Axion Dark Matter Search Using the Storage Ring EDM Method

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Yannis and − a − h rvosypooe trg igEMexper- EDM ring storage proposed previously The I EOAC FAININDUCED AXION OF RESONANCE II. rcsinfeuny hsmto sexpected is method This frequency. precession 2 ag f10 of range SILTN D WITH EDM OSCILLATING rqec,w utcnrladtn tt be to it tune and control just we frequency, 2 RCSINI TRG RINGS STORAGE IN PRECESSION d DC − d 9 rainlk atce (ALPs). particles like axion or ( t zt 0 H a npicpebe principle in can MHz 100 to Hz dEfilst rdc resonance a produce to E-fields nd and = ) 2 xo akmte nue EDM induced matter dark axion n301 eulco Korea of Republic 34051, on 13 nKim, On d d DC AC GeV 13 GeV + r h antdso h static the of magnitudes the are ≤ d AC ,2 1, f ≤ a ilto ogreatly to cillation rea cos( ≤ f ,2 1, a 10 ≤ m 30 a 10 t e,the GeV, + 30 g 3 − ϕ GeV. ,testored the s, − ax 2 9 ) SPIN , zu to up Hz ϕ (3) ax and B fields is expressed by the T-BMT equation In this idea of resonant axion induced EDM with Eq. (4)-(6) [22, 23]. g − 2 spin precession, one can utilize the strong effective electric field E~∗ = E~ + cβ~ × B~ , which comes from the B field due to particle motion, as expressed ~ω = ~ωa + ~ωd, (4) in Eq. (6). In this case, the effective electric field is about one or two orders of magnitude larger than the applied external E field which can be up to 10 MV/m and has an apparent technical limitation in e 1 β~ × E~ ~ω = − aB~ − a − , (5) strength. a m γ2 − 1 c " #

e η E~ III. SENSITIVITY CALCULATION ~ωd = − + β~ × B~ , (6) m " 2 c !# The statistical error in the EDM for proton or where a = (g − 2) /2 is the magnetic anomaly with deuteron can be expressed with the following equa- a = −0.14 for deuterons. Here, other terms are tion [19, 20]. omitted by assuming the conditions β~·E~ = β~·B~ = 0. The parameter η shown in the equation is related 2~s ~ σd = ∗ , (7) to the electric dipole moment d as d = ηe /4mc. P AE NcκTtotτp Since we are dealing with a time varying EDM due to the oscillating axion background field, η is also a p where P is the degree of polarization, A the analyz- function of time. The ~ωa is the angular frequency, ∗ i.e., 2π times the g −2 frequency, describing the spin ing power, E the effective electric field that causes precession in the horizontal plane relative to the mo- the EDM precession, Nc the number of particles mentum precession. stored per cycle, κ the detection efficiency of the polarimeter, τ is the polarization life time, and T The term ~ω is due to the EDM and the corre- p tot d the total experiment running time. The s in the sponding precession takes place in the vertical plane. numerator is 1/2 for protons and 1 for deuterons. For a time independent nucleon EDM, the spin vec- One can easily calculate the sensitivity of nucleon tor will precess vertically for the duration of the stor- EDM measurement using this formula with the cor- age time if the horizontal spin component is fixed to responding experimental parameters. the momentum direction [20]. This condition can be achieved by setting the E and B fields properly, and In this study, we used the following method to cal- is called the frozen spin condition. culate the sensitivities of the axion EDM measure- With a nonzero g − 2 frequency, the average EDM ment including the oscillation effect. First, we chose precession angle becomes zero for the static EDM the target axion frequency and then calculated the case because the relative E field direction to the spin corresponding E and B-fields for the particle stor- vector changes within every cycle of g−2 precession. age, which give the same g − 2 frequency as the cho- For example, the spin tilts in one direction (up or sen axion frequency. Then, the axion oscillation and down) due to the EDM within one half cycle and g − 2 spin precession will be on resonance and the then tilts in the opposite direction for the other half EDM precession angle in the vertical plane can keep cycle, resulting in an average accumulation of zero. accumulating during the measurement time. With The presence of a static EDM will only slightly tilt the chosen axion frequency and the axion quality the g − 2 precession plane away from the horizon- factor Qax, we estimated the statistical error and tal plane, without a vertical spin accumulation. In the resulting error was used to calculate the ex- contrast, for an oscillating EDM, when the axion periment sensitivity along with the effective electric frequency (ωax) is the same as the g − 2 frequency field. This method can be used not only for nucleons with the appropriate phase, the precession angle can like deuterons or protons but it can also be used for be accumulated in one direction. This is possible be- other leptonic particles like muons, provided there cause the EDM direction flips every half cycle due to is a coupling between the oscillating θQCD induced the axion filed oscillation and the relative direction by the background axion dark matter field and the between the E field and the EDM d always remains particle EDM. the same. As shown in Eq. (6), the EDM part of the preces-

