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

The Peccei-Quinn solution to the strong CP problem requires the existence of a pseudo-scalar Goldstone boson called an axion[1]. The axion is a strongly motivated particle beyond the Standard Model and is also a plausible candidate for cold dark matter[2, 3, 4]. While most axion search experiments seek to observe the axion-photon interaction using the resonant cavity method, it has been suggested that axion-gluon coupling in the strong interactions may result in an electric dipole moment (EDM) from a hadron oscillating at the axion frequency[5, 6]. A non-zero EDM of a hadron would require $CP$-violation in strong interactions, so there have been many efforts to measure the EDM of neutrons[7, 8, 9] and plans to measure the EDM of protons and/or deuterons using a storage ring method[10, 11, 12, 13]. Extending the experimental approach to the axion-induced oscillating EDM, a new axion-like dark matter search experiment was proposed using the storage ring method in the presence of an oscillating EDM[14, 15, 16]. Another recent study also proposed that the storage ring EDM method can be exploited to probe dark matter and dark energy[17].

The present paper also proposes using the storage ring method to search for axions, but with a different scheme. We introduce an RF Wien Filter (WF) and resonate the spin by applying the RF at the sidebands of the axion frequency and the $g-2$ frequency. The application of the WF to measure the EDM was studied in Ref. [18], but its target was a conventional static EDM, while the present study seeks to observe an oscillating EDM induced by an axion field. Also, by applying the WF at a frequency other than just $g-2$ frequency, the experiment can be freed from the severe systematics arising from beam and spin dynamics and WF misalignment issues.

The spin of a particle in a storage ring precesses as

where the angular spin frequency $\bm{\upomega}_{s}$ is given by the Thomas-BMT equation[19, 20]:

where $\bm{\upbeta}=\mathbf{v}/c$ is the particle velocity vector. Here $G\equiv(g-2)/2$ is a magnetic anomaly and $\eta$ is a unitless EDM that plays the same role as the $g$-factor in the magnetic dipole moment. The magnetic dipole moment $\mu$ and the electric dipole moment $d$ can be written in forms:

where $q$, $m$ and $S$ are the charge, mass and spin of the particle, respectively.

We work in an accelerator coordinate system $(x,y,s)$, where $x$ is an in-plane radial distance from the design orbit, $y$ is an out-of-plane vertical distance from the center and $s$ is along the arc length of the storage ring. The spin angular frequency has only transverse components $(x,y)$ under a homogeneous and uniform vertical dipole magnetic field and/or radial electric field in the storage ring. Furthermore, we employ a paraxial accelerator approximation $\bm{\upbeta}\cdot\mathbf{B}=\bm{\upbeta}\cdot\mathbf{E}=0$ only for the analytical estimations, but those terms are kept in the numerical simulations and the results were consistent. Accordingly, the radial and vertical components of the spin angular frequency are given by

Here we used the time-varying EDM for a hadron as a result of axion-gluon coupling:

For the analytical calculations of beam and spin dynamics for non-reference particles with non-zero EDM in storage rings, see Refs. [21, 22, 23].

The RF Wien Filter (WF) is a perfect candidate to drive the spin resonance without affecting the beam betatron oscillations, since it exerts no Lorentz force on particles with a specific momentum. The EDM term $\eta(t)$ only contributes to the radial component of the spin angular frequency, so we set the electromagnetic field of the WF as follows.

An artificial spin resonance driven by a WF in the presence of a static non-zero EDM has been well studied[18, 24, 25, 26]. We extend this idea to the oscillating component of the EDM. It is intuitive to expect that the vertical spin component will accumulate in one direction when the oscillation frequency of the EDM, $\omega_{\text{axion}}$, is the same as the WF frequency. Actually, it turns out that it is resonant with the sidebands of the axion and $g-2$ frequency: $\omega_{\text{WF}}=\omega_{g-2}\pm\omega_{\text{axion}}$. It is also true that its sidebands with a cyclotron frequency, for instance $\omega_{c}-(\omega_{g-2}\pm\omega_{\text{axion}})$, are also resonance frequencies, because the WF is normally located in a specific position in the azimuth and the coherent spin motion with respect to the WF will include aliased Fourier components. However, in this paper we will assume the WF is continuously located in the azimuth to simplify the spin equation and solve it analytically.

