versus time, when an rf Wien filter of $1\mathrm{MV}/\mathrm{m}$ electric field strength was applied at the higher sideband $f_{\text{axion}}+f_{g-2}$ (blue) and lower sideband $f_{\text{axion}}-f_{g-2}$ (orange). The Wien filter is continuously located on the storage ring in the simulation.

Figure 4

Vertical spin component versus time for a proton with a momentum $1\mathrm{GeV}/\mathrm{c}$, when an rf Wien filter of $1\mathrm{MV}/\mathrm{m}$ electric field strength was applied at the higher sideband $f_{\text{axion}}+f_{g-2}$ (blue) and lower sideband $f_{\text{axion}}-f_{g-2}$ (orange). The Wien filter is continuously located on the storage ring in the simulation.

Figure 5

Vertical spin component versus time for different Wien filter azimuthal occupancy: continuously located along the storage ring (blue), taking only 10% of the ring azimuth (orange) and 1% (green).

Figure 6

The electromagnetic field from the rf Wien filter, tilted by an angle $\theta$. Ideally, the electric field of the Wien filter designed for this storage ring should only have a radial component (${\widehat{e}}_x$) and the magnetic field should only have a vertical component (${\widehat{e}}_y$). But this small angle $\theta$ might be there because of the misalignment.

Figure 7

The vertical spin slope driven by the WF misalignment, obtained from the spin tracking simulation (blue circles) and by the analytical expression in Eq. (10) (red curve). Both the dc and ac EDM were set to 0 for the spin tracking. The misalignment angle ${\theta }_m$ is $1\mu \mathrm{rad}$.

Figure 8

The vertical spin component versus time, when there is only the ac EDM component (${\eta }_{\mathrm{ac}}={10}^{-6}$) and the rf Wien filter frequency is the lower sideband $f_{\text{axion}}-f_{g-2}$. The Wien filter is assumed to be perfectly aligned (blue), misaligned by $1\mu \mathrm{rad}$ (orange), $10\mu \mathrm{rad}$ (green) and $100\mu \mathrm{rad}$ (red). The tilted direction is the same in all cases, following Fig. 6.

Figure 9

The fluctuations of the vertical spin component versus time in presence of the random field errors. The blue curve is the reference plot without the field errors. The magnetic field errors up to the octupole ($k=4$), 8th azimuthal harmonics ($N=8$) were implemented, where the relative strength $b_{k,N}/B0$ are chosen to be 1 (orange) and 5 (green) parts-per-million (ppm), respectively. 10 independent series of simulation were conducted for each category, and the fluctuation between the maximum and the minimum values of the vertical spin component at each time bin is plotted.

Figure 10

The projected sensitivity to the axion-gluon coupling strength in the axion parameter space for axionlike dark matter. The colored dashed lines indicate the present study, depending on the spin coherence time ${\tau }_p$: ${10}^3\mathrm{s}$ (red), ${10}^4\mathrm{s}$ (green) and ${10}^5\mathrm{s}$ (blue). It assumes the given values in Eq. (12) for the proton. The integrated measurement time at each frequency is one year. The filled regions that were already excluded by the neutron EDM (nEDM) experiment [32] and the big bang nucleosynthesis [33] are shown as references, as well as the QCD axion band in Eq. (13).

Figure 11

The projected sensitivity to the ALP-nucleon EDM coupling strength ($g_d$) in the ALP parameter space for axionlike dark matter. The colored dashed lines indicate the present study, depending on the spin coherence time ${\tau }_p$: ${10}^3\mathrm{s}$ (red), ${10}^4\mathrm{s}$ (green) and ${10}^5\mathrm{s}$ (blue). It assumes the given values in Eq. (12) for the proton. The integrated measurement time at each frequency is one year. The filled regions that were excluded by excess cooling in SN1987A (light blue) and the static EDM measurement (orange) are adapted from Ref. [6], with proper extension to the range in the present parameter space.

Figure 12

Figure of merit for the statistical sensitivity, the inverse of the statistical uncertainty of ${\omega }_d$, as a function of the single measurement time $T$ divided by the spin coherence time ${\tau }_p$. Two methods are shown; “linear fit” determines the statistical uncertainty from ${\chi }^2$-minimization fit with a linear fit model, and “take-all” defines the slope by taking all data at a specific time $T$. The figure of merit is different for (blue) single measurement and (red) repeated measurement in a given experimental period.