2 sion rate can be rewritten as Eq. (8) R ≡ N(ψ, 0) = nNA∆ΩζI0(ψ)[1 − PyAy(ψ)], (13)

dθ d ∗ respectively. After simple algebra one can obtain the ωd = = − E , dt s~ (8) left-right asymmetry ǫ for the vertically polarized ∗ E = E + cβB, beam as follows

where s is 1/2 for protons and 1 for deuterons. Ac- L − R s~ ǫ(t)= (t)= P Aθ(t), (14) cordingly, d = E∗ ωd, and the error for the EDM d L + R can be written as Eq. (9). where A is the analyzing power, θ(t) is the accu- s~ mulated EDM precession angle in the vertical plane σ = σ , (9) d E∗ ωd and L and R are the number of hits on the left and right detectors, respectively. As mentioned earlier, where σω is the error for ωd. It can be obtained d one can get the EDM precession rate ωd from the from the fit of the vertical precession angle θ as a asymmetry ǫ(t). function of time. For error estimation, we assumed an axion qual- The time variation of θ(t) will be obtained from 6 ity factor of Qax = 3 × 10 . From the Qax value, the asymmetry ǫ(t) measurement (explained below) one can calculate the possible measurement time using a polarimeter [24–28]. There is currently no tm = Qax/fax (coherence time between the axion nondestructive way to measure the spin direction of oscillation and g − 2 spin precession), where fax is particles with large sensitivity. For the hadronic par- the axion frequency [11]. If this time tm is larger ticle case, one can utilize the nuclear interactions be- than the polarization life time tpol (or spin coher- tween the spin polarized particles and target nuclei. ence time, SCT), then we set the measurement time In this case, the spin-orbit interaction between the to tpol. Otherwise, the measurement time is set to spin polarized incident particle and target nucleus is Qax/fax. Furthermore, in some models the oscil- one of the major reactions that gives asymmetrical lating axion field is monochromatic with a quality scattering of the incident particle in the azimuthal factor in excess of 1010; see references [30, 31] and angle [29]. Carbon is one of the most efficient tar- references therein. As shown in Table I, with this get materials with large analyzing power for both high Qax value one can reach very high sensitivities deuterons and protons. up to 10−31∼−32 e·cm under the said assumptions. For example, the scattering cross section for a spin In resonance microwave cavity experiments, the 1/2 polarized particle (e.g., a proton) can be written conversion power of axion to photon is limited by as Eq. (10) [25] the cavity quality factor QL. Therefore, there is no large benefit from the high axion Qax value if the I(ψ, φ)= I0(ψ) [1 + pyAy(ψ)] , (10) cavity QL is smaller than the axion Qax value. The where I0(ψ) is the cross section for an unpolarized current cavity experiment assumes the axion Qax 6 particle scattered into the angle ψ, Ay(ψ) is the ana- to be about a few times 10 and the cavity QL is lyzing power of the reaction, and py is the transverse usually smaller than this. However, the proposed component of the beam polarization. From Eq. (10), storage ring method can obtain a large benefit from the number of hits recorded in a detector located at the large Qax value since the spin tune stability in (ψ, φ) can be written as follows the storage ring becomes very large and the sensitivity is even more enhanced with higher Qax values, N(ψ, φ)= nNA∆ΩζI(ψ, φ), (11) as shown in Tables I,II. For example, it was experi- mentally measured at COSY that the spin tune was where n is the number of particles incident on the controllable at the precision level of 10−10 for a con- target, NA is the number of target nuclei per square tinuous 102 s accelerator cycle time [32]. centimeter, ∆Ω is the solid angle subtended by the detector and ζ is the efficiency of the detector. If we assume identical detectors which are placed at A. Pure magnetic ring symmetrical locations on the left and right of the beam direction, the counts recorded on the left and The effective electric field is an important param- right are eter in the sensitivity estimation. As shown in Eq. (6), the E~ ∗ is the vector sum of the radial E~ field L ≡ N(ψ, 0) = nNA∆ΩζI0(ψ)[1 + PyAy(ψ)], (12) and ~v × B~ . For this study, we started with a pure