To see the resonance condition $\omega_{\text{WF}}=\omega_{g-2}\pm\omega_{\text{axion}}$ explicitly, let us solve the spin equations for a reference particle that travels the storage ring in the reference orbit. Let $E_{0}$ and $B_{0}$ be the magnitudes of a constant radial electric field and a vertical magnetic field, respectively, needed to store the particle in the storage ring. Substituting the spin components in Eq. (1) with Eq. (4) and using the electromagnetic field $\mathbf{E}=E_{0}\hat{e}_{x}+\mathbf{E}_{\text{WF}}$ and $\mathbf{B}=B_{0}\hat{e}_{y}+\mathbf{B}_{\text{WF}}$ from Eq. (6) yields

where

is the $g-2$ frequency, and

is the spin angular frequency component driven by the WF fields. Here $a_{\text{WF}}$ is a scaled WF field strength in units of the angular frequency. Finally, $\omega_{\eta}$ is the EDM-related term, namely

where $E^{*}\equiv E_{0}-vB_{0}$ is the effective electric field, which is proportional to the EDM signal. With a highly relativistic beam $v\approx c$, a vertical magnetic field of 1 T provides an effective electric field of roughly $300$ MV/m by itself.

Given the reasonable assumptions $|\omega_{\eta}|\ll|a_{\text{WF}}|$ and $|\omega_{\eta}|\ll|\omega_{g-2}|$, we adopt the strategy used in the section II of Ref. [18]. First we write down the exact solution for the radial and longitudinal spin components without the EDM, then plug those solutions into Eq. (8) to obtain the approximate time derivative of the vertical spin component. Without the EDM term, the radial and longitudinal spin components are given in forms:

where the initial polarization is assumed to be longitudinal $(S_{s}(0)=1)$. Then plugging this $S_{s}$ into Eq. (8) yields

by setting $\phi_{\text{WF}}=0$ for a moment. This equation immediately shows the working principle of the frozen-spin method to measure the static EDM. When the WF is absent ($a_{\text{WF}}=0$) and the spin is frozen ($\omega_{g-2}=0$) then it leads to $\dot{S_{y}}\propto\eta_{\text{DC}}$, therefore, the vertical spin component accumulates with a slope proportional to the DC EDM. The argument is similar when the WF is present. We will use this equation separately for the DC and AC EDM terms to reveal the resonance conditions more explicitly. Recalling the EDM $\eta(t)$ in Eq. (5), it follows that

for the DC EDM term. One can see the resonance condition is $\omega_{\text{WF}}=\omega_{g-2}$, as studied in Ref. [18]. To see it more manifestly, take $\omega_{g-2}t$ in the cosine bracket as a sawtooth wave with a period $2\pi/\omega_{g-2}$. The remaining term of the argument is a sine function with a period $2\pi/\omega_{\text{WF}}$. If the periods of the two terms are not identical, then the phase argument uniformly sweeps from 0 to $2\pi$, which leads to $\langle\dot{S}_{y}\rangle=0$ on average. The resonance happens when the two periods are identical, thus $\omega_{\text{WF}}=\omega_{g-2}$.

Similarly, the AC EDM term reads

Again we set $\phi_{\text{axion}}=0$ for clarity. Restoring the WF phase and the axion phase in the above equations are straightforward, but of no importance at this moment. Using the same argument we made for the DC EDM term, we can clearly see that there are two resonance conditions: $\omega_{\text{WF}}=\omega_{g-2}\pm\omega_{\text{axion}}$. When the WF is absent, the resonance happens when $\omega_{g-2}=\omega_{\text{axion}}$ which is used in Ref. [14].

We point out that in the presence of the WF, the average slope of the vertical spin component $\left\langle\dot{S}_{y}\right\rangle$ in the resonance condition is multiplied by the following factor:

where we dropped the subscript WF in the right hand side for a moment. Interestingly, it shows that $C_{\text{WF}}=-J_{1}(a/\omega)$ where $J_{1}$ is the Bessel function of the first kind. See App. A for the derivation. The maximum absolute value of $C_{\text{WF}}$ is therefore roughly 0.59 when $a/\omega\approx 1.84$. Eventually, we have the following expression for the vertical spin slope (EDM signal) on resonance.