3 Momentum (GeV/c) for r= 10 m B field (T) 3 1

2.5 0.8 y 2 0.6

1.5

B field (T) 0.4 E*=E-vB 1 Momentum (GeV/c) 0.2 B 0.5

× 3 × 3 0 10 0 10 0 200 400 600 800 1000 0 200 400 600 800 1000 g-2 frequency, f g-2 frequency, f g-2 g-2 E Kinetic Energy (MeV) Sensitivity (e.cm) 1600

− z,v 10 28 x 1400 1200 (a) The coordinate of B, E and v 1000 − 10 29 800 g-2 frequency (P=1 GeV/c, B=0.38 T) ×103 600 Sensitivity (e.cm) 400 Kinetic Energy (MeV) − 10 30 400 200

0 0 0.5 1 1.5 2 2.5 3 105 106 Momentum (GeV/c) g-2 frequency, f g-2 300

FIG. 1: Deuteron sensitivity vs. g − 2 frequency 200 (Hz) g-2 (or axion frequency). Purely magnetic ring only f with ring bending radius r = 10 m. 100

×103 magnetic ring first. We assumed the ring bending 0 2000 4000 6000 8000 10000 radius to be r = 10 m. In order to tune the g − 2 E (V/m) frequency fg−2 (or axion frequency on the resonance (b) g − 2 frequency vs E field condition), the B field was varied and the momentum was also changed accordingly to keep the ring FIG. 2: E/B combined ring for g − 2 frequency bending radius unchanged. The bottom right plot tunning in Fig. 1 shows the sensitivity as a function of the g − 2 frequency and Eq. (7) is used for the calculation. As can be seen in the plot, the experiment is more sensitive at high frequencies. This is because and an exponential function reflecting polarization the larger B field provides a larger effective E field decay. From the fit, we obtained the error for the precession frequency, σ . The fit error was inserted by ~v × B~ . Below ∼ 105 Hz, the sensitivity decreases ωd − into Eq. (9) to calculate the error of the EDM d. beyond ≥ 10 29 e· cm. We decided to use the E and B field combined ring for the low frequency region An example of the simulation result with the fit to improve the sensitivity in that range. is shown in Fig. 3. In the simulation, 40% of the particles were extracted in the early 10% of measurement time (or storage time) and another 40% was extracted in the late 10% of measurement time. B. E/B combined ring The remaining 20% of the particles was extracted in the middle 80% measurement time. The middle 80% The B field was set to 0.38 T and the E field was of time can be used to measure the g-2 frequency. applied to the radial direction (radially outwards in- The calculated sensitivities are also shown in Table dicates positive direction) as shown in Fig. 2a. Fig. I for the deuteron case and are all about 10−30 e·cm 2b shows the g −2 frequency as a function of the ap- or less. plied E field. To increase the frequency, the E field The polarimeter efficiency and the average ana- has to be reduced. However, with this E field change, lyzing power used for the sensitivity calculation were the ring radius changes as well. In order to keep 2% and 0.36, respectively, for the deuteron case (Ta- the ring radius unchanged (r = 10 m for deuteron ble I). The numbers are for the elastic d-C reaction in this example), the momentum is adjusted accord- (exclusive reaction) measured at the deuteron en- ingly. Some examples of the experimental conditions ergy of 270 MeV (p=1042 MeV/c) [33]. The ana- are listed in Table I. lyzing power is a function of particle energy and one Using the parameters shown in Table I, we gener- should avoid the energies that have small analyzing ated the asymmetry data and the data was fit to a powers. For a frequency of 1 MHz, the required mo- function which is a combination of a linear function mentum is about 2.8 GeV/c (T ∼ 1.5 GeV) for a

4 TABLE I: Examples of experimental parameters for frequency tuning, and results of sensitivity calculation (Deuteron). The analyzing power was assumed to be A =0.36 if the momentum P was below 2 GeV/c and A=0.15 was used for the momentum P>2 GeV/c. The ring bending radius was 10 m. The polarimeter eﬃciency was assumed to be 2% and initial polarization was 0.8. The axion quality factors: 6 10 Qax1 =3 × 10 , Qax2 = 10 .

Sensitivity (e·cm) 3 4 5 B (T) P (GeV/c) fg−2 Er (V/m) E* (V/m) SCT = 10 s SCT = 10 s SCT = 10 s Qax1 Qax2 Qax1 Qax2 Qax1 Qax2 0.3800 0.9428 1.E1 8.82E6 4.23E7 9.9E-31 9.9E-31 3.1E-31 3.1E-31 9.9E-32 9.9E-32 0.3800 0.9429 1.E2 8.82E6 4.23E7 9.9E-31 9.9E-31 3.1E-31 3.1E-31 1.4E-31 9.9E-32 0.3800 0.9433 1.E3 8.80E6 4.24E7 9.9E-31 9.9E-31 4.3E-31 3.1E-31 3.8E-31 9.9E-32 0.3800 0.9473 1.E4 8.65E6 4.27E7 1.4E-30 9.9E-31 8.3E-31 3.1E-31 1.2E-30 9.9E-32 0.3800 0.9880 1.E5 7.05E6 4.60E7 3.5E-30 9.1E-31 3.5E-30 2.9E-31 3.5E-30 9.1E-32 0.3800 1.0345 2.E5 5.06E6 5.00E7 4.6E-30 8.4E-31 4.5E-30 2.7E-31 4.5E-30 9.8E-32 0.3800 1.1326 4.E5 3.47E5 5.85E7 5.5E-30 7.2E-31 5.5E-30 2.3E-31 5.5E-30 5.2E-32 0.3800 1.2386 6.E5 -5.47E6 6.82E7 5.8E-30 6.2E-31 5.3E-30 2.0E-31 5.8E-30 1.1E-31 0.3800 1.3546 8.E5 -1.26E7 7.93E7 5.7E-30 5.3E-31 3.9E-30 1.7E-31 5.7E-30 1.0E-31 0.3800 1.4836 1.E6 -2.14E7 9.20E7 5.5E-30 4.6E-31 3.5E-30 1.4E-31 5.5E-30 1.0E-31 0.8000 2.5124 1.E6 -9.13E6 2.01E8 1.6E-30 1.5E-31 2.5E-30 6.6E-32 2.5E-30 4.6E-32 0.9198 2.7574 1.E6 0.0 2.28E8 5.3E-30 4.4E-31 3.4E-30 1.4E-31 5.3E-30 9.7E-32 9.1977 27.5740 1.E7 0.0 2.75E9 3.3E-30 3.7E-32 4.4E-30 2.5E-32 4.4E-30 2.4E-32

TABLE II: Examples of experimental parameters for frequency tuning, and results of sensitivity calculation (Proton). The analyzing power was assumed to be A =0.6 for the momentum P<1 GeV/c and A =0.25 was used for the momentum P>1 GeV/c. The ring bending radius was 52 m for the E/B combined ring, and r = 10 m for the pure magnetic ring. The polarimeter eﬃciency used was 2% and initial polarization 6 10 was 0.8. The axion quality factors: Qax1 =3 × 10 , Qax2 = 10 .