An overview of the various methods used to measure EDM in storage rings, including this work, is provided in Table 1.

In this section a spin tracking simulation for the reference particle was performed as a proof-of-principle. A deuteron, whose magnetic anomaly is $G_{d}=-0.143$, was simulated in a purely magnetic ring with a radius of 30 m. The reference momentum was given to be 1 GeV/c. The corresponding $g-2$ frequency is around 121 kHz. Also, the axion frequency was arbitrarily assumed to be 180 kHz, which corresponds to an axion mass of 0.7 neV/c². The initial setup parameters are summarized in Table 2.

First, the simulation was conducted without applying the WF. Fig. 1 shows the vertical spin component ($S_{y}$) of the reference particle in the presence of the EDM, with blue when there is only the DC component of the EDM (with $\eta_{\text{DC}}=10^{-6}$) and orange for the AC component only ($\eta_{\text{AC}}=10^{-6}$). Fig. 2 shows their Fourier spectra. One can immediately see $\eta_{\text{DC}}$ is responsible for the $g-2$ frequency and $\eta_{\text{AC}}$ is for its sidebands with the axion frequency: $f_{\text{axion}}\pm f_{g-2}$. The peak at the higher frequency sideband, $f_{\text{axion}}+f_{g-2}\approx 300$ kHz, clearly looks much smaller than the other sideband, $f_{\text{axion}}-f_{g-2}\approx 60$ kHz. The reason is quite straightforward. As implied in Eq. (18), without the WF ($a_{\text{WF}}=0$), one gets the time derivative of the vertical spin component asymptotically proportional to $\cos(\omega_{\text{axion}}t)\cos(\omega_{g-2}t)$. Integrating this expression, one obtains two sidebands whose amplitudes are divided by each frequency. In this case, the ratio between the higher and lower sidebands is roughly 5, which is the magnitude ratio of peaks for the two sidebands in Fig. 2.

Next, we want to observe the vertical spin resonance in the presence of the WF operating at one of the sidebands of the axion and $g-2$ frequency. The WF was assumed to occupy all parts of the storage ring continuously, and its electric field strength was set to 1 MV/m. As Fig. 3 represents, the spin vertical component accumulates in one direction when the WF is applied at either one of the two sidebands. The directions of the accumulation of the spin vertical component for two cases are opposite; they depend on the sign of $\omega_{\text{axion}}\pm\omega_{g-2}$ and $\omega_{\text{WF}}$. And the slope depends on the sideband frequency like the previous argument as well: $\langle\dot{S}_{y}\rangle\propto 1/\omega_{\text{WF}}$.

So far we have used a deuteron for the simulation. It is worth trying a proton as well, which has a magnetic anomaly $G_{p}=1.793$. Its mass is around 938 MeV/c², almost half that of the deuteron. Assuming the same reference momentum $p=1$ GeV/c and a WF electric field strength of 1 MV/m, the corresponding $g-2$ frequency for the proton becomes 3 MHz, which is 25 times larger than that of the deuteron. As a result, the vertical spin component growth rate for the proton is around an order of magnitude smaller than that of the deuteron, as shown in Fig. 4.

We also have briefly tested the WF taking only a fraction of the storage ring azimuth, instead of continuously located along the ring. Figure 5 shows the vertical spin component on resonance with different WF azimuthal occupancy, which is consistent with the natural expectation; the slope scales the same with the WF occupancy.

The statistical sensitivity of the oscillating EDM has been derived in a rigorous manner in App. B.

where $P_{0}$ is the initial beam polarization, $A$ is the analyzing power, $E^{*}$ is the equivalent electric field in the storage ring, $C_{\text{WF}}$ is the coefficient determined by the WF performance, $\kappa$ is the polarimeter efficiency, $N_{\text{cyc}}$ is the number of stored particles in one cycle (single measurement), $T_{\text{exp}}$ is the total experimental period and $\tau_{p}$ is the spin coherence time. Plugging in the typical experimental numbers in the ideal situation, one obtains

for the case of the proton. This calculation is done assuming we can obtain a large $C_{\text{WF}}$ for all targeting frequencies in the axion parameter space. For the realistic case, $C_{\text{WF}}$ is more restrictive depending on the WF performance. For instance, $a_{\text{WF}}$ should be proportional to the azimuthal fraction the WF occupies in the storage ring. But we do not cover the technical difficulties and details to achieve the maximum $C_{\text{WF}}$ in this paper.