Sensitivity (e·cm) 3 4 5 B (T) P (GeV/c) fg−2 Er (V/m) E* (V/m) SCT = 10 s SCT = 10 s SCT = 10 s Qax1 Qax2 Qax1 Qax2 Qax1 Qax2 0.00011 0.6984 1.E1 -8.0E6 8.02E6 1.6E-30 1.6E-30 5.0E-31 5.0E-31 1.6E-31 1.6E-31 0.00010 0.6984 1.E2 -8.0E6 8.02E6 1.6E-30 1.6E-30 3.6E-31 3.6E-31 2.2E-31 1.6E-31 0.00008 0.6982 1.E3 -8.0E6 8.01E6 1.6E-30 1.6E-30 6.9E-31 5.0E-31 6.1E-31 1.6E-31 -0.00015 0.6960 1.E4 -8.0E6 7.97E6 2.2E-30 1.6E-30 1.9E-30 4.1E-31 1.9E-30 1.6E-31 -0.00243 0.6747 1.E5 -8.0E6 7.57E6 6.4E-30 1.7E-30 4.9E-30 5.3E-31 6.4E-30 1.7E-31 -0.00495 0.6519 2.E5 -8.0E6 7.15E6 9.6E-30 1.8E-30 9.5E-30 5.6E-31 6.7E-30 2.0E-31 -0.01523 0.7103 4.E5 -1.1E7 8.24E6 1.2E-29 1.1E-30 1.2E-29 4.8E-31 1.2E-29 2.3E-31 -0.02002 0.6711 6.E5 -1.1E7 7.51E6 1.6E-29 1.7E-30 1.4E-29 5.3E-31 1.6E-29 2.9E-31 -0.02666 0.6643 8.E5 -1.2E7 7.38E6 1.8E-29 1.7E-30 1.8E-29 5.4E-31 1.8E-29 3.4E-31 -0.03327 0.6583 1.E6 -1.3E7 7.27E6 2.1E-29 1.7E-30 2.1E-29 5.5E-31 2.1E-29 3.8E-31 0.36587 1.0968 1.E7 0.0 8.33E7 4.4E-29 3.6E-31 4.4E-29 1.9E-31 4.4E-29 2.4E-31 3.65868 10.9684 1.E8 0.0 1.09E9 2.3E-29 3.9E-32 3.4E-29 5.8E-32 3.4E-29 5.8E-32 magnetic field of 0.92 T, as shown in Table I. In the well and the results are shown in Table II. For the literature [34] one can find corresponding analyzing proton, we assumed the ring bending radius to be powers for the inclusive d-C reaction to be maximum 52 m. This is the ring radius value suggested for the about 0.15 within the angle range we are interested static EDM measurement by the storage ring proton in, 5o −20o. This value is still practically useful, but EDM collaboration [20]. The proton has a larger beyond this momentum the analyzing power might magnetic anomaly (G=1.79) than the deuteron and be too small to be used for the deuteron polarization its mass is half that of the deuteron. Therefore, its analysis. spin precession rate in the magnetic field is about 26 times more sensitive than the deuteron case. For We repeated the estimations for the proton case as

5 5 − Axion Frequency : 10 Hz on the relative phase between the initial spin and ×10 6 3 h1 Entries 1000 axion phase ϕax. Fig. 4 shows the eﬀect of initial Mean 19.91 2 Std Dev 7.028 phase on the vertical spin precession (EDM eﬀect). χ2 / ndf 1066 / 999 1 The parameters used in the simulation are for an Prob 0.06976 5 p1 −1.257e−07 ± 8.779e−10 axion frequency of 10 Hz, which is shown in Table 0 I. The spin tracking was done by integrating the fol- −1 L-R

L+R lowing two equations [22, 23] for spin and velocity, −2 respectively.