In a search for axionlike dark matter, we can exclude the axion-gluon coupling parameter space with a sensitivity proportional to $\sigma_{d}$. The QCD axion has the relationship[31]

where $C_{G}$ is a model-dependent dimensionless coefficient of the axion-gluon coupling Lagrangian and $f_{a}$ is the symmetry breakdown scale. Exploiting this relation, the sensitivity to $(C_{G}/f_{a})$ that we can exclude from the parameter space if the EDM is not discovered with the uncertainty $\sigma_{d}$, is given as

where $d_{n}^{\text{QCD}}\approx 9\times 10^{-35}\cos(m_{a}t)$ [$e\cdot$ cm] holds for the QCD axion[6].

The projected sensitivity on $C_{G}/f_{a}$ is shown in Fig. 10, indicated by the colored dashed lines. These were obtained from Eq. (33) using Eq. (31) for the given values within, except the spin coherence time. The minimum axion frequency that one can scan is determined by a single measurement time, as long as it is free from systematic effects. Therefore, a higher spin coherence time not only improves the sensitivity at a given frequency but also widens the scanning area. The optimum single measurement time to minimize the statistical uncertainty of the repeated measurement is shown to be $\tau_{p}/2$, as derived in App. B. There are kinks, after which the projected sensitivity starts to decrease fast. This is because the single measurement time $T$ has to be smaller than $\tau_{p}/2$ after this point, as the axion phase decoheres faster than $\tau_{p}/2$. Explicitly, $T$ is given by

where $Q_{\text{axion}}$ is the axion quality factor, which was assumed to be $10^{6}$[32, 33].

On the other hand, the parameter space for the new coupling between the axion-like particles (ALPs) and the nucleon, $g_{d}$, can be scanned. This coupling, which is directly responsible for the oscillating nucleon EDM, appears in the Lagrangian[6]

where $a$ is the ALP field interacting with the nucleon $N$. Assuming the ALP makes up all of the local dark matter with a density $\rho_{\text{DM}}\approx 0.3\text{ GeV/cm${}^{3}$}$, the nucleon EDM is given by[6]

Figure 11 shows the projected sensitivity to the EDM coupling $g_{d}$. The constraints from the excess cooling in SN198A and the static EDM measurement are provided in Ref. [6], where we have set the lower bound of the constraint from the static EDM to 1/130 Hz as the experimental time for a single shot was $t_{\text{shot}}=130$ seconds[34].

Recently, there was a study offering new predictions of the ALPs coupling, called ALPs cogenesis[35]. Its projected region in the parameter space was above the QCD axion band, strengthening the motivation for scanning the axion parameter space above the QCD axion band.

Employing an RF Wien Filter in the storage ring EDM method provides a powerful method for probing an oscillating EDM. We revealed that vertical spin resonance happens when the Wien Filter is operating at the frequency of the sidebands of the axion and the $g-2$ frequency, $f_{\text{axion}}\pm f_{g-2}$. The approximated analytic solution for the spin resonance agreed well with the simulation results. Scanning the axion frequency would be straightforward, easily performed by tuning the Wien Filter frequency. This method avoids the large systematic effect that arises when the Wien Filter frequency is close to the $g-2$ frequency, as confirmed by both analytical calculations and spin tracking simulations. Even though when the Wien Filter frequency is close to the $g-2$ frequency to search for low-mass axion ($f_{\text{axion}}<1$ Hz), the systematic effect from Wien Filter misalignment is well understood and can be corrected precisely.

A systematic effect from random field errors was also studied, which did not show a critical influence on the vertical spin component at least up to 5 ppm. Nonetheless, further intensive numerical studies are necessary under more realistic lattice and beam conditions to understand the details of all systematic effects before conducting an experiment.

This particular idea of introducing an RF Wien Filter to look for the axion-induced oscillating EDM might be of interest for frozen-spin proton and deuteron EDM experiments in storage rings, because it allows physics probing data to be taken simultaneously from the static DC EDM and the oscillating AC EDM. This method can be applied to existing storage rings, e.g., the muon $g-2$ experiment at Fermilab[36], by storing polarized proton or deuteron beams.