−3

−4 d~s e g γ − 1 −5 = ~s × − B~ dt m 2 γ 0 5 10 15 20 25 30 " Time (s) g γ − − 1 β~ · B~ β~ 2 γ +1 FIG. 3: Simulation result of asymmetry vs. time g γ β~ × E~ with fit. − − 2 γ +1 c η E~ γ β~ · E~ this reason, small magnetic fields have to be used + β~ × B~ + − β~ , 2 c γ +1 c for a low axion frequency scan. Because of the small ! # magnetic field (contributing to effective electric field (15) E∗), the calculated sensitivities are not as good as in the deuteron case. However, the resulting sensi- ~ ~ ~ · ~ tivities are still comparable to the static EDM case dβ e ~ ~ E β E ~ −29 = β × B + − β . (16) (∼ 10 e·cm). Instead of varying the electric field dt γm " c c # strength (as was done for the deuteron case), the magnetic field was changed to modify the g − 2 fre- As can be seen in Fig. 4a, depending on the initial quency. We changed the momentum as needed to axion phases, the accumulated vertical EDM preces- keep the same ring radius. However, we kept the sion rates are different and random. However, this proton kinetic energy at around 200 MeV (P ≥ 650 random axion phase issue can be resolved using two MeV/c) to keep a large analyzing power from the po- (or four) orthogonally set spin polarizations. larimeter detector. The average analyzing power for An example of four orthogonally set polarizations a proton energy of about 200 MeV is about 0.6 [35] are presented in Fig. 4c. Fig. 4b shows the ef- and this value was used in the sensitivity estimation fect of four orthogonally set spin polarizations on as shown in Table II. the ωEDM for the case of ϕax = π/8. From the The last two rows in Table II are for the presence measurement of individual spin polarization, one can of B-field only and the ring bending radius used was calculate the total EDM precession rate using the re- r = 10 m. High frequency (≥ 107 Hz) can be eas- 2 2 lationship, ωEDM = ω + ω , where ily reached at small B-fields. However, for a con- EDM,S1 EDM,S2 ω and ω are two orthogonally set po- stant ring bending radius, the momentum has to be EDM,S1 EDM,Sq2 larizations. The corresponding axion phase can be changed. When the proton case is combined with the ωEDM,S2 obtained by ϕax = arctan( ). As can be seen deuteron results shown in Table I, one can perform ωEDM,S1 measurements from 10−9 Hz to 100 MHz using the in the example shown in Fig. 4b, the S1 and S3 −7 same storage ring with a bending radius of r = 10 states have large precession rates (3.2 × 10 rad/s) m. for the axion phase of ϕax = π/8, and the precession directions are opposite to each other. On the other hand, note that the other two polarizations S2 and S4 have smaller ωEDM than the two counter parts IV. AXION PHASE EFFECT of the polarizations (S1, S3). In any case, the actual precession rate ωEDM can be calculated using the Since the initial phase of the axion field is un- formula shown above. known, the phase ϕax that appears in Eq. (3) can- Fig. 5 shows a simulation result for deuteron spin not be controlled in the experiment. However, the precession in a E/B combined ring with 100 kHz of rate of the EDM precession angle strongly depends g-2 frequency. The electric and magnetic fields used

6 3.5 -11

0.0 S1

-7 2.8 -11 rad/s -12 = /8

-7.2x10 rad/s S2 x10 ax

3.2x10 x10 2.1 S3

-0.6

-7

1.4 rad/s

1.3x10

-1.2

0.7

-2.5x10

- 7 ax Sy Sy 0.0 -1.8

rad/s

= /8

-1.3x10

ax - 7

-3.2x10

-0.7 rad/s

= /4

-2.4 ax

- 7

-3.5x10

-1.4 rad/s

= /2 S1=(0,0,1) -3.2x10

- 7 - 7

S2=(1,0,0)

-3.0 -2.1 rad/s r a d / s

S3=(0,0,-1)

S =S1=(0,0,1) -2.8

0 S4=(-1,0,0)

-3.6

-3.5

-10 0 10 20 30 40 50 60 70 80 90 100 110 -10 0 10 20 30 40 50 60 70 80 90 100 110

Time ( s) Time ( s) (a) Axion phase dependence of EDM (b) Spin polarization dependence of EDM precession rates. precession rates. Initial axion phase ϕax = π/8

FIG. 4: Axion phases and EDM precession rates.

in the simulation were 7.05×106 V/m and 0.38 T, respectively. The initial spin direction was set to the +z direction (0,0,1) and the total precession time shown in the ﬁgure is 100 µs. As can be seen, the vertical spin component (Sy) is accumulated while the horizontal spin precession takes place at the g-2 frequency. As mentioned before, the vertical precession rate depends on the initial axion phase.

The θQCD for a proton is reported to be three times larger than that of the deuteron [18, 19]. This means that the proton EDM described by Eq. (1) is FIG. 5: Deuteron spin precession in E/B combined three times larger than the deuteron case. However, ring. in this sensitivity estimation, we used the same EDM d for both deuteron and proton.

7 -6 −6 faxion=10 Hz For high frequencies, for example >1 MHz, the ×10 χ2 / ndf 3.68e−16 / 8 Prob 1 resonance method can be used. Each run is done p0 −2.405e−09 ± 2.06e−09 p1 2.464e−07 ± 2.912e−09 at a ﬁxed frequency with the resonance conditions. 0.2 p2 6.283e−06 ± 0 In this case, the axion coherence time is used as the p3 ± 3.943 0.01185 measurement time for each storage time. We used 0.1 half of the axion width, ∆fax/2, as the scan step and ∆fax = fax/Qax. Assuming measurement time

(rad/s) 0 to be one axion coherent time for each frequency

EDM and 1011 particles per storage, SCT = 104 s and ω −0.1 6 Qax = 3 × 10 , the sensitivities were calculated to be about 10−26 − 10−25 e·cm. The total scan time −0.2 can also be estimated using the measurement time

×103 for each frequency multiplied by the number of steps. 0 200 400 600 800 1000 For example, total scan steps for the frequency range Time (s) of 1-100 MHz was ∼ 2.7 × 107 and total scan time was ∼ 8×107 s which corresponds to about 4 years of FIG. 6: ωEDM (t) measurement with frozen spin condition. The data is ﬁt to the sine function for measurement time. For this estimation, we assumed the sensitivity calculation. the minimum measurement time to be a machine cycle time of 3 s.

V. SCANNING METHOD VI. AXION GLUON COUPLED EDM SEARCHES AND THE SENSITIVITY OF The sensitivities presented in Table I and II are THE EXPERIMENTS based on the assumption that we know the axion mass (frequency) and perform the measurement at The current experimental limit comes from the ul- the same frequency for 8 × 107 s. However, since tra cold neutron trap method (nEDM) [36–40]. One we don’t know the axion mass yet, all the possi- can compare the sensitivities between the nEDM ble frequency ranges have to be searched for. We method and the storage ring EDM method. For used different scan methods for different frequency example, the statistical error of the nEDM for one regions. day’s measurement is reported to be 6 × 10−25 e·cm For very low frequencies, fax < 100 µHz, one can (see [38] for details). This result is based on the use the frozen spin method, which was used for the experimental parameters such as the number of par- static EDM search, parasitically. With this method, ticles per storage (13,000 neutrons), free precession one can take data repeatedly from the frozen spin time of 130 s and electric field 450 kV/m. Com- condition for a limited storage time. The storage pared with the proposed storage ring EDM method, time has to be smaller than the axion coherence time the number of particles can reach up to 1011 per and spin coherence time. For example, assuming a storage, the polarimeter efficiency can be about 2% spin coherence time of 104 s, one can use 104 s as the for the proton or deuteron case, the effective elec- maximum storage time, and with this storage time tric field can be over 1 GV/m, and the measurement one can test up to 100 µHz. time, which is limited by the spin coherence time, Fig. 6 shows an example of simulation results for can be more than 104 s. For example, for the 105 an axion frequency of 1 µHz. Each point corresponds Hz deuteron case with a measurement time of one 4 −28 to ωEDM (t) whose data was taken for 10 s. The day, the sensitivity is estimated to be ∼ 10 e·cm. data points are fit to the sine function and sensitiv- From this comparison, one can tell the storage ring ity is calculated from the fit results. The sensitivity method is more sensitive than the ultra-cold neu- calculated with this method was about 10−31 e·cm tron trap method, by roughly more than 3 orders of for frequencies < 100 µHz. This method can be used magnitude. for higher frequencies up to mHz if one uses a shorter Fig. 7 shows the experimental limits for the axion- measurement time (< 103 s). No extra measurement gluon coupled oscillating EDM measurements. For is required for the axion search in this frequency re- comparison, we include the nEDM results, which gion. One can do the static EDM experiment with were taken from Fig. 4 in reference [36]. Based frozen spin conditions and do the analysis for axion on the calculation results shown in Tables I and signal search afterwards. II, the projected limit of the storage ring EDM

8 Oscillation Frequency (Hz) ion searches using the nuclear magnetic resonance (NMR) method [41, 42]. They utilized the resonance

Resonance method of nuclei spin precession with the oscillating axion

short-time base ﬁeld. With the NMR method, one can search the high frequency region when a strong magnetic ﬁeld nEDM )

-1 is used.

Storage Ring EDM projection

Semi-frozen spin method The axion ﬁeld is proportional to the square root (GeV

long-time base a

/ f / of the local axion dark matter density. For this cal- G

C local 3 culation, we used ρDM ≈ 0.3 GeV/cm as the local

QCD Axion cold dark matter density. Furthermore, it can take Frozen spin method advantage of the recently proposed local dark matter enhancement factors from the focusing eﬀects due to planetary motion [43, 44].

Axion Mass (eV)

FIG. 7: Experimental limits for the axion-gluon coupled oscillating EDM measurement. The nEDM results are included for comparison. See reference [36] for detailed nEDM results. The limits for three VII. SUMMARY AND CONCLUSION different frequency regions are indicated with different colors of broken lines. Average values are As a candidate for dark matter, the axion has been used for each frequency region. See the column for the target of extensive searches using microwave cav- SCT = 104 s and Q =3 × 106 in Table I and II ax1 ities and other methods. The fact that the axion- for the relevant numbers. If 1010 is used as the gluon coupling can produce an oscillating EDM in axion quality factor, the C /f can be improved by G a nucleons led to the novel idea of measuring the os- 1 or 2 orders of magnitude in the high frequency cillating EDM in hadronic particles like the proton region. Estimation of the limit plot for storage ring and deuteron. We propose using the storage ring EDM method is made by assuming that the axion technique to measure the axion induced oscillating mass (frequency) is known and the measurement is EDM at the resonance conditions between the axion performed for 8 × 107 s at that frequency. However, frequency and g − 2 spin precession frequency. for very low frequency ranges (fax < 1 mHz), the axions can be searched for with high sensitivity In this study, we calculated the electric field and using the frozen spin method without knowing the magnetic field that are required for the resonance axion frequency. See the text for details. conditions. With the experimental conditions, we estimated the achievable sensitivities, and the result shows the experiment is more sensitive than the −29 · method is drawn by the red dotted line for the res- planned static EDM measurement (10 e cm) by at least one order of magnitude, ≤ 10−30 e·cm. This onance method (1 MHz < fax < 100 MHz), the blue dotted line for the semi-frozen spin method (100 sensitivity is achieved if we assume that we know the axion frequency and spend all the experimental µHzaxions over a wide frequency range.

9 ACKNOWLEDGEMENT give a talk at Stanford University in 2013, by Peter Graham, Surjeet Rajendran, and Savas Dimopoulos. This work was supported by IBS-R017-D1-2018- They suggested to him that the oscillating theta idea a00 of the Republic of Korea. This idea was devel- might be applicable to the storage ring EDM method oped when one of the authors (YkS) was invited to and we thank them for it.

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